DISCRETE ELEMENT METHOD APPLIED TO THE VIBRATION … · 2018-04-20 · BEHZAD MAJIDI DISCRETE ELEMENT METHOD APPLIED TO THE VIBRATION PROCESS OF COKE PARTICLES Mémoire présenté

BEHZAD MAJIDI

DISCRETE ELEMENT METHOD APPLIED TO THE

VIBRATION PROCESS OF COKE PARTICLES

Mémoire présenté à la Faculté des études supérieures et postdoctorales de l’Université Laval dans le cadre du programme de maîtrise en Génie Matériaux et Métallurgie

pour l’obtention du grade de Maître ès sciences (MSc.)

DÉPARTEMENT DE GÉNIE DES MINES, DE LA MÉTALLURGIE ET DES MATÉRIAUX FACULTÉ DES SCIENCES ET DE GÉNIE

UNIVERSITÉ LAVAL QUÉBEC

2012 © Behzad Majidi, 2012

http://www.fsg.ulaval.ca/

Résumé

Les propriétés physiques, mécaniques et chimiques des matières premières ont un effet majeur sur

la qualité des anodes en carbone pour le procédé de production d’aluminium. Ce travail tente

d’étudier la faisabilité de l’application de simulation de la Méthode des Élément Discrets (DEM) à

la technologie de production d’anodes. L’effet de la forme des particules et de la distribution de

leurs tailles sur la densité apparente vibrée (VBD) d’échantillons de coke sec est étudié. Les

particules de coke sont numérisées en utilisant des techniques d’imagerie à deux et trois

dimensions. Ces images donnent les formes et les aspects réels des particules qui sont utilisées pour

les modèles DEM pour les tests VBD pour le codage de flux de particules (PFC). Le coefficient de

friction interne des particules de coke est estimé par la méthode de mesure d’angle au repos. Les

résultats ont montrés comme attendu, que la VBD des échantillons de coke est affectée par la forme

et la distribution de taille des particules. Les simulations à deux dimensions ont confirmé qu’en

général, les échantillons formés de particules de tailles poly-dispersées ont une VBD plus haute que

ceux dont la taille des particules est mono-dispersée. De plus, la VBD des échantillons augmente

lorsque la fraction de grosses particules augmente. Cependant, la présence de 10 % massique de

particules fines est nécessaire pour remplir les pores entre les grosses particules. De même pour la

simulation 3D, le modèle suit la tendance des données expérimentales montrant que dans une

éprouvette de 2,9 cm de diamètre, l’augmentation de la quantité de particules de - 4+6 mesh (de

3,36 à 4,76 mm) engendre une augmentation de la VBD. En conclusion, un modèle DEM approprié

est capable de prédire le réarrangement des particules et l’évolution de la densité pendant le

processus de vibration.

iii

Abstract

Physical, mechanical and chemical properties of raw materials have considerable effects on quality

of carbon anodes for aluminium smelting process. The present work attempts to investigate the

feasibility of application of Discrete Element Method (DEM) simulations in anode production

technology. Effects of coke particles shape and size distribution on vibrated bulk density (VBD) of

dry coke samples are studied. Coke particles are digitized using two-dimensional and three-

dimensional imaging techniques and real-shape particles are used in DEM models of VBD test in

Particle Flow Code (PFC). Internal friction coefficient of coke particles were estimated by means of

angle of repose tests. Results showed that, as expected, VBD of coke samples is affected by shape

and size distribution of the particles. Two-dimensional simulations confirmed that in general,

mixed-sized samples have higher VBD than mono-sized cokes and as the fraction of coarse

particles increases vibrated bulk density increases. However, existence of 10 wt.% of fine particles

to fill the pores between coarse particles is essential. For 3D simulations also, the model follows the

trend of experimental data showing that in the container of 2.9 mm diameter, as the content of -4+6

mesh (3.36-4.76 mm) particles increase, VBD increases. It can be concluded that a well-tailored

DEM model is capable of predicting the particle rearrangement and density evolution during the

vibration process.

“To my kind mom and the memory of my dear father. Without their encouragements and love

I never would have been able to achieve my goals.”

v

ACKNOWLEDGEMENTS

The author would like to express his sincere thanks to Dr. Houshang Alamdari for his

invaluable advices and kind supports during the course of this study.

The author gratefully acknowledges Prof. Mario Fafard for his supportive direction and

important advices on different aspects of the problem.

Advices, direction and encouragements of Dr. Donald Ziegler and Mr. Gilles Dufour from

Alcoa Inc. were very important and are acknowledged.

This project is financially supported by National Science and Engineering Research

Council (NSERC) and Alcoa Inc. which are gratefully appreciated. Financial support of

“Fonds de recherche du Québec – Nature et technologies (FQR-NT)" through "Le Centre

de recherche sur l'aluminium (REGAL)" is appreciated. A part of this research is

financially supported by Centre québécois de recherche et de développement de

l'aluminium (CQRDA) which is gratefully acknowledged.

The author expresses his appreciation of the valuable assistance provided by Mr. Pierre

Mineau and Mr. Luc Grondin at Aluminerie de Deschambault, ALCOA, for the

experiments. Kind helps, friendship and ideas of Regal-Alcoa research group members

specially Dr. Donald Picard, Dr. Reza Aryanpour, Guillaume Gauvin, Kamran Azari,

Francois Chevarin, Geoffroy Rouget and Ramzy Ishak are appreciated.

Table of Contents

Résumé ..................................................................................................................................... ii Abstract ................................................................................................................................... iii Table of Contents ..................................................................................................................... vi List of Tables .......................................................................................................................... vii List of Figures .........................................................................................................................viii 1. Introduction ..................................................................................................................... 1

1.1. Aluminium Production............................................................................................... 1 1.2. Carbon anodes in aluminum industry .......................................................................... 4 1.3. Coke ......................................................................................................................... 6 1.4. Pitch ......................................................................................................................... 8 1.5. Problem statement and project objectives .................................................................... 9 1.6. Thesis Outline ......................................................................................................... 10

2. Particle Packing ............................................................................................................. 11 2.1. Basic Considerations................................................................................................ 11 2.2. Vibrated Bulk Density Test ...................................................................................... 15

3. Discrete Element Method ............................................................................................... 17 3.1. Introduction ............................................................................................................ 17

3.1.1. Continuum and discrete modeling approaches ....................................................... 17 3.1.2. History of Discrete Element Method ..................................................................... 19

3.2. Fundamentals of DEM modeling .............................................................................. 20 3.2.1. The basic theory .................................................................................................. 21 3.2.2. Force-displacement law ....................................................................................... 22 3.2.3. Law of motion ..................................................................................................... 24 3.2.4. Time integration in DEM ..................................................................................... 25 3.2.5. Critical time-step ................................................................................................. 26 3.2.6. Time-step determination in PFC ........................................................................... 26

3.3. Discrete Element Method simulation applied to irregular-shape particles..................... 28 4. Methods and procedure for determination of Material Properties ....................................... 31

4.1. Introduction ............................................................................................................ 31 4.2. Methodology ........................................................................................................... 32 4.3. Balls stiffness approximation.................................................................................... 33 4.4. Friction coefficient estimation .................................................................................. 41

5. Image processing ........................................................................................................... 47 5.1. Introduction ............................................................................................................ 47 5.2. Shape Analysis ........................................................................................................ 47 5.3. Two-dimensional particle modeling .......................................................................... 53 5.4. Three-dimensional particle modeling ........................................................................ 57

6. Vibrated Bulk Density Simulations ................................................................................. 60 6.1. Revisions in the experimental method ....................................................................... 60 6.2. Two-dimensional simulations ................................................................................... 62 6.3. Three-dimensional simulations ................................................................................. 68

Conclusion .............................................................................................................................. 72 References .............................................................................................................................. 75

vii

List of Tables

Table 1.1. Typical properties of industrial pre-backed anodes [10] ............................................... 5

Table 1.2. Typical green and calcined coke properties [14] .......................................................... 8

Table 4.1. Parameters used to define the material (coke particles) in the DEM model and the

method of parameter determination ................................................................................... 31

Table 4.2 Particle size distribution of the coke samples used for 2D simulations. Weight percentage

of each size range has been given...................................................................................... 32

Table 4.3. Particle size distribution of the coke samples used for 3D simulations. Weight

percentage of each size range has been given. .................................................................... 33

Table 5.1. Definitions of sphericity and circularity in the literature ............................................. 48

Table 5.2. Results of measurements of sphericity and comparison of the results with calculations

by two 2-dimensional methods ......................................................................................... 53

Table 6.1. Experimental and 2D simulation results of VBD for mono-size samples ..................... 64

Table 6.2. Comparison of experimental and 2D simulation results of VBD for mixed samples ..... 65

viii

List of Figures

Figure 1.1. Composition of earth's crust [1] ................................................................................ 1

Figure 1.2. Aluminum, from extraction to products and recycling [7] ........................................... 3

Figure 1.3. Schematic cross section of Hall-Héroult reduction cell with pre-backed anodes ........... 3

Figure 1.4. Aluminum smelting process costs [9] ........................................................................ 6

Figure 1.5. Changes in supply and sulfur content in U.S green coke [11] ...................................... 6

Figure 1.6. Delayed coking process, reproduced from [14] .......................................................... 7

Figure 2.1. Random packing structure of non-spherical particles with their sizes represented by

equivalent packing diameter [24] ...................................................................................... 12

Figure 2.2. Packing of binary mixtures of spherical particles [39]............................................... 12

Figure 2.3. Packing of ternary mixtures of particles (diameter and length of cylinder 1 are 19.374

and 1.384 mm respectively and those of cylinder 2 are 4 and 30 mm as diameter and length

respectively [39] .............................................................................................................. 13

Figure 2.4. Comparison between experimental and calculated results of void fraction with tapping

numbers [40] ................................................................................................................... 14

Figure 2.5. Final correlation of experimental results of void fraction with the ones estimated by

equation (2.1) [40] ........................................................................................................... 14

Figure 2.6. VBD test setup ...................................................................................................... 16

Figure 2.7. VBD of calcined cokes as function of the HGI of corresponding green cokes calcined

[43]................................................................................................................................. 16

Figure 3.1. Continuum and discrete models [45] ....................................................................... 18

Figure 3.2. FEM and DEM analogy [46]................................................................................... 19

Figure 3.3. A simple DEM model of an assembly of spherical particles. ..................................... 21

Figure 3.4. Calculation cycle of explicit discrete element method [59] ........................................ 22

