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CALIBRATION OF DISCRETE ELEMENT PARAMETERS AND THE MODELLING OF BULK MATERIALS HANDLING

Corné Coetzee1

ABSTRACT

The Discrete Element Method (DEM) is useful for modelling granular flow such as bulk materials

handling. The accuracy of DEM modelling is dependent on the model parameter values used.

Determining these values remains one of the main challenges. In this study a method for determining

the parameters of cohesionless granular material is presented. This includes the particle size and

shape distribution, the density, the contact stiffness and the contact friction. This was done by

conducting laboratory experiments followed by equivalent numerical experiments and iteratively

changing the parameters until the laboratory results were met.

Using the calibrated material properties, several industrial processes were modelled and where

possible the results were compared to experimental measurements. Dragline bucket filling, silo

discharge and a conveyor transfer point were modelled. A simple wear prediction model was

implemented in the DEM code and the results were compared to experiments.

The dragline model accurately predicted the orientation of the bucket. The model also accurately

predicted the drag force over the first third of the drag, but predicted drag forces too high for the latter

part of the drag. In the silo model, the flow patterns and flow rates were compared to experimental

measurements and observations. A good qualitative agreement with the observed flow patterns were

achieved and high flow rates could be predicted accurately with a maximum error of 7%. The

prediction of lower flow rates was less accurate.

DEM contact data, experimental results and empirical relations were used in the wear model to predict

the wear rates and wear patterns caused by particles on a structure. This model needs refinement

and can be used in future to predict wear rates and wear patterns on structures such as chutes,

conveyor belts and dragline buckets.

1 Department of Mechanical and Mechatronic Engineering, Stellenbosch University

021 808 4239 ccoetzee@sun.ac.za

Conveyors and Bulk Materials Conference, 6 & 7 August 2008, Midrand, South Africa

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1. INTRODUCTION

It is estimated that about 40% of the capacity of industrial plants are wasted because of bulk handling

problems (Tijskens et al. 2003). According to Nordell (1997), the modelling of granular behaviour

using Discrete Element Methods (DEM), has the potential to be one of the most important scientific

advancements to the mining industry. Over the last ten years, advances in computer technology have

made the modelling of full scale industrial processes possible (Groger and Katterfeld, 2006).

Processes that can be modelled and optimised include: drilling, blasting, block caving, digging,

crushing, flow regulation, storage, segregation and mixing.

Conveyor and chute designs can benefit from DEM modelling in the following ways (Nordell, 2003):

decreasing belt wear, less dust generation, improved belt capacity and centre loading, less spillage

and blockage, lower noise emission, minimising material degradation, minimising power consumption

and improved safety. Mustoe et al. studied the safety of gravity ore passes using DEM. The

modelling could be used to predict forces on the ore pass system and could aid in the design of

improved geometries to enhance safety and performance. A DEM model provides the engineer with

unique detailed information to assist in the design of transfer points in large conveyor systems

(Hustrulid and Mustoe).

A DEM model can only provide reliable and accurate answers if the input parameters are correctly

specified by the user (engineer). The most difficult part is to specify the material properties used in a

DEM simulation. In this paper a calibration process is proposed which can be used to determine the

material properties. The calibration process is presented, followed by examples of DEM modelling.

The examples include dragline bucket filling, silo discharge, the modelling of a transfer point and the

prediction of wear. The commercial DEM code, PFC3D (Itasca), was used.

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2. THE DISCRETE ELEMENT METHOD

Discrete element methods are based on the simulation of the motion of granular material as separate

(discrete) particles. DEM was first applied to rock mechanics and soils by Cundall and Strack (1979).

Calculations performed during a DEM simulation cycle alternate between the application of Newton’s

second law (applied to the particles) and a force-displacement law (applied at the contacts between

particles). Each particle has material parameters (micro properties) that influence the particle and

hence the bulk behaviour (macro properties).

Using the soft particle approach, each contact is modelled with a linear spring in the contact normal

direction (secant stiffness kn) and a linear spring in the contact tangential direction (tangent stiffness

ks) as depicted in Figure 1.

Frictional slip is allowed in the tangential direction with a friction coefficient µ. The particles are

allowed to overlap and the amount of overlap is used in combination with the spring stiffness to

calculate the contact force components. The contact force in the normal direction is given by

∑= nnn UkF (1)

where nU is the overlap in the contact normal direction. The contact shear force is given by

Figure 1 – Typical DEM contact model

Friction µ

kn cn

ks

cs

4

( )⎪⎩

⎪⎨⎧

≥∆

<∆=

∑∑

nssn

nssss FFUF

FFUkF

µµ

µ

forsign

for (2)

where sU∆ is the displacement increment in the contact tangential direction.

In PFC3D, the user can choose between two damping models. The first model makes use of two

viscous dampers, cn and cs in the contact normal and tangential directions respectively as shown in

Figure 1. This model dissipates energy at the contacts. The second model is named local damping.

In this model the following damping force is applied to each particle,

( )xsignFFd&

balance-of-outα−= (3)

where α is the damping factor (α < 1), balance-of-outF is the resultant force vector acting on the particle

due to all contacts and x& is the particle’s velocity vector. This damping model has the advantage that

is can bring a dynamic system to static equilibrium quickly, but it can not be physically justified.

