Particle Mixing Study in Different Mixers - UNSWorks

The University of New South Wales

Faculty of Science

School of Materials Science and Engineering

Particle Mixing Study in Different Mixers

Thesis by

Musha Halidan

Submitted in Partial Fulfilment of the Requirements of the Degree of

Doctor of Philosophy in

Materials Science and Engineering

January 2014

PLEASE TYPE

Surname or Family name: Halidan

First name: Musha

THE UNIVERSITY OF NEW SOUTH WALES Thesis/Dissertation Sheet

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Abbreviation for degree as given in the University calendar: PhD

School: Material Science and Engineering Faculty: Science

Title: Particle Mixing Study in Different Mixers

Abstract 350 words maximum: (PLEASE TYPE)

Particle m1xmg is an essential unit operation in the manufacture of many granular products, for example, in pharmaceutical, food and chemical engineering industries. A cylindrical mixer, ribbon mixer and rotating drum were investigated for the predictability of the mixture quality using the discrete element method. The mixture quality is known to be affected by variables such as particle properties, operational parameters and geometrical parameters. The particle (or mixture) properties considered are: particle size, particle density, volume fraction and cohesion, of which the predictability is sought only in relation to the first three variables. The effects of these three variables on the mixing behaviour of binary particles can be predicted for all three mixers with the rotating drum being operated in the rolling mode. The prediction equations for the mixers have a similar form but different in the coefficients. Further, the size and density interactions were interpreted using the mechanism of the driving-force generation due to size differences and its completion with the particle weight in the case of a cylindrical mixer.

The effect of cohesion on particle mixing was investigated in micro and macro ribbon mixer systems by creating geometrically and dynamically similar conditions by matching Froude and Bond numbers. A similarity was observed between the two systems in each of the results for mixing index, particle velocity and particle contact forces in the case of non-cohesive mixtures. The quality of the cohesive and noncohesive mixtures each improved with an increase in the shaft speed up to 1 OOrpm after which it deteriorated. A four-bladed ribbon impeller was found to be more effective for mixing cohesive particles and use at a higher fill level. A parametric study on the efiects of impeller geometry showed that the blade pitch, width and clearance significantly affects the particle flow pattern, mixing rate and homogeneity of cohesive particle mixtures. In the case of a four-bladed impeller, the outer blade angle affects the mixing behaviour, but the inner blade angle has no significant effect.

Overall, the present work has used the discrete element method successfully in the study of particle mixing behaviours in three mixers.

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'I hereby grant the University of New South Wales or its agents the right to archive and to make available my thesis or dissertation in whole or part in the University libraries in all forms of media, now or here after known, subject to the provisions of the Copyright Act 1968. I retain all proprietary rights, such as patent rights. I also retain the right to use in future works (such as articles or books) all or part of this thesis or dissertation. I also authorise University Microfilms to use the 350 word abstract of my thesis in Dissertation Abstract International (this is applicable to doctoral theses only). I have either used no substantial portions of copyright material in my thesis or I have obtained permission to use copyright material; where permission has not been granted I have applied/will apply for a partial restriction of the digital copy of my thesis or dissertation.'

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person, or substantial proportions of material which have been accepted for the

award of any other degree or diploma at UNS W or any other educational institution,

except where due acknowledgement is made in the thesis. Any contribution made to

the research by others, with whom I have worked at UNSW or elsewhere, is

explicitly acknowledged in the thesis. I also declare that the intellectual content of

this thesis is the product of my own work, except to the extent that assistance from

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iii

ACKNOWLEDGMENT

I would like to express my sincere gratitude to the following people who made it possible for me to complete my research project:

Prof. Aibing Yu, for giving me an opportunity to perform research on the mixing of particles and also for his invaluable guidance on the topic. This work would not have been possible without his direction, inspiration and support.

Dr Rohana Chandratilleke my co-supervisor for providing me support and suggestions during the research for my PhD.

Prof. Sammy Chan my co-supervisor who accepted me into the University of NSW.

Dr Kejun Dong who provided me with his DEM code.

Prof. John Bridgwater of Cambridge University for his invaluable suggestions and comments.

Dr. Ruiping Zou, who paid close attention to the administrative aspects of my work.

The staff and students in the Lab for Simulation and Modelling of Particulate Systems (Simpas)who have helped me in numerous ways during the completion of my research.

The Chinese Scholarship Council and the University of NSW for providing the financial support necessary for this work and their commitment to scientific research.. My Husband Ahmat and my son Ankar for their support during my PhD work.

My parents and parents in law who gave me encouragement, especially my mother in law Nusrat Hajim who provided tremendous support by looking after my 9 month old daughter whilst I was completing my PhD.

My best friends’ Susan and Nick Manousaridis who encouraged me during my PhD work.

iv

ABSTRACT

Particle mixing is an essential unit operation in the manufacture of many granular

products, for example, as in pharmaceutical, food and chemical engineering industries.

However, the predictability of the mixing behaviour of granular materials is limited at

present and case by case studies need to be carried out to find the mixing outcomes. A

cylindrical mixer, ribbon mixer and rotating drum were chosen for the study, and the

predictability of the mixture quality for the mixers was investigated. The mixture

quality is generally well known to be affected by many variables such as particle

properties, operational parameters and geometrical parameters. The particle (or mixture)

properties considered are: particle size, particle density, volume fraction and cohesion,

of which the predictability is sought only in relation to the first three variables. The

operational parameters considered are the shaft speed and fill level; and the geometrical

parameters are pith ratio, blade width, blade clearance, blade angle and blade number.

The discrete element method was used to investigate these effects on the particle mixing

behaviour.

In the first part of the study, a vertically-shafted cylindrical mixer was used and the

effects of particle size, density and volume fraction on the mixing behaviour of binary

mixtures were investigated. The mixture quality showed an improvement if the larger

particles are heavier and small ones lighter. With respect to variations in the size and

volume fractions, each mixture quality showed a peak value. A correlation was

established for predicting the effects of size, density and volume fraction on mixture

quality in a wide range of each variable. The correlation predicts a global peak value at

an optimum condition of size, density and volume fraction. Further, the size and density

interactions were interpreted using the mechanism of the driving-force generation due to

size differences and its completion with the particle weight. A cylindrical mixer can be

used as a standard method in studies of particle mixing with further extensions to the

correlation developed here to include the effects of parameters such as the operating

conditions and material properties.

In the second part of the study, a horizontal ribbon mixer was used and the effect of

cohesion on particles was investigated in micro and macro systems by creating

geometrically and dynamically similar conditions using matched Froud numbers and

v

Bond numbers. There was a similarity in the mixing index, velocity and forces of the

particles in the two systems in non-cohesive mixtures. The effects of the shaft speed and

filling level on cohesive and non-cohesive mixtures were also investigated using light

particles in the large-sized mixer in an attempt to simulate powder mixing behaviour in

the mixer. The quality of the cohesive and non-cohesive mixtures each improved with

an increase in the shaft speed up to 100rpm after which it deteriorated. A four-bladed

impeller was found to be more effective for mixing of cohesive particles and at a higher

fill level. In the parametric study of impeller geometry effects, it was found that the

blade pitch, blade width and blade clearance affect the flow pattern, mixing rate and

homogeneity of cohesive particle mixtures significantly at different cohesion levels. The

outer blade angle of a four-bladed impeller affects the particle mixing but the inner

blade angle has no significant effect. A correlation was established for predicting the

effect of particle size, density and volume fraction on the quality of binary particle

mixtures in the ribbon mixer. The equation has a similar form to that developed for a

cylindrical mixer, suggesting the presence of some similar mixing mechanisms in both

mixers.

In the third part of the study, a rotating drum was used and the segregation flow patterns

investigated for binary particle mixtures of 0.5 volume fraction in the rolling mode of

operation. Depending on the size and density ratios and volume fraction, the mixing and

segregation patterns changed and a correlation was established for predicting their

effects on the mixture quality.

The study shows that the effects of the size and density ratios and volume fraction on

the mixing behaviour of binary particles can be predicted for a cylindrical mixer, ribbon

mixer and rotating drum which is operated in the rolling mode. The prediction equations

for the mixers have a similar form suggesting similar mixing mechanisms in operation

in the mixers. However, differences in coefficients and interacting terms suggest

different degrees of influence of those mixing mechanisms in each mixer. A ribbon

mixer shows a better mixture quality than other mixers for the same conditions when

mixing binary particles, thus can be considered as an effective mixer for binary particle

mixing. Using a proper loading method, the mixing rate can also be improved in ribbon

mixers in addition to its ability to mix cohesive particles effectively. Hence, ribbon

mixers can be considered as a versatile mixer. Overall, the present work has used the

vi

discrete element method successfully in the study of the effect of material properties of

particles on particle mixing in different mixers; the effects of operational and

geometrical parameters of ribbon mixer on mixing have been established; and

predictability of particle mixing behaviours have been identified for different mixers.

vi

TABLE OF CONTENTS Page

Title Page Copyright and Authenticity Statements i Originality statement ii Acknowledgments iii Abstract iv Table of Contents vi List of Figures x List of Tables xvii

CHAPTER 1 Thesis Overview 1-1 CHAPTER 2 Literature Survey 2-7 2.1. Introduction 2-8 2.2. Applications of Powder mixing 2-9 2.3. Issues (segregation) 2-11 2.4. Overcoming the issues 2-12 2.5. Types of powder mixing processes 2-12 2.6. Solid mixing 2-13 2.7. Characterization of mixture 2-14

2.7.1. Qualitative approach 2-14 2.7.2. Quantitative approach 2-15

2.7.2.1. Definition of mixture status 2-15 2.7.2.2. Mixing index 2-16 2.7.2.3. Segregation index 2-16 2.7.2.4. Particle scale mixing index 2-17 2.7.2.5. Powder sampling (in practice) 2-19

2.8. Factors affecting solids mixing 2-21 2.8.1. Size or density differences 2-21 2.8.2. Combination of size and density differences 2-23 2.8.3. Cohesive particle mixing 2-24

2.8.3.1. Types of cohesion 2-24 2.8.3.2. Van der Wall force model 2-26

2.9. Effect of the particle cohesion on mixing in a uniform binary system 2-27 2.10. Effect of the particle cohesion on mixing in a non-uniform binary system 2-29 2.11. Operational conditions (Shaft speed and fill level effect) 2-32 2.12. Types of Mixer Types 2-32 2.13. Helical ribbon mixer 2-37 2.14. Ribbon mixers 2-44

2.14.1. Effect of shaft speed 2-45

vii

2.14.2. Effect of fill level 2-46 2.14.3. Effects of mixer geometry 2-47

2.15. Simulation method 2-49 2.16. Discrete Element Method 2-49

2.16.1. Force model 2-50 2.16.2. Implementation of the DEM 2-52 2.16.3. The application of DEM 2-53

2.17. Summary and Research Proposal 2-57

CHAPTER 3 Prediction of The Mixing Behaviour of Binary 3-60

Mixtures of Particles in a Bladed-Mixer 3.1 Introduction 3-61 3.2 Numerical Method 3-63 3.3 Mixing quantification method 3-65 3.4 Simulation conditions and procedure 3-65 3.5 Results and Discussion 3-68

3.5.1 Effects of density, size and volume fraction 3-68 3.5.2 Particle mixing mechanism 3-71 3.5.3 Mixing trends: Effects of rs and xl 3-78 3.5.4 Formulation of a predictive equation 3-80

3.6 Conclusions 3-82

CHAPTER 4 Effects of Particle Cohesion on Mixing in a Ribbon Mixer 4-86

4.1 Introduction 4-87 4.2 DEM Simulations 4-89 4.3 Mixing quantification method 4-90 4.4 Simulation Conditions 4-91 4.5 Results and Discussion 4-94

4.5.1 Comparison of micro and macro systems 4-94 4.5.1.1. Mixing Index 4-95 4.5.1.2. Coordination number 4-96 4.5.1.3. Velocity field 4-96 4.5.1.4. Quantification of velocity and force 4-99

4.5.2 Effects of cohesion on the homogeneity of the particles mixing 4-102 (Macro-system)

4.5.2.1. Effect of material density 4-103 4.5.2.2. Particle scale mixing index and total coordination number 4-104 4.5.2.3. Cohesion effect on the velocity field and porosity of the mixture 4-106 4.5.2.4. Cohesion effects on the radial, tangential and horizontal velocity 4-108 4.5.2.5. The stress analysis of particles with different cohesion 4-110


viii

CHAPTER 5 Mixing Performance of a Ribbon Mixer: Effect of Operational Parameters 5-115

5.1 Introduction 5-116 5.2 Methods to be Evaluated 5-118 5.3 Simulation Conditions 5-119 5.4 Results and Discussion 5-121

5.4.1 Blade Motion and Mixing Dynamics 5-121 5.4.2 Effect of impeller speed 5-122

5.4.2.1 Mixing performance 5-122 5.4.2.2 Velocity and Flow Pattern 5-124 5.4.2.3 Contact forces 5-129

5.4.3 Effect of fill level 5-132 5.4.3.1 Mixing index and mixing rate 5-132 5.4.3.2 Velocity field and quantification of velocity 5-134 5.4.3.3 Contact forces 5-138

5.4.4 Comparison of mixer performances: Ribbon versus cylindrical mixers 5-140 5.5 Conclusions 5-141

CHAPTER 6 DEM Simulation of Powder Mixing in a Ribbon Mixer: Effects of Impeller Geometry 5-144

6.1 Introduction 6-145 6.2 Numerical Method 6-146 6.3 Mixing quantification 6-147 6.4 Simulation conditions and procedures 6-148 6.5 Results and Discussion 6-151

6.5.1 Geometrical effects of type 1 ribbon impellers 6-151 6.5.1.1 Blade number or blade pitches effect 6-151

6.5.1.1.1 Mixing index and mixing rate 6-151 6.5.1.1.2 Coordination number 6-153 6.5.1.1.3 Flow pattern 6-155 6.5.1.1.4 Probability density of velocity and forces 6-156

6.5.1.2 Effect of blade width at different Ha 6-159 6.5.1.2.1 Mixing index and mixing rate 6-159 6.5.1.2.2 Particle flow 6-160 6.5.1.2.3 Velocity and force analysis 6-162

6.5.1.3 Effect of blade clearance at different Ha 6-165 6.5.1.3.1 Mixing quantification 6-165 6.5.1.3.2 Spatial-averaged velocity 6-167 6.5.1.3.3 Velocity and force components 6-169

6.5.1.4 Effect of blade angle 6-173 6.5.2 Comparison of type 1 and type 2 impellers 6-174

6.5.2.1 Blade number (or pitch ratio) effect 6-174 6.5.2.2 Blade width 6-177

ix


CHAPTER 7 A Study of Binary Particle Mixtures in a Ribbon Mixer 7-183

7.1 Introduction 7-184 7.2 Method of Analysis 7-186 7.3 Simulation Conditions 7-186 7.4 Results and Discussion 7-190

7.4.1 Effect of density on binary particle mixing 7-190 7.4.1.1 Mixing performances 7-190 7.4.1.2 Velocity and force analysis 7-191

7.4.2 Effect of size ratio rs on particle mixing 7-194 7.4.2.1 Mixing index 7-194 7.4.2.2 Force components and velocity fields 7-196

7.4.3 Prediction Equation 7-198 7.5 Conclusions 7-201

CHAPTER 8 Radial Segregation of a Binary Mixture in a Rotating Drum 8-204 8.1. Introduction 8-205 8.2. Analytical Methods 8-207

8.2.1 DEM Model 8-207 8.2.2 Simulation Conditions 8-208 8.2.3 Segregation index 8-209

8.3. Results and Discussion 8-212 8.3.1. Effect of density ratio on binary particle mixing 8-212

8.3.1.1. Segregation index 8-212 8.3.1.2. Force analysis 8-214

8.3.2. Effect of size ratio rs on particle mixing 8-215

8.3.2.1. Segregation patterns 8-215 8.3.2.2. Force components 8-217

8.3.3. Combined size and density effect 8-217 8.3.3.1 Size ratio r

s and density ratio r

d decrease at the same time 8-217

8.3.3.2 Size ratio rs is decreased and density ratio r

d increased 8-219

8.3.4. Prediction equation 8-220 8.4. Conclusions 8-222

CHAPTER 9 Summary and Future Work 9-225

References 231

List of Publications 241

x

List of Figures

Figure 2.1 Schematic representation of particle contacts: (a) a group of particles in the neighbourhood of particle ‘i’, which can be a particle of W-type or B-type; and (b) shows the contact condition according to which particle X is in contact with Y but not with Z

2-18

Figure 2.2 Sample variance of a fully-random mixture as a function of sample

2-20

Figure 2.3 (i) Effect of sample number on variance, (a) 100, (b) 200 and (c) 500 samples; plots are normalized variance frequency distributions; effect of particle number on variance, (a) 200 pps, (b) 400 pps, and (c) 600 pps, where pps is particles per sample

2-21

Figure 2.4 (a) effect of cohesion on mixing;(b) This Graph is in conjunction with A, plotting the log (intensity of segregation) with revolutions

2-28

Figure 2.5 Effect of cohesion in the smaller particles; (b) Intensity of segregation with time as a function of cohesion within smaller blue particles

2-30

Figure 2.6 The variation of intensity of segregation with time as a function of cohesion within bigger red particles

2-31

Figure 2.7 Intensity of segregation with time as a function of adhesion 2-31 Figure. 2.8 Typical Mixer 2-35 Figure 2.9 (a)Draught tube and screw mixer(b) Nauta mixer with orbiting

screw 2-36

Figure 2.10 Gravity-flow (hopper) mixers 2-36 Figure 2.11 fluidization bed 2-36 Figure 2.12 Helical ribbon impeller. 2-37 Figure 2.13 Ribbon mixer used in the industry 2-44 Figure 2.14 Effect of the blade speed for layered magnesium stearate and

59%fill level 2-46

Figure 2.15 Effect of fill level (59 and 100%) on the mixing performance of the blender with the 3-spoke ribbon blade operated at 20 rpm with an off-center spot injection of magnesium stearate.

2-47

Figure 2.16 Effect of fill level on the mixing performance of a blender with the 5-spoke blade operated at 20 rpm.

2-47

Figure 2.17 Evolution of Metzner and Otto factor with the pitch size for helical ribbon agitators found in literature

2-48

Figure 2.18 Evolution of mixing time number with clearance wall for helical ribbon agitators found in literature

2-49

Figure 2.19 (a) Neighbor region of particles I; and (b) zoning the Neighborhood region

2-53

Figure 3.1 Schematic illustration of : (a) mixer configuration; and (b) initial loading of particles

3-67

Figure 3.2 Mixing curves and steady state values: (a) mixing index M versus time for different rd at rs=0.5 and xl= 0.5; and (b) effect of rd on steady-state values of M for rs =0.5. The solid lines are the predictions based on Eq.(3.6), which will be discussed later.

3-68

xi

Figure 3.3 Mixing curves and steady state values: (a) effect of size ratio rson mixing index M for rd=0.22 at xl= 0.1 with l = 6040 kg/m3; and (b) steady-state values of mixing index M as a function of rs. The solid lines are the predictions based on Eq.(3.6).

3-70

Figure 3.4 Effect of xl at different levels of rs when rd =0.22 with l = 6040 kg/m3; the solid lines are the predictions based on Eq.(3.6).

3-71

Figure 3.5 Instantaneous normalized blade relative velocities of particles in horizontal sections at three heights Z measured from vessel base for different rd with rs=0.875 and xl=0.9: blue represents velocity vectors of small particles, and red, those of large particles.

3-72

Figure 3.6 Instantaneous normalized blade relative velocities in cylindrical sections at steady-state for different rd with rs=0.875: blue represents vectors of small particles, and red, those of large particles.

3-73

Figure 3.7 Instantaneous normalized blade relative velocities of particles in horizontal sections at three heights for different rswhen rd=0.22 at xl=0.9; blue arrows represent velocity vectors of small particles, and red, those of large particles.

3-74

Figure 3.8 The driving force on particles in a uniform system particle of d=5mm, ρ=2500kg/m3 and 17500 particles, laid initially in top-bottom arrangement: (a), instantaneous group-averaged driving force, and (b) time–averages of the driving forces in Figure 3.10(a).

3-76

Figure 3.9 The driving forces on the two types of particles as a function of time for binary particle mixtures with rs =0.5 and xl =0.9: (a), instantaneous group-averaged driving force for rd=0.22 and 0.75; and (b), the time-averaged vertical forces for the cases of rd=0.22, 0.41, 0.579, and 0.75

3-77

Figure 3.10 Snapshots showing the buoyancy and percolation effects of particles t= 5.11s; (a) heavy –large particles penetrate into small particles over blade (buoyancy); and (b) small particles percolation; arrow shows the direction of blade motion

3-78

Figure 3.11 Predicted effect of size ratio on steady-state mixing index at different values of volume fractions xl when rd =1.

3-81

Figure 3.12 Contour maps using Eq.(3-6), demonstrating the equivalence of size and density effects at different volume fractions: (a) xl=0.1, (b) xl=0.5 and (c) xl=0.9.

3-81

Figure 3.13 Equivalence of size and density effects at a given mixturequality M and volume fraction xl

3-82

Figure 4.1 Impeller configuration and initial particle deposition layout 4-94 Figure 4.2 Comparison of micro and macro system in terms of mixing

curves at different cohesion 4-95

Figure 4.3 Coordination number comparison of the micro and macro system at Bo number 0.02

4-96

Figure 4.4 Instantaneous velocity of particles in micro and macro systems 4-97

xii

in a vertically-cur vessel segment between z=300 and 350 for Bo=0.02

Figure 4.5 Instantaneous velocity of particles in micro and macro systems in a vessel segment between z=300 and 350 for Bo=2

4-98

Figure 4.6 Instantaneous velocities of particles in micro and macro systems in the longitudinal segment of the vessel between x=-20 and 20 at Bo=0.02.

4-98

Figure 4.7 Probability density distributions of instantaneous velocity of particles in micro and macro systems at 39th rev.

4-100

Figure 4.8 (a) and (b): Cumulative probability distributions of scaled contact force components of particles in micro and macro system at 39th rev; and (c) the distribution averages

4-101

Figure 4.9 Effect of particle density in the case of Bo=0.2 for side-by-side (SBS) initial arrangements: (a) Axially side-by-side arrangement;(b) Radially side-by-side arrangement

4-104

Figure 4.10 Particle Scale Mixing Index: (a) Particle scale mixing index as a function of shaft revolutions; and (b) Average steady state mixing index at different Bo numbers.

4-105

Figure 4.11 Overall coordination number at different Bo numbers: (a) overall coordination number as a function of shaft revolutions; and (b) overall coordination number at steady-state as a function of Bo number.

4-106

Figure 4.12 Figure 4.13 Figure 4.14 Figure 4.15 Figure 4.16 Figure 4.17

Instantaneous velocities of particles in vessel cross sections XY; Z= 350-400 mm, at three time instances with increasing of bond number: red and blue colour represents vectors of two type particles. Instantaneous velocity fields in the longitudinal section at different Bond numbers: red and blue colour represents vectors of two types of particles Average velocity and porosity of particles at t=25-30s, in a longitudinal segment between x=-20 to 20 and y= -255 to 255. Probability density distributions of velocity components: (a) tangential, (b) radial and (c) axial velocity Time averaged mean velocity components as a function of Bo number: radial velocity Vr, tangential velocity Vt and axial velocity Vz. Probability density distributions of average normal stress in the vessel at steady-state for different Bo numbers

4-107 4-107 4-108 4-109 4-109 4-111

Figure 5.1 Geometry of ribbon impellers: (a), 2-bladed impeller, and (b), 4-bladed impeller.

5-120

Figure 5.2 Snapshots depicting blade motion, axial transport and circumferential motion of particles in two-bladed mixer at 25% fill level (Bo=0.2).

5-122

Figure 5.3. Effect of impeller speed: Comparison of mixing behavious of cohesive and non-cohesive particles at 25% fill-level.

5-123

Figure 5.4 Effect of impeller speed on cohesive mixing: Macroscopic index (Bo=0.2 and 25% fill-level)

5-124

xiii

Figure 5.5 Velocity fields in vessel central segment (between z=350 and 450mm) for non-cohesive and cohesive particles mixing at impeller speeds of 50, 100, and 200 rpm from the top to bottom, respectively, at t=25 s.

5-125

Figure 5.6 Comparison of velocity fields in the two halves of the 2-bladed mixer (at 50 rpm).

5-126

Figure 5.7 Velocity field in a longitudinal section between x=-20 and 20mm at impeller speeds of 50, 100, and 200 rpm from top to bottom, respectively at time t=25s, x being measured horizontally from the shaft-axis perpendicular to it.

5-126

Figure 5.8 Probability density distributions of velocity components at different blade speeds for non-cohesive particles at steady state.

5-127

Figure 5.9 Probability density distributions of velocity components at different shaft speeds for cohesive particles (Bo=0.2) at steady state.

5-128

Figure 5.10 Force-network diagrams at different impeller speeds for cohesive and non-cohesive particles in the vessel segment between the axial positions, z= 350 and 450 mm at t=37-39 s; impeller speeds are 50, 100, and 200 rpm from the top to bottom, respectively; and colours and corresponding ranges of contact forces f are: blue, f < 0.235 N; green, 0.235 < f < 0.936 N; light green, 0.936< f < 1.17 N; and red, 1.17 < f <1.62N

5-129

Figure 5.11 Probability density distributions of force components for non-cohesive mixture at steady state.

5-130

Figure 5.12 Mean values of cumulative probability distributions of the force components for cohesive mixture at steady state

5-131

Figure 5.13 Effect of fill level on the mixing performance of a 2-bladed mixer at different Bo numbers (shaft speed = 100 rpm).