Figure 3.5. Ball-ball contact [59].............................................................................................. 23

Figure 3.6. Single mass-spring system [59]............................................................................... 27

Figure 3.7. Multiple mass-spring system [59] ........................................................................... 27

Figure 3.8. Clump of an arbitrary shape made by overlapping circles. Highlighted areas show the

inactive contacts and overlaps between the balls comprising the particle. ............................ 29

Figure 4.1. Particle packing with and .......................................................... 35

ix







Figure 4.8. Particle packing with and - ......................................................... 38

Figure 4.9. Variations of the volume of the particle assembly by change of stiffness ................... 39

Figure 4.10. Two-dimensional overlaps visualization, and ............................ 39




Figure 4.14. Angle of repose test on coke particles .................................................................... 42

Figure 4.15. Pile of coke particles in angle of repose test ........................................................... 42

Figure 4.16. 3D simulation of angle of repose test ..................................................................... 43

Figure 4.17. Estimation of friction coefficient between the plate and coke particles; a) experiment,

b) simulation ................................................................................................................... 44

Figure 4.18. Pile of the coke particles in AOR test, ....................................................... 45

Figure 4.19. Pile of the coke particles in AOR test, ....................................................... 45

Figure 4.20. Pile of the coke particles in AOR test, ..................................................... 45

Figure 4.21. Pile of the coke particles in AOR test, ..................................................... 46

Figure 5.1. Digitized 3D pattern of coke particles...................................................................... 48

Figure 5.2. Optical micrograph of coke particles within the size range of -6+14 mesh; a)non-

processed image; b) processed image to measure sphericity................................................ 49

Figure 5.3. Sphericity calculation based on non-polished particles for the size range of -30+50

mesh ............................................................................................................................... 50

Figure 5.4. Sphericity calculation based on non-polished particles for the size range of -14+30

mesh ............................................................................................................................... 51

Figure 5.5. Sphericity calculation based on non-polished particles for the size range of -6+14 mesh

....................................................................................................................................... 52

Figure 5.6. Micrograph of coke particles of the size range of -6+14 mesh ................................... 54

Figure 5.7. Skeletonization of the particle shape, step-1............................................................. 54

Figure 5.8. Fine skeletonization of the particle shape, step-2 ...................................................... 55

Figure 5.9. Edge detection to obtain the shape of the particles .................................................... 55

x

Figure 5.10. Filling the area and obtaining the shape in a black and white image ......................... 56

Figure 5.11. Coke particles as 2D clumps ................................................................................. 56

Figure 5.12(a) and (b). Samples of 3D particle modeling with spheres. The models were used in

PFC3D as clumps ............................................................................................................ 59

Figure 6.1. Vibration time effect on the volume of coke particle column in VBD test for different

sample sizes, -6+14 mesh size range ................................................................................. 61


sample sizes, -14+30 mesh size range ............................................................................... 61


sample sizes, -30+50 mesh size range ............................................................................... 62

Figure 6.4. Initial state of the system in 2D models of VBD test................................................. 63

Figure 6.5. Equilibrium state, after particles settlement in the container in 2D simulations ........... 64

Figure 6.6. Simulation of particle packings; a) sample M1, filling the surface roughness of coarse

particles by fines are shown by rectangles b) sample M3, the indicated areas show the filling

of the space between large particles by fines ...................................................................... 66

Figure 6.7. Simulation of particle packings; a) Regions near the container wall in sample S3,

filling the surface roughness of coarse particles by fines are shown by rectangles; b) Regions

near the container all in sample M3................................................................................... 67

Figure 6.8. Initial state of the particles in 3D simulation of VBD test. Sample M5 (composed of

50% of -6+14 and 50% of -4+6 mesh particles) ................................................................. 68

Figure 6.9. Equilibrium state before application of vibration; Sample M5 (composed of 50% of -

6+14 and 50% of -4+6 mesh particles) .............................................................................. 69

Figure 6.10. Experimental VBD test results of mixed-size samples; with standard method of 100g

and also 10g samples ....................................................................................................... 70

Figure 6.11. Experimental and simulation results of VBD test for mixed samples of 10g ............. 70

Figure 6.12. 3D simulations of VBD test; a) sample M5 with 50% of -4+6 particles; b) sample M7

with 80% of -4+6 particles ............................................................................................... 71

1. Introduction

1.1. Aluminium Production

Aluminium is the third most abundant element on the earth (after Oxygen and Silicon), making up

around 8.2 percent of it [1]. Composition of earth’s crust is shown in figure 1.1. However, due to its

high electronegativity, aluminium is never found as a pure metal, but combined with other

elements. The name of the metal comes from the Latin name for alum, alumen. In 1761 L.B.G de

Moreveau proposed the name alumina for the base in alum, and a few years later in 1787, the

French chemist Antoine Laurent Lavoisier showed that alumina is the oxide form of an unknown

metal [2]. The name aluminum, proposed by Sir Humphery Davy in 1807 for this metal and then it

changed to aluminium so to agree with spelling of most of the elements which are ended with

“ium”. It was the Danish physicist and chemist, Hans Christian Oersted who, for the first time, in

1825 developed a method to produce metallic aluminium [3].

Figure 1.1. Composition of earth's crust [1]

Overall chart of aluminum extraction and recycling processes is shown in figure 1.2. Extractive

metallurgy of aluminum involves two main and independent steps to obtain high purity metallic

aluminum from the ore (Bauxite). The first process, which is called “Bayer process”, outputs pure

alumina from bauxite. The first plant utilizing this method to purify bauxite started to produce pure

alumina in 1893 [4]. In this process, the aluminum-bearing minerals in the ore are selectively

extracted from the insoluble components by dissolving them in sodium hydrate solution. As an

instance, the chemical reaction for aluminium hydroxide, Gibbsite, is as follows:

(1.1)

2

The insoluble phases are separated from the aluminum-containing liquor by means of filtration.

Then in a process which is called precipitation, crystalline aluminum trihydroxide is obtained from

the following reaction:

(1.2)

The hydrate crystals are classified into size fractions and fed into a calcination kiln in which water

is driven off the material (reaction 1.3):

(1.3)

The refined aluminum oxide then goes to aluminum smelting plants to be reduced to metallic

aluminum.

Aluminium reduction is performed by electrolysis of dissolved alumina in a molten salt. This

process is called Hall-Héroult process. American chemist Charles Martin Hall and the Frenchman

Paul Héroult independently and simultaneously invented this method of reduction of alumina to

aluminum in electrolysis cells in 1886. The first large scale aluminum production plant was opened

by Hall in 1888 in Pittsburgh, Pennsylvania which later became as Aluminum Company of America

or Alcoa corporation. In this process, alumina is dissolved in a bath of molten cryolite (sodium

aluminum fluoride) at about 960˚C and an electric current of around 200 up to 350 KA is passed

through the electrolyte at low voltage. The anode of the cell is made of carbon. Molten aluminum is

denser than molten cryolite and it sinks to the bottom of the cell where it is periodically siphoned

off. The pool of already produced aluminum acts as cathode. Carbon anode and alumina are

consumed by the chemical reactions during the process and must be added to the cell accordingly.

The overall chemical reaction can be written as:

(1.4)

The worldwide annual aluminum production was around 44’100 Million tons by the end of 2011

[5]. All the smelters in the world use Hall-Héroult process which has remained almost unchanged

for a century. A schematic illustration of alumina reduction process has been shown in figure 1.3.

The process is carried out in electrolysis cells which are called ‘pots’. Refined aluminum oxide,

obtained from Bayer process, is dissolved in a bath of electrolyte which is composed of mainly

molten cryolite ( ) and additives such as aluminum fluoride ( ) and calcium fluoride

( ) at 960⁰C. Modern pots are steel containers large enough to produce 2 tons/day aluminum

[6]. The cell is lined with carbon cathode blocks and refractories. Anodes of the system are carbon

blocks which are partly immersed in the bath. Direct electrical current with high amperage and low

voltage passes through the anodes.

3

Figure 1.2. Aluminum, from extraction to products and recycling [7]

Figure 1.3. Schematic cross section of Hall-Héroult reduction cell with pre-backed anodes

4

Alumina is consumed and so is frequently fed to the cell. Cryolite is also required added to the pot

by the feeders. Molten aluminum sinks at the bottom of the cell and is extracted from the pot into

large crucibles which transports it to the casthouse. The entire pot is covered and reaction gases

(mainly and some ) are collected.

1.2. Carbon anodes in aluminum industry

As the main chemical reaction of the reduction process (equation (1.4)) shows, the anodes are

consumed during the process. The pot is designed to have two anodes to be replaced with new ones

each day. However, a controlling system adjusts the vertical position of the studs (see figure 1.3) to

keep the required anode-cathode distance (ACD). The production capacity of the cell is proportional

to the electrical current. Thus, high amperage current is necessary to obtain an economical metal

production rate. However, some limitations such as maximum current density for the anodes

practically restrict the production rate of the cell. In the most of the plants the current density of the

anodes is around 0.8 A/cm2 and it is limited to 1 A/cm

2 [6].

There are two types of anodes; self-backing Söderberg and pre-backed anodes. In Söderberg design,

anodes are continuously fed to the cell and backed in the cell using the heat generated for Hall-

Héroult process. Shortcomings of Söderberg anodes which include high environmental pollution

and lower compaction (which means higher consumption rates) led the industry toward pre-backed

anodes. Pre-backed carbon anodes are backed in a separate kiln and periodically replace the

consumed anodes.

Carbon anodes are a kind of composite materials composed of coal tar pitch as the binder and

petroleum coke as filler. In electrolysis process it is not possible to consume the whole anode and so

the reminder of the anode, butts, are recycled, washed and added to the recipe of the anode paste

[8]. Typical pre-backed anode is composed of around 65% coke, 15 % pitch and 20% of anode

butts. Coke particles are crushed and sized to the required size distribution and are mixed with

granulated recycled butts to form the dry aggregate recipe. The mixed powder is then heated to

about 160˚C and mixed with pitch at 150-180˚C. The obtained mixture is called ‘green paste’. The

green paste is then formed by means of either vibro-compaction or pressing to anode blocks. Green

anodes are then cooled and stored to be backed at 1100˚C. The obtained backed anodes have high

density and mechanical strength and can be used as electrodes in the Hall- Héroult cells. Typical

properties of pre-backed anodes have been given in table 1.1.