The model described above is the basic linear contact model as found in most DEM codes. Non-

linear models (Hertz, 1882; Mindlin and Deresiewicz, 1953; Walton and Braun, 1986) are also

available, but are usually computationally less efficient. In general, cohesion can be modelled using

bonds at contacts. In this study only cohesionless material is considered and bonds are not needed.

For a detailed description of DEM, the reader is referred to Cundall and Strack (1979), Cleary and

Sawley (1999), Hogue, (1998) , Zhang and Whiten (1996) and Itasca.

3. MATERIAL CALIBRATION

The accuracy of DEM modelling is dependent on the micro property values used. Determining these

values remains one of the main challenges. The micro properties should be specified such that the

flow of thousands of particles behaves in the same way as the real granular material (macro

properties). Laboratory experiments or in-situ tests are necessary to determine these parameters

before any useful modelling and predictions can be made (Asaf et al., 2005).

Very few researchers have tried to determine the micro parameters experimentally. Vu-Quoc et al.

(2000) measured the coefficient of restitution in soybeans by performing simple drop tests from

various heights. Data from high-speed video recordings was used to determine the particle properties.

Tanaka et al. (2000) conducted bar penetration tests and compared the results with those obtained

from DEM simulations. The material density could be measured, but the contact stiffness was chosen

without any experimental validation. By comparing the movement of the particles during the

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experiment with the movement of the particles during the simulation, the friction coefficient that gave

the best results could be determined.

Recently, Franco et al. (2005) proposed an inverse calibration method to determine the micro

parameters for modelling soil-bulldozer blade interaction. In their approach, the material was

assumed cohesionless. Based on DEM results, the particle friction coefficient and stiffness were

determined from energy principles and direct shear tests. The draft force and vertical force acting on

the blade, as predicted by DEM, were then compared with McKeys’s (1985) calculation model. The

calibration process was based on direct shear test DEM simulations and blade-soil DEM simulations

and experimental validation was not performed.

Asaf et al. (2005, 2006) proposed an in-situ method for determining the micro parameters. Their

method was based on wedge penetration tests and a non-linear optimisation scheme. The

methodology was validated using DEM simulation results instead of real in-situ tests. Landry et al.

(2006a,b,c) modelled machine-manure interaction using the discrete element method. In their

approach, data from numerical shear tests, values from literature and simple analytical approaches

were used to determine/estimate the material parameters.

In this paper, a new method for determining micro parameters for frictional material is proposed. The

method is based on two to three laboratory experiments, a shear test and/or angle of repose test and

a compression test (oedometer). There is assumed to be no cohesion or adhesion at particle-particle

and particle-wall contacts respectively. The parameters to be determined are the particle size and

size distribution, particle shape, particle density, particle stiffness, particle-particle friction coefficient

and particle-wall friction coefficient.

In this paper two materials were used namely, crushed rock (gravel) and corn grains. The rock was

used in the dragline bucket filling experiment and the wear measurements. The corn grains were

used in the silo discharge experiment. The calibration process is presented with a mix of results from

the rock and the corn grains.

3.1 Particle Shape and Shape Distribution

Most DEM codes make use of spherical particles because it simplifies contact detection and contact

force calculations. Using spherical particles, the internal friction of the material (assembly of particles)

is usually too low when compared to real granular material like crushed rock. Non-spherical particles

are needed to increase the particle interlocking effect and one solution is to make use of clumped

particles (Abbaspour-Fard, 2005). In PFC3D clumps can be formed by adding two or more spherical

particles together to form one rigid particle, i.e. particles comprising the clump remain at a fixed

distance from each other (Itasca). Particles within a clump can overlap to any extent and contact

forces are not generated between these particles. Clumps can not break up during simulations

regardless of the forces acting on them. Asaf et al. (2005) investigated the effect of particle structure

on internal friction angle. They used two equally sized discs, clumped together, and biaxial tests to

investigate the effect of particle centre distance on the internal friction angle. Results showed that the

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internal friction increases with an increase in the centre distance with an asymptotic behaviour. It was

noticeable that beyond a certain centre distance, the value of the internal friction remained relatively

constant although the centre distance is increased. Wang et al. (2007) made use of X-ray tomography

to accurately build complex particle shapes from spherical particles. Although their approach is

successful, it still needs up to 1000 spheres to accurately represent a physical particle. This will

increase simulation time tremendously and is not considered in this paper.

The material used in the dragline bucket experiments (Section 4) is shown in Figure 2. It was crushed

rock from a roller mill with a 25 mm aperture between the rollers. A random sample of 300 rocks was

taken and the rocks could be classified into four distinct particle shapes. These shapes were kept as

simple as possible, while ensuring that every rock in the sample belonged to one of the particle

shapes. The particle shapes were (Figure 3):

• Particle 1: long rectangular particle comprising of three aligned spheres (length to width ratio of

clumped particle greater than 2)

• Particle 2: pyramid shape particle comprising of four spheres

• Particle 3: flat particle comprising of four spheres (length to width ratio of clumped particle less than

two, and ratio between length and height greater than 2)

• Particle 4: spherical particles comprising of one sphere.