5-133

Figure 5.14 Effect of fill level on the mixing performance of a 4-bladed mixer at different Bo numbers. (shaft speed = 100 rpm).

5-133

Figure 5.15 Mixing rates k for the two-bladed and four-bladed mixers (shaft speed = 100 rpm).

5-134

Figure 5.16 Velocity fields at different fill levels in the two bladed and four bladed mixers at the 80th revolution for the two-bladed and four-bladed mixers in cases of non-cohesive and cohesive mixtures (shaft speed = 100 rpm).

5-135

Figure 5.17 Average velocity and porosity of particles in a longitudinal section between x=-40 and 40 mm at the 30 revolution for non-cohesive mixture in the 2-bladed mixer, x being measured at right angle to the shaft from the shaft axis (shaft speed = 100 rpm).

5-136

Figure 5.18 Probability distributions of velocity components and time-averaged mean velocities at different fill levels for the non-cohesive mixtures in 2-bladed mixer at 100 rpm shaft speed

5-137

Figure 5.19

Probability distributions of velocity components and time-averaged mean velocities at different fill levels for non-cohesive mixtures in the 4-bladed mixer at 100 rpm shaft speed.

5-138

xiv

Figure 5.20 Probability distributions of contact force components of particles of the non-cohesive mixture at different fill levels for the 2-bladed and 4-bladed mixers at 100 rpm shaft speed.

5-139

Figure 6.1 Impeller configurations: Top-row, type-1 impeller and bottom-row, type-2 impeller

6-150

Figure 6.2 Effect of blade number or pitch ratio on mixing at different Hap for type-1 impellers

6-152

Figure 6.3 Dependence of mixing rate on pitch ratio or blade number at different particle cohesion.

6-153

Figure 6.4 Comparison of total coordination number: (a) Total coordination number as a function of Hap; (b) Total coordination number as a function of pitch ratio S/D

6-154

Figure 6.5 Snapshots of particle flow in the mixer for 1.12 pitch ratio at different cohesion (at 40th rev.)

6-154

Figure 6.6 Snapshots of the particle flow in the mixer at pitch ratio 0.75 at different cohesion at 40th rev

6-154

Figure 6.7 The averaged velocity in the mixer at a different pitch ratio S/D and Hap; x =-40–40, y= -260– 260, z= 0 –1000, rev 30th

6-155

Figure 6.8 Probability density function velocities of particles in the mixer at different pitch ratio S/D rev 30 , Hap = 5.54 x 10-18 J

6-157

Figure 6.9 The Probability density function of forces of particles in the mixer at different pitch ratio S/D, rev 30, Hap = 5.54 x 10-18 J

6-158

Figure 6.10 Effect of blade width at different Hap for type-1 impeller 6-159 Figure 6.11 Effect of blade width on mixing rate for a type-1 impeller: (a)

mixing rate as a function of Hap; and (b) mixing rate as a function of blade width W.

6-160

Figure 6.12 Time and cell averaged particle velocities in the mixer at different W and Hap in the longitudinal section, where x values range between -40 and 40 (at 30th rev.)

6-161

Figure 6.13 The Probability density distributions of particle velocity components in the mixer at different blade widths W at Hap = 5.54 x 10-18 J (Bo=0.02) and at the 30th rev for type 1 mixer.

6-163

Figure 6.14 Cumulative probability density distributions of velocities of particles in the mixer at different blade width W, Hap = 5.54 x 10-18 J and rev 30th for type-1 mixer.

6-164

Figure 6.15 The effect of blade clearance at different cohesion (type-1 impeller).

6-166

Figure 6.16 Comparison of mixing rate: (a) Mixing rate as a function of Hap;(b) Mixing rate as a function of clearance C.

6-166

Figure 6.17 The averaged velocity of particles in the mixer at different blade width W and Hap ; x =-40-- 40, y= -260– 260 z= 0 --1000 , 30th rev.

6-168

Figure 6.18 The Probability density function of velocities of particles in the mixer with different blade clearance C, Hap = 5.54 x 10-18 J, 30th rev.

6-170

Figure 6.19 The Probability density function of forces of particles in the mixer at different blade clearance C, Hap = 5.54 x 10-18 J, rev 30th

6-171

Figure 6.20 Mixing index at 30rev as a function of blade clearance, width , pitch ratio and Hap

6-172

xv

Figure 6.21 Effect of outer rake angle (inner blade angle is fixed at 45°): (a) Mixing index variation with shaft revolutions; (b) Mixing rate k as a function of outer blade angle

6-173

Figure 6.22 Effect of blade number (or pitch ratio S/D) on mixing performance for non-cohesive mixtures. (a) type 2 impeller, (b) type 1 impeller (c) mixing rate k as a function of pitch ratio.

6-175

Figure 6.23 Average velocity and porosity of the particle mixture, snap shot at revolution at 38, t=13s,X =-20-20, at ZY plane in type-1 impeller

6-176

Figure 6.24 Velocity fields and snapshots of the mixing states at steady-state for the type-2 impellers of different pitch ratios, blue and red represent the two types of particles initially laid in the side-by-side arrangement

6-177

Figure 6.25 Effect of blade width on mixing behaviour at a pitch rratio of 1.12.

6-178

Figure 6.26 Mixing behaviour for different blade widths for type-1 impeller design. Top: Steady-state velocity fields and spatial distribution of void fraction in the longitudinal central plane; and bottom: snapshot of particles at 80th revolution (t=25s) in the longitudinal central plane, with blue and red representing two types of particles.

6-178

Figure 6.27 Velocity fields and snapshots of the mixing states at steady-state for the type-2 impellers of W=20, 40 and 60 mm: blue and red represent the two types of particles initially laid in side-by-side arrangement

6-179

Figure 7.1 Geometry of ribbon impeller 7-188 Figure 7.2 Particle scale mixing index as a function of revolutions at

different volume fractions when rs=0.66 : (a) xl=0.1; (b) xl=0.5; (c) xl=0.9

7-190

Figure 7.3 Effects of rd and xl on M at rs=0.66: (a), effect of rd at different xl ; and (b), effect of xl at different rd

7-191

Figure 7.4 Effects of rd on contact forces at rs=0.66: (a), normal force ; and (b), tangential force for a volume faction of 0.5.

7-192

Figure 7.5 Effects of rd on velocity field at rs=0.6 at different cylindrical height for a volume faction xl of 0.5

7-193

Figure 7.6 Instantaneous probability density distributions of particle velocity components at different density ratios when size ratio is fixed at rs=0. 6

7-194

Figure 7.7 Particle scale mixing index as a function of revolutions at different values of rs with rd=0.33: (a) xl=0.1; (b) xl=0.5; and (c) xl=0.9

7-194

Figure 7.8 Effect of rs and xl on steady-state values of mixing index (or mixture quality) for rd = 0.33 and l = 6040 kg/m3: (a) effect of rs at different xl; and (b) representation of results in (a) as an effect of xl at different rs

7-195

Figure 7.9 Effects of rs on contact forces when xl =0.9 7-196

xvi

Figure 7.10 Effect of rs on velocity field when rd = 0.33 and xl =0.9 7-197 Figure 7.11 Probability density of velocity components at different size

ratio when rd = 0.33, xl =0.9 at t=50s 7-198

Figure 7.12 Contour maps using the prediction equation, demonstrating the equivalence of size and density effects at different volume fractions, xl of 0. 1, 0.5 and 0.9.

7-200

Figure 7.13

Comparison of the effects of rs, rd and xl on particle mixing behaviour in the cylindrical mixer and ribbon mixer; M is steady-state values of mixing index, solid line is representing prediction: (a) the effect of size ratio when xl =0.9 for cylindrical mixer rd=0.33; for ribbon mixer rd=0.22; (b) density effect when xl =0.1 for cylindrical mixer rs=0.6; for ribbon mixer rs=0.5; (c)volume fraction effect: for cylindrical mixer rs=0.875, rd=0.22; for ribbon mixer rs=0.8, rd=0.33.

7-200

Figure 8.1 Figure 8.2

Initial loading pattern of particles in the rotating drum Effect of density ratio on mixing: (a) Segregation index as a function of time and revolutions when rs=0.66; and (b) steady-state segregation index as a function of density ratio

8-210 8-213

Figure 8.3 Effect of rd on the segregation flow pattern at rs=0.66; blue represents small particles, and red the large ones

8-213

Figure 8.4 Effects of rd on contact forces at rs=0.6: (a) normal force ; and (b) tangential force

8-214

Figure 8.5 Effect of size ratio: (a) Segregation index as a function of time and revolutions when rd=0.33; and (b) Average segregation index as a function of size ratio; the solid line is prediction value.

8-215

Figure 8.6 Effects of rs on the segregation flow pattern when rd = 0.33 8-216 Figure 8.7 Effect of rs on contact forces: (a), normal force ; and (b),

tangential force 8-217

Figure 8.8 Segregation index as a function of the time whenrs and rd are both decreased

8-218

Figure 8.9 Evolution of the segregation flow pattern when rs and rd

decreased; particle condition similar with Fig 8.3 8-218

Figure 8.10 Segregation index as a function of time when rs decreases and rd increases at the same time

8-219

Figure 8.11 Evolution of the segregation flow pattern when rs decreases and rd increases simultaneously.

8-220

Figure 8.12 Contour maps of segregation index using the prediction equation, to demonstrate the effects of size and density on the mixing.

8-222

xvii

List of Tables

Table 2.1 Main finding in the helical and horizontal ribbon impellers 2-38 Table 3.1 Formulae for contact forces and torques 3-64 Table 3.2 Particle details for varying rd at three levels of xlwith rs fixed. 3-66 Table 3.3 Particle details for varying rs at different levels of α with rd

fixed. 3-66

Table 3.4 Particle details for varying rd at different levels of rs with xlfixed

3-67

Table 4.1 Formulae for contact forces and torques 4-90 Table 4.2 DEM Input variables and their values 4-93 Table 4.3 Input values for the macro and micro systems 4-93 Table 4.4 Bond number for a single contact at different values of

Hamaker constant 4-94

Table 5.1 Formulae for contact and non-contact forces and torques 5-118 Table 5.2 Simulation Input variables and their values 5-119 Table 5.3 Bond number for a single contact at different values of 5-120 Hamaker constant

6-170

Table 6.1 Equations used to calculate forces in the DEM simulations 6-147 Table 6.2 Particle material Properties 6-149 Table 6.3 Geometry parameters of ribbon mixer 6-150 Table 7.1. Formulae for contact forces and torques 7-187

Table 7.2 Input variables and their values 7-188

Table 7.3 Particle information for density effect cases 7-189

Table 7.4 Particle information for size effect cases 7-189

Table 7.5 Comparison of mixing index of simulation and mixing index from equation

7-199

Table 8.1 Formulae for contact forces and torques 8-208

Table 8.2 Input variables and their values 8-210

Table 8.3 Particle information for size effect cases 8-211

Table 8.4 Comparison of segregation index and prediction 8-221

CHAPTER 1 Thesis Overview

1-1

Chapter 1

Overview of Thesis


1-2

1. INTRODUCTION Powder mixing is a widely used process in pharmaceutical, powder metallurgical and

food industries, to name a few. Assuring the homogeneity of powder mixtures is

essential to improve the quality of products of those industries. Mixing behaviour of

powders is significantly affected by impeller geometry, operational conditions of the

mixer and properties of particles to be mixed. The particle properties of the mixture can

also have a significant effect on the mixing behaviour, the most significant apart from

the size difference being the density difference. An increase in either the size or density

differences results in increased segregation tendencies in cylindrical mixers, for

example (Stewart, Bridgwater et al. 2001, Zhou, Yu et al. 2003). The segregation

mechanism for cylindrical mixers has been investigated (Zhou, Yu et al. 2004).

According to this mechanism, particles in a cylindrical mixer are segregated due to the

generation of vertical forces on particles when there is either a size or density difference

or both. Nevertheless, the scope of the study was mainly limited to the examination of

particle segregation due to size and density differences. The predictability of size and

density effect is still unknown with the exception of rotating drums, where the condition

for transition from mixing to segregation can be predicted. Consequently it is important

to investigate the predictability of the effects of material properties such as size, density

and volume fraction on particle mixing and their mechanisms in different mixers.

The selection of a mixer for a mixing operation depends on the mixture

homogeneity and many other factors (Poux, Fayolle et al. 1991). Ribbon mixers are

considered to be suitable for mixing of dry powders as well as free-flowing granular

material (Poux, Fayolle et al. 1991). It is also reported that a ribbon mixer can produce

an improved uniformity in powder mixing due to large shear stresses in the mixer as

well as that it can handle mixing of different size particles (Muzzio, Llusa et al. 2008).

The impeller speed of the mixer and fill level are parameters that can affect the

performance of the mixer, with regard to mixture quality and stresses on particles. It is

reported that a high impeller speed can have a negative effect on the mixture uniformity

(Muzzio, Llusa et al. 2008). A low speed will reduce shear stresses on particles, but may

result in a poor homogeneity for cohesive mixtures (Muzzio, Llusa et al. 2008). A shaft

speed in the range of 50-70rpm appears to be a favourable speed for obtaining uniform


1-3

mixing (Sanoh, Arai et al. 1974). However research on the effects of operational

parameters and geometrical parameters on powder mixing in ribbon mixer are few in

the literature. Further, a mixture of particles with large density and size differences can

also be mixed in a ribbon mixer (Poux, Fayolle et al. 1991). Mixture quality is

significantly affected by size differences of powder particles, and an increase in either

the size or density differences results in increased segregation tendencies (Fan, Gelves-

Arocha et al. 1975, Stephens and Bridgwater 1978). However, research in this regard

using a complicated mixer such as the ribbon mixer are few, and the prediction of the

effects of particle size and density on the mixer performance has not been clearly

established.

In rotating drums, the radial particle segregation flow pattern and segregation

mechanism have been widely investigated due to its simplicity. The particle size

induced radial segregation (Clément, Rajchenbach et al. 1995)(Makse 1999, Eskin and

Kalman 2000) (Thomas 2000) and density induced segregation (Ristow 1994, Ottino

and Khakhar 2000) exist in rotating drums, and the size segregation can be counter-

balanced by density segregation by varying the density of small particles with the size

ratio of a binary particle mixture is fixed (Dury and Ristow 1999). It has been reported

that the transition from mixing to segregation due to the effect of the combination of

size and density differences can be predicted for rotating drums (Alonso, Satoh et al.

1991). Mostly, the percolation and buoyancy effects are combined to enhance the

segregation when smaller particles are heavier. At other times, the peocolation and

buoyancy effects can oppose each other and the segregation reduced (Liu, Yang et al.

2013). The feasibility of prediction of segregation in rotating drums in a wide range of

particle properties still needs to be investigated, and the segregation mechanisms

compared against other mixers.

Here, we are concerned with the particle mixing behaviour in a cylindrical mixer,

ribbon mixer and rotating drum. The discrete element method is used to investigate the

effect of material properties, operational properties and geometrical parameters on the

mixture quality. A particle scale mixing index (Chandratilleke, Yu et al. 2012) based on

the coordination number was primarily used to quantify the mixture quality. The

parameters of interest in the study are as follows:


1-4

The material properties:

Particle size ratio

Particle density ratio

Volume fraction

Particle cohesion

The operational parameters are:

Rotational speed

Vessel fill level

The geometry related parameters:

Impeller pitch ratio

Blade width

Blade clearance

Blade angle

Blade number

Types of mixtures:

Mono-sized particles of uniform density,

Binary particle mixtures of different size and density at different volume

fractions,

Cohesive particle mixtures with uniform size and density.

The thesis consists of the following chapters and contents.

Chapter 2 summarizes a literature survey on the particle mixing studies, which include

types of powders, characterization of mixers and factors affecting mixing such as

material properties, geometrical parameters and operational parameters and types of

mixers.

Chapter 3 investigates the effect of size, density and volume fraction on non-cohesive

binary mixtures in a cylindrical mixer. The mechanism of mixing improvement was

examined for binary mixtures. A correlation was established to predict the effect of size,


1-5

density and volume fraction. Such an equation can be used to study the effects of such

as particle properties on mixing behaviour in a cylindrical mixer.

Chapter 4 investigates the effect of particle cohesion on mixing behaviour in a ribbon

mixer. The particle mixing behaviours in micro and macro systems were compared

keeping the dimensionless numbers Froude number and Bond number fixed for the two

systems. The two systems show the similarity in the mixing behaviour, velocity field,

contact forces and coordination number in a low-cohesive mixture. The effect of

particle cohesion on mixing behaviour is investigated using large lighter particles in the

macro system.

Chapter 5 investigates the effects of two operational parameters at different mixture

cohesion using different ribbon impellers. The effect of shaft speed on particle mixing

rate, mixture quality, particle velocity and contact forces were investigated in cohesive

and non-cohesive mixtures using a two-bladed ribbon impeller. The effect of the mixer

fill level of particle mixing behaviour was examined for mixtures with different

cohesion both in two-bladed and four-bladed ribbon mixers.

Chapter 6 examines the effect of geometrical parameters of a ribbon impeller for

different cohesive particle mixtures. The effects of pitch ratio, blade width, blade

clearance, blade angle and blade numbers on the mixing behaviour were investigated for

different cohesive mixtures using mixing rate, mixture quality, particle velocity and

contact forces.

Chapter 7 investigates the particle mixing behaviour in a ribbon mixer for non-

cohesive binary mixtures. The effects of particle size, density ratio and volume fraction

on binary particle mixtures in a ribbon mixer were studied. An equation for predicting

steady-state mixture quality at different size and density ratios and volume fraction was

established.

Chapter 8 examines the radial segregation flow pattern of binary mixtures in a rotating

drum. The size and density induced segregation mechanism in the rolling mode was

investigated. An equation for predicting steady-state mixture quality at different size


1-6

and density ratios and volume fraction was established. The predictability of the effects

of size and density of particles at volume fraction 0.5 was confirmed.

Finally, Chapter 9 gives an overall summary of the thesis on mixing of particles.

Possible future research is also suggested.

Chapter2

Literature Review

CHAPTER 2 Literature Review

2-8

2.1 Introduction

Powder mixing is essential to control the quality of products manufactured from

granular mixtures in many industries. For example, tablets and capsules in

pharmaceuticals (Hilton and Cleary 2013) and high hardness composite material for

cutting in powder metallurgy industries (Fernandez, Cleary et al. 2011). Plastic

materials (Saberian, Segonne et al. 2002; Metzger and Glasser 2012) and cosmetic

materials (Delaney, Cleary et al. 2012). Hence, the knowledge of the mixing behaviour

of powders is important in processing industries. The size and density differences of the

particles are the most important properties that effect the mixing behaviour of particles

(Fan, Gelves-Arocha et al. 1975; Stephens and Bridgwater 1978; Fan, Chen et al. 1990).

Segregation is enhanced due to an increase in either the size or density differences as

reported in the following cases: a rotating drum (Alonso, Satoh et al. 1991; Metcalfe

and Shattuck 1996; Eskin and Kalman 2000; Xu, Xu et al. 2010; Jayasundara, Yang et

al. 2012); a cylindrical mixer (Zhou, Yu et al. 2003; Chandratilleke, Yu et al. 2011) and

a vibrated granular system (Rosato, Strandburg et al. 1987; Shinbrot and Muzzio 1998;

Yang 2006). It is stated that a prediction equation to quantify the combined effects of

density and size differences would enable us to predict the transition from mixing to

segregation by percolation due to size difference and by buoyancy effect induced by

density differences (Alonso, Satoh et al. 1991). Such a relationship would be valueble to

quantitatively predict the mixing states of not only dry particles, but also wet particles at

low cohesion when the particle flow is in continuous regime in rotating drums (Liu,

Yang et al. 2013). However, predicting the size and density effects in the size-range

where percolation effects are negligible has not been a theme in the reported works.

A ribbon mixer is known to possess a combination of mixing mechanisms such as

convection, diffusion and shear when mixing granular matter. Ribbon mixers are widely

used in practice because they are capable of providing high speed convective mixing. At

present, there is only a limited understanding of the cohesive effects on the mixing

behavior of dry fine particles in the mixers used in practice (Chaudhuri, Mehrotra et al.

2006), particularly in relation to ribbon mixers in the pharmaceutical industry where

they are used for powder mixing purposes (Muzzio, Llusa et al. 2008). The impeller

speed of the mixer is a parameter that can affect the performance of the mixer, with

regard to mixture quality and stresses on particles. The fill level of the blender is the


2-9

next most important operational parameter. The efficiency of such operations and

quality of the resulting mixtures will depend on the impeller geometry. It is important to

understand the effects of the impeller geometry and use the right geometry for the

product to be manufactured based on industry requirements, which can be different

from one industry to another. Some examples of different industries being

pharmaceuticals, food, cosmetics and powder metallurgy industries. Lack of knowledge

about all of the issues we mentioned above provide us the motivation to conduct

systematic research on the effects of size, density and volume fraction in cylindrical

mixers, ribbon mixers and rotating drums as well as the effects of cohesion, operational

parameters and geometry in ribbon mixers.

2.2 Applications of Powder mixing

The mixing quality affects the product quality. For example, in the

manufacturing of polyvinyl chloride (PVC) products, mixing of PVC particles with

appropriate additive particles can modify and enhance the properties of PVC (Saberian,

Segonne et al. 2002). Preparation of ceramics, mixtures for glass manufacture, materials

for high-strength cutting-tools and pharmaceuticals are some further examples. Below, a

brief description is given of how mixing is being used in each of the three typical

industries.

Pharmaceutical industry

The pharmaceutical industry uses a variety of mixers in the preparation of

powder mixtures for subsequent granulation, compaction or encapsulation. Many

mixers used include a variety of tumbling mixers, such as cube mixers, cone mixers, V-

mixers and Y-mixers or ribbon mixers (Hersey). In this industry, quality control of the

mixed material is very important. However, it is said that mixing processes are carried

out largely in an empirical manner and in accordance with arbitrary regulations. If the

product variation is beyond specified limits, a batch of products will be discarded, to

comply with regulations. Such procedures are needed to protect consumers from

possible medical health risks since poor quality can cause different drug-release rates

with possible side-effects. In addition, production costs may rise if tablets are prone to

crumbling in the manufacturing process because of non-uniformity in the structure.

Therefore, the overall quality of a mixture is understood to be crucial for all the parties


2-10

concerned. Similar consideration is applicable to many other applications. For example

micro-scale mixing is shown to improve the strength of a tool-making compound, which

consists of alloyed WC- Co particles and a mixture of TiC and Al2O3 coating particles.

Correspondingly, particle scale mixing has been shown to be important in the cosmetic

industry for improving the efficiency of sun-screen lotions while avoiding skin

irritation (Liang, Ueno et al. 2000).

Food industry

Many mixing devices and agitators are used in the food processing industry

today. Dry food materials that are mixed include flour, sugar, salt, flavouring materials,

flaked cereals, dried milk, and dried vegetables and fruits. Solids mixing or blending of

ingredients is an extensive processing operation used for the preparation of animal feeds,

fertilizers, seed stocks, insecticides, fertilizer, and packaged foods. Solids may be

mixed to facilitate reactions in the preparation of cereal products (Lindley).

Powder metallurgy

The methods of reinforcement used in powder mixing influence the physical and

mechanical properties of the matrix composites. There are three methods for

reinforcement such as powder metallurgy, liquid metal particulate mixing and in-situ

production of dispersions. The powder metallurgy method has some advantages in

obtaining net shaped products but the high cost and limitations for the homogeneity of

the material are still an issue (Parashivamurthy, Kumar et al. 2001). They have shown

that the state of the interface between particles and matrix contributes to the

enhancement of the elastic modulus, yield strength and wear resistance. The metal-

matrix interface coherence and the particle distribution govern the strengthening

mechanism. In terms of powder metallurgy, the mixing quality (uniformity) of the

particles and volume fraction of reinforcement strongly influenced the mechanical

properties of the matrix composites. It is found that the strength of particle-reinforced

matrix composites is higher than that of monolithic material. With a decrease in particle

size of the reinforced particles, 0.2% proof stress and tensile strength tend to increase.

However, the toughness and ductility of the material with the reinforced particulate

mass have decreased (Doel and Bowen 1996). The hardness of the reinforced alloy

matrix has increased with higher volume fraction of reinforcement (Srivatsan and


2-11

Auradkar 1992). The morphology of the TiC reinforcement particle influenced the

properties of the Ti-TiC composites were investigated. Refinement of the secondary

dendrite arm spacing of TiC particles in 3-D was found to dramatically improve the

ultimate tensile strength (UTS) and ductility of the Ti-TiC composites (Lin, Zee et al.

1991). Mechanical properties of in-situ synthesized titanium matrix composites with

2.11 vol.% TiB reinforcements shows the highest tensile strength and lowest steady

state creep rate. Morphology of TiB whiskers was essential to mechanical properties of

high temperature titanium matrix composites (Zhang, Qin et al. 2010).

2.3 Issues

Mixing behavior of powders is significantly affected by several particle

properties of a mixture and they can have a significant effect on the mixing behaviour.

One of the most significant effects (apart from the size difference) being the density

difference. An increase in either size or density differences results in increased

segregation tendencies. On the other hand, the smaller the difference in particle size, the

higher the mixing rate. An increase in either the size or density difference causes an

increase in the rate of segregation in a vertically-shafted cylindrical bladed mixer. The

particles are segregated due to the generation of a vertical force on particles when there

is either a size or density difference or both. However, research shows that the

combination of the size and density ratios would minimize the free surface segregation.