5

Table 1.1. Typical properties of industrial pre-backed anodes [10]

Anode Property Unit Typical Range Typical 2σ Method

Green Apparent Density kg/dm3

1.55-1.65 0.03 Dimension

Baked Apparent Density kg/dm3 1.50-1.60 0.03 ISO N 838

Baking Loss % 4.5-6.0 0.5 Dimension

Specific Electrical Resistance

µΩm 50-60 5 ISO N 752

Air Permeability nPm 0.5-2.0 1.5 RDC 145

Compressive Strength MPa 40-50 8 DIN 51910

Flexural Strength MPa 8-14 4 ISO N 848

Static Elastic Modulus GPa 3.5-5.5 1 RDC 150

Dynamic Elastic Modulus GPa 6-10 2 Grindosonic

Coefficient of Thermal Expansion

10-6

/K 3.7-4.5 0.5 RDC 158

Fracture Energy J/m2

250-350 100 RDC 148

Thermal Conductivity W/mK 3.0-4.5 1.0 ISO N 813

High electrical conductivity, high mechanical strength and homogeneity, and also low reactivity

towards air and carbon dioxide are very important quality indices of carbon anodes [10]. As given

in figure 1.4, it has been shown [9] that anode cost represents about 17% of the cost of the

aluminum smelting process belongs to the anodes. However, since the anodes are consumed during

the process any deviation from the required quality ranges affects the anode consumption rate and

increases the process costs. Both raw materials properties and process parameters influence the final

properties of anodes.

Process parameters could be tailored according to changes in raw materials properties (as an

instance softening point of pitch). Nowadays, smelters obtain the raw materials from several

sources which may come with different properties. Distinct changes in properties of available coke

and pitches in the market have been observed. For example, content of impurities such as sulfur and

vanadium in petroleum coke has been increased [11] and for pitch, drop in QI content and raise in

softening point have been reported [12]. Figure 1.5 shows the changes in supply, and sulfur content

changes in U.S green coke from 1982 to 2000.

6

Figure 1.4. Aluminum smelting process costs [9]

Figure 1.5. Changes in supply and sulfur content in U.S green coke [11]

1.3. Coke

Calcined coke is the most important raw material of anode manufacturing process. Calcined coke is

obtained by calcination of green coke which is a by-product of crude oil making about 6 volume %

of oil refinery output [13]. Depending on the process used and operating conditions, petroleum coke

can be of different types. Delayed coking and fluid coking are two means of green coke production.

Delayed coking is a thermal cracking process in which petroleum residuum is upgraded and

converted into liquid and gas products and a concentrated carbon material (petroleum coke)

7

remains. Delayed coking process has been summarized in figure 1.6. In fluid coking, fluidized bed

of coke particles is maintained at 500⁰C and 20-40 psi and heavy oil is sprayed into the vessel. The

feed vapors are cracked while forming a liquid film on the coke particles. A portion of the

remaining coke is returned to the fluidized bed and the balance is withdrawn as product.

Fuel grade cokes are sold as green coke, but cokes used in aluminum industry to produce anodes

must be calcined at around 1300⁰C. Table 1.2 compares the properties of green and calcined coke

produced with different methods. Consumption of coke for aluminum smelting process is 0.4-0.5 kg

per kg of produced metallic Al. Aluminum industry was using around 10 million tons of calcined

coke per year during 80s [14] and due to increase of global aluminum production in the last twenty

years, this consumption level is now more than 20 million tons per year.

Figure 1.6. Delayed coking process, reproduced from [14]

8

Table 1.2. Typical green and calcined coke properties [14]

characteristics From delayed coking Needle coke

Crude Calcined Crude Calcined

Moisture, % 6-10 0.1 6-10 0.1

Volatiles, % 8-14 0.5 4-7 0.5

Sulfur, % 1.0-4.0 1.0-4.0 0.2-2.0 0.5-1.0

SiO2, % 0.02 0.02 0.02 0.02

Fe, % 0.013 0.02 0.013 0.02

Ni, % 0.02 0.03 0.02 0.03

Ash, % 0.25 0.4 0.25 0.4

V, % 0.015 0.03 0.015 0.02

Bulk density, g/cm3

0.720-0.800 0.673-0.720 0.720-0.800 0.673-0.720

Real density, g/cm3

2.06 2.11

1.4. Pitch

Coal tar pitch is a residue produced by distillation or heat treatment of coal tar. Pitch is a complex

material composed of polycyclic aromatic compounds [15] with a broad molecular weight

distribution. This structure provides binder properties to pitch and also the potential to have a

transformation into graphitizable carbon by pyrolysis [16]. Due to its properties, coal tar pitch is

widely used in different industries such as production of anodes for aluminum industry and

electrodes for electric arc furnaces [17].

Crude tar composition, the distillation method and efficiency and further treatments of pitch are the

factors influencing the physical and chemical properties of coal tar pitch. Due to structure, pitch has

no melting pint and shows a thermoplastic behavior. Thus, it starts to soften at temperatures lower

than the melting point of its individual components [18]. Softening point of pitch is one the

parameters which are used as indicators of pitch quality. Carbon anode plants use pitches with

mettler softening point between 100 and 120⁰C [19]. Coking value, density, quinoline insolubles

(QI), toluene insolubles (TI) and viscosity are other important parameters which are used as quality

indices of pitch in aluminum industry [20].

9

1.5. Problem statement and project objectives

Quality and properties of anodes have a considerable effect on the efficiency of electrolysis cells.

Carbon anodes are produced by mixing granulated calcined petroleum coke and coal tar pitch,

followed by either pressing or vibro-compaction, and baking steps. Physical and chemical

properties of raw materials influence the quality of the anodes. Porosity of calcined coke is an

important parameter in carbon anode production which is used in determination of pitch demand.

As an instance, increasing the bulk density of coke enhances the baked apparent density and also

reduces the air permeability of the anodes. Parameters which affect the bulk density of coke include

the shape, size, and size distribution of coke particles. Vibrated Bulk Density (VBD) of calcined

cokes is also related to properties of green coke such as Hardgrove Grindability Index (HGI) and

volatilies content. However, exact effects of shape and size distribution of coke particles especially

in the range that are widely used in anode plants are not fully understood and need further

investigations.

On the other hand, the presence of several parameters (including raw materials properties and also

process parameters) affecting the final properties of the baked anodes, makes it difficult to tailor the

best parameters matrix to obtain high quality anodes.

Computer simulations, in such cases, can potentially provide valuable information. The materials

which are used to produce carbon anodes are granular and using a discrete method instead of a

continuum approach can suit better with the physics of the materials. Moreover, applying a discrete

numerical modeling in carbon anode production process can provide more information about the

micromechanisms of packing, compaction and flow of the anode paste.

In this project the feasibility of application of Discrete Element Method in carbon anodes

production process is studied. Discrete Element Method is coupled with image processing

techniques to model the coke particles. This method allows one to evaluate the effects of coke

particles characteristics such as shape, size and size distribution on the bulk density of coke

assemblies. In this study both two-dimensional and three-dimensional DEM simulations are applied

to the Vibrated Bulk Density (VBD) test of coke particle samples. Application of DEM to simulate

the settlement and vibration of coke particles in VBD test is studied. Using coke particles with

different shape and size distribution and comparing the experimental data of VBD with simulation

results, predictions of DEM models are verified. The specific goals of this study are summarized as

following;

10

Study the feasibility of application of DEM simulations in carbon anodes production

technology;

Develop an approach by combining DEM with image processing to evaluate the shape

factor effect in particle packing;

Study the effects of coke particles shape, size, and size distribution on the vibrated bulk

density of industrial coke samples using numerical simulations.

1.6. Thesis Outline

This thesis is organized in eight chapters as follow:

Chapter I introduces the Aluminum smelting process, carbon anodes, and motivations of

this study followed by the objectives and the outline of the dissertation.

Chapter II presents particle packing problem and its importance in anode production.

Chapter III contains the history and fundamentals of the numerical modeling technique

which is used in this study.

Chapter IV presents the methodology of the present research work, and the approaches

used to estimate the micro-properties of the material.

Chapter V contains the image processing techniques used to digitize and model the coke

particles in both 2D and 3D simulations.

Chapter VI presents the simulation results of VBD test, comparison with experimental data

followed by discussions on the results.

Chapter VII includes the conclusions of this study and final recommendations of this work.

11

2. Particle Packing

2.1. Basic Considerations

Granular materials are now widely used in different industries, from pharmaceutical industries to

soils and industrial powders [21]. Understanding the mechanisms of granular flow, however, is very

important for the industries dealing with granular materials. Packing density of granular assemblies

is also an important centuries-old problem for different industrial processes [22]. For example the

packing density in ceramic industry is believed to have a considerable effect on the shrinkage and

density during the sintering process and as a result on the final properties of the products [23].

The packing characteristics of granular assemblies can be influenced by several parameters such as

particle size distribution, particle shape, friction and surface chemistry [24]. It is believed that the

particle size distribution is the most important parameter affecting the packing density [41] (see

figure 2.1). There have been considerable research works on this subject in the last century [25-33].

However, the case for more complex systems such as multi-component and non-spherical

assemblies still needs more investigations.

In 1981, Ouchiyama and Tanaka [34] developed a theoretical model to predict the packing density

of particulate systems from the size distribution. This model showed a limited efficiency to predict

the porosity of binary, ternary and quaternary spherical systems [35, 36]. Some other researchers

such as Furnas [37] and Westman [38] used discrete methods to deal with packing problems. As an

instance, Westman [38] in 1936 proposed a simple model which can fairly predict the porosity of

binary mixtures. Ai-Bing et al. [39] in 1992 published an interesting work on the packing of ternary

nonspherical particles. By an experimental investigation, they showed that there is a similarity

between the packing systems of spherical and nonspherical particles. Figure 2.2 gives the packing

density of binary mixtures of spherical particles which shows the importance of having a well-

tailored mixture of fine and coarse particle to obtain the maximum density.

The variations in the packing density of ternary mixtures of spherical and cylindrical particles with

different proportions are shown in figure 2.3. As it can be seen, more spherical particles in the

mixture result in higher packing density. Cylinder 2 is very elongated and it is the reason why the

maximum packing density happens at low fractions of cylinder 2. However, cylinder 1 is a disk

with diameter of 19.374 mm and length of 1.384 mm. It is also interesting that the maximum

12

Figure 2.1. Random packing structure of non-spherical particles with their sizes represented by

equivalent packing diameter [24]

Figure 2.2. Packing of binary mixtures of spherical particles [39]

13

Figure 2.3. Packing of ternary mixtures of particles . (diameter and length of cylinder 1 are 19.374 and

1.384 mm respectively and those of cylinder 2 are 4 and 30 mm as diameter and length respectively [39]

packing density (0.66 g/cm3) doesn’t belong for the sample with 100% spheres but for a mixture of

spheres and the cylinders.