The rock sample was classified according to the above shapes. The number of particles belonging to

each shape was recorded and a particle shape distribution was obtained (Figure 4). This data was

then used to ensure that the same distribution between particle shapes was maintained when

generating particles for the DEM model.

Figure 2 – Crushed rock used in dragline experiments

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Particle shape 1 Particle shape 2

Particle shape 3 Particle shape 4

Figure 3 – Simplified rock particle shapes used in dragline DEM model

Figure 4 – Measured rock particle shape distribution

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3.2 Particle Size and Size Distribution

A sample of each particle shape was taken and the size distribution obtained using a sieve sorting

machine (mesh sizes 9.5 mm, 13.2 mm, 19 mm, 26.5 mm and 37.5 mm). The results for each particle

shape can be seen in Figure 5. A MATLAB code was written to generate clumps using the particle

shape distribution and the particle size distribution. The sorting sieve classifies the particles in

discrete sizes, and the distribution between these sizes was assumed to be linear. The linear

distribution used to generate the clumps is also shown in Figure 5.

3.3 Particle Density and Bulk Density

A container with a known volume was filled with a rock sample. The sample was weighed and divided

by the volume of the container to obtain the bulk density. An identical container was generated

numerically in PFC3D. Particles with the previously determined shape and size distribution were then

generated and allowed to fill the container. Once the particles had settled under gravity, the mass of

the particles occupying the container was calculated together with the occupied volume. The density

of the particles was then adjusted using the ratio of the measured bulk density to the numerically

calculated bulk density. The system was allowed to reach static equilibrium and the process was

Figure 5 – Measured rock particle size distribution and linear distribution used in the DEM model

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repeated until the measured bulk density was achieved. Within three to four iterations, an accuracy of

0.1% could be achieved.

3.4 Normal and Shear Stiffness

In the confined compression test (also called the oedometer test), stress is applied to the specimen

along the vertical axis, while strain in the horizontal directions is prevented. Shear stresses and shear

strains as well as compressive stresses and volume changes occur in this test, but since the material

is prevented from failing in shear, compression is the dominant source of strain. Usually this test is

used to determine soil consolidation ratios. Here it is used to determine the elastic stiffness of the

material, similar to a uniaxial tensile test.

A cylindrical container was used. The system was slowly loaded, unloaded and reloaded and the

force-displacement curve recorded as shown in Figure 6. After a couple of loading-unloading cycles,

a stable hysteresis loop was obtained with the slope of the loading curve the same for each cycle.

The stiffness was defined as the gradient of the loading curve between two points as shown in Figure

6 (note that the load force was converted to a pressure using the cylinder diameter and the

displacement to axial strain). The process was repeated numerically, and the stiffness compared to

the measured stiffness. A series of numerical experiments were done using different contact stiffness

and contact friction coefficients. It was found that the friction coefficient had no significant effect on

the numerical results, and the particle stiffness the only unknown parameter with a major influence on

the results. From Figure 7 it can be seen that the relation between the particle stiffness and the

oedometer stiffness is linear. The measured bulk stiffness was numerically matched by changing the

particle stiffness.

Figure 6 –Typical oedometer load-displacement curve for rocks

0 0.5

2

6

10

Displacement [mm]

Load

[kN

]

∆s

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

16

0

4

8

12

14

∆F

k

10

3.5 Friction

The internal friction is comprised of two components, namely, the particle interlocking and the contact

friction between individual particles. The particle interlocking is affected by the particle shapes and

size distribution.

A shearbox can be used to measure the material’s internal friction angleφ . The shearbox can only be

used if the size of the particles relative to the box size is small. In the case of corn grains, a box with

dimensions of 375 × 375 mm with a total height of approximately 175 mm was used (this box is

adequate for testing particle sizes up to 12 mm, Head, 1998).

Although the ratio of particle normal stiffness to particle tangential stiffness, kn/ks, has an influence on

the materials Young’s modulus and Poisson’s ratio (Itasca, Landry et al., 2006a), this ratio was taken

equal to one, i.e. the particle normal and tangential stiffness were assumed to be the same.

Asaf et al. (2005) have shown that the internal friction angle is dependent on the normal stress applied

to the shear test. The internal friction angle decreases with an increase in normal stress and it is

important to apply an expected range of normal stress during testing. In this study, the normal

0 50 100 150 200 250 300 350 400 450 500

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Oed

omet

er s

tiffn

ess

[MP

a]

Particle stiffness kn = ks [kN/m]

Experiment

µ= 0.10

µ= 0.12

µ= 0.20

µ= 0.30

Figure 7 –Influence of particle stiffness and friction coefficient on the bulk stiffness of corn grains

11

stresses imposed during the shear tests were chosen to be within the range experienced during the

experiments (dragline bucket, silo discharge and wear prediction).

A numerically equivalent DEM model with the same dimensions as the physical apparatus was

constructed and tests were performed under the same conditions. The average shear stress between

2.1% and 4.8% strain (8 and 18 mm of displacement in Figure 8) was used to calculate the friction

angle, which was defined as the average shear stress divided by the applied normal stress.