Differences in both size and density increase the rate of mixing and segregation. It is

reported that the powder mixing can be improved under certain size and density

combinations in the case of mixing in rotating drums (Alonso, Satoh et al. 1991).

Although the mechanism of size and density effect is explained using phenomenal

description, it is not fully understood based on a fundamental study. The predictability

of size and density effects is still unknown.

Another issue is that the selection of a mixer for a mixing operation depends on

the product uniformity required and many other factors (Poux, Fayolle et al. 1991). The

impeller speed of the mixer is a parameter that can affect the performance of the mixer

with regard to mixture quality and stresses on particles. It is reported that a high

impeller speed can have a negative effect on the mixture uniformity. A low speed will

reduce shear stresses on particles, and may result in a poor homogeneity for cohesive

mixtures. A shaft speed in the range of 50-70rpm appears to be a favourable speed for


2-12

obtaining uniform mixing. There is very little research on a complicated mixer such as

the ribbon mixer and the relationship of mixture quality to inter-particle forces and

impeller speed has not been quantitatively established. A ribbon mixer displays a

combination of mixing mechanisms such as convection, diffusion and shear when

mixing granular matter. Ribbon mixers are widely used in practice because they are

capable of providing high speed convective mixing. The efficiency of such operations

and quality of the resulting mixtures will depend on the impeller geometry. Thus, it is

important to understand the effects of the blade geometry and use the right geometry for

product manufacture in many industries such as pharmaceuticals and powder metallurgy

industries.

Thus, lack of understanding of the effects of material properties, operational

parameters and geometrical parameters on the powder mixing process in the ribbon

mixer will lead to unstable product quality in industry.

The rotating drum is well known for its simplicity to study the segregation

mechanism such as a percolation and buoyancy due to particle size and density

differences. But the predictability of the segregation or mixing have not been fully

understood so far.

2.4 Overcoming the issues

DEM has been used in order to acquire information on the performance of

different mixers that is difficult and/or expensive to obtain using traditional

experimental approaches (Cleary and Sinnott 2008). Controlling the mixing time can be

used to reduce segregation for example due to size or density effects(Chandratilleke, Yu

et al. 2012). For the mixing time to be controlled, we need to know the mixing

behaviour of particles beforehand for each type of mixtures. Thus the establishment of

mixture quality characteristics (peak values and steady-state ones) will be important. In

order to do this, a mixing index that is not dependent on sample size or number needs to

be used. One solution to this problem is the use of the Particles Scale Mixing index

(Chandratilleke, Yu et al. 2012).

2.5 Types of powder mixing processes

Mixing is an essential process of many processes in the food, pharmaceutical,

paper, plastics, ceramics and rubber industries. There are several types of mixing


2-13

processes, for example: solid-liquid mixing; gas-liquid mixing; three phase mixing; and

solid mixing (Nienow, Harnby et al. 1997).

Solid-liquid mixing

In operations such as crystallization or solid catalysed liquid reactions, it is

necessary to suspend solid particles in a relatively low viscous liquid. This can be

achieved in mechanically agitated vessels where the mixer is used to prevent

sedimentation of solids and to provide conditions suitable for good liquid-solid mass

transfer and chemical reaction. If the agitation is stopped the solids will settle out of the

liquid and float to the surface, depending upon the relative densities of the solid and

liquid phases. At the opposite extreme it may be required to disperse very fine particles

into a highly viscous liquid.

Gas-liquid mixing

Several major industrial operations, e.g. oxidation, hydrogenation, and

biological fermentation, involve the contacting of gases and liquids. The objective of

such processes is to agitate the gas-liquid mixture, generating a dispersion of gas

bubbles in a continuous liquid phase. Mass transfer then takes place across the gas-

liquid interface which is created.

Three-phase contacting

In some penetrations (e.g. hydrogenation, froth flotation and evaporative

crystallization), it is necessary to achieve contact between three phases.

Solid mixing

A feature which tends to be present only in solid mixing is segregation. This is

the tendency of particles to separate out according to size and/or density.

2.6 Solid mixing

The solid mixing is an operation by which two or more solid materials in

particulate form are scattered randomly in a mixer among each other by random

movements of particles. The important aspects of the solid mixing study include: the


2-14

mixing mechanism of the solid mixing; the homogeneity of the mixture; the

characterization methods for uniformity of solid mixing; types of solid mixer; the effect

of material properties of the solids on the solid mixing; and the effect of the operational

and geometrical parameters of the mixer.

Mixing mechanisms

The mixing of powders takes place by three main processes (Poux, Fayolle et al.

1991).

• Mixing by convection

This type of homogenization is characterized by the motion of groups of

particles within the mixture. The components are subdivided into clumps. They are

displaced relative to one another and their size is reduced. This motion creates contact

area between different components and carries out mixing on a large scale.

• Mixing by Ddiffusion

Homogenization is created by motion of individual particles, to ensure mixing

on a fine scale. Diffusive mixing is caused by the random motion of powder particles.

The mechanism has a slow rate of mixing compared to the convective mixing. But the

diffusive mixing is critical for homogenization of a mixture at microscopic or particle

scale.

• Shear mixing

Mixing occurs by slipping of particle planes within the whole volume. This third

mechanism is often considered a combination of the two previous processes, but a

specific mechanical action needs to be considered.

2.7 Characterization of mixture

When is a mixture well mixed? This question can be clarified in two ways:

qualitatively and quantitatively (Harnby 1997).

2.7.1 Qualitative approach

The characterization could be improved by a better quality of mixture

determined by the scale of segregation within the mixture. The “scale of segregation” of


2-15

a mixture is a measure of the size of the region of segregation within the mixture. The

smaller the scale of segregation results in a better mixture. The divergence from the

mean composition is measured using intensity of segregation. Alternatively, the

intensity of segregation can be regarded as the amount of dilution that has happened

within the segregation area. The lower the intensity of segregation means the better the

mixture.

2.7.2 Quantitative approach

The mean composition value usually used to control overall content of key

component. For measuring the quality of the mixture, the standard deviation of

distribution is used. The low standard deviation means a good quality of mixture.

2.7.2.1 Definition of mixture status

The mixture is operated to get a distribution in which each particle of a

constituent is near a particle of another constituent. It is referred to as a perfect mixture.

• Perfect random mixture

The probability of finding a particle of a constituent of the mixture is the same

for all points in the mixture (Poux, Fayolle et al. 1991).

• Random mixture

It requires particles of equal size and weight with little or no surface effects

(Poux, Fayolle et al. 1991). Ordered mixture does not require equally sized or weighted

particles but rather interaction.

• Ordered mixture

This can be observed when a small portion of fine particles adhere to the coarse

particles of different type of materials.

• Homogeneous mixture

The compositions of all constituents are uniform within the whole mixture (Fan,

Chen et al. 1990).


2-16

• Segregation

This occurs within a mixture when differences in particulate properties cause a

preferential movement of particles to certain regions of the mixer. The differences of

particle size is a main reason for causing segregation (Harnby 1997).

2.7.2.2 Mixing index

For characterization of uniformity of a solid mixture, a mixing index is generally

used to measure the degree of mixedness. Over 40 different mixing indexes have been

proposed by various authors. The number of the criteria of the degree of mixedness

shows the complexity of the mixing process (Poux, Fayolle et al. 1991) and the

difficulties in estimating homogeneity (Poux, Fayolle et al. 1991). The index concerned

with a binary mixture is introduced here. Lacey‟s mixing index is defined as follows:

(Lacey 1954)

(2.1)

where, 02

and R2 are sample variances of fully-segregated and fully-random

states respectively as above, and 2 is the sample variance of a mixture at a transition

state between the two states - reference states. This mixing index compares the sample

variance of the actual mixture with respect to the two reference mixing states, the fully-

segregated state, and fully-mixed (or randomly mixed) states of the mixture.

2.7.2.3 Segregation index

Intensity of segregation has been defined by (Missiaen and Thomas 1995) similar

to Lacey‟s mixing index , which is given below.

(2.2)

Here, (2 - R

2) is the residual variance of the actual mixture with respect to the

uniform (or fully-random) mixture, and (02

- R2 ) is the variance of the transition state

with respect to the uniform state. In contrast to Lacey‟s mixing index, I = 0 for the fully

mixed state, and I = 1 for the fully-segregated state. The intensity of segregation

decreases when the mixing proceeds from the unmixed state to a mixed state. It is

220

220

R

M

220

22

R

RI


2-17

reported that I should be independent of the sample size N at any state of the mixing for

I to be considered as an intrinsic characteristic of the actual mixture (Missiaen and

Thomas 1995). The efforts on the relationship between the residual variance 2 - R2)

and sample size N. The composition variations within samples can be neglected when

the residual variance 2 - R2 ) varied in proportion to N (N-1) (Carley-Macauly and

Donald 1962). It is state that the focus of such works is segregation occurring at large

scales (Missiaen and Thomas 1995). On the other hand, Yamane used a slightly

different definition for intensity of segregation in their DEM (Discrete Element Method)

simulations as follows, which uses standard deviations (Yamane 2004):

(2.3)

Further, Danckwerts used the ratio of the variances as the intensity of

segregation as follows (Danckwerts 1952; Danckwerts 1952; Poux, Fayolle et al. 1991;

Muzzio, Robinson et al. 1997):

(2.4)

The use of variance over standard deviation may be more appropriate because the

standard deviation can be both positive and negative.

2.7.2.4 Particle-scale mixing index

Chandratilleke, Bridgewater et al (Chandratilleke, Yu et al. 2012)recently

defined a particle-scale degree of mixing as a particle i in a binary mixture as shown in

Figure. 2.1. It can have several particles, in contact or near-contact with it, and we

consider those particles, including particle „i‟ as a sample at the particle scale. A contact

sphere is defined for each particle i with a diameter of 1.05 d, where d is the maximum

diameter of the particles in the mixture (see Fig.2.1 (b)). If this sphere, belonging to

particle i interacts with another particle the two particles are said to be in contact with

each other. Particle i can either be a particle of W-type or B-type, being a particle of a

binary mixture.

0

I

20

2

I


2-18

Figure 2.1 Schematic representation of particle contacts: (a) a group of particles in the neighbourhood of particle „i‟, which can be a particle of W-type or B-type; and (b) shows the contact condition according to which particle X is in contact with Y but not with Z.

If the number of particles of B-type in the sample is NB, then the number fraction pi

of the B-type key particles to the total number of particles in the sample

is (Chandratilleke, Bridgwater et al.):

(2.5)

where Cni is the total coordination number of particle i. If particle i is of B-type, NB=

Cn B(B) +1, and if it is of W-type, then NB= Cn B(W), where CnB(B) and CnB(W) are B-type

contacts with particle i, which can be either B-type or W-type particle respectively as

indicated by brackets.

Based on the above definition, the particle fraction pi of a target type of particles

could be calculated in a particle-scale sample around each particle i in the mixture, at

time t. by the way a frequency distribution curve for pi at time t is established. Next, it

could settle the average value for this instantaneous frequency distribution as in

Eq.(2.6). Finally, using , we can calculate the variance St2 of the instantaneous

frequency distribution of pi as in Eq.(2.7), where St is the standard deviation of the

instantaneous particle fraction distribution at time t, and N is the total number of

particles.

1

i

Bi Cn

Np

tp

tp

(a) (b)


2-19

(2.6)

(2.7)

The particle-scale degree of mixing M identical to Lacey‟s mixing index are

defined as follows (Chandratilleke, Bridgwater et al.):

(2.8)

where, So2 is the particle scale variance of the fully segregated state, and Sr

2 is

the variance for the fully-mixed state. The population variance at segregated state for a

mixing ratio of p is given by , and S02 is made equal to 0

2 for the reason

that 02 does not depend on the sample size. Sr

2 is not known for a binary sized mixture.

Therefore, to account for the case of mono-sized as well as binary sized particles, we

choose the reference well-mixed state for Eq.(2.8), as the well-mixed state of mono-

sized particles, at the same mixing ratio p as that of the binary mixture; after all, no

better mixing can be obtained than those of mono-sized particles. Therefore, Sr2 is

defined as follows (Chandratilleke, Bridgwater et al.):

(2.9)

Here, n is a particle-scale sample size, and is chosen as 1 + the average total

coordination number.

2.7.2.5 Powder Sampling (In practice)

Powder sampling is an essential procedure that determines the quality of mixture.

Sampling methods used in assessing homogeneity of powder mixtures are affected by

parameters such as sample size, sample shape and sample number.

Ni

iip

Np

1t

1

Ni

ii pp

NS

1

2t

2t

1

220

220

r

t

SSSSM

pp 120

n

ppSr

12


2-20

A small volume of material is analysed to assess the quality of the mixture and

to calculate its composition. A sample can be taken from mixture by probes or on-line

during the mixing operation. The analysis of the composition can be performed using

the Lacey Mixing Index.

The size of the sample must be suitable to the dimensions of the powder

materials. It is obvious that the large volume and small volume of material both led to

error in accordance with the mixing index. Indeed, the volume of the mixture sample

form must have an appropriate scale of size (Poux, Fayolle et al. 1991) . We can see

that when the sample size is very large, the variance of the sample is very small,

indicating that the mixture is in a fully mixed state. However, if we reduce the sample

size, the sample variance starts increasing even when the mixture is at a fully-mixed

state, such that at this time we begin to see the non-uniformities of the mixture.

Therefore, to quantify the homogeneity of the mixture, we have to define a sample size.

Figure 2.2 Sample variance of a fully-random mixture as a function of sample

size

In terms of the number of samples, Poux et al suggest that twenty to forty samples

should generally suffice to obtain correct information on the homogeneity of a

mixture (Poux, Fayolle et al. 1991); evidently, a large number of samples can cause

structural instabilities in a mixture. They point out that even with a small number of

samples; a distribution such as a Student-distribution can be used to obtain a mean value

with certain confidence limits. Although various researchers have attempted to establish

a relationship between the coefficient of variation and sample size, there appears to be

no common consensus yet (Poux, Fayolle et al. 1991).

However, Portillo et al. more recently addressed the issue of sample size and

sample number using a compartment modeling approach (Portillo, Muzzio et al. 2006).

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0 50 100 150

Sam

ple

varia

nce

Sample size

Fully-random

Fully-segregated


2-21

In the compartment modeling approach, the mixer system is discredited into a number

of homogeneous subs-sections each containing a fixed number of particles. Particles are

allowed to enter and exit each compartment using experimentally determined fluxes of

particles between compartments, and changes in the concentrations in the compartments

are then captured, although exact particle positions cannot be determined. Using this

method, a large number of particles can be handled (Portillo, Muzzio et al. 2006).

Figure 2.3 (i) Effect of sample number on variance, (a) 100, (b) 200 and (c) 500

samples; plots are normalized variance frequency distributions; (Siiria and Yliruusi)

effect of particle number on variance, (a) 200 pps, (b) 400 pps, and (c) 600 pps, where

pps is particles per sample (Portillo, Muzzio et al. 2006).

The effect of a sample number is shown in Figure 2.3(i). It shows that increasing

the number of samples makes the distribution narrower. The distribution approaches a

2 (or a chi-squared) distribution; the skewness of the distribution has increased with the

increased sample number. Although increasing either the particle number per sample or

samples number reduces the variance, increasing the particles per sample (pps) is most

effective in reducing the variance as seen from Fig. 2.6 (Siiria and Yliruusi).

2.8 Factors affecting solids mixing

2.8.1 Size or density differences

Mixing behavior of particles is significantly affected by particle size differences,

and an increase in either size or density differences results in increased segregation

tendencies. (Fan, Gelves-Arocha et al. 1975; Stephens and Bridgwater 1978). Further,

according to Fan et al. (1990), several other particle properties of a mixture can also

have a significant effect on the mixing behavior, the most significant apart from the size

difference being the density difference. On the other hand, Eskin et al. (2000)


2-22

investigated that the smaller the difference in particle size, the higher the mixing rate.

Zhou et al. (2003) reported that an increase in either the size or density difference

causes an increase in the rate of segregation in a vertically-shafted cylindrical bladed

mixer. They also showed that the particles are segregated due to the generation of a

vertical force on particles when there is either a size or density difference or both. They

observed that the light particles are mainly collected on the top and outer regions of the

particle bed while heavy (Ahmad and Smalley) particles are collected in the inner and

bottom regions of the mixer. However, the scope of the study was mainly limited to the

examination of particle segregation due to size and density differences.

The segregation phenomena and it‟s mechanism have been studied for many

years. The segregation is inevitable when heavy particles of a binary mixture of light

and heavy equal-size particles of a binary mixture are located near the centre of mass of

particles in the drum(Ristow 1994). Radial segregation will occur in avalanches and

continuous flow regimes in half-filled rotating drum due to the size differences of the

disks(Cantelaube and Bideau 1995). The smaller particles will disperse into the centre

of the mixture while larger ones dwell on the edge of the rotating drum(Clément,

Rajchenbach et al. 1995). It is investigated that the percolation primarily occurs in the

rapid flow layer formed on the bed surface(Cantelaube, Bideau et al. 1997). Khahar

found that the size of the core region increases with an increase of more denser

particles(Khakhar, McCarthy et al. 1997). It is stated the size segregation will counter-

balance the density segregation as size ratio of a binary particle mixture fixed, and

varying the density of small particles (Dury and Ristow 1999). The smallest and

roughest grains being found at the center of the drum when segregation occur due to

different size and surface properties (Makse 1999). Segregation decreased with a

decrease of size difference of the particles(Eskin and Kalman 2000). The flow-induced

segregation occurs due to small differences in either size or density. The denser particles

or smaller particles migrate towards the core of the cylinders(Ottino and Khakhar 2000).

The large beads segregate at the surface as small size ratio, but for high size ratios, the

large beads segregate inside the mixtures(Thomas 2000). Ternary mixtures with

different size and density particles segregate due to size and density driving forces

which may complement or oppose each other(Hajra and Khakhar 2011). Large particle

size ratio or density ratio induce segregation, though segregation can be deteriorated

due to percolation in a ternary or multi sized system (Xu, Xu et al. 2010). The small


2-23

particles in ternary mixtures exhibit reverse segregation as in binary mixtures.

Segregation is nearly independent of the sizes of the medium size and large particles.

Ternary mixtures with particles of different sizes and densities segregate due to size and

density generated driving forces which may complement or oppose each other. (Hajra

and Khakhar 2011). The radial segregation is driven by a density segregation flux,

which results in heavier particles tending to come to rest deeper in the bed, as a result of

their larger mass (Pereira, Sinnott et al. 2011).

2.8.2 Combined effect of size and density differences

The free surface segregation could be deteriorated by an optimum combination of the

size and density differences of particles in the case of a rotating drum (Alonso, Satoh et

al. 1991). Generally, the segregation occurs in the periphery of the tumbling container

for a mixture of particles with uniform density but different size. On the other hand, the

denser particles move to the center of the tumbling container when a mixture of

particles with uniform size but different density. Interestingly, for a binary mixture

where large particles are denser, both the size and density segregation effects come into

play at the same time, cancelling the segregation forces and resulting in good mixing

(Metcalfe and Shattuck 1996). Nitin et al. (2005) also showed that powder mixing can

be improved under certain size and density combinations in the case of mixing in

rotating drums. In particular, when the size ratio rs=dheavy / dlight is substantially larger

than the density ratio rd=ρheavy / ρlight where d and ρ are particle diameter and density

respectively, an improvement in the mixing occurred.

An attempt has been made in a previous work to describe the particle segregation

mechanism in terms of the vertical (buoyancy) forces on particles generated due to

either size or density differences or both (Zhou, Yu et al. 2003). Further, the segregation

mechanism of particles when size or density differences or both exist, has been

described phenomologically as driven by percolation or buoyancy forces (Nitin, Julio et

al. 2005). However, there are only a few studies dealing with the mechanisms of mixing

or segregation when size and density differences of particles coexist.

It is stated that a prediction equation to quantify the combined effects of density and

size differences would enable us to predict transition from mixing to segregation by

percolation due to size difference and buoyancy effect induced by density differences


2-24

(Alonso, Satoh et al. 1991). Such a relationship is promising to quantitatively predict

the mixing states of not only dry particles, but also wet particles at low cohesion when

the particle flow is in continuous regime (Liu, Yang et al. 2013). However, the

equivalence between size and density effects in the size-range where percolation effects

are negligible has not been considered in the present theories.

2.8.3 Cohesive particle mixing

2.8.3.1 Types of cohesion

Both Fan and Bridgwater classified mixtures into two groups such as

cohesionless (free flowing) mixtures and cohesive mixtures. Cohesionless mixtures

include cohesionless (free flowing) particles. Cohesionless powder is a completely

random mixture that they do not agglomerate. A mixture of free flowing particles is

relatively less homogenous due to incomplete mixing and segregation (Bridgwater ; Fan,

Chen et al. 1990). However a cohesive mixture contains one or more cohesive powders

which have strong inter-particulate forces. Mechanism of mixing of cohesive particles

will be introduced in the following section.

Wet, dry and electrostatic and overall studies

A cohesive mixture commonly has a good mixing quality in a wider region,

however there is still some segregation in local areas. The nature and strength of the

inter-particulate forces determine the mixing quality and intensity of segregation. The

inter-particulate forces are determined by moisture, electrostatic charging as well as van

der Waal‟forces (Harnby 1997). The bonding caused by the overlapping of absorbed

layers of neighbouring particles and its strength. The strength of the structure is affected

by packing densities, particle shape and size, as well as particle roughness. The two

solids in mixing will charge each other electrostatically. The particles tend to bond to

the mixer wall or dissimilar particles and produce mixing or segregation depending on

the sign of charge.

Van der Waal‟s force is a natural attractive force between neutral atoms or

molecules which separated by a distance. The Van der Waal‟s force will decrease with

the separation distance according to a power law. It increases because of the transient


2-25

polarization of the atom or molecule. Absorbed moisture increases the Van der Waal‟s

forces when the absorbed layer is considered as a part of the particle, and it decreases

with the inter-particulate distance. The electrostatic forces will be reduced when

humidity causes air to be more conductive and therefore the particle to discharge. The

mechanical forces such as friction and interlocking are reduced by absorbed moisture.

Li experimentally and theoretically studied the cohesive mixing and segregation

in an annular shear cell (Li and McCarthy 2006). The effect of liquid-bridge induced

cohesive forces on particle mixing under shear was analysed using Collision number

which is a newly developed heterogeneous characterization tool. Granular Bond

Number is a main parameter since the pseudo-static system is a no flow-related forces

such as shear forces and drag forces. However in the dynamic system the

collisional/shearing forces begin to play a more important role. Another Collision

number is needed where the collision force is comparable to the cohesion force but

larger than the particle weight. Collision number is equal to the ratio of maximum

capillary force over to collisional force.

The influence of adding liquid into sheared granular flows on mixing process is

studied. The cohesive force and viscous force exist in the wetted granular flows and the

fluctuations, the shear rates and the self-diffusion coefficients are smaller. The self-

diffusion coefficient is increased with decrease liquid volume. Also it linearly increased

with granular temperature and shear rate along the upper wall(Hsiau, Lu et al. 2008).

Also Yang studied the effects of small amounts of added liquid on the segregation

behaviour of granular system under vertical vibration by DEM simulation(Yang 2006).

Fluidized beds have been used in industries with gas-solid system. Various types

of fluidized beds have emerged. Cohesive particles tend to be agglomerate due to strong

interaction forces. Geldart (Geldart1973) classified powders into four groups: A for

aerated, B for bubbling, C for cohesive particle. Vibration in fluidization is one of the

good methods for improving the fluidity of fine cohesive particles. The vibration

decreases the voidage and minimum fluidization velocity. The minimum air velocity

and vibration intensities are essential to have good fluidization(Marring, Hoffmann et al.

1994). The intermittent channel breakage upper limit gas velocity was higher with

increasing gas velocity method than decreasing gas velocity method. The gas velocity


2-26

and bed void fraction are reached maximum from a relationship between gas velocity

and bed pressure job(Mawatari, Tsunekawa et al. 2005).

There are other several methods for improving the fluidization quality of

cohesive particles except for the vibration method which was discussed above. For

example baffles, acoustic field, magnetic field, adding particles and modifying surface

of particles, etc. It is difficult to fluidize cohesive particles (Geldart‟s group C)

compared to the group A particles due to the strong inter-particle force. The fluidization

behaviour is very different due to the particle properties such as density, size, shape,

size distribution etc. Zhou investigated that the fluidization behaviour of cohesive

particles were strongly affected by size, superficial gas velocity and additive particles.

SiC of average size larger than 10um can be fluidized with increasing gas velocity,

however the 5um SiC can‟t be fluidized. Adding right particle in cohesive particles may

decrease the inter-particle cohesive force and improve fluidization behaviour(Zhou and

Li 1999).

The effect of particle size on fluidization quality was studied. The minimum

fluidization velocity decreased for powder C, when powder A is almost constant under

some experimental conditions. The bed expansion ratio is lowered, and the range of

vibration strength and the gas velocity for bed rotation becomes narrower with

decreasing particles diameter. The gas velocity for channel breakage was decreased with

increasing vibration strength. It is supposed that decrease in particle diameter leads to a

more complex bed structure including various sizes of agglomerates, which makes it

difficult for vibration(Mawatari, Koide et al. 2002).

2.8.3.2 Van der Waals force model

Van der Waals force model was used in analysing the solid mixing. Attractive

forces exist between neutral atoms or molecules which are separated by a distance.

These are known as Van der Waals‟ forces. They decrease with the distance apart

according to the reduced power and increase because of transient polarization of an

atom or molecule which will act on the surrounding area to produce spontaneous

fluctuations elsewhere.