Packings of nonspherical particles need more investigations to elucidate the effects of shape and

size of the particles on the packing characteristics. Miyajima et al. [40] used image analysis to

characterize the shape parameters (circularity and smoothness) of silica, and glass and mullite bead

particles. They used a tapping apparatus to measure the packing density of the particles in a

cylindrical container. They proposed the following equation to express the void fraction of the

particles;

(2.1)

where is the initial void fraction (before tapping), is when the tapping is finished (300000

times) and is the void fraction of the system at any time. The parameters in the above equation can

be calculated as follows;

14

where , and are the average circularity, the average smoothness and the average diameter of

the particles (by sieve analysis) respectively and is true density of the particles.

Using equation (2.1) Miyajima et al. [40] calculated the void fraction for different particle

assemblies. The obtained results have been shown in figures 2.4 and 2.5. It seems that equation

(2.1) can fairly predict the void fraction in compaction process of particle assemblies.

Figure 2.4. Comparison between experimental and calculated results of void fraction with tapping

numbers [40]

Figure 2.5. Final correlation of experimental results of void fraction with the ones estimated by

equation (2.1) [40]

15

However, application of experimental equations like equation (2.1) is not always easy and useful.

For the case such as coke particles in the present work or aggregates in concrete mixes in which

particles have irregular shapes with wide ranges of sphericity, roughness and etc, providing an

equation to fairly predict the packing density can be very difficult. Numerical approaches and in

particular discrete methods, which are in accordance with physics of particulate systems, are

considered as powerful tools to study the packing mechanisms and predict the flow and/or packing

density of particle assemblies. In the present work, discrete element simulations are used to study

the packing density of particulate systems (coke samples).

2.2. Vibrated Bulk Density Test

Effects of calcined coke characteristics on the properties of carbon anodes for aluminum industry

have been well researched [41]. One important quality factor of coke is its porosity which is used to

determine pitch demand in making anode paste. Since 1960s researchers in carbon and aluminum

industries have been working to develop a method to measure the porosity of coke samples. It was

finally in 1983 that ASTM D4292, Standard Test Method for Determination of Vibrated Bulk

Density of Calcined Petroleum Coke was issued. Vibrated Bulk Density (VBD) test is considered as

an easy and rapid method to estimate the coke packing density [41].

VBD test apparatus has been shown in figure 2.6. This setup contains three parts. Part 1, as shown

in figure 2.6, is a vibrating funnel which transfers the powder to the main container. According to

the standard ASTM D4292, the transfer time from the funnel into the graduate must be 70 to 100

seconds. Part 2 of the setup is the graduated cylinder of 250 ml size with the inside diameter of

approximately 37 mm made of glass. The third part is the vibrator which must be capable of

vibrating a 215 g graduated cylinder containing 100g coke sample at a frequency of 60Hz and

amplitude of 0.20 to 0.22 mm. A quantity of 100.00 ±0.1g of coke fraction is charged in the funnel

and the vibrating table vibrates the cylindrical container for 5 minutes at 60Hz. Then, by measuring

the height of the particle column the volume of the sample is calculated according to ASTM D4292

and the VBD value can be calculated:

16

Figure 2.6. VBD test setup

Packing density of calcined coke and also VBD of coke samples are affected by many production

variables [42]. VBD of calcined coke can be predicted (by a limited extent) from green coke

properties such as Hardgrove Grindability Index (HGI) and the volatiles content [43]. As an

instance, as shown in figure 2.7, VBD of calcined coke decreases by increasing the corresponding

green coke HGI [43]. Shape and size [42-44] of the particles are considered as important parameters

affecting the VBD of the coke samples.

In the present work, coke samples from the same source with the same chemical properties have

been used and only the effect of particles shape and size distribution on the VBD of coke samples is

studied.

Figure 2.7. VBD of calcined cokes as function of the HGI of corresponding green cokes [43]

17

3. Discrete Element Method

3.1. Introduction

3.1.1. Continuum and discrete modeling approaches

In numerical modeling of engineering problems, there are some problems which are modeled using

a finite number of well-defined components. The behavior of these components are either well

known of independently calculated using mathematical methods. Interactions between the

individual components (elements) define the global behavior of the problem. Such systems are

termed discrete.

However, in some problems definition of such independent components may require an indefinite

subdivision of the problem domain. In such case, implying an infinite number of components, the

problem is treated using the mathematical fiction of an infinitesimal. Such systems are termed

continuous and have infinite degrees of freedom. In these systems, the problem domain is

subdivided into finite number of sub-domains. Simple mathematical equations describe the behavior

of these sub-domains (elements). A continuous system with infinite degrees of freedom is here

approximated using a discrete system of elements with finite degrees of freedom. Another

characteristic of continuous and discrete systems which provide two distinct and different

approaches in numerical modeling is the way ‘discontinuities’ are defined in a computational

model. Continuum models may include discontinuities in the medium explicitly or implicitly.

However, in discrete models discontinuities are represented explicitly. For a particular system, it is

the size of the discontinuities with respect to the size of the problem which provides an idea of the

need to use a continuum or discontinuum method.

This can be more elucidated using a qualitative guideline as shown in figure 3.1. Figure 3.1(a)

shows an opening in a continuous medium. The displacement field here is continuous and thus

using a continuum method is recommended. The medium in figure 3.1(b) is divided into a number

of small continuous regions. The displacement field is continuous inside each region but may be

discontinuous across the regions. However, still a continuum model should be able to describe the

displacement. The medium in figure 3.1(c) is composed of discontinuities such that the scale of

regions is within the scale of the opening. In this case, the displacement is by rotation of the blocks

and also slip along the discontinuities and so a discontinuum method is more appropriate. The

blocks making up the medium in figure 3.1(d) have a size much smaller than the size of the

opening.

18

Figure 3.1. Continuum and discrete models [45]

In this case the displacement field is pseudo-continuous and a continuum model is more

appropriate. In all discrete element method (DEM) models, the problem is mathematically stated

by the following equation which is solved at discrete time intervals (damping has not been

considered here);

(3.1)

where M is the mass matrix, a is the acceleration vector, K is the stiffness matrix, is the

displacement vector and is the force vector. This equation is similar to those used in finite

element modeling of continua. Thus, an analogy can be drawn between discrete and finite element

frameworks. Finite element nodes correspond to the discrete element particles and finite elements

correspond to the inter-particle contacts [46] (see figure 3.2). In a FEM model, material properties

are assigned to the elements and the medium is discretized and the equations are numerically solved

at the nodes. Correspondingly, in DEM models, the global properties of the material are defined by

assigning appropriate models to the contacts. However, since the material properties inputs for

DEM models are micro-properties of the material, they need to be calculated or approximated

separately.

19

Figure 3.2. FEM and DEM analogy [46]

Efficiency and suitability of a discrete or a continuous model depends on the individual problem of

interest and both approaches have their own advantages and drawbacks. As an instance, in studying

the fracture in rock mechanics, DEM fails to provide a known uncertainty about the geometry of the

fracture system [47] and as a result use of some statistical analyses on the fracture population is

often required.

Another well-known shortcoming of DEM models is the calculation time. Discrete element method

a proposed by Cuncall and Strack [48] uses an explicit centered- finite difference scheme for time

integration. However, the stability of the numerical solution is conditioned to the size of the time-

steps. Thus, using a time-steps as small as 10-8

s is sometimes inevasible.

However, DEM has significant advantages specially for studying the flow and fracture in granular

systems. Using this method, macroscopic deformations are better understood through the studying

of the microscopic mechanisms.

3.1.2. History of Discrete Element Method

Discrete Element Method (DEM) introduced for the first time by Cundall [49] in 1971 and then

developed by Cundall and Strack [48] in 1979, is now considered as a powerful approach to study

the micromechanics of materials. Although DEM can be used to simulate deformations in

continuum problems, it is specially very comprehensive and versatile for studying the flow and

deformation [50] in granular materials. Range of applications of discrete element method as a

numerical modeling tool is expanding rapidly and it now covers engineering processes in different

industries such as pharmaceutical, powder metallurgy, geological and construction industries.

20

In DEM models, microscopic particulate system interactions are used to derive the macroscopic

constitutive laws of granular materials [51-53] using so-called molecular dynamics [54]. There are

two separate modeling approaches in molecular dynamics, hard sphere and soft sphere methods.

In hard sphere molecular dynamics which is also called Event-Driven (ED) approach, a collision

matrix determines the collisions as well as momentum change of the particles. Particles are

considered as perfectly rigid spheres and so collisions occur instantaneously and change the

velocity and (as a result) energy of the spheres [55-57]. Dynamic equations of motion, here, are

solved using an explicit time-stepping scheme by finite difference discretization.

The second numerical method in molecular dynamics is so-called soft sphere method. In this

method, spheres can have overlaps and a pre-defined contact model determines the propagated

contact forces which are used to calculate the acceleration of the particles based on Newton’s

second law.

There are two solution schemes for the soft sphere approach;

1- Explicit solution based on finite difference discretization. This method is often called

Distinct Element method.

2- Implicit method in which a finite element scheme discretizes the body and matrix equation

governs both the motion and deformation of the elements. This method is called implicit

DEM or discontinuous deformation analysis (DDA) [58].

In this project an explicit DEM scheme using Particle Flow Code (PFC) is used to study the packing

density of coke particles. Thus, the solution approach of the explicit DEM is explained here in detail

which is mainly based on the theory explanation in PFC user manual.

3.2. Fundamentals of DEM modeling

In the discrete element method using a dynamic process of interactions between elements, a

complex behavior of a material can be simulated. Interactions between the elements follow the pre-

defined contact constitutive models. A simple three-dimensional DEM model can be seen in figure

3.3. Movements can occur due to either an external force or motion of a discrete element and

equilibrium state develops whenever there is a balance in internal forces [59].

21

3.2.1. The basic theory

The basic concept of DEM modeling is simple in which deformation or flow of a material is

simulated by successively solving the law of motion for each element and force-displacement law

for each contact. In this dynamic process, a centered finite difference scheme solves the equations

by a time-stepping algorithm assuming that the time-step is well small such that the velocities and

accelerations are constant within each time-step.