A series of shear tests were performed using a range of particle stiffness and friction coefficient

values. The shear simulations showed that the internal friction angle depended on both the particle

stiffness and friction coefficient, especially in the low stiffness range.

The shear test results are shown in Figure 9. All particle friction values µ resulted in a higher internal

friction value φ. For example, with the particle friction µ = 0.3, all the calculated internal friction angles

were greater than tan-1(0.3) = 16.7º. This is a direct result of the interlocking phenomenon and was

also observed by Asaf et al. (2006b). The interlocking friction angle can be found by calculating the

internal friction angle based on a shear test simulation where the particle friction coefficient µ is set

equal to zero. According to Asaf et al. (2005) the interlocking angle is dependent on the particle size,

size distribution, porosity and particle stiffness.

2 4 6 8 10 12 14 16 18

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Displacement [mm]

Shea

r stre

ss [k

Pa]

0

DEM

Experiment

Figure 8 – A typical shearbox result using corn grains

12

From Figure 9, it is clear that both the particle stiffness and friction coefficient had an influence on the

material internal friction angle, φ. For stiffness values below 100 kN/m the internal friction angle is

strongly dependent on the particle stiffness, but for higher stiffness values the dependence decreases,

and more so for lower friction coefficients. Thus, the direct shear test can not be used to determine a

unique set of parameters. It is clear from Figure 9 that more than one combination of the particle

friction coefficient and stiffness values can result in an internal friction angle similar to the measured

value (indicated by the dotted line). However, using the results from the oedometer test, the stiffness

is known and can the friction coefficient be determined. In this case a particle stiffness of 450 kN/m

was used based on the results from Figure 7. With the stiffness known, Figure 9 could be used to

determine the friction coefficient µ = 0.12 which closely matched the measured internal friction angle

φ = 23°.

In all the above shearbox results corn grains were used. The rocks used in the dragline bucket

experiment were too large to test in the existing shearbox apparatus. In this case the friction

coefficient was calibrated using an angle of repose simulation. It is known that for frictional

cohesionless granular material, the angle of repose is a good indication of the internal friction angle

(Lambe and Whitman, 1969).

0 100 200 300 400 500 600 10

15

20

25

30

35

40

µ = 0.10

µ = 0.15

µ = 0.20

µ = 0.30

Particle stiffness kn = ks [kN/m]

Inte

rnal

fric

tion

angl

e φ

[deg

]

µ = 0.12

Figure 9 –Shearbox numerical results using corn grains with different stiffness and friction coefficients

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A sample of the material was taken and dropped from a given height through a funnel with a known

diameter. This process was continued until a given mass of material had been allowed to flow though

the funnel. The angle of repose was then measured to be 41° (Figure 2).

The dimension of the experimental setup was then used to construct an identical numerical model.

The same mass of material was allowed to flow though the funnel and the angle generated was

measured. This process was continued, varying the inter-particle friction coefficient, until the same

angle was obtained.

3.6 Summary of Material Properties

If one parameter is changed, all the other parameters need to be re-calibrated. The particle shape

and size were kept fixed, and a couple of iterations were used to determine final values for the particle

density, particle stiffness and particle friction coefficient (Table 1). Note that the shear stiffness is

assumed to have the same value as the normal stiffness. This is an assumption often made in

discrete element modelling (Itasca).

Table 1 – Calibrated material properties

Parameter Rock (dragline) Corn (silo)

Particle density ρ 1518 kg/m3 855 kg/m3

Contact stiffness kn/s 1·75×107 N/m 4·5×105 N/m

Friction coefficient µ 0.53 0.12

Angle of repose φr 41° 24°

Hartl and Ooi (2006) and Groger and Katterfeld (2006) decreased the stiffness by a factor of 10 to

100. This was done to decrease simulation time since the stable time step is inversely proportional to

the square root of the stiffness. Hartl and Ooi (2006) showed that decreasing the stiffness by a factor

100, had no influence on the limiting shear stress as modelled in a direct shear test. It did, however,

influence the initial response as the stiffness was much lower. Decreasing the stiffness by a factor

1000 influenced the results significantly and it no longer matched experimental measurement. In this

study, the stiffness had an effect on the internal friction angle and therefore the stiffness reduction is

not used. The reason for the discrepancy in observations might be due to the particle used (corn

versus glass beads) and the stiffness range tested. This however needs further investigation.

According to Groger and Katterfeld (2006) no experiment is known that could be used for the

calibration of the damping constants, except for the measurement of the rebound height of a dropped

spherical particle. Practically relatively high contact damping coefficients are required. In all the

simulations performed here, a viscous critical damping ratio ofξ = 0.6 to 0.8 was used.

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4. DRAGLINE BUCKET FILLING

In open cast mining the overburden needs to be removed in order to mine the ore below. This

overburden can vary from topsoil to hard rock. The bulk of the costs involved in open cast mining can

be attributed to overburden removal.