2-27

bRHaFv 111

6 2 (2.10)

Here, Ha is known as Hamaker constant, and is a material dependent quantity to

be determined by Eq.(2.11) using the respective Ha values, Ha1 and Ha2 for the pair of

objects under consideration. Similarly, R is an equivalent radius for the pair of objects,

using respective radio r1 and r2 of the object pair; for a flat plate, the radius is infinite

and thus the limiting value for R obtained from Eq.(2.12) should be used. is the

separation distance between the two objects. b is a constant equal to 5.32. is dipole

interaction wavelength (= 10-7m).

21 aa HHHa= (2.11)

21

21

r r rrR=

(2.12)

2.9 Eeffect of the particle cohesion on mixing in a uniform binary system

The effect of cohesion and non-cohesion on particle mixing in uniform binary

system was studied (Chaudhuri, Mehrotra et al. 2006). The computational model system

for this case study consists of 10,000 “green” and 10,000 “red” identical particles of

2mm diameter, loaded initially side-by-side along the axis. The mixing drum in the

simulation has a diameter of 9cm, length 1cm and is rotated at 20rpm。The cohesive

bond number is varied from K=0 to 120 to estimate the effect of cohesion. The end caps

of the drum are made frictionless. The static friction coefficients for particle–particle

and particle–wall contacts are chosen to be 0.4 and 0.7, respectively. Four values of

cohesion are simulated with our model system. The snapshots are taken at time T=0, 1,

3 and 5 revolutions. The free-flowing mixture (K=0) blends faster than the moderately

cohesive (K=60) and highly cohesive (K=90) cases, where we see red particles move in

unison forming a stretchable band. Interestingly, the very mild cohesive case (K=0.1)

shows faster mixing than the free-flowing case. However, the difference is not large;

mixing is only slightly enhanced by the introduction of a small amount of cohesion.


2-28

The intensity decreases and we observe best mixing at K=0.1 but segregation is more

pronounced with further increase in cohesion as shown the Fig 2.4.

Figure .2.4(a) effect of cohesion on mixing;(b) This Graph is in conjunction with

A, plotting the log (intensity of segregation) with revolutions

The effect of blender rotation speed, another commonly studied parameter, is

examined (Chaudhuri, Mehrotra et al. 2006). The homogeneity of the cohesive mixtures

A - -

k=O \

• ~ '-' ~ ~-

~ k=O.l

~ ~ ~ - .,

~ .,..-- ---., -- --

k=60 -.~ .. ...... - . ,.,..---. , -

k=90 • • T = 0 Rev T = 1 Rev T = 3 Rev T = 5 Rev

8

-o.s

-+- K=1 20 -r K=90

-+- K:60

-+ Kdl.1

"*' NC

-!.l .S+-------~---~---~----1 0.00 1.00 2.00 3.00 4.00 5.00

t RQVOiuUons


2-29

is shown to be a function of blender rotation speed, facilitating better mixing at higher

rotation speeds. This result is in agreement with experimental results and in contrast

with previous findings for non-cohesive materials, for which the mixing rate is

independent of the rotation speed (Sudah, Coffin-Beach et al. 2002). A detailed study is

done on the effect of “dry” cohesion/ adhesion in non-uniform binary system

(constituting particles of different sizes). It is shown that the effect of varying cohesion

in small/big particles is more complex than originally thought; mixing rate is optimum

for an intermediate value of cohesion.

Cohesion plays a key role in the mixing/segregation of granular systems.

However, high values of cohesion (typical of commonly used pharmaceutical powders)

show slower mixing, which is attributed to the formation of plug flow in the cascading

layer, causing sluggish thinning of striations of similar particles.

2.10 Effect of the particle cohesion on mixing in a non-uniform binary system

The effect of cohesion in the smaller particles on the homogeneity of binary

mixture of different sizes was studied(Chaudhuri, Mehrotra et al. 2006). The

computational system includes 8000 blue particles (radius1mm) and 2370 red particles

(radius1.5mm). Red particles are considered to be free flowing and there is no adhesive

force acting between the particles.

The smaller particle (blue) as cohesive particles Kbb is varied from 0 to150. Krb

and Krr were fixed 0. It is seen that mixing is optimum for an intermediate value of

Kbb=50 which is better than free flowing materials (Kbb ,Krb and Krr equal to 0 ) as

shown in Fig 2.5.

Subsequently, the effect of cohesion in bigger particles (red) on homogeneity

was investigated. The value keeps Krb =0 and Kbb =50. The value of Krr will vary from

5 to 100. The results showed that the best mixing will be obtained in the value Krr =5

shown as the Fig 2.6.


2-30

Figure 2.5(a) Effect of cohesion in the smaller particles; (b) Intensity of

segregation with time as a function of cohesion within smaller blue particles


2-31

Figure 2.6. The variation of intensity of segregation with time as a function of

cohesion within bigger red particles

Finally, the effect of adhesive interaction between two species on uniformity of

a binary system with different particle sizes was investigated. The value keeps Krr =0

and Kbb =50, The value of Krb will vary from 5 to 100.

Figure 2.7. Intensity of segregation with time as a function of adhesion.

These simulation results show that mixing is more effective as the adhesive

force is increased. It is observed that as Krb is increased, the intensity of segregation

decreases, indicating better mixing, and all the curves lay below the non-cohesive case

as shown Figure 2.7.


2-32

2.11 Operational conditions (Shaft speed and fill level effect)

In terms of the effect of impeller speed and fill level and blade number of the

different mixers, the research shows that the axial diffusion coefficient increased with

an increase of blade speed, however, it decreased with an increase of fill level

(Bridgwater 2003). Recent studies of the effect of the fill level and blade speed in the

mixer with single or multiple blades provide us with an excellent reference for our study

(Laurent and Bridgwater 2002). The deteriorated mixing was obtained with a high blade

speed for mixing of free-flowing particles in a cylindrical mixer (Chandratilleke, Zhou

et al. 2010). However, the speed effect on the homogeneity and flow pattern of the

solid from microdynamic aspects have not been fully understood.

2.12 Types of Mixers

Particle motion in a mixer is three dimensional and random. All mixers should

display one of the following three mixing mechanisms (Venables and Wells 2001;

Bridgwater 2012): The mixer can be categorized based on the mixing mechanisms(Poux,

Fayolle et al. 1991).

(1) Convective mixing happens when circulation patterns are set up in a mixer,

due to which particles are conveyed along these pattern paths. Examples of mixers with

a predominating convective mechanism are ribbon mixers, high speed cylindrical

mixers, and nauta mixers.

(2) Shear mixing occurs along with convective mixing due to velocity gradients

present in the mixture. This is the predominant mechanism of pan mixer and mill.

(3) Diffusive mixing is the mixing at a particle scale and is responsible for

mixing once the convection and shearing effects fade away.

Mixer can also be classified into two groups based on the segregation or non-

segregation or based on the shell rotation or not. Here we classify the mixer into two

classes based on the shell movements.

First, the mixer shell rotates and the material slopes around inside (Poux,

Fayolle et al. 1991; Bridgwater 2012). Example of this type of mixers are:


2-33

The cylindrical drum: the drum rotation causes both axial and radial mixing. The

rotating drum is easier to use for slightly cohesive or non-cohesive materials,

and suitable for free flowing or for granulation.

The off-centre drum: the mixing is improved in this mixer by particles sloping

backwards and forwards in the axial direction.

The double cone mixer: this mixer is made of two conical sections, and the

material rolls and folds, breaking the agglomerates by diffusion and shear

mechanism.

The V mixer: These kinds of mixers are suitable for mixing weakly cohesive

powders, but small agglomerates or aggregates of an ingredient might remain

intact in the mixture as shown in the Figure 2.9. This segregation may be

minimized by adding an internal impeller. The degree of fill of the mixer also

affects the mixture quality, and forms a limit on the batch operation performance

of the mixer. The mixer has an advantage in that it gives good access for both

cleaning. Tumbler mixers can handle free-flowing and cohesive powders but not

pastes or dough, and the quality of the mixture can be a problem. Free-flowing

powders can segregate relatively easily on the tumbling surface; the emptying

process also frequently leads to segregation in the mixture. In these mixers,

mixing takes place relatively fast in the radial direction, but slowly in the axial

direction, which is therefore the rate-controlling factor for the mixer.

Second main classes of the mixer, the internal rotor or internal rotor fitted blades

rotating, whilst the shell is stationary. Example of these types (figure 2.8) are:

The centrifugal mixer with horizontal axis: the material in the mixer pushed

circumferential, and displaced axially by the blade moves in the low speed.

At higher speeds, the material is centrifuged.

The centrifugal mixer with vertical axis: the blade pushes particles around

the mixer on low speed. At high speed, the material forms a toroid next to

the wall.

Ribbon mixer: the mixer consists of one or two screws or blades in the

opposite direction. The material is rolled, folded, reversed in direction and

radially dispersed. The wide range of materials can be mixed.


2-34

Planetary mixer: material is rotated on a vertical axis, being brought into the

zone of action of a mixing blade rotating at high speed about an offset

vertical axis. The mechanism is convection and shear. It is not suited for

very cohesive materials.

The draught tube and screw mixer (figure 2.9 a): the material is conveyed to

the free surface by vertical screw contained within a tube, and then the

material falls down the free surface and is recycled back to base of the screw.

Orbiting screw mixer (figure 2.9b): the mixer consists of a screw attached at

the base of the cone confined to the batch mixer. The screw rotates about its‟

own axis while at the same time processing about the vertical axis of the

cone. The material is conveyed upward by the screw and the material is

distributed onto the surface. The mechanisms of this mixer are convection

and diffusion. Suitable for improving mixing purpose.

With a hopper mixer, the particles flow under the influence of gravity, and

this flow can be used for mixing particles at the outlet of the hopper without

any external energy. To avoid dead-zones within the silo, a central cone is

usually installed so that a pronounced velocity gradient is produced in the

vertical direction (see Fig. 2.10) (Brown and Nielsen 1998). To avoid

arching across the annulus and at the outlet, the hopper diameter and section

widths of sections should be chosen appropriately. This design is known as

Binsert (developed by Janike and Johanson Inc.), and can be used for mixing

cohesive bulk solids as well. Depending on the required degree of

homogeneity, the particles may have to be recycled externally, thus causing

considerable axial mixing (Fan, Chen et al. 1990). Due to percolation,

segregation is likely to occur in this type of mixer, both on the free surface

of the hopper and within the bulk of the material (Fan, Chen et al. 1990).

Fluidized mixer: The powders in a fluidized bed mixer receive energy from

gravity and convective effects as shown in Figure 2.11. The gas flow carries

the particles against their weight which is counter balanced by the buoyancy.

The good mixing will be produced by mobility of particles and turbulence if

the gas flow rate is sufficient (Fan, Chen et al. 1990).


2-35

Figure 2.8 Typical Mixer(Bridgwater 2012)

Cy!indricif drum Cfi·c~nue <:rum

•

'+i ¢ f (t'l!ln $hell;

Rotating shells

Centrifugal mlxor with horizontal axis

Vert:>: <- C•irection of rotation

_.---r--._-'\ /

(r-~/F· -~~77'-i-"'"'=+-=' :..v ""

Centrifugal m i xer with verti(;al axis

---· ~ ·,.. Ribbons

Ribbon mtxet Planetart mixer

Plan.:tary 119 ~1StCf


2-36

Figure 2.9 (a) Draught tube and screw mixer(b) Nauta mixer with orbiting

screw(Bridgwater 2012)

Fig. 2.10 Gravity-flow (hopper) mixers Fig.2.11 fluidization bed


2-37

Figure 2.12 Helical ribbon impeller(Masiuk, Lacki et al. 1992)

2.13 Helical ribbon mixer

Mixing of highly viscos fluids is a key step in most chemical and food industries.

Helical ribbon mixers enable keeping the entire vessel contents circulating, which is

very suitable for mixing high viscous liquids, and recognized to be a very efficient

system as shown in Figure 2.12. For comparing performances and understanding the

main findings of the studies of helical ribbon mixer and ribbon mixer, Table 2.1 is

established showing main findings and material and mixer conditions.


2-38

Date author and Title Material and conditions

Main findings

(Bortnikov, Pavlushenko et al.

1973),

Design of an apparatus with a helical ribbon

mixer

Fluid molassesdensity T = 1410 kg/m 3 and viscosity g = 3.14 kg. sec/m2. The flat-bottomed vessel helical ribbon mixer.

Axial forces are 1.6-1.7 times lower for single

helix mixer compare to the dual helix mixers.

Doubling the ribbon width increases axial forces by 40-50% while power consumption

for mixing remains practically constant.

(Novák and Rieger 1975)

Homogenization Efficiency of Helical Ribbon and Anchor

Agitator

Aqueous solutions of corn syrup, glycerol and distilled water were used in the experiments. Three types of agitators : Double helical ribbons Anchor agitators Pitched blade anchor

Helical ribbon agitator is suitable for homogenization of liquids in the creeping flow regime. This is about 7 times higher than for screw agitators with a draught tube. Anchor agitators are inadequate for homogenization of highly viscous liquids in the creeping flow regime. The best efficiency in the creeping flow

regime is with screw agitators with a draught tube. It is very efficient in the region over the creeping flow regime and suitable for mixing of liquids whose viscosity varies considerably during the course of the process.

(Sawinsky, Havas et al. 1976)

Power requirement of

anchor and helical ribbon impellers for the

case of agitating Newtonian and

pseudo-plastic liquids

Newtonian liquids Achor impeller and Helical-ribbon impeller.

Rheological behaviour of pseudo-plastic liquids, the relationship between the shearing stress and the shear rate can be described by means of the Ostwald-de Waele equation.

the prediction equation was established for power consumption in the laminar flow range of pseudo-plastic –liquids

(Patterson, Carreau et

al. 1979)

Mixing with Helical Ribbon Agitators

Newtonian fluids Helical Ribbon Agitator

HRA is admirably suited to low Reynolds number mixing process.

Derived model to predict power consumption of HRA.

( l asi s i and z ys i 1980)

Power requirements of helical ribbon mixers

Fluids in laminar region Re<100 Helical Ribbon Agitator

Presented the equations which enable mixing power to be calculated for helical ribbon mixers within the laminar region of mixing

Table 2.1 Main finding in the helical and horizontal ribbon impellers


2-39

(Le Cardinal, Germain et al. 1980)

The design of stirred batch polymerization

reactor

Laboratory-scale tank reactors Methods used: decolarisation and conductivity measurements

Scaling method proposed Screw mixer with four baffles is best mixer.

(Ottino and Macosko

1980)

An efficiency parameter for batch mixing of viscous

fluids

Newtonian Fluids Six bladed turbine, three inclined blade paddles, helical screw, helical ribbon, propeller in draught tube, anchor

A rational ranking of the mixers, in the sense of criteria (efficiency as a function of time), has been provided using the concept of intermatenal are a generation An efficiency of mixing, applied here for Newtonian liquids, can be extended to other constitutive relations. Generalization for power law fluids is trivial, but other constitutive relations should be managed with care.

Characterization of helical impellers by

circulation times

Newtonian and non-newtonian fluids Helical ribbon impellers

The shear thinning properties of non-newtonian fluids do not affect the helical impellers‟ circulation capacities. In highly shear thinning fluids, the presence of important stagnant zones causes much longer mixing times which consequently do not correlate with circulation parameters.

The wide blade is more efficient on the helical-ribbon impellers but is less efficient than the screw impeller in a draft coil.

(Rieger, Novák et al.

1986)

Homogenization Efficiency of Helical

Ribbon Agitator

Newtonian Fluids Helical ribbon agitators

A greater clearance and has a lower power

consumption but a longer time of

homogenization

With the pitch s = d exhibit higher power

consumptions but shorter homogenization

times.

Agitator with wider blades exhibit higher

power consumption and shorter mixing time.

With an increase of blade number, the power

consumption increases significantly but the

homogenization time decreases considerably.

(Cooker and Nedderman 1987)

A theory of the

Powder Helical ribbon agitators

Stress analyses, using appropriate modifications of Janssen‟s method, performed for both the rotating core and the material between the blades of the helix.

Torques on the vessel wall and the helical


2-40

mechanical of helical ribbon powder agitator

ribbon calculated. Proposed equation can predict the

circulation, geometry and frictional properties.

(Ryan, Janssen et al.

1988)

Circulation time prediction in the scale up of polymerization reactors with helical

ribbon agitators

Newtonian fluids Double helical ribbon impeller

A fundamental model of fluid circulation in

polymerization reactors agitated by helical

ribbon impellers is presented.

An expression is developed for the circulatory

flow generated by a double helical impeller in

a reactor with a draught tube.

The circulation time and ribbon rotational

speed is predicted to be a function only of

geometric variables for Newtonian fluids in the

viscous flow regime.

The circulation of Newtonian fluids in an

experimental reactor is investigated to verify

the model. There is good agreement between

the model and experimental results for

Reynolds numbers less than 500.

At higher Reynolds numbers, inertial effects

increase fluid circulation times. Preliminary

data are also presented for non-Newtonian

fluids.

The implications of these results in the scale-

up of polymerization

(Tanguy, Lacroix et al.

1992)

Finite element analysis of viscous mixing with a helical ribbon screw

impeller

Newtonian fluid Helical ribbon screw impeller

The mixing pattern of viscous Newtonian

fluid in a helical ribbon screw impeller using

a finite element method.

The circulation time and torque were

predicted and compared these with

experimental results.


2-41

(Masiuk and Lacki 1993)

Power consumption and mixing time for Newtonian and non-Newtonian liquids mixing in a ribbon

mixer

Newtonian and non-Newtonian fluids Helical ribbon agitator

The shape of ribbon agitator has a significant

influence on the mixing energy required for

mixing liquid.

Best mixer type presented in terms of mixing

efficiency for the non-Newtonian liquids.

For practical purpose, the general correlation equation proposed for calculating the power consumption, mixing time, and mixing energy.

(Tanguy, Thibault et al.

1997)

Mixing performance induced by coaxial flat

blade-helical ribbon impellers rotating at

different speeds

Newtonian and non-Newtonian fluids Dual impeller mixer composed of a disc turbine and a helical ribbon impeller

The mixing performance (pumping,

dispersion capabilities, power consumption)

at different speeds is investigated.

the dual impeller mixer outperforms the standard helical ribbon in terms of top-to-bottom pumping when the fluid rheology evolves during the process.

The power consumption of this new mixer is also studied which allows to derive a generalized power curve

(Delaplace, Leuliet et al. 2000)

Circulation and mixing times for helical ribbon impellers. Review and

Experiments

Different type of helical ribbon agitator

Homogenization in lamina regime is

controlled by amount of shear imparted in

the gap by agitator and renewal fluids in the

gap.

The circulation time measured by

conductivity and thermal methods are

similar. Circulation time is independent of

the pumping direction.

Helical ribbon agitators better efficiency of

the mixing process if blade width ratio were

more adapted to clearance-wall size.

Increase of blade width ratio enables to improve mixer efficiency.

(Kaneko, Shiojima et al. 2000)

Numerical analysis of

particle mixing characteristics in a

single helical ribbon agitator using DEM

simulation

100000 1mm polypropylene homopolymer DEM method and experimentally

Vertical mixing of particles rather poor during upward and downward flows through the blade and core regions,

In the core region, particles flow in the manner of funnel flow.

In the blade region, the vertical velocity decreases toward the wall and reduces to zero near the boundary between the blade and core regions.

Particles carried up to the bed surface by the


2-42

helical ribbon flow down into the center of the core, and those not carried completely to the surface flow into the outer periphery of the core.

If the bed height is lower than the blade top, particles mix rapidly because of the high fall of particles from the blade top. If the bed height is higher than the blade top, the velocity of particles in the bed surface region is much lower than that of the other region

(Kaneko, Shiojima et al. 2000; Dieulot,

Delaplace et al. 2002) Laminar mixing

performances of a stirred tank equipped with helical ribbon

agitator subjected to steady and unsteady

rotational speed

Highly viscous Newtonian fluids Non-standard helical ribbon impeller

The use of the time dependent rotational speed during the mixing process allows energy savings. For unsteady stirring approaches tested, energy saving can reach up to 60% compared to the energy required to obtain the same mixing time with constant impeller rotational speed.

(Shekhar and Jayanti 2003)

Mixing of

pseudoplastic fluids using helical ribbon

impellers

Pseudoplastic fluids of high viscosity Helical ribbon impeller

Shear rate varies within mixing vessel. Linear relationship between the impeller

speed and local shear rate near the tip of the impeller.

The constant associated with this linear relation dependent on the geometric parameters of the system, but independent of the flow behavior index

(Niedzielska and Kuncewicz 2005)

Heat transfer and

power consumption for ribbon impellers mixing efficiency

fluids Helical ribbon impeller

The heat transfer efficiency of particular impellers was calculated.

Geometrical dimensions of the most efficient ribbon impeller is the impeller with a diameter of d/D=0.94 and ribbon pitch p/d=0.25

(Delaplace, Guerin et

al. 2006)

An analytical model for the prediction of

power consumption for shear thinning fluids

with helical ribbon and helical screw ribbon

impellers

Fluids Helical ribbon and helical screw impeller

A new model based on effective viscosity taking into account the shear-thinning behavior is proposed. The model predicts the value of Ks as a function of the power constant Kp and the geometrical parameters of the helical ribbon mixing systems (wall clearance, blade width) and the flow behavior index of the pseudoplastic fluid.

The model is able to explain the variation of the experimental value of Ks with the flow behaviour index.

The model can predict well the effect of wall clearance on the power demand for both helical ribbon and helical screw ribbon impellers.


2-43

(Robinson and Cleary

2012)

Flow and mixing performance in helical

ribbon mixers

A single helical ribbon mixer creates an axially symmetric circulation cell that moves fluid down the outside of the tank and upward near the center.

Smaller circulation cells near the inside edge of the ribbon superimpose a chaotic mixing flow over this primary flow

The horizontal struts supporting the ribbon generate strong circumferential mixing, but this is only within a narrow horizontal plane with thickness comparable with strut diameter.

(Masiuk 1987)

Power consumption, mixing time and

attrition action for solid mixing in a ribbon

mixer

Ribbon mixer Solid particles

Power consumption in an agitated vessel varies linearly with rotational speed of helical ribbon mixers.

The loading ratio has a strong effect on powder consumption in the mixing of particulate materials.

(Côté and Abatzoglou

2006) Powder and other

divided solids mixing. Scale up and

parametric study of a ribbon blender used in

pharmaceutical powders mixing

Ribbon mixer Mixing endpoint not reached at shaft filling height.

Impeller rotation speed to play important role in the range of conditions.

The mixing time does not play a major role in the specified range.

(Masiuk and Rakoczy

2006) The entropy criterion

for the homogenisation process in a multi-

ribbon blender

Multi ribbon blender

The new form of the entropy criterion estimating the current state of a mixture is proposed.

A mathematical model to describe the variations of the informational entropy during process is developed. For blending process the experimental validation for this model conducted.

(Muzzio, Llusa et al.

2008) Evaluating the mixing

performance of a ribbon blender

Ribbon mixer The loading methods (layering and off-center spot injection) are explored. Layering method create greater homogeneity of the blends, and the reduction of mixing time, blend exposure to shear, and risk of over lubrication.

The fill level effect 3 –spoke blade is significant. Lower fill level yield more homogeneous blends.

For 5-spoke blade, the fill level is not insignificant.

Higher blade speeds increase shear and the risk of over-lubrication.


2-44

2.14 Ribbon mixers

Powder mixing is an essential unit operation for manufacturing processes, for

example, in ceramic, food, pharmaceutical, chemical and agricultural industries, where

a wide variety of high quality products are produced. The ribbon impeller has one or

two helical blades starting from each end, but spiraling in opposite directions as shown

Figure .2.13.

Figure 2.13 Ribbon mixer used in the industry

This ribbon mixer with two helical screws as blades pushes particles in one axial

direction close to the centre and simultaneously the other screw pushes in the opposite

direction close to the wall. Material is rolled, folded, reversed in direction and radially

displaced. The screw makes it easy to push the product to an exit although it can make

cleaning more difficult. The mixer can be used for a wide range of materials from dry

powders to pastes. Radial mixing is good, axial mixing is less so (Metzger and Glasser

2013). A ribbon mixer displays a combination of mixing mechanisms such as

convection, diffusion and shear when granular matter. The ribbon mixer is a good

(Musha, Dong et al. 2013)

Mixing behaviour of cohesive and non-cohesive particle

mixtures in a ribbon mixer

Ribbon mixer Mixing rate increases with impeller speed for both cohesive and non-cohesive mixture up to certain speed.

Particle contact forces increases with an impeller speeds in non-cohesive particles. But for cohesive mixture increase with impeller speed, and then decreases after a certain speed.


2-45

choice for aerating materials. The selection of a mixer for a mixing operation depends

on the product uniformity required and many other factors (Poux, Fayolle et al. 1991).

Ribbon mixers are considered to be suitable for mixing of dry powders as well as free-

flowing granular material (Poux, Fayolle et al. 1991). It is also reported that a ribbon

mixer can produce an improved homogeneity in powder mixing because of large shear

stresses in the mixer. In addition it can handle mixing of different sized particles

(Muzzio, Llusa et al. 2008). Therefore, a ribbon mixer with a horizontal shaft will be the

focus of study here. The literature survey of the effect of operational parameters,

geometrical parameters of a ribbon mixer with a horizontal shaft on the powder mixing

is conducted below.