Figure 3.3. A simple DEM model of an assembly of spherical particles

The calculation cycle has been summarized in figure 3.4. At the beginning, the position of all

elements and walls (boundary conditions) are known. The algorithm tries to detect the contacts

according to the known positions of the elements and so the magnitude of the possible overlaps

between the elements is detected. Then by applying the force-displacement law, the propagated

contact forces are calculated. The forces then inserted in law of motion for each ball and the

velocity and acceleration of the balls are calculated. According to the obtained values, updated

positions of all balls and walls in the current time-step are determined. This cycle of calculations is

repeated and solved at each time-step and thus the flow or deformation of the material is simulated.

22

3.2.2. Force-displacement law

In the present work, the commercially available discrete element code, Particle Flow Code, PFC3D

and also PFC2D developed by Itasca Consulting Group Inc. are used. The basic assumptions in

particles flow modeling with PFC and other distinct element method code are as flowing;

a) The elements are circles (in 2D) and spheres (in 3D) and considered as rigid bodies.

b) The contact area is very small compared to the size of the balls (i.e. at a point)

c) The balls can have overlaps due to the applied forces and the magnitude of the overlaps is

related to the contact force via the pre-defined contact model.

d) Irregular shape particles can be created with combination of balls.

Figure 3.4. Calculation cycle of explicit discrete element method [59]

The basic equation of DEM models is Newton’s equation of motion which can be expressed as

(3.2)

where , and are linear acceleration, velocity and displacement vectors respectively. M is the

mass of the ball, D is damping coefficient (if any), R is internal restoring force and F is external

force.

For the ball-ball contact, as shown in figure 3.5, the unit normal vector that defines the contact

plane is given by;

Update particle & wall position

Force-displacement law

for all contacts

contact force

law of motion for all particles

23

(3.3)

in which

and

are the position vectors of the elements and d is the distance between the

elements centers. The location of the contact point can be also determined easily as;

(3.4)

The contact force can be decomposed into the normal and shear components at the contact point.

(3.5)

in which and

are the normal and shear components of the contact force. The normal and shear

contact forces can be calculated by the following equations;

Figure 3.5. Ball-ball contact [59]

(3.6)

(3.7)

where and are the normal and shear stiffnesses of the contact respectively, is the overlap,

is the unit normal vector of the contact plane and is the increment of shear displacement.

It should be noted that the shear component of contact force vector, is incrementally computed. At

the beginning, by detecting the contact, the shear contact force is defined as zero, but the motion of

24

the contact adds an increment to the elastic shear force at each time-step. Motion of the contact is

monitored and measured by updating and

. Thus by calculating the rotation about the line of

intersection of the new and old contact planes and also the rotation about the new normal direction,

the value and direction of shear contact force is calculated.

For the linear contact model, normal secant stiffness as well as shear tangent stiffness is given by

the following equations;

(3.8)

(3.9)

3.2.3. Law of motion

Particles move due to the active forces and moments. Translational and rotational motion of the

center of mass of the particle can describe the motion vectors. The following equation relates the

translational motion to the resultant force;

(3.10)

where is the sum of all externally applied forces on the particle, is the mass of the particle,

is the acceleration vector, and is the gravity acceleration vector.

The equation of rotational motion is expressed as;

(3.11)

in which is the total moment acting on the particle and is the angular momentum. The

coordinate system for this equation is the local system, centered at the particle center of mass.

However, if it is assumed that the orientation of this local coordinate system lies along the principal

axes of inertia of the particle, equation (3.11) can be reduced to the Euler’s equation of motion [60]

as follows;

(3.12)

, and are the principal moments of inertia, , and are the angular accelerations

about the principal axes, and , and are the components of the resultant moment.

Since the particles are spherical here and so the center of mass coincides with the sphere center, the

principal moments of inertia are equal to one another. Moreover, any local coordinate system

attached to the center of mass will be a principal axis system. As a result, equation (3.12) for a

spherical particle can be simplified and referred to the global axis system;

25

(3.13)

in which R is the radius of the sphere.

Equations of motion are now mathematically defined (3.10 and 3.13). These equations are

discretized using a centered finite difference method with a time-step of and then are integrated.

It should be noted that the values of velocities (translational and rotational) are computed at the

mid-intervals of while the other quantities (position, accelerations, resultant force and

moment) are computed at the primary intervals of . The equations for the accelerations are

expressed as

(3.14)

(3.15)

The values for the velocities can be solved by inserting equations (3.14) and (3.15) into (3.10) and

(3.13).

(3.16)

(3.17)

Using the values of the velocities, the updated position of the center of mass of the sphere is

calculated as

(3.18)

Since at the beginning the values of

,

,

,

and

are known,

and

are obtained according to equations (3.16) and (3.17). Then

is calculated by

equation (3.18). The values of

and

which are used in the next cycle, calculated

from the force-displacement law.

3.2.4. Time integration in DEM

Time integration scheme is an important consideration for any discrete element model. The choice

between an implicit or explicit method of time integration mostly depends on the type of the partial

differential equations as well as smoothness of data [61].

The important feature of implicit method of time integration compared to the explicit method is the

enhanced stability of the solver. It is believed that even for (at least some certain) nonlinear

problems the method is unconditionally stable [61]. However, the unconditional stability of the

26

implicit method comes with significantly computationally-expensive equations. In fact, in an

implicit method asset of nonlinear algebraic equations needs to be solved at each time-step. These

equations are usually solved using Newton-Raphson’s method.

Centered finite difference method is one of the most popular explicit solutions in computational

mechanics. The advantage of explicit method is that it is easily implemented and more importantly

the numerical algorithm seldom fails. However, the stability of the solution is conditioned. The

condition is the size of the time-step which needs to be smaller than a critical value, otherwise the

errors in the calculations propagate unboundedly and the final solution will be erroneous. [47,61].

Since in this project, a discrete element method code utilizing an explicit scheme is used to simulate

the packing of coke particles, critical time-step calculations is briefly discussed here.

3.2.5. Critical time-step

It has been shown [62,63] that the maximum time increment ( ) is a function of the eigenvalues

of the current stiffness matrix. As it has been given [46] for a linear undamped system, the critical

time-step can be expressed as;

(3.19)

in which is the maximum frequency which is related to the maximum eigenvalues ( ) of

the matrix (M= mass matrix, K= stiffness matrix).

√ (3.20)

However, since calculation of maximum eigenvalues of a large matrix is difficult, the following

formula is often used to avoid large calculations [46];

(3.21)

where is the maximum eigenvalues of the matrix for element “e”. The critical time

increment is then estimated by applying equations (3.20) and (3.21) and knowing .

3.2.6. Time-step determination in PFC

As mentioned earlier, PFC as a distinct element method code utilizes an explicit finite difference

scheme to integrate the equations of motion. As a result the solution stability is conditioned to the

size of time-steps. In PFC a simple procedure estimates the critical time-step at each cycle and the

actual time-step is taken as a fraction of the estimated critical value.

The procedure of the critical time-step estimation can be explained using a simple example of a

one-dimensional mass-spring system, as shown in figure 3.6, the motion of the point mass is

mathematically expressed by the following equation;

27

Figure 3.6. Single mass-spring system [59]

(3.22)

It has been shown [48] that the critical time-step for the second order finite difference solution of

this equation is;

, √ (3.23)

where m,k and T are the mass , stiffness of the spring and the period of the system respectively.

Now, as shown in figure 3.7, a system with infinite series of point masses and springs is considered.

The smallest period of such a system will be when the masses have an opposite synchronized

motion such that there is no motion at the center of each spring. The equivalent system shown in

figures 3.7 (b) and (c) can describe the motion of a single point mass. The critical time-step for this

system is given by [59];

Figure 3.7. Multiple mass-spring system [59]

28

√ √ (3.24)

However, DEM simulations in PFC2D and PFC3D can be complicated assemblies of discrete

particles and springs and each particle and spring may have different mass and stiffness. Thus, to

estimate , equation (3.24) is separately applied to each degree of freedom of each body. The

minimum of all , estimated for all degrees of freedom of all bodies, is considered as the

critical time-step of the system.

3.3. Discrete Element Method simulation applied to irregular-shape particles

As mentioned in the chapter 3.1, basic elements of any DEM model are circle (2D) or spheres (3D).

However, any arbitrary shape of particles can be generated using combination of the elements.

These super-particles are called clumps. Figure 3.8 shows a simple example of a clump composed

of circles.

In Particle Flow Code (PFC), clumps are considered as rigid bodies. During the calculation, only

the external contacts of a clump are active (and so calculated) and the code skips the internal

contacts between the balls comprising the clump. Balls within a clump may have overlaps of any

extent (see figure 3.8) and contact forces will not generate and the relative position of balls within a

clump will remain unchanged (clumps are rigid). This option provides an opportunity to introduce

any irregular shape particles to a DEM model which can help to have more realistic simulations of

particle assemblies and also to evaluate the shape factor effects on flow and packing characteristics

of particulate systems. A clump is defined with the number, radius and coordinates of its balls.

The basic mass properties of a clump are the total mass (m), center of mass (

), and moments and

products of inertia ( and ). These properties can be mathematically expressed by the following

equations [59];

∑

(3.25)

∑

(3.26)

∑ { (

)(

)

}

(3.27)

∑ { (

)

}

(3.28)

29

Figure 3.8. Clump of an arbitrary shape made by overlapping circles. Highlighted areas show the

inactive contacts and overlaps between the balls comprising the particle.

Moments and products of inertia are calculated with respect to a coordinate system which is

attached to the center of mass of the clump and aligned with the global axis system.

The resultant force, which is the sum of all externally applied forces acting on the clump, can be

expressed as;

∑

∑

(3.29)

in which is the externally applied force on the clump,

is the externally applied force acting

on particle (p), and

is the force acting on particle (p) at contact (c).

The resultant moment about the center of mass of the clump is calculated by

∑

(

)

∑

(3.30)

in which is the externally applied moment acting on the clump,

is the externally applied

moment acting on particle (p),

is the resultant force acting on the centroid of particle (p), and

is the force acting on particle (p) at contact (c).

Rotational motion of a clump is given by the vector equation of , where is the time rate

of change of the angular momentum of the clump [59]. can be written as

30

(3.31)

where .