Draglines are very economical and the most popular of all overburden removal devices. They are

usually used in combination with scrapers to reduce the amount of re-handle. A dragline is a crane-like

structure with a huge bucket of up to 100 m3 in volume, suspended by steel ropes (Figure 10). The

bucket is dragged through the overburden and once the bucket has been filled, it is hoisted up. The

base of the dragline swivels and the overburden is dumped elsewhere. The overburden usually needs

to be blasted before the dragline can be used.

Drag ropes

Drag chains

Hoist chains

Hoist ropes

Spreader bar

Dump rope

Figure 10 – A typical walking dragline and the bucket rigging (VR Steel)

Bucket

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Draglines are designed to operate twenty four hours a day for three hundred and sixty days a year.

The cost of the loss of production due to dragline down time has been estimated at 8000 Australian

dollars an hour (Dayawansa and Price, 2004). Many dragline breakdowns can be attributed to the

design of the dragline bucket. The buckets either fail or overload the machine and causes failures in

the dragline boom and main structure.

The filling of a bucket is a complex granular flow problem. Instrumentation of equipment for measuring

the operation is difficult and expensive. It is possible to use small-scale (usually 1/10th scale)

experimental rigs to evaluate bucket designs (Esterhuyse, 1997; Rowlands, 1991) but they are

expensive and there are questions regarding the validity of scaling (Poschel et al., 2001a,b). To

scale-up results from model experiments is problematic since there are no general scaling laws for

granular flows (Cleary, 1998).

Numerical models and simulations have become an important design tool in engineering industries.

Although numerical simulations seldom totally replace experiments, they allow designers to investigate

a far wider range of options in a relatively short time and usually at much lower cost than using

experimental investigations. A sensitivity analysis can be done more easily and comprehensively.

Expensive experiments can then be used after numerical investigations to “fine tune” the design

(Landry et al., 2006a).

In order to accurately model dragline bucket filling, both the granular material and the bucket dynamics

must be accurately modelled. A dragline bucket is suspended by ropes and the motion of the bucket

is influenced by the forces acting on it (gravity, rope forces and the interaction force with the soil). The

path followed by the bucket and the bucket’s orientation is not known in advance or manipulated by

the operator, as in the case of hydraulic excavators for example. The dragline operator positions the

bucket on the ground and then the bucket is dragged in the general direction of the drag ropes at

constant rope speed.

In a DEM model, structures such as buckets are modelled using walls. DEM codes calculate the

behaviour of the particles based on the forces acting on them, but most DEM codes do not solve the

equations of motion for the walls. In some DEM codes, the translational and rotational velocity of the

walls can not be changed during the simulation, or the velocity can only be pre-programmed as a

function of time. In dragline bucket modelling the velocity of the walls (bucket) needs to change

according to the forces acting on it and it can not be pre-programmed.

The DEM codes by Itasca give the user access to almost all internal variables via the powerful built-in

programming language, FISH. This feature makes it possible to obtain the resultant force and moment

caused by particles on a wall. The resultant force and moment vectors can then be used to solve the

equations of motion for each wall and update the velocity of each wall accordingly. In this paper, this

option was used to model the dynamic behaviour of a dragline bucket. A dynamic model was

developed and implemented in FISH, taking into account the rope forces, the bucket’s mass, the

bucket’s moments of inertia and the interaction with the soil.

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4.1 Experimental Setup

A scale dragline model was build to obtain data which could be used to validate the numerical results.

The drag forces, the bucket trajectory (path) and the bucket’s orientation during a filling cycle were

recorded.

The bucket was a 1:18 scale model of a 61m3 bucket (VR Steel). The bucket had a length and width of

roughly 300 mm and a fill height of 175 mm. The drag bed made provision for sensor attachment and

was designed to be inclined which allowed different drag angles to be tested (Figure 11).

The bucket was dragged at constant rope speed using a hydraulic cylinder in conjunction with a servo

valve. The cylinder speed was measured by means of a linear variable differential transducer. The

drag force was measured using a calibrated commercial load cell on each rope.

Three ASM W12 cable displacement sensors were used to determine the three-dimensional position

of a point on the bucket. The body of each sensor was attached to the rig frame, with the end of each

cable attached to the centre of the bucket arch (Figure 11). Using a triangulation algorithm, the

position of the arch point could be determined. The three sensors were placed in a plane parallel to

the drag bed and positioned to maximize the angle between sensors. The bucket’s orientation (angle)

was measured using a Micro-Strain 3DM-G three axis inclinometer. This sensor was also attached to

the bucket arch. Data was collected at 25 Hz using a HBM spider 8-30.

Figure 11 – Dragline experimental setup showing the position of the sensors

Drag ropes with load cells

Position sensors

Arch attachment point

Inclinometer position

17

4.2 DEM Model

The modelling of large systems is still a challenge (Tijskens, 2006). Even with parallel processing

becoming available, clever modelling can reduce the computation times without loss in accuracy.

Reducing the number of particles and walls in the model is the most obvious and most effective

solution to reduce computation time. Reducing the number of particles and walls reduces the number

of calculations performed per cycle. A drag bed is considered optimised when its volume is as small

as possible, without allowing boundary effects to unduly influence the simulation. Experimental data

revealed that the bucket follows a parabolic trajectory while filling. The drag bed was modelled using

this information and simplified to reduce the number of balls. The drag bed can be seen in Figure 12.