2.14.1 Effect of shaft speed

Ribbon mixers are widely used in practice because they are capable of providing

high speed convective mixing. The impeller speed of the mixer is a parameter that can

affect the performance of the mixer, with regard to mixture quality and stresses on

particles. Inspection of velocity fields shows that many local recirculation flows exists

in the case of non-cohesive particle mixing, preventing the overall mixing as shown in

Figure 3.15. By contrast, in the case of the cohesive mixture, there exists a

circumferential motion about the shaft and a convective motion in the horizontal axial

direction, improving the particle mixing (Musha, Dong et al. 2013). The Mixing

deteriorated with an increment of blade speed in a continuous ribbon mixer for

obtaining uniform mixing (Sanoh, Arai et al. 1974). The power consumption increased

with an increase of blade speed in a ribbon mixer (Masiuk 1987). Energy usage can be

reduced by up to 60% compared to the energy required to obtain the same mixing time

with constant impeller rotational speed in a ribbon mixer (Dieulot, Delaplace et al.

2002). The uniformity of the mixture was never reached at a high level at the high

impeller rotational speed (Côté and Abatzoglou 2006).


2-46

Figure 2.14 Effect of the blade speed for layered magnesium tstearate and

59%fill level(Muzzio, Llusa et al. 2008)

It is reported that a high impeller speed can have a negative effect on the

uniformity of a mixture; the shear stresses on particles will reduce at a low speed; and

may result in a poor homogeneity for cohesive mixtures as shown in Figure

2.14(Muzzio, Llusa et al. 2008).

2.14.2 Effect of fill level

The fill level of the blender is the next most important operational parameter

(Côté and Abatzoglou 2006). Pascal reported that for an entire batch mixer, the mixing

did not reach the end point when the blender is filled to the shaft level. The mixing

reached the end point when the outer shaft fill level at a blade speed of 20rpm. Muzzio

et al. experimentally investigated the mixing performance of a ribbon mixer (Muzzio,

Llusa et al. 2008) with regard to the effect of loading (layering and off-center spot

loading method) and layering method showed faster mixing and a better homogeneity.

The effect of the fill level is significant for 3 spokes 2-bladed ribbon impeller. The

mixer demonstrates that a lower fill level is more homogeneous, and the effect of fill

level on 5 spokes 2-bladed ribbon impeller is not as significant.


2-47

Figure 2. 15 Effect of fill level (59 and 100%) on the mixing performance of the blender with the 3-spoke ribbon blade operated at 20 rpm with an off-center spot injection of magnesium stearate.

Figure 2.16 Effect of fill level on the mixing performance of a blender with the 5-spoke blade operated at 20 rpm.

2.14.3 Effects of mixer geometry

A ribbon mixer displays a combination of mixing mechanisms such as convection,

diffusion and shear when mixing granular matter. Ribbon mixers are widely used in

practice because they are capable of providing high speed convective mixing. The

efficiency of such operations and quality of the resulting mixtures will depend on the

impeller geometry. Thus, it is important to understand the effects of the blade geometry

and use the right geometry for product manufacture in many industries such as

pharmaceuticals and powder metallurgy industries.

Research on the effect of geometry of a helical ribbon on mixing has been

reported for liquid mixing widely (Masiuk and Lacki 1993). For example, Masiuk et al

(1992) reported that the helix pitch and width have a perceptible influence on the


2-48

mixing time and power consumption for mixing liquids. It is investigated that the axial

forces has increased by 40-50% by doubling the helical ribbon blade width while the

power consumption has remained practically constant (Bortnikov, Pavlushenko et al.

1973). Furthermore, doubling the helical ribbon width has resulted in an increase of 10%

in the power required for mixing ( l asi s i and zys i 1 0) . The lower power

consumption required for a helical ribbon mixer with a greater clearance has, but it

takes a longer time for the homogenization. On the other hand, an agitator with a

smaller clearance has exhibited a greater efficiency (Rieger, Novák et al. 1986). Thus,

the design of a blade has an essential impact on the homogenization of mixtures.

Masiuk et al. (1993) also reported that the shape of the ribbon agitator has a significant

influence on the energy required for mixing the liquid. Takahashi et al. (1988) reported

that even though the primary circulation patterns are approximately the same, the

impeller geometry can strongly affect the secondary circulation flows. The following

parametric effects have been reported for liquid mixing with a helical blade. When the

pitch of the helical blade is decreased with the wall clearance fixed, the shear rate (γ =N

Ks , Ks is a constant that is dependent only on the mixing system geometry; N is

rotational speed (rev/s)) and becomes higher as shown Figure 2.17(Delaplace, Leuliet et

al. 2000). The mixing time (t (s)) obtained for various helical ribbons agitators with

different clearance ratios c/d between blades and wall. It is clear from Figure 2.17

which shows that the mixing process is the most rapid when the narrowest clearance is

not used as shown Figure 2.18 (Delaplace, Leuliet et al. 2000).

Figure 2.17 Evolution of Metzner and Otto factor (1957) with pitch size for helical ribbon agitators found in literature(Delaplace, Leuliet et al. 2000)

Muzzio et al (2008). experimentally investigated the mixing performance of a

ribbon mixer regarding the effect of loading (layering and off-center spot loading


2-49

method). It is reported that the layering method is faster in mixing and produces better

homogeneity. There are significant effects of fill level on 3 spokes 2-bladed ribbon

impeller as shown in Figure 2.15. However the effect of fill level on the mixing in a 5

spokes 2-bladed ribbon impeller is not significant as shown in Figure 2.16. However,

there is limited knowledge in the literature regarding this effect of geometrical

parameters of horizontal ribbon on granular mixing (Muzzio, Llusa et al. 2008). In

addition, there are limited studies of the effects of geometrical parameters on the mixing

of cohesive particles in a horizontal ribbon mixer.

Figure 2.18 Evolution of mixing time number with clearance wall for helical ribbon agitators found in literature(Delaplace, Leuliet et al. 2000).

2.15 Simulation method

There are several discrete modelling techniques which have been developed

such as the Monte Carlo method, cellular automata, smooth particle

hydrodynamics(SPH) and discrete element method(Berthiaux, Mosorov et al.).

2.16 Discrete Element Method

The Discrete element method (Berthiaux, Mosorov et al.) is commonly used for

investigating particle system. DEM simulation can provide dynamic information related

to particle motions such as trajectories, velocities of particles and forces acting on each

individual particle(Cundall and Strack 1979). There are two types of particle/particle

contact modeling method used in DEM, soft particle and hard particle approach. The

soft particle method was developed three decades ago. In this method, particle presumed

overlap, and the overlaps treated as deformations of particles. By the way the

interparticle forces such as elastic, plastic and friction forces are calculated. The motion

of particle is described by Newton‟s law of motion. The soft-sphere models are able to


2-50

describe multiple particle contacts simultaneously. However, the hard-particle processes

collisions sequentially, one collision at a time. The collisions being instantaneous, the

forces between particles are not explicitly considered. The soft-particle methods have

been used to study various phenomena, such as particle packing, transport properties,

heaping or pilling processes, hopper flows, mixing and granulation process (Zhu, Zhou

et al. 2007).

2.16.1 Force model

The model uses two momentum conservation equations to describe the translational and

rotational motion of particle i in a system at time t subjected to the gravity g and

interactions with the neighbouring particles, blade and walls:

ii N

jijv

k

jijdijci

ii m

dtdm

1,

1,, FFFgV

(2.13)

ik

jijij

ii d

dI

1tMT

(2.14)

Fc elastic contact force, Fcn,ij and Fct,ij

Fd damping force, Fdn, ij and Fdt, ij

FV cohesive force

Tij torque

M rolling friction torque

mi, Ii, Vi, and I are mass, moment of inertia, translational and rotational velocity of

particle.

Normal contact force nF ˆ)2()1(3

23212, niijcn RE

(2.15)

Normal damping force n nVF ˆˆ)()1(2

321

212,

ijn

inijdn R

Emc

(2.16)

Tangential contact force

tt

t

i

ijcnijct

F

23

max,

max,,,

,min11

tsF (2.17)


2-51

Tangential damping force ijtt

ttijcnsitijdt Fmc ,

21

max,

max,,,

1 6 VF

(2.18)

Rotational torque ijdtijctiij ,, FFRT (2.19)

Rolling friction torque ιijcnrij F ̂,M (2.20)

The non-contact force due to the van der Wall force F,ij (Thornton and Zhang 2010)

in Eq(2.13) explained as following Equations:

ijjijiji

jijivdwij RRhRhRhhRhRh

RRhRRHa nF ˆ)422()22(

)(646 2222

33

(2.21)

iji

ivdwi hRhh

RHa nF ˆ)2(

23 222

3

(2.22)

)( jiji RRRRR (2.23)

||/)(ˆ jijiij RRRRn

The formulae Eq(2.21) used for particle-particle contacts; and Eq(2.22) used for

particle-wall contacts. Ha in the van der Waals force formulae is the Hamaker constant,

which is material dependent constant. If the two contacting surfaces are of different

materials, Ha is considered as the geometric mean of the Hamaker constants, Ha1 and

Ha2 of the two contacting objects, i.e. Ha= 21 HaHa (Kruusing 2008). R is the

effective radius for a pair of spheres of radii Ri and Rj coming into interaction with each

other, which is determined by the harmonic mean, as given in Eq(2.23). For particle-to-

wall or blade contacts, the same formulae can be used with the radius of the contacting

surface Rj being set to infinity.

JKR model is used for obtaining adhesive force. It is reported that JKR model is not

good enough to describe interaction forces at the micro-scale and in addition that such

forces can be derived from the van der Waals forces (S.Alvo , Lambert et al. 2010).


2-52

2.16.2 Implementation of the DEM

DEM simulation can simulate the motion of every particle in the flow and

modelling each collision between the particles and between particles and their shell of a

mixer.

The collisional forces are determined by the overlap, normal and tangential

velocities by contact force law. There are a number of possible contact force models to

approximate collision dynamics such as a linear model based on the Hook type relation,

non-linear model based on the Hertz theory and non-slip solution of theory developed

by Mindlin and Deresiewicz (Di Renzo and Di Maio 2004). We use a linear spring –

dash pot model. The normal force consists of a linear spring to provide the repulsive

force and a dashpot to dissipate a proportion of relative kinetic energy. The maximum

overlap between particles is calculated by stiffness of spring in normal direction.

Average overlaps are 5% in these simulations. The normal damping coefficient Cn is

chosen to give required coefficient of restitution ε. The total tangential force is limited

by Coulomb frictional limit μFn , at which point the surface contact shears and the

particles begin to slide over each other.

The implementation of the DEM as follows: the particles in contact need to be

determined at each time step. The two strategies such as formation of neighbour list and

use of boxing or zoning methods used for reducing the computation time(Asmar,

Langston et al. 2002). In a neighbour list method, the neighbour list is a single list that

records neighbors at each particle in the bed at any given time using particle indexes.

The particle j is considered as a neighbour of the i particle and recorded in the inventory

of particles i if the particles j falls within a sphere of ro=1.5ri as shown in Figure 2.19

The computation region is divided into cubic cells of ro width, the particle indexes are

recorded under the cell index, maintaining a single cell list(CL) for the bed in addition

to the neighbour list in the zoning method. The particles have moved by a pre-fixed

amount when the CL and NL need to be updated. The particles contacts determined

when the particles recorded in the NL, thus the duplication of contact force calculation

can be avoided(Asmar, Langston et al. 2002).


2-53

(a) (b)

Figure 2.19(a) neighbour region of particles I; and (b) zoning the

neighbourhood region

The forces of each pair of colliding particles or boundary objects are calculated

using spring-dashpot model and then transformed into the simulation frame of reference.

All the forces and torques on the particles and objects are summed and the resulting

equations of motion are integrated. The trajectories of all the particles can be calculated

by integrating the Eel (2.13) and Eel (2.14) with assumption that acceleration is constant

over the time step ∆t.

2.16.3 The application of DEM

The Discrete particles simulation has been used in the particle packing, particles

flow and particle-fluid flow research for many years due to advantages of DEM that the

dynamic information such as trajectories of transient forces acting on individual

particles can be obtained which is difficult in traditional experimental techniques. The

DEM application in fundamental research and applied research of the particle flow is

reported here.

Fundamental research

The fundamental research on particle flow by DEM can be generalized. That is,

simple geometry is used to understand the confined and non-confined flow. Confined

flow, is extensively investigated experimentally and numerically to understand the

quasi-static rheological behaviour of granular materials(Zhu, Zhou et al. 2008). To

measure the bulk strength of materials, the direct shear testers are usually used. The

most popular shear tester is the Janice cell, which is one of the oldest direct shear testers

used in DEM simulation. The particle flow can be measured using annual cell which is

important shear test to measure the shear strength of soils(Howell, Behringer et al.


2-54

1999). The vertical flow is always experienced in the industrial process such as in

hoppers. It is also an important fundamental study due to it is simplicity (Zhou, Yu et al.

2003; Zhou, Yu et al. 2004).

In contrast with confined flow, the unconfined flow studies can help understand

the fundamentals governing the related complex phenomena. The plane shear flow of

uniform particles in a periodic domain investigated at different cohesion(Aarons and

Sundaresan 2006). The fluctuation of the top location of a sand pile was investigated.

For the small interval, avalanches occur continuously either on the left or right side

slope of a sand piles, and the slope on which avalanches take place switches

intermittently(Urabe 2005). The vibrating flow was also investigated in terms of

convection. For example, the flow of convection cells of glass beads in two dimensional

vibrated granular flow investigated by means of DEM. The mass flow rate increased

with bed velocity in power law relation, decreased with fixed bed vibration

acceleration(Yang and Hsiau 2000).

Applied research

The Applied research on the systems considered are more related to the

operations in practice in the applied research of granular flow by means of DEM. The

particle movements are driven by gravity or external mechanical forces as a result of

interaction between particles and devices in this system. The driving force of particle

movement in Hooper is gravity force; for flow in rotating drum, the sliding friction

between particles is a driving force; the vertical force is a driving force in the cylindrical

mixer.

Hoppers are widely used in the industry due to their capacity to enhance flow

conditions for granular materials. The wall stress or pressure, discharge rate and internal

properties in Hopper have been studied by means of DEM. The orifice size and wall

roughness of hoppers, and frictional and damping coefficients between particles has

been investigated. There are four different zones in the hopper flow: a stagnant zone, a

plug flow zone, a converging flow zone, and a transition zone from plug flow to

converging flow. It is reported that the Beverloo equation can describe the relationship

between discharge rate and orifice size; however, the constants in the equation may vary

with the wall friction coefficient, particle friction and damping coefficients. It is


2-55

investigated that the flow and force structures of particles in the hopper are spatially

non-uniform. In particular, porosity is high in the region near the orifice and low in the

upper part and around the bottom corner of the hopper. The particles around the bottom

corner experiences large contact forces, whereas there are small forces in the upper part

and the region near the orifice. However, there is a region above the orifice where

particles experience the maximum total interaction forces between particles; the forces

gradually propagate from this region into the bed and have a minimum value in the

central upper part. The velocity distribution, flow and force structures are affected by

the geometric and physical parameters of the hopper and particles(Zhu and Yu 2004).

Flow in the mixer

It is essential to understand reliable design of mixer and control the mixing

process for obtained controlled quality of product in the industry. DEM method is a

promising approach in this field. The particle flow, mixing, and heat transport in

granular flow system in rotating calciners and impregnators were investigated by means

of DEM. To understand the effect of granular flow and heat on the dry and calciner

performance, their properties are considered in the simulation. The heat transfer and

temperature uniformity of granular bed for both calciner and impregnator increase with

a decrease of rotation speed(Chaudhuri, Muzzio et al. 2006).

Particle mixing and transport pattern investigated both in the tumbling blenders

such as double cone and V-blender using DEM methods. The dynamics of blending

differ significantly between two tumblers, flow in the double-cone is nearly continuous

and steady, while flow in the V-blender is intermittent(Moakher, Shinbrot et al. 2000).

The influence of DEM simulation parameters such as restitution coefficient, normal

stiffness, friction coefficient on the particle behaviour in a V-mixer investigated. The

prediction and experimental results of restitution coefficient value, internal friction

coefficient values and wall friction coefficient value are quite close(Kuo, Knight et al.

2002).

In the toe blender, the mixing and segregation of spherical and free-flowing

material is investigated by using DEM. The simulation results are similar with the

experimental results. The effects of blender geometry on particle velocity and flow

pattern were investigated. The presence of hopper, bin section and axial offset greatly

improved the axial mixing rate(Sudah, Arratia et al. 2005). The DEM method is also

used to investigate qualitative visualization of the particle patterns and quantities


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diagnosed analysis in terms of kinetic energy, rotational energy and collisional granular

pressure within the two simplified Mechanofusion devices(Chen, Dave et al. 2004).

The DEM simulation flow of granules in a bladed mixer compared with the

experimental results of Positron emission particle tracking. PEPT is based on the

detection of gamma rays given off as a result of positron decay. The three dimensional

flow data from both experiment and simulation, made possible comprehensive

quantitative comparisons(Stewart, Bridgwater et al. 2001). The flow in a vertical blade

form a heap in front of blade and then either download on the bed surface over the blade,

and there is a recirculating zone in front of the blade(Zhou, Yu et al. 2003).

Six flow patterns of particles named slipping, slumping, rolling, cascading,

cataracting and centrifuging in a rotating drums, some of them are simulated by DEM

method. The granular flow dynamics in different six regimes were investigated by

varying the rotational speed and particle-wall sliding friction in DEM. It is investigated

that the angle repose of moving particles increased with the rotation speed in cascading

and cataracting regimes and slightly depends on the rotation speed in the slumping and

rolling regimes(Yang, Yu et al. 2008).

The power draw, linear wear rates, linear stresses and energy spectra were

investigated by DEM and compared the experimental results. DEM well predicted the

breakage rates, mill throughput, and equilibrium particle size distribution of

charge(Cleary 2001).

The flow pattern, mixing pattern and power draw, flow velocity and force field

investigated at different flow properties in the IsaMill by DEM method. The effects

relating to particle material such as sliding friction coefficient and damping coefficient;

operational parameter, rotational speed and fill level, were investigated. The flow

velocity and compressive force decreased with an increase of sliding friction. However,

the flow velocity, compressive force and power draw of mill increase with an increase

in rotational speed and loading(Yang, Jayasundara et al. 2006).

The three-dimensional motion of particles in a single helical ribbon agitator was

carried out by means of DEM and compared with experimental results. It is investigated

from DEM simulation that the vertical mixing of particles was deteriorated during

upward and downward flows through the blade and core regions. Particle flow manned

as funnel flow in the core region. The vertical velocity in the blade region decreases

toward the wall and it becomes zero near the boundary between the blade and core


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region. Particles were carried up to the bed surface by helical ribbon and flow down

into the center of the core. The bed height was found to be the most important factor for

mixing and circulation. The particles circulate rather regularly when the bed height is

appropriate and when the blade height is properly adjusted. The particles have higher

velocity when the blade top higher than the bed height due to high falls of particles from

blade top. However, the particles velocity in bed surface region is much lower than that

of the other region(Kaneko, Shiojima et al. 2000). There was no literature found in

relation to the particle flow analysis in ribbon mixer studied by DEM.

The DEM method was originally proposed by Cundall and Stack(Cundall and

Strack 1979), and became widely used to simulated the granular flows. This method

simulated the individual dynamics of all particles in an assembly by numerically

integrating their acceleration resulting from all the contact forces. Particle flows related

to the various industrial problems have been studied by DEM to elucidate the

mechanisms governing the dynamics of granular materials. The DEM is successfully

used in fundamental and application research as well. The advantages of DEM

simulation is obvious, as the flow pattern, velocity field, solid fraction, granular

temperature and force structure can be readily obtained compared to the conventional

studies. The effect of geometry, operation properties of the system and material

properties on the dynamic behaviour of particles flows under different conditions has

been quantified by means of DEM simulations. The limitation of the DEM is that it

cannot deal with large numbers of particles, and most of the studies have focused on

spherical particles.

2.17 Summary and Research Proposal

From the literature survey above, one can conclude that the material properties,

the operational parameter and the geometrical parameters are significantly affecting the

degree of particle mixing. When the particles mixing held with different sizes and

densities, segregation mechanisms will work along with the mixing mechanisms. It is

not very clear how to suppress the segregation mechanisms to make the mixing a

dominant one. One possibility is using different volume fractions of the mixing

materials. Another feasibility is using particles that have a combination of both size and

density differences. In the literature, there is evidence to show that correct combinations

of density, size and volume fractions can counter de-mixing processes. As shown in the

first topic if the predictability of particle mixing is based on the size and density


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differences of the particles in the cylindrical mixer is possible, this is necessary to

control the quality of particles based products in the industry.

The literature review shows that the research of the particle mixing in ribbon

mixer is limited. The knowledge of the effect of operational parameters on final

mixture quality is important in industrial operations. The parameters of interest here are:

operational parameters such as shaft speed and filling level; the geometrical parameters

such as pitch ratio, blade width, blade clearance, blade angle and blade number; material

properties such as particle size, density, volume fraction; and particle cohesion.

Cohesiveness plays a key role in the mixing /segregation on the granular system.

From the literature review it is evident that the effect of cohesiveness on powder mixing

is significant. However, there is almost no published work on the effect of cohesiveness

in the powder mixing in a ribbon mixer. The cohesive force between particles has been

simulated using a square-well potential method and a numerical model have been used

to characterize flow and mixing properties in a tumbling blender. This method has

ended as a demonstration of the effects of cohesiveness between particles but is rather

difficult to relate the results to a real system. Presently we are using the Hamaker

constant to represent the cohesive forces between particle pairs, which is much more

realistic than the square-well approach for dry powders. Some effects of cohesive

particles have already been demonstrated in the case of a cylindrical mixer

(Chandratilleke 2010). When the particle size decreases (or Hamaker constant increases)

the mixing has become difficult to simulate due to the increase of the simulation time.

To simulate cohesive effects in DEM in a limited time, we propose using macro particle

system to represent the micro particle system keeping dynamical similarity using

Froude number and Bond number as done under the second topic.

The literature survey shows some work has been done regarding the effects of

blade speed and fill level on the particle mixing in different mixers. The improved

mixing is obtained with an increase of blade speed and filling level up to a certain value

in cylindrical mixer. However, there is limited research about the the effects of the

operational parameters on the mixing in ribbon mixers. Third topic will be the effect of

operational parameters such as a blade speed and filling level at different cohesive

particles.

The effect of geometrical parameters on powder or liquid mixing are widely

conducted in helical ribbon mixer, experimentally and numerically, but there is limited


2-59

research on the effect of geometrical parameters in horizontal ribbon mixer, though it is

used widely in the industry. Therefore, as the fourth topic, the investigation of the

geometrical effect will contribute to the knowledge of the ribbon mixer for mixer

designers.

It is well known that size, density and volume fraction affect the mixing

behaviour in different mixers, but how they will affect particle mixing in the ribbon

mixer is unknown to date. As the fifth topic, examination of predictability of the particle

mixing in a ribbon mixer will be confirmed as in the first topic.

Radial segregation and its mechanism due to the particle size and density

differences in rotating drum have been studied for many years. It is well known that the

percolation mechanism is due to density induced segregation, and buoyancy mechanism

is due to particle size differences, and these effects can counterbalance each other when

an optimum combination of size and density ratio is used. However, the predictability of

the size and density effects in a rotating drum is still not confirmed. As the sixth

research topic, the predictability of particle mixing of binary mixtures will be confirmed

as in the cases of the cylindrical and ribbon mixers.

As the tool of the investigation, we use Discrete Element Method(Cundall and Strack

1979), which is demonstrated to offer insight into mixing at the particle scale. The

following are the topics of interest in the PhD project:

Investigation of the effects of particle size, density and volume fraction on

particle mixing and establishing a correlation for predicting mixture quality in a

cylindrical mixer;

Effect of particle cohesion on mixing in a ribbon mixer;

Effect of blade speed and fill level on different cohesive mixtures in a ribbon

mixer;

Effect of geometrical parameters on particle mixing at different cohesion in a

ribbon mixer;

Investigation of the effect of size, density and volume fraction on the binary

mixture in the ribbon mixer;

Predictability of binary particle mixing in a rotating drum.

CHAPTER 3 Prediction of The Mixing Behaviour of Binary Mixtures of Particles in a Bladed Mixer

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Chapter 3

Prediction of The Mixing Behaviour of Binary Mixtures of

Particles in a Bladed-Mixer


3-61

3.1 Introduction

It is important to control the quality of products manufactured from granular

mixtures in many industries, some examples of such products being tablets and capsules

in pharmaceuticals (Hilton and Cleary, 2013); high hardness composite materials for

use in cutting tools in powder metallurgy industries (Fernandez et al., 2011), plastic

materials (Metzger and Glasser, 2012; Saberian et al., 2002) and cosmetics (Delaney et

al., 2012). A proper understanding of the mixing behaviour of powders will be very

important for designing unit operations used in the manufacture of such products. There

are several particle properties that can significantly affect the mixing behaviour of

particles, the most important being the particle size and density difference (Fan et al.,

1990; Fan et al., 1975; Stephens and Bridgwater, 1978).