The equations of motions are integrated using a centered finite difference scheme (the same way of

discrete balls as explained earlier). For rotational motion equation, according to equation (3.31), one

can write

(3.32)

and it can be written in matrix form as

{ } { } { } (3.33)

where

{

}

{

}

[

]

{

} {

}

Equation (3.33) is a differential equation which is solved using an iterative procedure. First we set

, then

is set to the initial angular velocity and equation (3.33) is solved for

. Then, a

new angular velocity is determined as;

. The estimation for can now be

revised by

)/2. Then, we put and the procedure goes to

equation (3.33) to solve it for

. It has been shown [59] that the solution converges after four

iterations.

31

4. Methods and procedure for determination of

Material Properties

4.1. Introduction

Since the algorithm of DEM simulations is based on finite difference formulation of particle

motions, the input material properties are micro-properties instead of macro-properties. Micro-

properties of a material with known macro-properties and with a simple packing arrangement of

particles can be predicted from its commonly measured macro-properties [64]. However, for a

material with an arbitrary packing arrangement (the case in the present work), micro-properties,

such as particle stiffness and friction coefficient, need to be estimated by micro-properties

calibration procedure. Table 4.1 gives the material parameters as well as the methods used to

determine them in this project. Apparent density of coke samples is already known and it was used

as the density of balls comprising the clumps in DEM models. However, the other material

properties, as given in table 4.1, need to be determined. Sphericity was used as the parameter

defining the shape of the particles. Thus, sphericity values for different samples were obtained using

image analysis.

To estimate the friction coefficient and particle stiffness, a calibration procedure is required. It is

worthy to note that in this specific application of DEM simulation (vibrated bulk density test) there

is no deformation in coke particles. Thus, the effect of particle stiffness is limited to the extent of

overlaps between coke particles and also energy dissipation in vibration process.

Table 4.1. Parameters used to define the material (coke particles) in the DEM model and the method of

parameter determination

Property Determination method

Apparent density Known

Friction coefficient Micro-properties calibration

Particle stiffness Micro-properties calibration

Particle shape (sphericity) Image processing

Particle size distribution Image processing

32

On the other hand, stiffness of the balls has a direct effect on the size of the time-step of discrete

element method simulations and as the stiffness increases a larger time-step is required to have a

stable numerical solution. Therefore, the stiffness of the balls in the models needs to be decreased to

the minimum value by which realistic overlaps develops between the particles.

4.2. Methodology

In this study, Conoco calcined coke samples which are currently used in Alcoa Deschambault

smelter for making prebaked anodes, have been selected as the coke source. Vibrated bulk densities

of the samples with different size distributions are studied.

For the first step of the work, coke samples within the size ranges of -30+50, -14+30 and -6+14

mesh have been selected for the study. It is noteworthy that these size ranges are common ranges in

carbon plants. Three mono-size range coke samples (S1, S2 and S3) and three mixed-size samples

(M1, M2 and M3), as given in table 4.2, examined for the VBD tests as well as DEM simulations.

VBD of these samples was obtained according to ASTM D4292. Then, two-dimensional discrete

element method was used to simulate the VBD tests using Particle Flow Code (PFC2D) software

developed by Itasca Consulting Group Inc. Experimental and simulation results were then

compared.

Since coke particles have irregular shapes and the shape of particles has considerable effect on the

packing density of particle assemblies, morphological studies on each sample have been performed

using 2D image analysis method by optical microscope powered by Clemex software. Shape

parameters such as circularity and roughness and also size distribution of the particles at each size

range were obtained and were used to model the samples.

Table 4.2 Particle size distribution of the coke samples used for 2D simulations. Weight percentage of

each size range has been given.

Samples Mesh Size range

-30+50 -14+30 -6+14

S1 100% - -

S2 - 100% -

S3 - - 100%

M1 50% 40% 10%

33

M2 30% 40% 30%

M3 10% 40% 50%

In the second step, two size ranges of 6+14 mesh and -4+6 mesh were selected for experimental and

simulation studies. Samples as presented in table 4.3 were prepared and VBD values were obtained

experimentally as well as numerically by means of 3D-DEM simulations. Details of the approaches

used to model the coke particles for 2D and also 3D simulations will be discussed in the next

chapter.

Table 4.3. Particle size distribution of the coke samples used for 3D simulations. Weight percentage of

each size range has been given.

Samples Mesh Size range

-4+6 -6+14

S4 - 100%

S5 100% -

M4 30% 70%

M5 50% 50%

M6 70% 30%

M7 80% 20%

4.3. Balls stiffness approximation

To estimate the minimum stiffness which can be assigned to balls comprising coke particles to have

realistic packings, number of simulations were run to calibrate the stiffness value. A simple 3-

dimensioal particle packing test was simulated in which spherical particles with the same size

distribution of coke particles fall due to the force of gravity and settle in a cylindrical container. It

should be noted that spherical particles have been used to simplify the test and to decrease the

number of elements. Normal ( ) and shear ( ) stiffnesses of the balls have been considered equal

and shear movements of the balls are mainly controlled by friction coefficient which is estimated

separately. Eight values were selected for normal ( ) and shear ( ) stiffnesses of the balls and

were assigned to the balls in seven separate simulations;

and

,

34

The system was then given a time to reach an equilibrium state. The equilibrium state in this study

means when the ratio of maximum unbalanced mechanical force to the average applied force is

equal or less than The column of particles with different stiffness values, settled in the

container, is shown in figures 4.1-4.8.Three horizontal walls with 1 millimeter difference in z

position have been added to the system after the equilibrium to have an easier visual comparison of

height of the particle assembly. The numbers beside the walls show the z coordinate of the wall. As

it can be seen, the stiffness value of causes erroneous results with unrealistic overlaps

between the particles and also between the particles and container walls. As the stiffness increases,

the height of the particle column increases. Since the density and so the weight of the particles are

the same in all tests, the contact forces are of the same magnitude. Thus, higher values of stiffness

causes lower overlaps. As a result particles occupy a higher volume in the container. Results of

simulations shown in figures 4.1-4.8 have been summarized in figure 4.9 which shows the

variations in volume of the particles assembly by change of stiffness. As it can be seen, there is no

changes in volume of the powders by changing the stiffness from to

. It shows that the stiffness is high enough to have a packing density independent of the

stiffness value. However, to approximate the appropriate value of stiffness, two-dimensional

simulations were the run to visualize the overlaps. The results have been shown in figures 4.10-

4.13.

Two-dimensional simulations were run to have a better approximation of ball stiffness. 3D works

showed that for the values between and the packing

density doesn’t change (see figure 4.9). However, the stiffness value as explained in chapter 3 has a

considerable effect on the time-step of the simulations and so on the time of calculations for VBD

tests simulations. Therefore, the minimum acceptable value should be chosen to increase the

efficiency of calculations. For this purpose, 2D works with stiffness values ranging from

to were performed and the results showed that is the minimum

value by which realistic overlaps develops between the particles. Therefore, this value was chosen

as the stiffness of balls comprising coke particles clumps in models of VBD test simulations.

35

Figure 4.1. Particle packing with and


36



37



38



39

Figure 4.9. Variations of the volume of the particle assembly by change of stiffness

Figure 4.10. Two-dimensional overlaps visualization, and

40



41


4.4. Friction coefficient estimation

Granular flow is inevitably affected by the friction between the particles [65, 66]. The effect of

internal friction coefficient on macroscopic properties of granular materials can be shown by angle

of repose test. Angle of repose, so is one of the most important macroscopic features of granular

materials [67,68]. In this test, the angle between the horizontal surface and the free surface in a pile

of solid particles is considered as the angle of repose [69].

In the present investigation, angle of repose test has been used to estimate the internal friction

coefficient of coke particles. Coke particles with the mass of 10g, as shown in figures 4.14 and 4.15,

were allowed to fall down from a funnel on a horizontal plate and the angle formed between the

elements of the cone and the base was measured. Ten tests were conducted and the mean value of

the recorded angles of repose was calculated for each coke samples. The mean value of angle of

repose for coke particles obtained in this study was 31 degrees.

Angle of repose test then simulated by a three dimensional DEM model using PFC3D software.

Funnel with the same geometry and scales was created in the model (as shown in figure 4.16) and

42

Figure 4.14. Angle of repose test on coke particles

Figure 4.15. Pile of coke particles in angle of repose test

10g of coke particles with the real size and shape parameters in the same way of the experimental

method were allowed to fall down from the funnel on the plate.

Before starting the calibration simulations for internal friction coefficient, the friction coefficient

between the plate and the particles was estimated. Coke particles placed on a horizontal plate and

then the inclination angle by which the particles start to slip was recorded. This process simulated in

the same way in PFC3D with different friction coefficients assigned to the plate. Thus, the friction

coefficient between the plate and particles determined by obtaining the same angle for onset of

particles slip in the experiments and simulations. The experimental method and simulation have

43

been shown in figure 4.17. Friction coefficient of 0.45 between the particles and the plate was

obtained using this method. This value then was applied to the horizontal plate in angle of repose

tests. General configuration of angle of repose test model and also some simulation results are

shown in figures 4.18-4.21. Internal friction coefficient of 0.27 (figure 4.20) resulted in the best

match between and the experimental (figure 4.15) and simulation results. Thus, it can be concluded

that using this method of calibration, the coefficient of friction between coke particles was

estimated and the obtained value ( ) was used in the simulation of vibrated bulk density test

as a property of material.

Figure 4.16. 3D simulation of angle of repose test

44

Figure 4.17. Estimation of friction coefficient between the plate and coke particles; a) experiment, b)

simulation

45

Figure 4.18. Pile of the coke particles in AOR test,



46


47

5. Image processing

5.1. Introduction

Image processing techniques are widely used in different areas of science and technology such as

medicine, physics, medical sciences and geotechnical fields [70-72]. Using image analysis,

microstructural features can be simply captured with a good accuracy.

In this work, optical microscopy powered by Clemex PE software was used to study the size

distribution and shape of coke samples. ImageJ and Matlab were also used to process the

micrographs and to model the particles with overlapping circles for 2D-DEM simulations.

5.2. Shape Analysis

Particulate materials are now widely used in many industries and so it is very important to

characterize the physical, chemical and mechanical features of the particles to ensure the

performance of the powders. Particles shape is considered as one of the most important features of

particulate assemblies [73,74]. There are different methods to assess and describe the shape of

particles which include from simple and common parameters such as sphericity, aspect ratio,

elongation ratio, shape factor, and convexity ratio [73, 75] to sophisticated methods such as use of

Fourier descriptors to describe the 2D section of the particles by mathematical equations. However,

sphericity is considered as a single parameter which can define the shape of the particle to a very

good extent with a simple method. Thus, in the present work, sphericity is used as the shape

defining parameter to characterize the particles in different size ranges.