This shape requires 25 percent less particles than a standard rectangular drag bed.

A CAD model of the bucket was created and converted to a STL model. A code was written to convert

the STL model to PFC3D wall commands. In order to decrease the total number of walls, the bucket

was simplified in the following ways.

• Rounds or fillets were replaced with chamfers or completely removed

• Giving the bucket basket a uniform thickness

• Removing the arch and top rail

• Closing the small gap that normal exists between the teeth and shrouds (Figure 13)

3000

600

200

100

600

Figure 12 –Drag bed design (dimensions in mm)

18

The simplifications were restricted to areas that would have little or no effect on the flow of material in

or around the bucket. The simplified bucket shape can be seen in Figure 13. The new bucket allowed

the number of walls to be reduced from 5000 to 520. Although these changes would affect the bucket

weight and moments of inertia, the original weight and moments of inertia were still used in the

dynamic bucket model.

4.3 Results

Each experiment was repeated three times to ensure the results were reliable and repeatable. The

bucket position, orientation and drag force were recorded for each test. The results showed that the

bucket behaviour is predominantly two-dimensional with very little rolling and yawing motion.

Rowlands (1991) also observed that the flow of material into the bucket is mostly two-dimensional and

the motion of a dragline bucket can be accurately modelled taking into account only the bucket

translation in the drag direction, translation vertical to the drag direction as the bucket digs into the soil

and pitching (roll and yaw can be ignored)

Figure 14 shows the bucket’s pitch angle versus drag distance. In both the experiment and the

simulation, the bucket was placed flat on the drag bed and the pitch angle set to zero. During the

experiment, the pitch angle increased to almost 3 degrees before it decreased as the bucket started to

dig into the soil. At a displacement of roughly 0.3 m (one bucket length), the pitch angle has

decreased to -6.5 degrees. The DEM model also predicts this behaviour with the pitch angle at -10

degrees after a displacement of one bucket length.

Figure 15 shows the total drag force (sum of the force in each of the two drag ropes) versus drag

distance. There is an almost linear increase in measured drag force with a final value of 500 N at a

drag distance of 1 m (three bucket lengths). The DEM model accurately predicts the drag force up to

a displacement of 0.3 m (one bucket length). Thereafter, DEM predicts higher drag forces and at the

Figure 13 – Original bucket and the simplified design

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end of the drag it is almost three times the measured value. The reason for this is still unknown and

needs further investigation.

0 0.1

-10

-5

0

Drag distance [m]

Pitc

h an

gle

[deg

]

DEM

Experiment

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

5

Figure 14 – Comparison between the measured and the DEM predicted pitch angle

0 0.1

100

300

500

700

900

1100

0

Drag distance [m]

Tota

l dra

g fo

rce

[N]

DEM

Experiment

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1300

1500

1700

1900

Figure 15 – Comparison between the measured and the DEM predicted drag force

20

Figure 16 illustrates one of the advantages of DEM simulations where the mass of material inside the

bucket can easily be determined as a function of drag distance. A FISH function was written which

checked all particles and simply added up the mass of all particles found to be inside the bucket. This

can easily be done since the position and orientation of the bucket is known. It is difficult to verify this

experimentally since the mass of material inside the bucket can not be determined without influencing

the experiment. This result also confirms that the bucket fills within 3 bucket lengths (1 m in drag

distance), a rule of thumb used by bucket manufacturers.

5. SILO DISCHARGE

Roughly one-half of the products and about three-quarters of the raw materials of the chemical

industry are in the form of granular material that is usually stored in silos or bunkers. In agriculture,

chemicals, minerals and seed grains are also stored in silos and hoppers. Although the silo is widely

used, definite theories for the flow of material in silos are not available (Yang and Hsiau, 2001).

5.1 Experimental Setup

A model silo was built to investigate the flow of material out of a rectangular flat-bottomed silo. The

focus was mainly on flow patterns and flow rate. The flow pattern in a silo is of great importance when

handling material that degenerates with time. Then it is important to achieve the first-in-first-out

storage principle (Karlsson et al., 1998). The model silo used for the experiments had dimensions of

310 mm wide, a maximum fill height of 600 mm and a depth (out-of-plane) of 730 mm. All the sides

were constructed of glass and the opening width could be varied. Coloured grains were used to

0 0.2 0

4

8

Drag distance [m]

Mas

s in

buc

ket [

kg]

0.4 0.6 0.8 1.0 1.2 1.4 1.6

12

2

6

10

Figure 16 – The DEM predicted soil mass inside the bucket

21

create layers within the material which could be used to compare flow patterns. During discharge, the

mass of the material within the silo was measured by suspending the silo from a load cell. The

material properties were determined using the procedures described in Section 3.

5.2 Results

Figure 17 shows the flow of corn out of the silo. There is a good agreement between the DEM and

experimental results regarding flow patterns. It can, however, be seen that the DEM discharge rate is

higher compared to the experiment. To make a quantitative comparison of the flow patterns is difficult.