An increase in either of these properties will result in an increase in the tendency

for particles to segregate as observed in mixers such as rotating drums (Alonso et al.,

1991; Eskin and Kalman, 2000; Jayasundara et al., 2012; Metcalfe and Shattuck, 1996;

Xu et al., 2010), vertically-shafted cylindrical mixers (Chandratilleke et al., 2011; Zhou

et al., 2003), as well as in vibrated granular systems (Rosato et al., 1987; Shinbrot and

Muzzio, 1998; Yang, 2006). Further, it is reported that an optimum combination of the

size and density differences can minimize the free surface segregation in the case of a

rotating drum (Alonso et al., 1991). Generally, a binary mixture of particles with

uniform density but different in the size will segregate so that large particles would

collect at the periphery of the rotating drum. On the other hand, a mixture of particles

with uniform size but different density would segregate so that the denser particles

would collect at the centre of the rotating drum. Interestingly, a binary mixture where

large particles are denser shows a different behaviour. Here, both the size and density

segregation effects compete against each other resulting in good mixing (Metcalfe and

Shattuck, 1996). Similarly, Nitin et al. (2005) also investigated that powder mixing can

be improved under certain size and density combinations in the case of rotating drums.

In particular, the mixing will be improved when the size ratio rs=dheavy / dlight is

substantially larger than the density ratio rd=ρheavy / ρlight; here, d and ρ represent the

particle diameters and densities of a binary mixture, respectively.

There are several publications in the literature addressing segregation

phenomena observed in granular media, the mechanisms being described as mainly due

to percolation and Brazilian nut effect, which occur at different particle sizes


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(Bridgwater et al., 1985; Hong et al., 2001; Rosato et al., 1987). In the phenomenon of

the Brazilian nut effect, larger particles move to the top of the container when a

container with two different types of particles is shaken (Rosato et al., 1987). Reverse

effect may also occur depending on the density of the particles; namely, the larger

particles can sink to the bottom (Huerta and Ruiz-Suárez, 2004; Shinbrot and Muzzio,

1998). In addition, it has been shown that percolation and buoyancy can oppose each

other if large-heavy particles and light-small particles are used in a rotating drum (Nitin

et al., 2005). Although the Brazil nut effect has been described as due to buoyancy

(Nitin et al., 2005), studies that illustrate the direct involvement of forces acting on the

particles in the segregation process are rare. For example, a previous study has shown

that the particle segregation mechanism in relation to a binary mixture in a vertically-

shafted cylindrical mixer can be likened to the vertical (buoyancy) forces acting on

particles, which are generated due to either the size or density differences or both (Zhou

et al., 2003). However, there are only a few studies that deal with the mechanisms of

mixing or segregation in vertically-shafted cylindrical mixers when both the size and

density differences of particles coexist.

It has been reported that the transition between states of mixing and segregation

due to size and density differences at a given volume fraction can be predicted for

rotating drums based on an expression (Alonso et al., 1991).The same relationship is

reported to be promising to quantitatively predict the mixing states of not only dry

particles, but also wet particles at low cohesion when the particle flow is in the

continuous flow regime (Liu et al., 2013). However, no such relationship is available in

general for cylindrical mixers. Nevertheless, a recent study has investigated separate

effects of the size and density differences on the mixing behaviour in a vertical bladed

mixer by considering the variation of only one of the differences at a time

(Chandratilleke et al., 2012). Based on the study, the two effects are interchangeable, or

the two effects produce similar mixing states, if the particle weight ratio rd rsis

matched. In other words, an equivalence between the two separate effects has been

found under the tested conditions. Nonetheless, it is not clear whether such a size and

density equivalence would generally exist or if an equation to predict these effects can

be established.

Accordingly, the combined effects of size ratio rs and density ratio rd on the

mixing behavior of binary particles are investigated here for a cylindrical bladed mixer


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with a vertical shaft, using the discrete element method (Cundall and Strack, 1979). The

objectives of the study are firstly to quantify the mixture quality of binary mixtures as a

function of rs, rd and volume fraction xl, and then to find a prediction equation that

quantifies the equivalence between the size and density effects and the mechanism of

the size-density interaction. Establishing such a correlation equation will make it

possible to predict the mixing outcomes beforehand, which has not been possible except

for a rotating drum. Further, if the equation includes the effects of other variables such

as particle size, blade speed and mixer dimensions, the mixing behavior of this mixer

becomes fully predictable and new opportunities will arise. For example, one can use

the cylindrical mixer as a standard mixer, for which the mixing behavior is fully

understood. If a different mixer shows a different mixing behavior in comparison to the

cylinders, one can conclude that different mixing mechanisms are dominant in that

mixer. The concept is investigated in a future work.

The work is organized as follows. First, a brief introduction is given in sections

3.2 and 3.3 on the simulation and quantification methods of mixing, which are then

followed by the simulation conditions and procedure in section 3.4. Results of the

present study are discussed in section 3.5 and subsections therein, the focus in section

3.5.1 being the quantification of the effects of particle density, size and volume fraction

of the mixing particles. In section 3.5.2, the particle mixing mechanism is explained. In

section 3.5.3, mixing trends of effects of size and volume fractions are presented. In

section 3.5.4, the effects of size, density and volume fraction are correlated. Finally, the

conclusions of the study are presented in section 3.6.

3.2 Numerical Method

The DEM model used here is essentially that by Zhou et al. used in their study of

particle mixing in a vertically-shafted cylindrical bladed mixer (1999; 2004). It uses two

momentum conservation equations to describe the translational and rotational motion of

particle i in a system at time t, subjected to the gravity g and interactions with

neighboring particles, blades and walls:


3-64

ik

jijdijci

ii m

dtd

m1

,, FFgV

(3.1)

and

ik

jijij

ii d

dI1t

MT (3.2)

where mi, Ii, Vi and i are the mass, moment of inertia, translational and rotational

velocities of the particle respectively; k is the number of particles in contact with

particle i, Fc, ij represents the contact force which is the summation of the normal and

tangential forces. Fd,ij represents the damping force, which is the summation of the

normal and tangential damping forces at the contact point with particle j. Tij and Mij are

the torque and rolling friction torque on particle i due to particle j. Expressions for the

forces and torques in Eqs. (3.1) and (3.2) are given in Table 1.

Table 3.1 Formulae for contact forces and torques

Forces and torques

Formula Associated definitions

Normal contact force

Normal damping force


Tangential damping force

Rotational torque

Rolling friction torque


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3.3 Mixing quantification method

A particle scale mixing index (PSMI) developed previously is used here for the

quantification of mixture quality because of its success in correlating the effects of

particle size and density differences (Chandratilleke et al., 2012). The index is based on

the use of the coordination number to define the particle fraction pi of one type of

particles (referred to as the target particles) in the immediate neighborhood of particle i

in a binary mixture, and the variance St2 of pi is obtained relative to the instantaneous

mean value of pi for the mixture. Next, St2 is used in the calculation of Lacey’s mixing

index M (Lacey, 1954),

)3.3(220

220

R

t

SSSSM

where ppS 120 and /12 ppSR are the variances of the fully-segregated

and fully-mixed mixed states, respectively, with p and ƞ respectively representing the

particle number ratio of the target particles to the total particle number in the binary

mixture and average particle-scale sample size for the mixture. The average sample size

is determined from the average value of the coordination number for the mixture. Here,

Ni

ii pp

NS

1

2t

2t

1 , where N is the total number of particles in the mixer, with tp

representing the average of pi at time t. In determining St2, one has to use a particle-

contact condition, which is taken as an inter-particle gap size of 5% of the small particle

diameter, to be consistent with our previous work (Chandratilleke et al., 2012).

3.4 Simulation conditions and procedure

The mixing equipment used is shown in Figure 3.1(a) and is a vertically-shafted

two-bladed cylindrical mixer. The blades have their wider surfaces vertical (or a rake

angle of 90°), and they have no clearance at the vessel base. The mixer is filled up to the

top edge of the blades as shown Figure 3.1(b), to keep the total volume of a mixture

constant. Here, 43 cases of numerical experiments are conducted, which can be

categorized into three types of experiments. In the first, density ratio rd is changed with

size ratio rs fixed at 0.5, at different values of volume fraction α (see Table 3.2). In the


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second type, rs is changed with γ fixed at 0.22 at different α (see Table 3.3). In the third,

rd is changed with α fixed at 0.9, at different rssee Table 3.4). Particles are deposited

in the mixer, layer by layer until the required number of particles is generated. Initial

loading arrangement of the particles is such that large (or heavy) particles are laid on

top of the small (or light) ones as shown in Figure 3.1(b).

Table 3.2 Particle details for varying rd at three levels of xlwith rs fixed.

Table 3.3 Particle details for varying rs at different levels of α with rd fixed.

1 4 3200 1300 8 3600 6040 0.22 0.5 0.9 0.92 4 3200 2500 8 3600 6040 0.41 0.5 0.9 0.93 4 3200 3500 8 3600 6040 0.58 0.5 0.9 0.94 4 3200 4500 8 3600 6040 0.75 0.5 0.9 0.95 4 3200 6040 8 3600 6040 1.00 0.5 0.9 0.96 4 15800 1300 8 2000 6040 0.22 0.5 7.9 0.57 4 15800 2500 8 2000 6040 0.41 0.5 7.9 0.58 4 15800 3500 8 2000 6040 0.58 0.5 7.9 0.59 4 15800 4500 8 2000 6040 0.75 0.5 7.9 0.5

10 4 15800 6040 8 2000 6040 1.00 0.5 7.9 0.511 4 28800 1300 8 400 6040 0.22 0.5 72 0.112 4 28800 2500 8 400 6040 0.41 0.5 72 0.113 4 28800 3500 8 400 6040 0.58 0.5 72 0.114 4 28800 4500 8 400 6040 0.75 0.5 72 0.115 4 28800 6040 8 400 6040 1.00 0.5 72 0.1

x lcase No

Small Particles Large particlesr d r s)( 3kgml)(mmd s sN )(mmd l lN)( 3kgms ls NN /

1 4 28400 1300 9 300 6040 0.22 0.44 94.67 0.12 5 14500 1300 8 400 6040 0.22 0.63 36.25 0.13 7 5300 1300 8 390 6040 0.22 0.88 13.59 0.14 8 3500 1300 8 400 6040 0.22 1.00 8.75 0.15 4 15800 1300 9 1300 6040 0.22 0.44 12.15 0.56 5 8000 1300 8 2000 6040 0.22 0.63 4.00 0.57 7 2900 1300 8 1900 6040 0.22 0.88 1.53 0.58 8 1900 1300 8 1900 6040 0.22 1.00 1.00 0.59 4 9500 1300 9 2000 6040 0.22 0.44 4.75 0.710 5 5000 1300 8 3000 6040 0.22 0.63 1.67 0.711 7 1700 1300 8 2700 6040 0.22 0.88 0.63 0.712 8 1200 1300 8 3000 6040 0.22 1.00 0.40 0.713 4 3100 1300 9 2500 6040 0.22 0.44 1.24 0.9

14 5 1600 1300 8 3500 6040 0.22 0.63 0.46 0.9

15 7 590 1300 8 3500 6040 0.22 0.88 0.17 0.9

16 8 400 1300 8 3500 6040 0.22 1.00 0.11 0.9

x lcase No

Small Particles Large particlesr d r s)( 3kgml)(mmd s sN )(mmd l lN)( 3kgms

ls NN /


3-67

Table 3.4 Particle details for varying rd at different levels of rs with xlfixed.

(a) (b)

Figure 3.1 Schematic illustration of : (a) mixer configuration; and (b) initial loading

of particles

After the particles have settled down, the impeller is rotated from the stationary

state at a constant acceleration until it reaches a pre-set speed of 20 rpm, at which point

it continues to rotate at that constant speed.

The particle properties used are as follows: Young’s modulus E= 1107 N/m2,

Poisson’s ratio ν= 0.3, damping coefficient (for both normal and tangential) c= 0.3,

static sliding friction coefficient µ= 0.3, and rolling friction coefficient= 0.001d, where

d is the diameter of small particles if particle sizes are different. Details of the binary

mixtures are listed in Tables 3.2, 3.3 and 3.4 for the above three types of experiments,

respectively. The size ratio rs is defined as ls dd / and density ratio rd as ls / , where

s and l refer to small and large particles respectively. Volume fraction xl is the ratio of

1 5 1600 2500 8 3500 6040 0.41 0.63 0.46 0.902 5 1600 3500 8 3500 6040 0.58 0.63 0.46 0.903 5 1600 4500 8 3500 6040 0.75 0.63 0.46 0.904 5 1600 6040 8 3500 6040 1.00 0.63 0.46 0.905 7 590 2500 8 3500 6040 0.41 0.88 0.17 0.906 7 590 3500 8 3500 6040 0.58 0.88 0.17 0.907 7 590 4500 8 3500 6040 0.75 0.88 0.17 0.908 7 590 6040 8 3500 6040 1.00 0.88 0.17 0.909 8 390 2500 8 3900 6040 0.41 1.00 0.10 0.90

10 8 390 3500 8 3900 6040 0.58 1.00 0.10 0.9011 8 390 4500 8 3900 6040 0.75 1.00 0.10 0.9012 8 390 6040 8 3900 6040 1.00 1.00 0.10 0.90

x lcase No

Small Particles Large particlesr d r s)( 3kgml)(mmd s sN )(mmd l lN)( 3kgms

ls NN /

X Y

Z

f249mm


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volume of large particles to the total particle volume. Note that xlrs, and rd are each

less than or equal to 1.

3.5 Results and discussion

The results of the numerical experiments are discussed as follows: in section 3.1,

the effects of rd, rs and xlare discussed; in section 3.2, the mechanism of mixing is

discussed; and in section 3.3, a predictive equation is developed to describe the trends

of the results.

3.5.1 Effects of density, size and volume fraction

Effect of density differences

The effect of rd on the mixture quality is investigated for xl= 0.1, 0.5 and 0.9, in

each case rd being varied with the size ratio rs=0.5 and density of large particles, l =

6040 kg/m3; these cases are listed in Table 3.2.

0.00

0.20

0.40

0.60

0.80

1.00

0 5 10 15 20 25

0 1 2 3 4 5 6 7

Parti

cle

Scal

e M

ixin

g In

dex,

M

Time (s)

Shaft revolutions

1

rd=0.22 0.41

0.57 0.75

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Stea

dy-s

tate

val

ue o

f M

Density ratio, rd

xl = 0.1

xl = 0.9

xl = 0.5

(a) (b)

Figure 3.2 Mixing curves and steady state values: (a) mixing index M versus

time for different rd at rs=0.5 and xl= 0.5; and (b) effect of rd on steady-state

values of M for rs =0.5. The solid lines are the predictions based on Eq.(3.6),

which will be discussed later.


3-69

Figure 3.2 (a) shows a typical example of the effect of rd on mixing using a

diagram of mixing index M as a function of time. A large value of l is chosen here so

that the weight of large particles will counter the vertically upward buoyancy force

which causes segregation as reported previously (Zhou et al., 2003). For this xl, M

improves with a decrease in rdwhen rs=0.5. Though not shown here, similar trends

were observed at the other two xl values tested. The time for M to reach the steady state

decreases with a reduction in rd, a result observed at other xl as well. The steady-state

values of M are shown in Figure 3. 2 (b) for the tested xl values; note that the solid lines

in the figure are trend lines based on a predictive equation developed in this work which

will be discussed later. Figure 3.2 (b) shows that the mixture quality improves with a

decrease in rdat each of the volume fractions for rs=0.5. This implies that when

rdreduced, the weight of large particles gradually increases, which increasingly

suppresses the upward driving force on large particles generated due to the particular

size difference considered. This increasing trend of M is different from that observed in

the case of mono-sized particles (Chandratilleke et al., 2011; Ford, 1972), where the

mixture quality deteriorates with a decrease in rd due to the absence of the size-

difference generated buoyancy force that will counter the increasing particle weight.

Figure 3.2(b) also shows that the mixing quality at xl=0.5 is superior to those at xl=0.1

and 0.9 for any rdThis is probably related to the large porosity of the mixture, which

changes with the value of xl. For example, for smaller xl values, large particle would

find it difficult to penetrate to the vessel base through the small particles at the bottom

of the mixture. On the other hand, larger xl values would allow large particles to move

down through the mixture competing with small ones, which would normally sink to the

bottom (Zhou et al., 2003).

Effect of size differences

Figure 3.3(a) shows a typical example of the effect of size ratio rs on mixing for

xl=0.1 with rd=0.22, the other parameters being listed case 1-4 in Table 3.3. The steady-

state values of mixing index M are shown in Figure 3.3(b) for the tested xl values; note


3-70

that the solid lines in the figure are trend lines based on a predictive equation developed

in this work which will be discussed later. Although all the results are not shown here,

the mixing index M reaches a peak value quite early in the mixing process for a given

volume fraction as seen from Figure 3.3(a) and then either remains at that value or

decreases slowly to reach a steady-state.

0.00

0.20

0.40

0.60

0.80

1.00

4 8 12 16 20

1 2 3 4 5 6

Parti

cle

Scal

e M

ixin

g In

dex,

M

Time (s)

Shaft revolutions

rs=1

0.4

0.875

0.6

0.00

0.20

0.40

0.60

0.80

1.00

0.4 0.5 0.6 0.7 0.8 0.9 1

Stea

dy-s

tate

val

ue o

f M

Size ratio, rs

xl= 0.1

xl = 0.9

xl = 0.5

xl= 0.7

(a) (b)

Figure 3.3 Mixing curves and steady state values: (a) effect of size ratio rson mixing

index M for rd=0.22 at xl= 0.1 with l = 6040 kg/m3; and (b) steady-state values of

mixing index M as a function of rs. The solid lines are the predictions based on

Eq.(3.6), which will be discussed later.

Figure 3.3(b) shows that mixture quality or steady-state value of M can be

improved with an increase of rs up to 0.76 at a given xl value, but it deteriorates as rs

gets closer to 1 because there is still a density difference, rd =0.22. The figure also

shows that the mixing behaviour is similar at any xl, and the condition of (xl =0.5 and

rs= 0.76) yields roughly the maximum for M. Regarding the optimum for xl at rd =0.22,

we examine further. Figure 3.4 shows the steady-state mixing index as a function of xl

at different fixed rs values for rd =0.22. It shows that xl =0.55 is the optimum volume

fraction at a given size ratio for obtaining a maximum for M. Therefore, combining the

results of Figures 3.3(b) and 4, one can conclude that the condition of (xl =0.55 and

rs=0.76) yields the maximum M for rd =0.22.


3-71

0.00

0.20

0.40

0.60

0.80

1.00

0 0.2 0.4 0.6 0.8 1

Stea

dy-s

tate

mix

ing

inde

x

Volume fraction, xl

rs = 0.4

rs = 1

rs = 0.6

rs= 0.875

Figure 3.4 Effect of xl at different levels of rs when rd =0.22 with l = 6040 kg/m3;

the solid lines are the predictions based on Eq.(3.6).

3.5.2 Particle mixing mechanism

Velocity field

Below, the effects of rd and rs on velocity field are investigated. First, to

understand the changes in the mixing behaviour that accompanies the reduction in

density ratio, the velocity fields are examined in the horizontal and co-axial cylindrical

sections. Flow patterns in a vertically-shafted cylindrical two-bladed mixer has been

well established for mono-sized particle mixing (Zhou et al., 2004). For binary particles,

mixing patterns of a two-bladed mixer have been investigated by simulations only for

two types of mixtures where: (i) rdwhile rs=1 and, rswhile rd=1 (Zhou, Yu et al.

2003). Therefore, to illustrate the interaction of density and size differences, we

investigate below the flow patterns of the three cases of rd=0.22, 0.579 and 1 at fixed

size and volume fractions, rs=0.875 and xl=0.9, the three cases being case 15 (of Table

3.3), cases 6 and 8 (of Table 3.4), respectively.

Figure 3.5 shows the instantaneous velocity fields of scaled blade relative

velocity in horizontal sections at three different heights Z, measured from the vessel

base. Further, Figure 3.6 shows the velocity fields in vertical cylindrical sections at 50,

80 and 100 mm radii of an annular width of 4dl, which are unrolled onto a vertical plane,

where dl is the large particle diameter(Chandratilleke et al., 2010).


3-72

Figure 3.5 Instantaneous normalized blade relative velocities of particles in

horizontal sections at three heights Z measured from vessel base for different rd

with rs=0.875 and xl=0.9: blue represents velocity vectors of small particles, and

red, those of large particles.

Note that the velocity vectors of small particles are coloured blue, while those of

large particles are coloured red in both diagrams. The velocity vector of a particle

represents the scaled particle velocity relative to the blade speed at a given radial

position, and is obtained by deducting the blade speed at that radial position from the

circumferential component of the particle velocity and scaling the resulting blade

relative speed by that blade speed. Figure 5 shows that the velocity vectors of the two

types of particles are mostly of equal length in each section, meaning that both particle

types are moving at mostly equal velocities. At Z=8mm, we observe that the number of

blue vectors reduces when rdis reduced from 1 to 0.22, implying that small particles are

few at the bottom, which is consistent with the mixing improvement observed with a

reduction in rd seen from Figure 3.2(b).

color

4

2

β= 0.875γ = 0.22

color

4

2

color

4

2

color

4

2

β= 0.875γ = 0.579

color

4

2

color

4

2

color

4

2

β= 0.875γ = 1

color

4

2

color

4

2

Z= 8 mm Z= 32 mm Z= 48 mm

γ= 0.22

γ= 0.579

γ= 1

(a) rd =0.22

(b) rd =0.579

(c) rd =1


3-73

Color

4

2

γ = 0.22

r= 100 mmr= 80 mmr= 50 mm

Color

4

2

Color

4

2

γ = 0.579

Color

4

2

Color

4

2

Color

4

2

Color

4

2

γ = 1

Color

4

2

Color

4

2

Figure 3.6 Instantaneous normalized blade relative velocities in cylindrical sections

at steady-state for different rd with rs=0.875: blue represents vectors of small

particles and red, those of large particles.

In this section at Z=8mm, particles mainly have circumferential motion except at

the shaft where there is some re-circulation flows for all rdvalues tested. At Z=32mm,

more blue vectors (small particles) are visible for rd=0.22 than for the other two cases.

Re-circulation flows are observed near the shaft in front of the blades, which change a

little with rd changedAt Z=48 mm also, blue vectors (small particles) are visible for

rd=0.22 than for the other two cases, consistent with the mixing results (Figure 3.2(b)).

There are no visible changes in mixing patterns at this height due to changes in rd.

From Figure 3.6, we can confirm that small particles (blue coloured vectors)

have risen to the top of the bed in most sections in the case of rd=0.22, but they are

mostly at the bottom in the case of rd=1, which are also clear from the velocity fields in

horizontal sections. In the case of rd=0.22, one can also observe higher heaps in front of

the blade as a result of small particles rising to the top of the bed. Therefore, a formation

of a weak re-circulating flow in the heap is also observed in this case, which contributes

to mixing.

Secondly, the effect of rs on the velocity field is investigated. Figure 3.7 shows

instantaneous normalized blade relative velocity vectors of particles in horizontal

sections at three heights Z=8, 32 and 48 mm for different rsand 0.875 mm)

(a) rd =0.22

(b) rd =0.579

(c) rd =1


3-74

Z= 8 mm Z= 48 mmZ= 32mm

β= 0.6

β= 0.875

β= 0.5

color

4

2

color

4

2

color

4

2

color

4

2

color

4

2

color

4

2

color

4

2

color

4

2

color

4

2

and fixed values of rd=0.22 and xl=0.9. In the case of rs=0.5 (case 1, Table 3.2), where

the small-light particles (represented by blue coloured vectors) are larger in the number

ratio, they are collected mainly at the base in the outer part of the vessel and in front of

the blades at other heights as seen from the top row figures in Figure 3.7.

Figure 3.7 Instantaneous normalized blade relative velocities of particles in

horizontal sections at three heights for different rswhen rd=0.22 at xl=0.9; blue

arrows represent velocity vectors of small particles, and red, those of large

particles.

When rsis increased up to 0.875 (case 15, Table 3.3), the number of small

particles becomes smaller compared to the large ones, and, thus they become distributed

more uniformly in all sections, indicating better mixing in consistent with the mixture

quality curves of Figure 3.4. As for the flow patterns, the circumferential motion of

particles becomes prominent due to the influence of the large particles which are larger

in number when rsis increased, under the conditions considered.

rs=0.5

rs=0.6

rs=0.875


3-75

Force analysis: effect of rd

To explain improvement in mixing with a reduction in rd, the vertical forces on

particles are analyzed, following Zhou et al. (2003) who used these forces to explain the

segregation of particles in binary mixtures. Eq(3.4) shows the group averaged

instantaneous value of the vertical forces on one type of particles which has a total of Np

particles, and Eq(3.5) the time average of the instantaneous vertical forces, Nt being the

number of data points of instantaneous average values in time t (Zhou et al., 2003).

zijd

k

jijc

N

ii

pz FFgm

Nf

ip

)]([1,

1,

1

(3.4)

tN

zt

tz f

Nf

1

1 (3.5)

To demonstrate the behaviour of the two types of averaged forces, we first

consider the case of a uniform particle system (i.e, rd=1 and rs=1) with d=5mm,

ρ=2500kg/m3 and a total of 17500 particles, in which particles of two different colours

in equal amounts (i.e., are laid in the top-bottom arrangement initially. The

particles in both groups begin to have a downward vertical force on them as shown in

Figure 3.8(a), but with the rotation of the blades, those in the bottom group receive a

larger net upward force or an upward driving force; the instantaneous driving force on a

particle is defined here as the difference between the vertical upward and downward

forces on a particle, the upward direction being positive. On the other hand, the particles

in the top group receive a downward force, and therefore, they will sink to the bottom

while the bottom ones rise to the top, resulting in gradual mixing. As Figure 3.8(a)

shows, the driving forces on the two particle groups each will reach a common value of

zero driving force. Figure 3.8(b) shows the time-averaged driving forces of Figure 3.8(a)

based on Eq(3.5), and the difference between the forces on the two particle groups still

persists within the time span investigated. However, it is anticipated that this difference

will eventually disappear, because the instantaneous values the forces in Figure 3.8(a)

approach each other at the steady state. The result supports the observation that a system

of uniform particles in equal amounts results in a perfectly mixed state in the steady

state (Chandratilleke, Yu et al. 2011).