Wadell [76] introduced the term sphericity in 1935. Sphericity in general states how close is the

particle geometry to a perfect sphere. It is worthy to note that this term is also used for 2D

projections of the particles and for that case the name circularity matches better. A number of

definitions of circularity and sphericity have been proposed in the literature which have been

summarized in table 5.1 [77].

The method proposed by Wadell [76, 78] is widely used in the literature and was used in this study

also as the definition of sphericity. According to Wadell sphericity is defined as

(5.1)

where is the surface area of the real particle and is the surface area of the equivalent

sphere with the same volume of the real particle.

48

As it was mentioned above, optical microscopy was used to study the shape of the particles and

calculate the sphericity of particles in this study. However, this method of shape analysis is a 2D

method. Thus, the correctness of data obtained by this method required to be examined. To do so,

45 random particles of -6+14 mesh size ranges and 15 particles from the other size ranges analyzed

by microdiamond size imaging technique (3D scanning) and the particle were digitized. Some

examples are shown in figure 5.1.

Table 5.1. Definitions of sphericity and circularity in the literature

Reference Symbol Expression *

Wadell [76, 78] Degree of Sphericity, Ψ

Wadell [79] Degree of Sphericity,Φ √

ISO [80] Circularity, C

Cho et al. [81] Sphericity, SCE

Sympatec [82] Sphericity, SQ √

* p is the perimeter, A is the area, and dcmin and dimax are the minimum and maximum axis of the particle

Figure 5.1. Digitized 3D pattern of coke particles

49

By having the 3D digital true design of the particles, sphericity of each particle was calculated

according to equation (5.1). Then the sphericity of the particles was calculated with optical

microscopy (2D method) with two methods for each size range; polished particles and non-polished

particles. The results of sphericity calculations for non-polished particles (see figure 5.2) are given

in figures 5.3-5.5. The mean values of sphericity obtained by different methods are given in table

5.2. As it can be seen, the results for non-polished particles are closer to the sphericity obtained by

calculations on 3D models. Calculation on the polished particles underestimates the sphericity.

Thus, for all size ranges in this study the sphericity value was measured by optical microscopy on

non-polished particles and this value was recorded as the shape controlling parameter. Since the

number of the particles used to calculate the sphericity is high (more than 400 for each size range) it

could be said that the obtained value for sphericity represents the shape of the coke particles.

Figure 5.2. Optical micrograph of coke particles within the size range of -6+14 mesh; a)non-processed

image; b) processed image to measure sphericity

50


51


52


53

Table 5.2. Results of measurements of sphericity and comparison of the results with calculations by two

2-dimensional methods

Method

Mean sphericity

-30+50

mesh -14+30 mesh

-6+14

mesh

True value by 3D particles 0.8042 0.8047 0.7780

2D method by polished particles 0.707 0.678 0.406

2D method by non-polished particles

0.760 0.774 0.709

5.3. Two-dimensional particle modeling

As it was mentioned in chapter 3, basic elements of any 2D-DEM model are circles. Thus, to

simulate the packing process of coke particles, first the shape of the particles need to be modeled

with circles. In the present work, the following procedure was used to capture the shape of the

particles and to model them with circles for two-dimensional simulations:

1- Take the micrograph of the particles;

2- Convert the image to black and white image;

3- Detect the particles within the desired size range;

4- Skeletonize the particles;

5- Detect and highlight the edges;

6- Fill in the areas (of the particles);

7- Crop the image and extract the shape of all the particles in the photo and save them as

separate images;

8- Insert the obtained image at step 6 for each particle into the “area tessellation algorithm” ;

9- Save the radii and coordinated of the circles used to cover the area of the particle obtained

at step 8;

10- Insert the data obtained at step 8 into PFC2D to define a particle as a clump.

An example of processing a micrograph of coke particles with the above procedure has been shown

in figures 5.6-5.10.

54

Figure 5.6. Micrograph of coke particles of the size range of -6+14 mesh

Figure 5.7. Skeletonization of the particle shape, step-1

55

Figure 5.8. Fine skeletonization of the particle shape, step-2

Figure 5.9. Edge detection to obtain the shape of the particles

56

Figure 5.10. Filling the area and obtaining the shape in a black and white image

The shapes obtained in step 6, as shown in figure 5.10, were then extracted from the photo and

separately inserted in the “area tessellation algorithm” in Matlab to be covered by circles. Some

examples of the obtained results are shown in figure 5.11.

Figure 5.11. Coke particles as 2D clumps

57

The particles shown in figure 5.11 are clumps (super-particles) composed of overlapping circles.

The radii and coordinates of circles required to model a particle (cover the area shown in figure

5.10) obtained from the area tessellation algorithm. The circles then inserted as information of

clumps in PFC2D and clumps such as those shown in figure 5.11 were created.

5.4. Three-dimensional particle modeling

For three-dimensional simulations, like two-dimensional works, coke particles need to be modeled

with spheres in PFC3D. Three-dimensional digitized and meshed shapes of coke particles (as shown

in figure 5.1) were obtained by 3D scanning. Number of 15 particles for the size ranges of -30+50

and -14+30 mesh and 45 particles for the size range of -6+14 mesh were sent for 3D imaging. Then,

Automatic Sphere-clump Generator (ASG) software [83] was used to approximate the 3D shapes

using spherical elements. Some samples of particle shape approximation with spheres are shown in

figures 5.12(a) and 5.12(b)

Two parameters were used to control the modeling efficiency; volume error and mass distribution

error. Volume error, which is the difference between the volume of the real particle and that of the

modeled particle, of all models compared to the original particles was kept less than 1 percent. Mass

distribution error states how the distribution of spheres around the principal axis of the shape

creates the same momentum. The maximum value of percentage of mass error was 1.5 percent for

all modeled particles. As it can be seen in figure 5.12, coke particles have a variety of irregular

shapes with a wide range of sphericity. However, the 3D modeling approach has successfully

modeled the shapes to an acceptable extent.

58

59

Figure 5.12(a) and (b). Samples of 3D particle modeling with spheres. The models were used in PFC3D

as clumps

6. Vibrated Bulk Density Simulations

6.1. Revisions in the experimental method

As explained in chapter 2, in VBD test 100g of coke particles are poured in a cylindrical container

with inner diameter of 37 mm and vibrated for 60 seconds. However, since the modeled particles

are composed of several balls (elements), DEM modeling of 100g of coke particles as clumps

requires more than 1’000’000 balls which is a huge number for DEM simulations. On the other

hand, the modeled problem here is a dynamic case with vibrations involved and so density scaling

cannot be applied to enlarge the time-steps. Thus, the true value of time-step, automatically

calculated by PFC according to the density and stiffness of the balls, were applied in calculations.

The time-step value was between 5.6e-5 to 2.0e-6 second for different cases in this study. This

means that for the time-step of 2.0e-6 second, solving 3.0e7 steps is required to complete a 60

seconds-long vibration. Since the used DEM code cannot be run on cluster computers, this volume

of calculations requires several months to be solved by a powerful desktop computer. Therefore,

considering the objectives of this research study (as stated in the chapter 1), VBD test in this work

revised in some parameters to save the calculation time for DEM simulations.

First, the mass of coke samples used in VBD test decreased to 10g and then to have a column of

particles, the container diameter also decreased to 29 mm. By these revisions the number of

elements for DEM models decreased to maximum of 120’000. However, vibration time was still an

issue and thus some experiments were designed to evaluate the effect of vibration time in VBD test.

Some results of experimental studies on vibration time effects have been summarized in figures 6.1-

6.3. As it can be clearly seen from the graphs, a considerable point in vibration process, is the time

t=10s. In all cases, the drop in the volume of the powder due to the applied vibration happens in the

first 10 seconds of vibration. Vibrating the sample for a longer time does not change the volume

and so the bulk density of the samples. Therefore, new method planned for VBD tests in which

coke samples are poured in a cylindrical container with diameter of 29 mm and then a vertical

vibration is applied to the system for a period of 10 seconds. VBD values for different samples

measured according to the new method and DEM modeling results were compared with these

values.

61

Figure 6.1. Vibration time effect on the volume of coke particle column in VBD test for different sample

sizes, -6+14 mesh size range



62



6.2. Two-dimensional simulations

Six samples of Conoco coke as presented in table 4.2, were examined for VBD test and the samples

modeled according the method explained in the chapter 5 in PFC2D. Coke clumps with the desired

shapes and sizes were introduced in PFC2D to make up the coke powder. Particles randomly placed

in the container (see figure 6.4) and then the force of gravity was applied to the system and particles

allowed to settle in the container. Vibration applied to the container after reaching the equilibrium

state. Figure 6.5 shows the system of particles and container in the equilibrium state and ready for

vibration.

Experimental results of the VBD test and those obtained from DEM simulations have been given in

tables 6.1 and 6.2. Table 6.1 gives the VBD values for mono-size range powders (S1, S2 and S3).

As it can be seen, changing the particle size from -30+50 to -14+30 mesh did not have a visible

effect on the VBD value (0.910 g/cm3 for both). However, increasing the particle size to the range

of -6+14 mesh reduces the VBD to 0.834 g/cm3. The diameter of the largest particles is 3.3 mm and

that of the smallest particle, which corresponds to -30+50 mesh range, is 0.3 mm. The exact reason

63

of the difference in the obtained values of the VBD is not clear. However, there are some

parameters that can affect the packing density of the particles. It has been shown that container wall

induces a region of low density with the size equal to around 10 particle diameters [84]. This effect

is more pronounced for large particles. The simulations of VBD tests for samples S1, S2 and S3 as

shown in table 6.1 follow the same trend and give the minimum value of VBD for the coarse

particles. On the other hand, microscopic studies on the coke particles showed that the mean

sphericity of particles decreases, as the particle size increases. The mean sphericity of particles in

S1, S2 and S3 samples was 0.804, 0.794 and 0.4. Therefore, lower VBD of S3 compared to S1and

S2 could be also due to particles shape (lower sphericity) within the size range of -6+14 mesh.

Results of simulation predictions and also experimental data for the mixed samples (M1, M2 and

M3) have been given in table 6.2. It can be seen that as the content of coarse particles increases the

VBD increases.