The flow rate can, however, be compared quantitatively.

Figure 18 shows the corn mass within the silo as a function of time for two openings: w = 45 mm

and w = 80 mm. The initial fill height is h = 500 mm. For both openings, DEM predicts a slightly

higher flow rate than the measured rates. Taking the linear slope between 60 kg and 20 kg, the

discharge rates are calculated as shown in the table in Figure 18. From this it is confirmed that

DEM predicts a slightly higher flow rate and is more accurate for larger silo openings where the

flow is less restricted.

DEM

t = 1 s t = 3 s t = 5 s t = 7 s

Expe

rimen

t

Figure 17 – Silo discharge patterns using corn grains

22

6. TRANSFER POINT

Transfer points can contribute to some of highest maintenance costs on a mine. Transfer points

should have the same level of importance as any other machinery in the minerals processing cycle

(Baller, 2008). In this section a transfer point is modelled as a demonstration and not for validation.

Figure 19 shows a transfer point designed for a South African mine. Platinum ore on the existing

conveyor is transferred via the chute to the shuttle conveyor. The existing conveyor is inclined at 8.9°

and the shuttle conveyor is running horizontally. The angle between the two conveyors is 35°. The

conveyor data is summarised in Table 2.

Table 2 – Conveyor data

Existing conveyor Shuttle conveyor

Belt speed 1.2 m/s 1.5 m/s

Belt width 1200 mm 1350 mm

Pulley diameter 820 mm 500 mm

Flow duty 460 t/h -

w = 80 mm w = 45 mm

DEM 30.8 kg·s-1 9.2 kg·s-1

Experiment 28.6 kg·s-1 7.0 kg·s-1

Error 7.1% 31.4%

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0

10

20

30

40

50

60

70

80

90

100

Time [s]

Mas

s in

silo

[kg

] Experiment

DEM

w = 80 mm w = 45 mm

Figure 18 – The mass inside the silo as measured and modelled

23

Existing conveyor

Shuttle conveyor

Top view

35º

Side view

8.9º

Figure 19 – The transfer point

24

6.1 Material Properties

Four different shapes were used to model the material. These are shown in Figure 20. Figure 21

shows the measured size distribution. Also included in the figure is the distribution used in the DEM

model. Only particles with a size between 8 mm and 255 mm were modelled. Particles were created

using the shapes defined above and the size was randomly chosen from the distribution shown in

Figure 21. The particle density (rock density) was reverse calibrated using the given bulk density. No

data was available on material stiffness. The stiffness of the particles was estimated using past

experience and information from literature. Analyses showed that the model is not sensitive to the

particle stiffness. Values chosen within the range kn = 5·104 N/m - 1·106 N/m had no significant effect

on the flow behaviour. An angle of repose simulation was used to determine the coefficient of friction

between contacting particles. Figure 22 shows the final result and the measured maximum and

minimum of 37º and 42º respectively. Table 3 summarises the material properties and the DEM

parameters used.

Table 3 – Material properties

Material DEM

Bulk Density 2700 kg/m3 2700 kg/m3

Material Friction Angle of Repose φ = 37º - 42º Friction coefficient µ = 0.35

Stiffness - kn = ks = 1·105 N/m

Material-Steel Friction φ = 29º - 37º (estimated) φ = 33º

Type A Type B Type C Type D

Figure 20 – Particle shapes used

25

Figure 21 – Particle size distribution

0

2

4

6

8

10

12

14

16

25518012790634531.522.41611.285.642.821.410.710.50.360.25

Size [mm]

%

DEMMeasured

37º

42º

Figure 22 – Result from angle of repose simulation

26

6.2 Results

Material was added to the existing conveyor at a constant rate of 530 ton per hour, 15% above the

specified rate of 460 tons per hour. The material moved through the chute and onto the shuttle

conveyor. At the end of the shuttle conveyor (modelled as 12 m long) the material was deleted and

the mass flow of material recorded.

Figure 23 shows the accumulative material mass deleted at the end of the shuttle conveyor. The

simulation is started with material on the existing conveyor and no material on the shuttle conveyor.

During the first stages, there is no flow of material recorded until roughly 8 seconds into the simulation.

The flow rate is calculated as the slope of the mass-time line, as indicated in the figure. The flow rate

quickly reaches a stable value of roughly 530 ton per hour which indicates that the flow through the

chute reached a steady state. This shows that there is no excessive build up of material in the chute.

Figure 24 and Figure 25 show the flow of material on the conveyor belts and through the chute.

Mas

s [k

g]

0 5 10 15 20 25 30 350

500

1000

1500

2000

2500

3000

3500

Time[s]

Figure 23 – Flow rate at the end of the shuttle conveyor

27

Figure 25– Flow at time = 20 seconds

Figure 24 – Flow at time = 2 seconds

28

7. THE MODELLING OF WEAR

Wear is a common problem in bulk materials handling. It occurs at transfer chutes on the steel

structures, the skirts and the conveyor belts and on the liners of tumbling mills (Kalala et al., 2008). In

dragline bucket design, special wear elements are added to the bucket to protect the structural

components from excessive wear. It is believed that abrasive wear is the typical type of wear

occurring in transfer points (Hustrulid).