3-76

-10.0

-8.0

-6.0

-4.0

-2.0

0.0

0 5 10 15 20

Tim

e A

vera

ged

driv

ing

forc

e / P

artic

le w

eigh

t

Time (s)

Top group

Bottom group

-10.0

-8.0

-6.0

-4.0

-2.0

0.0

2.0

0 5 10 15 20

Ave

rage

driv

ing

forc

e / P

artic

le w

eigh

t

Time (s)

Top group

Bottom group

(a) (b)

Figure 3.8 The driving force on particles in a uniform system particle of d=5mm,

ρ=2500kg/m3 and 17500 particles, laid initially in top-bottom arrangement: (a),

instantaneous group-averaged driving force, and (b) time–averages of the driving

forces in Figure 3.8(a).

In order to further understand the mechanism of mixing improvement in binary

particle mixtures, the vertical driving forces on particles are investigated from the start

of the mixing process for the cases, rd=0.22, 0.41, 0.578 and 0.75 when rs=0.5 and

=0.9 (Table 1: cases 1, 2, 3 and 4). The instantaneous driving forces on the two groups

of particles are shown in Figure 3.9 (a) just for the cases of rd =0.22 and 0.75 for clarity,

and the time-averaged upward driving forces for the two groups are shown in Figure3. 9

(b) for all the above cases of rd. First, Figure 3.9 (a) shows that the top group of (large-

heavy) particles initially have a larger driving force downwards than the bottom group.

As a result, these will penetrate into the small particle group eventually. The snapshot of

Figure 3.10 (a) shows that large-heavy particles have in fact penetrated into the small

particle group in the heap in front of the blade (see the circled area). In other parts of the

mixture as well, for example behind the blade, one can see small particles in the spaces

between the large ones as seen from Figure 3.10 (b), although the small particles were

initially at the bottom of the vessel. With the lapse of time, the difference between the

two forces gradually disappears, the scaled forces reaching a steady-state value of about

-0.995, which is mostly the same in both cases. In fact, the trace of the instantaneous

force for rd=0.75 is slighly above that for rd=0.22 when mixing is taking place; the

traces for top group, where the properties are not varied, are mostly identical.


3-77

-1.010

-1.005

-1.000

-0.995

-0.990

-0.985

0 4 8 12 16 20 24 28 32

Ave

rage

driv

ing

forc

e/ P

artic

le w

eigh

t

Time (s)

Top group (Heavy-large)

Bottom group (light-small)r

d=0.22

rd=0.75

rd=0.22

rd=0.75

-1.050

-1.040

-1.030

-1.020

-1.010

-1.000

-0.990

-0.980

4 8 12 16 20

rd=0.22

rd=0.41

rd=0.57

rd=0.75

Tim

e-av

erag

ed d

rivin

g fo

rce

/ Par

ticle

wei

ght

Time (s)

Top-group

Bottom-group

(a) (b)

Figure 3.9 The driving forces on the two types of particles as a function of time for binary

particle mixtures with rs =0.5 and xl =0.9: (a), instantaneous group-averaged driving force

for rd=0.22 and 0.75; and (b), the time-averaged vertical forces for the cases of rd=0.22,

0.41, 0.579, and 0.75 (see Table 3.2: cases 1, 2, 3 and 4).

The time-averaged curves of these two instantaneous group-averaged forces are

shown in Figure 3.9 (b), for example, by the solid upward triangles for rd=0.22. The

initial difference of the two forces, which provides the driving force for mixing, persists

in the averaged force curves although the instantaneous values have converged to a

common value of -0.995 in the steady-state. However, over a longer period of time, the

time-averaged forces of any rd will also converge to about the same value as the

instantaneous forces (i.e. -0.995); this can be stated because the instantaneous forces for

rd=0.41 and 0.57 also converge to about -0.995 although not shown here. It can be seen

from Figure 3.9(b) that the time-averaged forces on the bottom (or light-small) particle

group have almost reached the steady-state at larger times, but those on the top (or

heavy-large) particle group are still gradually approaching the steady-state value even at

20s. Therefore, the initial difference of the driving forces on the two groups persists

over a long period of time, confirming the fact that the initial driving force differences

probably have a strong influence on the mixing behaviour (Zhou et al., 2003).

Figure 3.9(b) shows that the bottom (small-light) particle group receive a larger

time-averaged upward driving force than the top (large-heavy) particle group initially,

promoting mixing. This can be confirmed from the snapshot of Figure 3.10(a), where


3-78

small particles are seen to have moved to the upper parts of the particle bed pushed by

the large ones. Note that smaller particles would have sunken to the bottom causing

segregation if the density ratio is unity as reported previously (Zhou et al., 2003). Thus

the interaction of size and density differences has caused the driving force on small

particles to be larger than those on the large particles.

(a) (b)

Figure 3.10 Snapshots showing the buoyancy and percolation effects of particles

t= 5.11s; (a) heavy -large particles penetrate into small particles over blade (buoyancy);

and (b) small particles percolation; arrow shows the direction of blade motion.

Further, it can be seen from the figure that the difference between the time-

averaged forces become larger compared to that for rd=0.75 when rdis reduced.

Therefore, the improvement in the mixing behaviour with a reduction of rd can be

generally explained using the driving forces, which is observed in mixing index results

and supported by the velocity fields. However, the number of small particles being

small at xl=0.9, the sensitivity of the time-averaged forces to changes in rdis not large

therefore we cannot observe much change in the difference between the initial time-

averaged forces with a change in rdfrom 0.22 to 0.57.

3.5.3 Mixing trends: Effects of rs and xl

Another trend yet to be explained is the change of mixture quality with the size

ratio at fixed values of rdand xl as shown in Figure 3.3 (b). When particle size ratio rs

is increased with xl fixed, the particle number ratio of small to large particles Ns/Nl

decreases as seen from Table 3.4 because the total particle volume is kept constant.


3-79

Therefore, the size effect of Figure 3.3(b) is a manifestation of the interaction of size

and density differences as well as the effect of changes in the particle numbers. At small

size fractions, the small particles are large in number, and as a result, large-heavy

particles (at a small rd such as 0.22) can penetrate into segregated small-particles

groups as seen from the density effect above, Thus, mixing is improved at small size-

ratios. However, when the size ratio increases, small particles become reduced in their

numbers and the weight of large particles becomes smaller, and thus the improvement

in the mixing due to large particles penetrating into segregated small particle groups

becomes weaker. Therefore, the mixing reaches a maximum value at some size ratios.

When the size ratio further increases and approaches unity, the effect of size difference

disappears and the density effect becomes dominant. Therefore, particles will tend to

segregate by the dominant density effect, and mixture quality decreases. The above is a

possible explanation of how the mixing index at steady-state shows a peak value against

the changes in the size ratio at a fixed volume fraction. Yet another trend to be

explained is the variation of mixture quality with xl when both rs and rd are fixed as

shown in Figure 3.4. It shows that there is a peak value when rs and rd are fixed. When

the volume fraction is increased under the above conditions, small particle numbers

decrease (see Table 3.4, for example). At low volume fractions, large particles will find

it difficult to penetrate through the small particle bed, which has smaller interstitial

spaces, and thus, mixing is poor. Conversely, when the volume fraction is high, small

particles are few in their numbers compared to the large ones, and thus, large particles

cannot penetrate in small particle regions, leading to segregation. However, at

moderately high volume fractions, the small particle number is large enough and large

particles with higher density (or for lower rd) can penetrate into small particle regions

improving mixing; the larger the value of rd the smaller the penetration of large

particles and thus, the mixture quality as seen from Figure 3.10. Thus, the observed

peak in the mixture quality with a change in volume fraction can possibly be explained

as above.


3-80

3.5.4 Formulation of a predictive equation

The above results show that xl, rs and rdaffect the mixing behaviour of binary

mixture of particles. It has been reported previously that it is possible to represent the

effect of rs for a binary system of rd=1 and effect of rd for rs=1 system by a single

characteristic curve using the particle weight ratio rd (rs3) in the case of xl=0.5. Here a

first assessment is made to seek a predictive equation for the effects of size, density and

volume fractions using the data points shown in Figures 3.2 and 3.3. A second order

polynomial of three variables with 7 unknown coefficients is used for fitting the dataset

using ‘FindFit’ function of Mathematical software, which results in the equation:

(3.6)

Note that Eq. (3.6) has been chosen so that it can represent the trends of steady-

state values of M (see, Figures. 3.2, 3.3 and 3.44) when either rd or rs or xl is changed

with the other two variables fixed. An interaction term rd rs has also been included to

account for the improvement in M due to the interaction between rd and rs when xl is

fixed, as seen from Figure 3.2(b).

The solid lines of Figures 3.2(b), 3.3(b) and 3.4 show the trends of the effects of

rd, rs , xl as predicted by Eq.(6) respectively. The figure confirms that the general trends

of the data can be described by the predictive equation. However, some deviation can

be observed at smaller values of rd (i.e. rd<0.4) as seen from Figure 3.2(b); for rd>0.4;

the predictions are reasonably good. In addition to the above comparisons, predicted

effect of size ratio rs on the steady-state mixing index in the case of uniform density

system, rd =1 is shown in Figure 3.11 at different volume fractions xl. In reality, mixture

quality at xl=0.5 for a uniform system (i.e. rd =1 and rs=1) should be unity, but Figure

3.11 shows that the mixture quality at rs=1 is slightly less. However, the figure confirms

that the mixture quality deteriorates at a given value of xl with a reduction in the size

ratio, which is a similar to the trend reported elsewhere(Chandratilleke et al., 2012).

Therefore, the developed predictive equation is successful in predicting not only size-

density interactions, but also mixture quality of a system with uniform density.

dssslldlds rrrrxxrxrrM 8.098.374.283.163.142.024.1),,( 222


3-81

0.00

0.20

0.40

0.60

0.80

1.00

0.4 0.5 0.6 0.7 0.8 0.9 1

Pred

icte

d st

eady

-sta

te m

ixin

g in

dex

Size ratio, rs

xl= 0.1

xl = 0.9

xl = 0.5

xl= 0.7

Figure 3.11 Predicted effect of size ratio on steady-state mixing index at different values

of volume fractions xl when rd =1.

To demonstrate the predictions of the size and density effects by Eq.(3-6) at a

given volume fraction, contour plots are drawn for M of Eq.(3-6) at three different

volume fractions as shown in Figure 3.12.

(a) (b) (c)

Figure 3.12. Contour maps using Eq.(3-6), demonstrating the equivalence of size and

density effects at different volume fractions: (a) xl=0.1, (b) xl=0.5 and (c) xl=0.9.

It is clear that one can choose at given volume fraction many size-density

combinations on a contour so that they will produce the identical M value. Therefore,

mixtures with such size-density combinations can be considered equivalent in that they

produce identical mixing states. The figure also demonstrates that the mixture quality is

larger when both rs and rd are close to 1 at a given xl value. The equation in this form


3-82

will be useful in predicting mixing behaviour of binary particles in a wide range of size

and density combinations for volume fractions ranging from 0.1 to 1 at the shaft speed

used in the study.

0.55

0.60

0.65

0.70

0.75

0.80

0.4 0.5 0.6 0.7 0.8 0.9 1

M=0.8; xl=0.5

M=0.5; xl=0.1

M=0.7; xl=0.9

r s

rd

Figure 3.13 Equivalence of size and density effects at a given mixture quality M and

volume fraction xl

Effects of variables such as material properties, blade geometry and vessel scale

on mixing behaviour also need to be addressed in future works. Figure 3.13 shows the

equivalence of the size and density effects at three different xl values for three different

mixture qualities. The curves are in fact the same as the contours of M=0.5, 0.7 and 0.8

in Figure 3.12 and can be obtained by solving Eq.(3.6) for either rs or rd for given

values of xl and M.

3.6 Conclusions

The effects of particle size, density and volume fraction on the mixing behaviour

of binary particles in a vertically-shafted bladed mixer have been studied by means of


3-83

the discrete element method. The mixing states of the mixtures are analyzed by a

particle scale mixing index to avoid sampling issues of conventional mixing indexes,

and the fundamentals underlying the effects on micro-dynamics are investigated

focusing on velocity fields and inter-particle forces of the mixtures. The following

conclusions can be drawn.

The effects of size and density are predictable and can be correlated within the

scope of this study. Binary particle mixtures show a maximum mixture quality at

an optimum size ratio when both density ratio and volume fraction are fixed.

Mixtures also show a quality improvement if the density ratio is reduced with

the size ratio and volume fraction fixed, for a given combination of size and

density ratio, a volume fraction of 0.5 gives the highest mixture quality for a

binary particle mixture.

Forces on particles play a major role in the mixing and segregation mechanism

in the mixer when there are differences in either the particle size or density or

both. Particles in a binary mixture will segregate when there is only either a size

or density difference. On the contrary, combinations of size and density

differences can improve mixing. For example, if the density of the large particles

is chosen large enough to counter the upward forces on them, the light-small

particles will receive a larger average vertical upward force consistently, pushing

them on top of the large ones, leading to an improved mixing behaviour.

A correlation has been established for predicting the mixing behaviour of binary

particles at different size and density ratios and volume fractions. The

relationship can predict well the effects of size and density differences and

volume fraction of binary particle mixtures. Particularly, this relationship is

suitable for use at density ratios larger than 0.4. The regression coefficient of the

equation is about 0.94.

The availability of a predictive relationship for mixing in the cylindrical mixer

makes it suitable to be considered as a standard mixer in studies of particle

mixing. The mixer also has the advantage that its geometry is simple and is


3-84

known to provide complete mixing for uniform particles. The correlation can be

used to study, for example, effects of particle properties on mixing behaviour.

Nomenclature d Particle diameter, (mm) dl Large particle diameter, (mm) ds Small particle diameter, (mm) E Young’s modulus, (N/m2) Fc,ij Contact force vector between i and j, (N) Fd,ij Damping force vector between i and j, (N)

zf Average vertical force on one type of particles (N) t

zf Time average of the instantaneous vertical force zf in time t, (N) g Acceleration due to gravity, (m/s2) Ii Moment of inertia of particle i, (kg m2) ki Number of particles in contact with particle i, (-) M Particle-scale mixing index defined in Eq.(3) , (-) Mij Vector of rolling friction torque on particle, (Nm) Mp Predicted mixing index at steady-state, (-) N Total number of particles in the mixture, (-) Nt The number of sample/data points of instantaneous average values in time t, (-) p Number ratio of the target type particles to all the particles, (-) pi Particle fraction of a target type particle in the neighbourhood of particle i, (-)

tp Average value of pi at time t for the entire mixture, (-) rs Size ratio ls dd / , (-) rd Density ratio ls / , (-) S0 Standard deviation of pi at fully-segregated state, (-) SR Standard deviation of fully-mixed state for uniform-sized particles of particle fraction of p, (-) St Standard deviation of pi with respect to tp at time t, (-) Tij Vector of rolling friction torque on particle i, (Nm) Vb Blade speed, (m/s) Vi Velocity of particle i, (m/s) xl Volume fraction, which is ratio of volume of large particles to total particle

volume (-)

z Height from vessel base, (mm)

Greek letters

Average particle-scale sample size for the mixture, (-)

ik Number of particles in contact with particle i Density of particles of a uniform system, (kg m-3)

l Density of large particle, (kg m-3)


3-85

s Density of small particle, (kg m-3) Shaft rotational speed, (rad/s) i Angular velocity vector of particle i, (rad/s)

CHAPTER 4 Effects of Particle Cohesion on Mixing in a Ribbon Mixer

4-86

Chapter 4

Effects of Particle Cohesion on Mixing in a Ribbon Mixer


4-87

4.1 Introduction

Generally, the cohesive particles can be divided into two groups, one with dry

particles where particle sizes is generally less than 10m and the cohesion is due to the

van der Waals force, and another with wet particles where the cohesion is originated

from capillary forces. Mixing processes involving dry fine solid particles play an

important role in manufacturing processes used in many industries that produce

products such as pharmaceuticals, semiconductors, food, ceramics, fertilizers,

petrochemicals and cosmetics. However, at present, there is only a limited

understanding of the cohesive effects on the mixing behaviour of dry fine particles in

industrial mixers (Chaudhuri, Mehrotra et al. 2006), particularly in relation to the

pharmaceutical industry where ribbon mixers are used for powder mixing purposes

(Muzzio, Llusa et al. 2008).

Discrete element method (Cundall and Strack 1979) has been used in the

investigation of cohesion effects on particle mixing. For example, effects of cohesion of

dry particles on the mixing rate and mixture uniformity have been investigated for

uniform and binary particle systems using rotating drums, with the assumption that the

cohesion can be represented by a square-well potential (Chaudhuri, Mehrotra et al.

2006). They found that some degree of cohesion can improve the mixture quality.

Further, cohesive effects on particle mixing have also been studied for a vertically-

shafted cylindrical mixer, which was scaled down to a miniature size to implement the

van der Waals force on particles of 100m diameter (Chandratilleke, Yu et al. 2009).

The study showed that the mixing can deteriorate at high cohesion between particles

and particles as well as particles and walls, when the blades would simply slice through

the particle bed without any mixing happening. Such results are also supported by the

experimental work of Knight et. al., who used the wider surfaces of horizontal blades in

a cylindrical mixer to avoid the occurrence of such bed separations (Knight, Seville et al.

2001).

In practice, a wide range of bladed batch mixers are used for powder mixing,

some such mixers being the orbiting screw (or Nauta) mixer, paddle mixer and ribbon

mixer (Poux, Fayolle et al. 1991). The ribbon mixer is of particular interest in this study

because it is capable of effectively performing a wide range of mixing processes

including blending of liquids, solids and solids-liquids (Cleary 2013). Muzzio et al.


4-88

(2008) described the ribbon mixer as an ideal mixer for cohesive mixing as it provides

large shear stresses in all three directions, axial, radial and tangential. Nevertheless the

effect of cohesion on particle flow has not been investigated for this mixer at a

microscopic level. However, some works related to cohesive particle behaviour are

available for other mixers in the literature. Generally, it is reported that the initial

loading pattern, fill level, mixing time, mixer geometry, material properties and impeller

speed significantly affect the quality of cohesive mixtures (Tsuji, Kawaguchi et al. 1993;

Cleary 2000; Sudah, Coffin-Beach et al. 2002).There are quite a few published works

related to mixing in vertical helical ribbon mixers regarding the effects of the fill level,

mixing time, power consumption, heat transfer and impeller geometry due to cohesion

in wet and dry particle systems (Ford 1972; Bortnikov, Pavlushenko et al. 1973; Rieger,

Novák et al. 1986; Masiuk 1987; Poux, Fayolle et al. 1991; Masiuk, Lacki et al. 1992;

Masiuk and Lacki 1993; Delaplace, Leuliet et al. 2000; Dieulot, Delaplace et al. 2002;

Shekhar and Jayanti 2003; Niedzielska and Kuncewicz 2005). Conversely, publications

on powder mixing in horizontal ribbon mixers regarding the effects of fill level, or blade

speed on the homogeneity are rare, one such publication being that of (Muzzio, Llusa et

al. 2008) who investigated experimentally the effects of mixing Magnesium Stearate

powder in a ribbon mixer. However, the effects of cohesion on the particle flow in

ribbon mixers have not been studied so far. The objectives of this study are the

quantification of the particle cohesion effects on the particle flow and mixing at a

particle-scale level for a ribbon mixer.

This paper is organized as follows. First, a brief introduction is given in section

4.2 on the simulation and in section 4.3 mixing quantification methods, which is then

followed by the simulation conditions and procedure in section 4.4. In section4. 5,

results of the present study are discussed, focusing on the comparison of micro and

macro system of the effects of particle cohesion on mixing in section 4.5.1. The

cohesion effect on particle mixing in macro system are investigated in section 4.5.2.

The cohesive particle flow behavior is explained using the particle scale mixing index ,

coordination number, velocity field and porosity of the mixture at different levels of

cohesion. The quantification of the radial, tangential and horizontal velocity at different

levels of cohesion is described. The shear and normal stress the particles encounter are

quantified. Finally, the conclusions of the study are presented in section4.6.


4-89

4.2 DEM simulation

The model of the discrete element method used here is based on an extension to

the original DEM model proposed by Cundall and Strack(1979) to account for the

rolling friction of particles, and is essentially the same as that previously developed and

validated by Zhou et al. (1999; 2004). The model uses two momentum conservation

equations to describe the translational and rotational motion of particle i in a system at

time t subjected to the gravity g and interactions with the neighboring particles, blade

and walls:

ii N

jijv

k

jijdijci

ii m

dtdm

1,

1,, FFFgV

(4.1)

ik

jijij

ii d

dI1t

MT (4.2)

Here mi, Ii, Vi and i are the mass, moment of inertia, translational and rotational

velocities of particle i respectively; ki is the number of particles that are in contact with

particle i, Fc,ij represents the elastic contact force which is the summation of the normal

and tangential forces. The tangential force Fd,ij represents the damping force, which is

the summation of the normal and tangential damping forces respectively at the contact

point of particle i with particle j. Tij and Mij are the torque and rolling friction torques on

particle i due to particle j. The contact model used here is the Hertz non-linear contact

model(Zhu, Zhou et al. 2007) and the expressions for the forces and torques in Eqs.(4.1)

and (4.2) can be found in Table 4.1. F,ij represents non-contact force due to the van der

Wall force, and the formulae used are tabulated in Table 4.1. Ha in the van der Waals

force formulae is the Hamaker constant, which is material dependent constant. If the

two contacting surfaces are of different materials, Ha is considered as the geometric

mean of the Hamaker constants, Ha1 and Ha2 of the two contacting objects, i.e. Ha=

21 HaHa (Kruusing 2008). R is the effective radius for a pair of spheres of radii Ri

and Rj coming into interaction with each other, which is determined by the harmonic

mean, as given in Table 4.1. For particle-to-wall or blade contacts, the same formulae

can be used with the radius of the contacting surface Rj being set to infinity.


4-90

Table 4.1 Formulae for non-contact forces

Force Equation

Particle-particle non-contact force

ijijijnnnnij RYRY nnvF ˆˆ~1~13

22

2/32

ssssnijs

sij ξFF ˆ/,min11 2/3

max,max,

ijjijiji


RRhRRHa nF ˆ)422()22(

)(646 2222

33

iji

ivdwi hRhh

RHa nF ˆ)2(

23 222

3

where: ||/)(ˆ jijiij RRRRn , )( jiji RRRRR ,

nss )~1(2~2max, , ||/ˆsss ξξξ

4.3 Mixing quantification method

A particle scale mixing index (PSMI) developed previously is used here for the

quantification of mixture quality because of its success in correlating effects of size and

density differences of particles despite the changes in the sample sizes (Chandratilleke,

Yu et al. 2011). The index is based on the use of the coordination number to define the

particle fraction pi of one type of particle (referred to as the target type particle) in the

immediate neighborhood of particle i in a binary mixture, and the variance St2of pi is

obtained relative to the instantaneous mean value of pi for the mixture.

2

1

2 )/( ppwwS iTi

N

it

(4.3)

where N is the total number of samples, which is here equal to the total number of

particles, and wi/wT is a weighting factor with wi = Ni/N and

N

iiT ww

1

, where Ni is

the total number of particles in a sample. Next, St2is used in the following formula of

Lacey’s mixing index, M to obtain the instantaneous mixing index for the mixture:

220

220

R

t

SSSSM

(4.4)


4-91

where, ppS 120 and /12 ppSR are the variances of the fully-segregated

and fully-mixed states respectively, with p and ƞ representing the particle number ratio

of the target type particles to the total particle number in the binary mixture and average

particle-scale sample size for the mixture, respectively. Where n is the instantaneous

average particle-scale sample size, which is equal to and rounded to the nearest number

the average sample size ƞ is equal to

N

ii NN

1

, at any given instant. In determining St2,

one has to use a particle-contact condition, which is taken as an inter-particle gap size

of 5% of the particle diameter to be consistent with our previous work (Chandratilleke,

Yu et al. 2011)

4.4 Simulation conditions and procedure

The ribbon mixer used in this simulation is as shown in Figure 4.1. The impeller

has a horizontal shaft and two helical blades starting from each end, but spiralling in

opposite directions. It is placed axially in a horizontal-axis cylindrical vessel, the

dimensions of which and other input values for DEM simulations are shown in Table

4.2. The dimensionless Bond number, Bo is used for quantifying the cohesiveness of the

particles, which is defined as the cohesive force divided by particle weight for particle-

particle or particle-wall contacts. Note that Bo given in the text is only a representative

value of cohesiveness because it has been calculated using a cohesive force for h=10-9 m,

h being the gap between the two interacting surfaces (see Table 4.3). The cohesive force

increases with a reduction in h, but is assumed to remain constant at a value

corresponding to h=10-9 m even if the two interacting surfaces form an overlap.

Cohesive particle mixing systems can be simplified for the purpose of

simulations in two ways: one is scaling down the mixer, so that the particle size

represents the actual particle size and a reasonable amount of particles can be

considered for the simulations. This system is named the micro system here. The

disadvantage of this method is that the time-step of the simulations can be very small

that the simulations take a long time to complete. The other method is to use the actual

mixing vessel size with the particles being considerably larger than the actual powder

particles. This system is named the macro system here. The advantage of this system is

that the time step can be increased and computation time reduced. The cohesiveness of

large particles based on van der Waals force is very large (see Eqs. in Table 4.1), but Bo


4-92

number becomes very small due to their large weight. Therefore, we match Bo number

of the macro system to that of micro system by adjusting Ha values of the macro-system,

where Ha is the Hamaker constant of materials involved.