Figure 6.4. Initial state of the system in 2D models of VBD test

64

Figure 6.5. Equilibrium state, after particles settlement in the container in 2D simulations

Table 6.1. Experimental and 2D simulation results of VBD for mono-size samples

Coke sample VBD (g/cm

3)

Experimental Simulation

S1 (-30+50 mesh) 0.91 0.91

S2 (-14+30 mesh) 0.91 0.904

S3 (-6+14 mesh) 0.834 0.842

The sample M3 has more coarse particles and less fines compared to M1 and M2 and it holds the

maximum VBD in this investigation. Table 6.1 shows that coarse particles have lower packing

density whereas table 6.2 suggests that increasing the content of coarse particles leads to higher

VBD. It should be noted that the results in table 6.1 are for mono-size particles while table 6.2

presents the data for the mixed samples.

65

Table 6.2. Comparison of experimental and 2D simulation results of VBD for mixed samples

Coke sample VBD (g/cm

3)

Experimental Simulation

M1 0.943 0.94

M2 0.962 0.945

M3 0.971 0.65

Images from the simulations may be helpful to understand the mechanisms of particle packing.

Images of coke particle assemblies for the mixed samples of M1 and M3, obtained from

simulations, are shown in Figure 6.6. For the sample M1 ( Figure 6.6(a)) coarse particles of -6+14

mesh range make up only 10 wt.% of the sample. However, for the sample M3, 50 wt.% are coarse

particles. Comparing figures 6.6(a) and 6.6(b) can explain the higher bulk density of M3 compared

to M1. Formation of inter-particle porosity between fines (as indicated by rectangular areas in

figure 6.6(a)) seems to be the reason for lower VBD of sample with higher percentage of fine

particles. As can be seen in figure 6.6(b), increasing the content of coarse particles leads to

formation of less inter-particle spaces. However, it should be noted that the size of fine particles

here is small enough to fill the surface roughness of coarse particles and also the voids between

large particles.

Porosity measurements in different regions of the particle assemblies showed that the mean porosity

value in areas with fine particles is around 15.4% while it is l0.8% for the same area around a large

particle. Although coarse particles have positive effect on the bulk density, it should be noted that

existence of fines to fill the gaps between large particles (as shown in Figure 6.6(b)) is necessary.

Comparing the data in tables 6.1 and 6.2 shows that the maximum value of VBD is not for S3

sample (having 100% of large particles) or sample S2 (with 100% of medium size particles).

Another parameter which affects the particle packing densities is the container wall effect. It is

believed that the ratio of container size to the particle size is also an important parameter in packing

density of granular materials [84]. The container wall induces a local low density region nearby

which reduces the total packing density of the material. This effect is more pronounced when the

particle size increases (in the same container). As can be seen in Figure 6.7(a), there are large voids

between the particles close to the container wall which can explain the lower density of S3

compared to S2 and S1. However, these voids have been filled by fine particles in sample M3

66

(Figure 6.7(b)).Therefore, it can be said that coarse particles have a positive effect on the vibrated

bulk density and increasing the coarse particle fraction increases the VBD, only if there are enough

fines to fill the gaps between large particles.

Figure 6.6. Simulation of particle packings; a) sample M1, filling the surface roughness of coarse

particles by fines are shown by rectangles b) sample M3, the indicated areas show the filling of the

space between large particles by fines

67

Figure 6.7. Simulation of particle packings; a) Regions near the container wall in sample S3, filling the

surface roughness of coarse particles by fines are shown by rectangles; b) Regions near the container

all in sample M3

68

6.3. Three-dimensional simulations

Three dimensional simulation of VBD test in PFC3D was done using the same method of two-

dimensional works. Coke particles were randomly placed in the container and then force of gravity

was applied to the system. Figure 6.8 shows the initial state of the system and the particles are

settled in figure 6.9 and system is in equilibrium state.

Figure 6.8. Initial state of the particles in 3D simulation of VBD test. Sample M5 (composed of 50% of -

6+14 and 50% of -4+6 mesh particles)

As it was explained, the size of samples for 3D simulations decreased to 10g to increase the

efficiency of the numerical model. The results here are compared with both standard VBD test with

100g of particles and also the modified test with small samples to see the effect of size of samples.

In figure 6.10, experimental results for VBD values of different samples obtained with standard

69

Figure 6.9. Equilibrium state before application of vibration; Sample M5 (composed of 50% of -6+14

and 50% of -4+6 mesh particles)

100g method and also modified 10g samples have been compared. As it can be seen in the graph,

mono-sized sample of -6+14 mesh has a higher VBD compared to mono-sized sample of -4+6 mesh

(0.835 compared to 0.715 g/cm3).

This is in accordance to 2D results in which increasing the size of the particles caused obtaining

lower values of VBD. Figure 6.10 also shows that for 100g samples as the content of larger

particles (-4+6 mesh) increases VBD drops. Two points need to be explained here. First, both sizes

in these tests are large compared to the container size (-4+6 mesh corresponds to 3.36- 4.76 mm and

-6+14 mesh corresponds to 1.41-3.36 mm). Therefore, mixing the particles did not result in an

70

increase in the bulk density (like what often happens and in 2D samples of this work it was also

shown) and the curve (100g sample) in figure 6.10 shows a linear drop in VBD by increasing the

Figure 6.10. Experimental VBD test results of mixed-size samples; with standard method of 100g and

also 10g samples

Figure 6.11. Experimental and simulation results of VBD test for mixed samples of 10g

71

content of -4+6 mesh particles. It means that in this case there are no fine particles to fill the gaps

between the large particles and so the VBD of the mixture doesn’t increases. Secondly, 10g samples

give a VBD value of slightly lower than 100g samples and it seems that the sample of 10g does not

clearly show the variations in VBD value. Addition of around 70 wt. % of -4+6 mesh particles to -

6+14 mesh particles does not change the VBD and the drop in VBD is just seen after 80 wt.%.

Simulation results have been presented in the graph of figure 6.11. The model for all samples

predicts a slightly lower density but simulations follow the trend of experimental data. Figure 6.12

compares two samples of M5 and M7 with different mixtures. Two samples have the same weight

of 10g but M7 with more large particles visibly occupies a higher volume which results in lower

bulk density.

Figure 6.12. 3D simulations of VBD test; a) sample M5 with 50% of -4+6 particles; b) sample M7 with

80% of -4+6 particles

72

Conclusion

The present work was the first step of the idea of application of discrete element method simulation

in anode production technology to study the effects of raw materials properties on final anode

quality and also to explore the possibility of utilizing new sources of raw materials. Packing

characteristics of dry coke particles by changing the pitch demand value of the paste affects final

anode properties. Vibrated Bulk Density (VBD) test is regularly used in anode manufacturing plants

to test the packing density of coke particles. Therefore, VBD test was selected as the point of

interest to examine the application of DEM simulations. Conoco coke samples with the size ranges

which are currently utilized in industry were used in this work.

Image processing techniques with ImagJ and Matlab used to capture the shapes of coke particles

and to model them with overlapping circles for two-dimensional simulations. For three-dimensional

works, coke particles shapes captured by Microdiamond Size Imaging (MSI) machine and the DEM

models obtained by Automatic Sphere-clump Generator (ASG) software.

Three mono-size samples with the size ranges of -6+14, -14+30 and -30+50 mesh (S1, S2 and S3,

respectively) and three mixed samples prepared and experienced a standard VBD test. Three

samples also prepared by mixing the different size ranges to evaluate the effects of size distribution

on the VBD value. For mono-size samples within the investigated size range, the maximum and the

minimum VBD values correspond to the samples in the range of -30+50 and -6+14 mesh

respectively. It is believed that the wall effect and lower sphericity of coarse particles (-6+14 mesh)

compared to S2 and S1 are the main reasons for the lower VBD value of S3. In general the mixed

samples have higher bulk density than mono-size ones. It could be concluded that in the

investigated size range, as the fraction of coarse particles increases, VBD increases only if there are

fines to fill the pores between large particles.

Three-dimensional simulations were performed on mono-size samples of -6+14 and -4+6 mesh and

four mixed-size samples. Results showed that again coarse particle (here -4+6 mesh) have lower

VBD compared to finer particles. For the mixed-size samples as the content of coarse particles

increases VBD decreases. Although 3D simulations are in accordance with experimental

observations, this result was in contrary to the samples used for 2D works and also 2D simulations.

However, it should be noted that the coarse particles size for 3D works is very large (3.36-4.76 mm)

and the container diameter is just 14.5 mm. On the other hand, the sphericity of -4+6 mesh is very

73

low (0.6). Thus, addition of -4+6 mesh particles to -6+14 mesh samples decreases the VBD because

of increasing the size of the voids between the particles while there are no fines to fill the gaps.

It can be concluded that discrete element method simulations coupled with image processing

techniques used in this research is capable of predicting the particle rearrangement and density

evolution during the vibration process.

It was shown that DEM simulations by having the ability to include the real shape of the particles in

simulations is very useful to study the effects of shape, size and size distribution in studying the

packing characteristics of coke samples (or any other particulate system).

Summary of the outcomes and recommendations for future works;

1- The present work confirms the effects of coke particles shape and size distribution of the

packing density of coke assemblies through DEM simulations.

2- Mixed-size assemblies hold higher VBD compared to mono-size samples. However, the

size distribution needs to be well-tailored in which there are some fine particles to fill the

voids between coarse particles.

3- Micro-properties of the materials for DEM simulations can be obtained through calibration

tests. In the present work, friction coefficient and stiffness of the elements for coke particles

DEM models estimated by running angle of repose and simple packing tests, respectively.

4- Discrete element method coupled with image analysis techniques is a powerful tool to study

the packing of particulate systems.

5- To have a more precise model of coke particles assembly, it is recommended that the

internal friction coefficient of the particles measured with a more precise method. The

shape of particles have been generated by a very good precision in the models but

determination method of stiffness and friction coefficient of the particles can be the

weakness of this work.

6- Parallelization of the problem and using a DEM code which is compatible with cluster

processors seems to be necessary to be able to include more elements and run the problem

for big samples. Number of elements and the issue of calculation time were the main

concern in this project.

7- This project was the first step of a research work to apply DEM simulations in studying the

behavior of anode paste under vibro-compaction. Results of this work show the power of

74

discrete element method and feasibility of its application in carbon anodes technology.

However, for modeling the anode paste coke particles should be mixed with a binder with

viscous properties which will make the problem more complicated. For this case it is highly

recommended to use parallel processing to decrease the solution time.

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DISCRETE ELEMENT METHOD APPLIED TO THE VIBRATION … · 2018-04-20 · BEHZAD MAJIDI DISCRETE ELEMENT METHOD APPLIED TO THE VIBRATION PROCESS OF COKE PARTICLES Mémoire présenté