DEM can be used to predict wear as shown by Kalala et al. (2008) and Hustrulid. In this paper, a

simple experiment was designed to measure wear. A identical DEM simulation was build and two

wear models implemented to predict the wear.

7.1 Experiment

A 50 mm diameter aluminium rod, 450 mm long was dragged through gravel and the wear on the rod

measured. The rod consisted of four different segments screwed together as shown in Figure 26.

Each segment could be weighed individually to determine the material mass removed due to wear.

The rod was repeatedly dragged at a speed of 25 mm/s over a distance of 450 mm and back. The

same rock used in the dragline experiment was used in this experiment. The segments were weighed

(accurate to 0.01 g) every two hours to determine the amount of wear. Figure 27 shows the

accumulative mass removed over a period of 28 hours. In this specific test, only segments 1 and 2

experienced any wear. It is clear that between the hours 2 and 28 the wear rate (0.05 g/hour) is

constant.

Figure 26– Aluminium rod for wear measurements

29

Cummulative Mass Removal

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 5 10 15 20 25 30

Time [hours]

Mas

s R

emov

ed [g

ram

s]

Part 1Part 2TotalCalculated Rate

7.2 DEM Model

During a PFC3D simulation, all internal variables are available to the user through a set of so-called

FISH functions. All the contacts, the entities in contact, the contact position, the entity velocity and the

contact forces are available and can be stored and/or manipulated during the simulation. Two wear

models were implemented: an impact model and a sliding model.

The impact model is based on the model by Finnie (1960). In this model, the amount of material

removed by a single impacting particle on a structure is given by

( )αασ

22

sin32sin4

−=p

iimVKM if °≤ 5.18α (4)

( )ασ

22

cos12 p

iimVKM = if °≥ 5.18α (5)

where iM is the material removed (kg), m is the particle mass (kg), V is the particle speed (m/s), pσ

(Pa) is the material yield stress and α is the angle at which the particle impacts. The constant iK is

introduced here to find a correlation between the measured wear and the wear predicted by the DEM

model.

Figure 27– Accumulative mass removed

30

The sliding model is based on a model implemented by Hustrulid, Figure 28. The volume of material

removed is given by

xshears AtVV ∆= (6)

where shearV is the relative velocity at the contact point in the shear direction, t∆ is the time step size

and xA is the cross sectional area of the particle in contact. Using a series expansion, the area can

be approximated as (Hustrulid),

5.15.08856.1 δRAx = (7)

where R is the sphere radius and δ the contact overlap.

To account for the different particle shapes, the conversion from volume to mass and the constant

1.8856, a constant sK is introduced and the mass of material removed given by

5.15.0 δRtVKM shearss ∆= (8)

For each contact in a DEM simulation the shear velocity, the ball radius, the contact depth (overlap)

δ and the time step size is known. Summing the effect of each contact over time, the total wear can

be predicted.

First the constants sK and iK were taken as unity and total wear calculated during a DEM simulation.

New contacts can be identified and they were used to calculate impact wear while existing contacts

were used to calculate sliding wear. It was found that the model also predicts a constant wear rate,

although different from the measured rate. Using constant values of sK = iK = 0.852, the predicted

rate could be set equal to the measured rate.

Harder material

Softer material Material removed

R

δ

time = t time = t + ∆t

Figure 28– Sliding wear model by Hustrulid

Contact area

shearV

31

In this initial phase of the project, the effect of impact wear and sliding wear was taken to be equal.

The effect of each needs further investigation so that the constants sK and iK can be determined

independently.

From the DEM data, the wear patterns can also be shown. In Figure 29 the wear patterns on the rod

is shown after on cycle. The front and rear sides are plotted separately and as expected, by far the

most wear occurs on the front of the rod. The predicted wear patterns also correlated with the actual

patterns seen on the rod.

From the DEM simulations, an energy balance can also be done. In Figure 30 the slip work (energy

loss due to sliding in the contact shear direction) as calculated on the front side and the rear side of

the rod is shown. The accumulative energy loss is almost linear and could be used to predict the wear

rates. This needs further investigation.

Figure 29 – Wear patterns as predicted by the DEM model

32

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7x 10-3 Accumulative Slip work

time [s]

slip

wor

k [J

]

Front surfaceRear surfaceBottom surface

8. CONCLUSION

A calibration process to determine the material properties of cohesionless granular material is

presented. Cohesion is not modelled and this aspect needs further research. The calibrated

materials are used in DEM simulations of typical industrial processes. It is shown that DEM can

accurately model dragline bucket filling and silo discharge. As demonstration, a transfer point with two

conveyors and a chute is modelled. A wear model is implemented in the DEM code and it is shown

that this model can be used to predict abrasive wear. Further research is however needed to improve

this model and to show that it is valid for any process and any structural geometry. In general, DEM

shows good promise and can be used to improve existing bulk materials handling processes and

equipment.

Figure 30 – Accumulative slip energy loss as predicted by DEM

33

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