The simulation conditions for the micro and macro systems are shown in the

Table 4.3. The particle diameter D of the macro system is chosen as 15mm to limit the

number of particles in the system, while D of the micro system was set to 100m to be

consistent with our previous study on cylindrical mixers (Chandratilleke, Yu et al.

2009). The micro-system is the 1/150 scaled-down model of the actual vessel and

particle sizes. The shaft speed is set to 100rpm in the macro-system, and that of the

micro system is calculated matching the Froude numbers (Fr= D2/g) of the two

systems, and is found to be 1220 rpm. Bo numbers used for the micro and macro

systems are listed in Table 4.3 for particle-particle (P-P) and particle-wall (P-W)

contacts. They are calculated as described in Section 2.1. Note that Bo for P-W contacts

is fixed at 0.2 instead of keeping the wall Ha constant, to avoid variation of wall

cohesion when particle cohesion is varied in the simulations. Therefore, the wall Haw

varies when Hap is varied.

The time step of the simulations for the macro-system is set to 2.9x10-6s based

on the time step criterion (Tsuji, Kawaguchi et al. 1993). On the other hand, the time

step for the micro-system is set to 10-7s. Note that the density of particle material for the

macro system has a reduced value of 417 kg/m3, which was selected after several trial

simulations to avoid momentum effects of the large particle collisions on the mixing

behaviour. Granular (hydro-static) pressure (Remy, Khinast et al. 2009) suggests that

material density of particles in the micro-system needs to be increased as much as the

scaling factor to keep the pressures of the systems equal. Maintaining similar pressures

in the two systems may be necessary to have a similarity in the particle motion in the

two systems. Thus, the material density of particles in the micro system was chosen as

8000 kg/m3 after some trial DEM simulations to investigate changes in time-variation of

the mixing index. To investigate the mixing performance of the mixer, particles of equal

numbers are deposited in the side-by-side arrangement in the axial direction Figure

4.1(b). The mixture quality is quantified by the particle scale mixing index M described

above. The particle properties used are listed in Table 4.1.


4-93

Table 4.2 Simulation Input variables and their values

Input variable Value Vessel diameter Vessel length

216 (mm) 970 (mm)

Blade pitch 484 (mm) Blade width 70 (mm) Shaft diameter 31.75 (mm) Rake angle 45°, 135° Blade gap 8.5 (mm) Particle number 18000 Particle dia., d 15 mm Shaft speed, w rpm 100 rpm Young’s mod. E 1108 N/m2 Poisson’s ratio, 0.29 s(P-P) (P-W) 0.3 R (P-P) (P-W) 0.002 Damping Coefficient 0.3 Time step 2.910-6(s) Particle material Thermoset Polyurethane Foam (unreinforced) Shear strength 16 MPa Tensile stress 0.24-103 MPa

Table 4.3 Input values for the macro and micro systems

Macro system Micro system

Hamaker Constant, Hap (J) 5.5E-18 5.5E-17 5.5E-16 3.94e-22 3.94e-21 3.94e-20

Hamaker Constant, Ha(w-p) (J) 8.8E-19 2.8E-18 8.8E-18 2.46e-23 7.79E-23 2.46E-22

Van der wall force (N) 1.7E-03 1.7E-02 1.7E-01 8.21E-10 8.21E-9 8.21E-8

Bo number 2.0E-02 2.0E+01 2.0E 0.02 0.2 2

Particle radius (m) 7.50E-03 5.00E-05

Particle density (kg/m3) 4.17E+02 8.00E+03

shaft angular velocity (rpm) 1.00E+02 1.22E+03

Mixer length (m) 9.70E-01 6.47E-03

Mixer radius (m) 2.16E-01 1.44E-03

h (m) 1.00E-09 1.00E-09


4-94

Table 4.4 Bond number for a single contact at different values of Hamaker constant.

Bond number (P-P)

Bond number (P-W)

Hamaker Constant, Hap (J)

Hamaker Constant Hap-w(J)

20.0 0.2 5.55E-15 2.77E-17 14.0

3.88E-15 2.32E-17

8.0

2.22E-15 1.75E-17 2

5.54E-16 8.76E-18

0.2

5.54E-17 2.77E-18 0.02

5.54E-18 8.76E-19

(a) (b)

Figure 4.1 Impeller configuration and initial particle deposition layout

4.5 Results and Discussion

A comparison of the particle mixing behaviours in micro and macro systems is

carried out here using mixing index, coordination number, velocity and contact forces in

the section 4.1. In section 4.2, cohesive effects are investigated using the macro system.

4.5.1 Comparison of micro and macro systems

It is essential to confirm the validity of using the macro system, where the

particle size is 150 times that of the micro system, for investigating the effect of particle

cohesion. The use of the macro-system has an advantage in that the time steps in DEM

simulations can be increased considerably, thus reducing the computation time. To

obtain the dynamic similarity between the two systems, we have imposed the

geometrical similarity while, at the same time, matching Froude numbers and Bo

numbers of the two systems. Below, we compare mixing index, overall coordination

number, flow pattern, particle velocities and contact forces of the two systems.

pitch

width

45°135°


4-95

4.5.1.1 Mixing Index

Figure 4.2 shows the mixing curves for the micro and macro systems at three

different Bo numbers, 0.02, 0.2 and 2. Figure 4.2(a) shows that the mixing index and

mixing rate of the micro and macro system are quite identical when the Bo=0.02, which

is almost non-cohesive. Note that particle densities of macro and micro systems are 417

and 8000 kg/m3 respectively. When the cohesion of particles is increased, the mixing

rate for macro system becomes slower and the difference between the mixing curves for

the two systems at a given Bo number increases as seen from the figure. However, the

steady-state values are mostly the same. The observation implies that the effect of

particle material density is different depending on the cohesion or Bo number, which

will be discussed later under velocity field.

Below, the case of Bo=0.02 is further investigated using coordination number

variations, particle flow, and velocity and force distributions. There may be differences

between the two systems if one compares the variation of a variable such as particle

velocity or contact forces, but average values for the particle bed should be similar.

0.000

0.200

0.400

0.600

0.800

1.000

0 20 40 60 80 100

Micro_systemMacro_system

Parti

cle

Scal

e M

ixin

g In

dex

Revolutions

0.000

0.200

0.400

0.600

0.800

1.000

0 20 40 60 80 100

Parti

cle

Scal

e M

ixin

g In

dex

Revolutions

Micro-system

Macro-system

0.000

0.200

0.400

0.600

0.800

1.000

0 20 40 60 80 100

Parti

cle

Scal

e M

ixin

g In

dex

Revolutions

Micro-system

Macro-system

(a) Bo=0.02 (b) Bo=0.2 (c) Bo=2

Figure 4.2 Comparison of micro and macro system in terms of mixing curves at

different cohesion


4-96

4.5.1.2 Coordination number

Averaged partial and overall coordination number variations are shown in

Figure 4.3 as a function of time for the case of Bo=0.02. It shows that the coordination

numbers and their variations are similar for both systems. Initially, B-B and W-W

contact numbers are about 4.8 and 5.3 respectively. On the other hand, B-W and W-B

contacts are both about 2. The average of the total number of contacts for a particle is

about 5.3. When blade rotates, W-B and B-W contacts increase, while the B-B and W-

W contact decreases as shown in figure 4.3. All four curves reach steady state in about

30~40 revolutions in the micro-system, but in about 20~30 revolutions in the macro-

system; however, their steady state values are identical. Such identical steady state

values are a result of the two types of particles; B and W are roughly equal.

2.5

3

3.5

4

4.5

5

5.5

6

0 20 40 60 80

CN_TotalCN_BBCN_WWCN_BWCN_WB

Tota

l Coo

rdin

atio

n N

umbe

r

Revolutions

2.5

3

3.5

4

4.5

5

5.5

6

0 10 20 30 40 50 60 70

CN_TotalCN_BBCN_WWCN_BWCN_WB

Tota

l Coo

rdin

atio

n N

umbe

r

Revolutions

(a) Micro-system (b) Macro-system

Figure 4.3 Coordination number comparison of the micro and macro system at Bo

number 0.02

4.5.1.3 Velocity field

The instantaneous velocity fields of the micro and macro systems are

investigated at different revolutions in the cross-section at the middle part of the vessel

when particle bond number is 0.02. Figure 4.4 shows that there are some differences in

the velocity field in the two systems. This is understandable to some degree, the

velocity fields being instantaneous ones. In addition, material properties such as friction

can also have different effects on the particle motion in the two systems when

considering dynamic similarity (Chandratilleke, Yu et al. 2012).


4-97

Figure 4.4 Instantaneous velocity of particles in micro and macro systems

in a vertically-cur vessel segment between z=300 and 350 for Bo=0.02

Figure 4.5shows the velocity fields in the vessel segment of the vessels of the

macro and micro systems in the case of Bo=2. The figure shows that particles in the

macro-system are lifted to the top part of the vessel and the bottom part is devoid of

particles. On the other hand, in the micro-system, particles are mostly at the bottom and

behave like in the lightly-cohesive case (see Fig. 4.5). Due to the matching of Fr

numbers, the shaft speed of the micro-system has increased by about 12 fold.

Therefore, the tangential velocity gradient, which is equal to Rx/R=has changed.

The velocity gradient is representative of the amount of shearing clustered particles

undergo. Therefore, it is likely that the cohesive forces are easily overcome by the shear

forces present at a high shaft speed in the case of the micro-system, and particles would

behave like non-cohesive particles. Thus, micro system will show a higher mixer rate

than the corresponding macro-system as seen from Fig. 4.2. The matching of Fr

numbers has not resulted in similar mixing curves for micro and macro systems in the

case of higher cohesion, except when cohesion is very low. It will be confirmed later

that this discrepancy is not due to the density difference between the particles of the two

systems. It is of interest to test the two systems at the same shaft speed for the cases of

higher cohesion.

82 Rev19 Rev 39 Rev

Frame 001 20 May 2013 Frame 001 20 May 2013 Frame 001 20 May 2013

Macro-system

Micro-system

Frame 001 13 Oct 2013 Frame 001 13 Oct 2013 Frame 001 13 Oct 2013


4-98

Figure 4.5 Instantaneous velocity of particles in micro and macro systems

in a vessel segment between z=300 and 350 for Bo=2

Figure 4.6 Instantaneous velocities of particles in micro and macro systems

in the longitudinal segment of the vessel between x=-20 and 20 at Bo=0.02.

Macro-system Micro-system

Frame 001 13 Oct 2013

Frame 001 13 Oct 2013

Frame 001 13 Oct 2013

Frame 001 13 Oct 2013

82 Rev

19 Rev

39 Rev

Frame 001 13 Oct 2013

Frame 001 13 Oct 2013

Macro-system

Frame 001 10 Jun 2013 Frame 001 10 Jun 2013 Frame 001 10 Jun 2013

82 Rev 19 Rev 39 Rev

Micro-system

Frame 001 06 Nov 2013 Frame 001 06 Nov 2013

Frame 001 06 Nov 2013


4-99

It is also of interest to test the two systems at the same Fr numbers but with reduced

coefficients of friction (i.e., p-p and p-w) in the micro-system. It is known that

similarity can fail because of friction (Chandratilleke, Yu et al. 2012). When the friction

coefficients are identical for the two systems, micro-system sees this friction as large

relative to the representative dimension, which may have also caused a large shear rate

in the micro-system, and thus a faster mixing rate.

Figure 4.6 shows the velocity in longitudinal section at different revolutions for

Bo=0.02. The local recirculation flow can be seen in the lower part of the vessel both in

micro and macro system at different revolutions. Once again, there exists some

differences in the instantaneous velocity fields.

4.5.1.4 Quantification of velocity and force

The radial, tangential and axial velocity components are investigated by means

of probability density function in micro and macro systems as shown in Fig 4.7. Figure

4.7 (a) shows that the probability density distributions of scaled radial velocity Vrad of

micro and macro system are quite similar; the average value of Vrad for both systems is

zero as expected. However, the tangential velocities of the particles in the micro system

are slightly higher on the average than those in the macro-system as shown in the Figure

4.7 (b); this can be confirmed also from Fig. 4.7(d), which shows the distribution

averages. The reason probably could be related to the energy dissipation on particle

collisions, which increases non-linearly with respect to particle radius and overlap. Thus,

in the macro system, the fraction of energy lost from the energy of a particle is larger

resulting in a loss of particle velocity. Figure 4.7(c) shows that the distributions of axial

particle velocities in both systems are centered at a value of zero as expected. As shown

in Fig. 4.7(d), the distribution average values are slightly different being instantaneous

values; the time average value should be close to zero.


4-100

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-1.375 0 1.375 2.75

Vr_MacroVr_Micro

Prob

abili

ty D

ensi

ty

Scaled Radial Velocity (-)

0

0.1

0.2

0.3

0.4

0.5

-1.875 0 1.875 3.75

Vt_MacroVt_Micro

Prob

abili

ty D

ensi

ty

Scaled Tangential Velocity, (-)

(a) (b)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-2 -1 0 1 2 3

Vz_MacroVz_Micro

Prob

abili

ty D

ensi

ty

Scaled Axial Velocity, (-)

-0.2

0.0

0.2

0.4

0.6

0.8

Macro Micro

Vrad

Vtan

Vaxial

Scal

ed A

vera

ge v

eloc

ity, (

-)

(c) (d)

Figure 4.7 Probability density distributions of instantaneous velocity of particles in

micro and macro systems at 39th rev.

To be quantitative, instantaneous cumulative probability distributions of the contact

forces are obtained here as shown in Figure 4.8. Figure 4.8 (a) shows the curve of

normal force in micro system slightly shifted to the right; it implies that the normal

force of particles in the micro system is slightly higher than that in the macro system,

which can be confirmed form Figure 4.8(c). However, the average shear force in the

two systems is similar as shown Figure 4.8 (c). The mean forces diagram shows that

mean tangential forces are same in the two systems, while the mean normal force of the

micro system slightly higher than that in the macro system.

.


4-101

0

0.2

0.4

0.6

0.8

1

10-4

10-3

10-2

10-1

100

101

Fn_MicroFn_Macro

Cum

ulat

ive

Prob

abili

ty

Scaled Normal Force, (-)

0

0.2

0.4

0.6

0.8

1

10-5

10-4

10-3

10-2

10-1

100

101

Ft_MacroFt_Micro

Cum

ulat

ive

Prob

abili

ty

Scaled Shear Force, (-)

(a) (b)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Macro Micro

NormalTangential

Ave

rage

con

tact

forc

e, (-

)

(c)

Figure 4.8(a) and (b): Cumulative probability distributions of scaled contact force

components of particles in micro and macro system at 39th rev; and (c) the distribution

averages

Because of the increased discrepancy between the two systems for mixing of cohesive particles,

the further research is needed for improving the scaling method. Dimensionless numbers such

as Froude number Fr, Reynolds number Re, Schmidt number (H.Willig 2001) and particularly

cohesion number (W.Alexander 2005) should be matched for the dynamic similarity while

maintaining the geometric similarity of the two systems.


4-102

4.5.2 Effects of cohesion on homogeneity of the particles mixing (Macro-system)

Above comparisons of the two systems showed that differences appear in the

mixing curves at larger cohesion or Bo numbers, with faster mixing in the micro-system

compared to the macro-system. The faster mixing may have resulted from the breakage

of cohesive bonding at the high shaft speed of the micro-system.

Here, further considerations are made of the diffusion and convection effects. It

is generally known that mixing in the axial direction (convection or diffusion) is slower

compared to that in the circumferential direction in horizontal-axis bladed

mixers(Laurent, Bridgwater et al. 2002; Muzzio, Llusa et al. 2008), and therefore, the

axial diffusion could be one factor governing the mixture quality differences between

the two systems. Fourier number for mass transfer by diffusion can be expressed as:

Fom= t / L2, where is mass diffusivity, t is a characteristic time, and L a

length scale. We can rewrite the above as follows: Fom= /(2 L2), where is shaft

speed (rad/s) and L is the mixer length over which diffusion occurs. If Fourier numbers

of the two systems are set the same for similarity, the mass diffusivity of the micro-

system becomes:

mi=ma mami(Lma/ Lmi)-2 where suffixes ‘ma’ and ‘mi’ represent the macro

and micro systems respectively. After substitution of appropriate values (see Sec. 4.3) in

the above, mi= 3.64x10-6ma. Thus, the diffusion in the micro-system is very small

compared to that in the macro-system.

To explain the initial rapid mixing-rate of the micro-system in the previous

section, the differences in the particle convection in the axial direction should be

considered for the two systems. The axial convection should be affected by many

factors for a given blade geometry: for example, by particle material density, cohesion,

particle size and shaft speed. Thus, studying cohesive effects is complex even if we use

a micro-system, where the mixer size is reduced to make particle size small. In such a

system, the particle size is still large compared to the mixer size just as in the case of a

macro-system, thus, affecting the velocity field and as a result, mixing. Below, material

density and its effect on axial convection and diffusion are investigated. Next, as a first

step, cohesive effects are investigated using the macro-system to speed up the

computations.


4-103

4.5.2.1 Effect of material density

To overcome the longer computation time associated with a large number of

particles, we used large particles with a reduced density to make the particles lighter as

in the case of small particles. However, it is necessary to evaluate the effect of particle

density on the mixing behaviour. Here, we use two particle densities, 2500 and 417

kg/m3, so that the effect of particle density can be clarified. Also two initial particle

layouts are tested: namely, side-by-side (SBS) layouts in axial and radial directions. In

the SBS-radial layout, the major mixing mechanism is radial mixing. In the SBS-axial

layout, axial mixing is the major mechanism.

Figure 4.9 shows the comparisons of mixing curves for the two densities under

the two initial layouts mentioned above. Figure 4.9 (a) shows that the density difference

has very little effect on the mixing behaviour in the macro-system for SBS-axial layout.

However, a close inspection shows that=2500 kg/m3 has a slightly slower initial

mixing rate than =417 kg/m3, but later it shows a better mixing state at the steady-

state. It is generally known that mixing is initially taking place by convection and later

by diffusion (Harnby 2000) (Dodds 1980). Therefore, =2500 kg/m3` has the better

ability to diffuse, but convection in axial direction is rather poor; this is clear for the

reason that for mixing to happen in the case of SBS axially, particles have to move in

the axial direction. Figure b shows that, for the SBS-radial layout,=2500 kg/m3`

still has a slower mixing rate initially, but later a better mixing state than=417 kg/m3`

as in the case of the other layout. The two figures also show that SBS-radial layout

produces faster mixing than SBS-axial layout, where radial mixing is the main

mechanism. Thus, one can say that the axial mixing is slower and a governing factor in

mixing for a ribbon mixer. The figure shows that both the axial and radial mixing rates

can be increased by using =417 kg/m3 or a lower density. It can be also argued that

the reduction of particle density reduces the inertia of particles to stay in orbits around

the shaft, thus improving axial motion. Thus, reducing density will not reduce the

mixing rate in the micro-system either, which was an issue for lager Bo numbers; a

reduction in shaft speed seems necessary in that case to reduce the mixing rate, which

implies that Fr number cannot be matched for cohesive particles. Furthermore, although

not shown here, =2500 kg/m3` failed to produce effect of particle cohesion in trial

simulations in macro-system implying that the momentum effects is too high to sustain


4-104

cohesive effects. Therefore, =417 kg/m3 is used here to investigate the effect of

particle cohesion as described below.

0.0

0.20

0.40

0.60

0.80

1.0

0 10 20 30 40 50 60 70 80

=2500 kg/m3

=417 kg/m3

Mix

ing

Inde

x

Revolutions

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80

=2500 kg/m3

=417 kg/m3

Mix

ing

Inde

xRevolutions

(a) (b)

Figure 4.9 Effect of particle density in the case of Bo=0.2 for two side-by-side (SBS)

initial arrangements: (a) Axially side-by-side arrangement; (b) Radially side-by-side

arrangement

4.5.2.2 Particle scale mixing index and total coordination number The effect of cohesion on mixture uniformity is investigated using particle

scale mixing index, coordination number, velocity field, porosity and stresses averaged

over cells as well as time. The results are shown in Figure 4.10, which shows that the

mixing rate and homogeneity of the mixture decrease with an increase in Bo number,

the particle mixing deteriorating considerably when Bond Number is 14 or 20. The

average steady-state particle-scale mixing index decreases sharply for Bo numbers

greater than 8 as seen from Figure 4.10(b).

The particle mixing is better at moderate and low cohesion of the particles

since the cohesive bonds between particles are easy to break with a decreased of

cohesion of the mixture. Figure 4.10(a) show the particles with Bo=0.02 reach the

steady state of mixing at revolution 20. However the increase of cohesion of the

particles slows down the time to reach the steady state. For example the mixtures with

Bo=0.2 and 2 reach steady state at revolution 40 and 70 respectively, which are two-

three times that mixture with Bo=0.02. The time to reach steady state mixing value of


4-105

the case with Bo=8 is slightly lower than that of the cases with Bo=0.02, 0.2 and 2. The

mixture with Bo=8 reaches steady state mixing index of 0.93 at revolution 80, while

other cases with Bo=0.02, 0.2 and 2 reach mixing indexes of 0.95 and 0.96 at the

revolutions 30 and 40 respectively. The steady state mixing index of the mixture with

Bo=14 and 20 are considerably low. They reach their steady state values of 0.11 and 0.2

at revolution 5 and 10 as shown Figure 4.10(a).

Figure 4.10(b) shows the steady state mixing index M as a function of Bo

number. M decreases slightly with an increase of Bo number up to 8, and exceeding

which M decreases sharply. The overall coordination number is shown in Figure 4.11(a)

as a function of shaft revolutions, and the steady-state values in Figure 4.11(b) as a

function of Bo number. The results show that overall coordination number decreases

steadily with the mixing time (or shaft revolutions) for the mixtures of Bo=0.02, 2 and 8.

However, at high cohesion (Bo=20), it decreases only slightly and reaches steady state

at a high value as shown in Figure 4.11(a). It means that it is difficult to break the bond

between particles at a high cohesion. Therefore, the particle mixing at high cohesion is

slower as shown in Figure 4.10. Figure 4.11(b) shows that the average coordination

number reduces at first with respect to Bo number variation and then steadily increases

with Bond number, which is a sign of deterioration of mixing.

0.000

0.200

0.400

0.600

0.800

1.000

0 20 40 60 80 100

Parti

cle

Scal

e M

ixin

g In

dex

Shaft revolutions

Bo =8

Bo =20 Bo =14

Bo =0.02

Bo =2

Bo =0.2

0.00

0.20

0.40

0.60

0.80

1.00

0 5 10 15 20

Ave

rage

Ste

ady

Stat

e M

ixin

g In

dex,

M

Bond Number (a) (b)

Figure 4.10 Particle Scale Mixing Index: (a) Particle scale mixing index as a

function of shaft revolutions; and (b) Average steady state mixing index at

different Bo numbers.


4-106

2.4

2.8

3.2

3.6

4

4.4

4.8

0 20 40 60 80 100

Ove

rall

Coo

rdin

atio

n N

umbe

r

Revolutions

Bo=2

Bo=20

Bo=8 Bo=0.02

2.00

2.50

3.00

3.50

4.00

4.50

0 5 10 15 20Ave

rage

Coo

rdin

atio

n N

umbe

r at S

tead

y-st

ate

Bo Number

(a) (b)

Figure 4.11 Overall coordination number at different Bo numbers: (a) overall

coordination number as a function of shaft revolutions; and (b) overall

coordination number at steady-state as a function of Bo number.

4.5.2.3 Cohesion effect on the velocity field and porosity of the mixture

To further understand the cohesive particle flow during mixing stage, the

effects of cohesion of the particles on the velocity and porosity are examined here. The

particle velocity fields are obtained for the section between Z= 350 and 400mm, at 10,

39 and 82 revolutions respectively, and are shown in Fig. 4.12. The figure shows that

the circumferential flow of the particles increases with an increase of bond number, but

the blue and red coloured particles are clearly separated at higher Bo numbers,

indicating the lack of mixing. There are recirculation flows for Bo=0.02 at 19 and 39

rev, which are reduced when the Bo number increases. Figure 4.13 shows the velocity

field in the longitudinal section obtained at different cohesion at 19, 39 and 82 rev. It

shows that particle-axial flows enable particles to move from both ends to the middle of

the vessel, where they mix. It is clearly seen that the axial flow is deteriorated, and

circumferential flow enhanced with increase of cohesion. Notably particles cannot mix

when bond number 20. The velocity fields in the longitudinal section show that the

more particles are involved in the convective motion with less cohesion between

particles, since they can easily break the cohesive bonding between particles, and be

involved in motions in the horizontal, radial and tangential directions of the mixer as

shown in Figure 4.13.


4-107

Figure 4.12 Instantaneous velocities of particles in vessel cross sections XY; Z= 350-400 mm, at three time instances with increasing of bond number: red and blue colour represents vectors of two type particles.

Figure 4.13 Instantaneous velocity fields in the longitudinal section at different Bond numbers: red and blue colour represents vectors of two types of particles

Bo=0.02 Bo=20Bo=2

82 Rev

19 Rev

39 Rev

Bo=8

Frame 001 20 May 2013

Frame 001 20 May 2013

Frame 001 20 May 2013

Frame 001 20 May 2013

Frame 001 20 May 2013

Frame 001 20 May 2013

Frame 001 10 Jun 2013

Frame 001 10 Jun 2013

Frame 001 10 Jun 2013

Frame 001 20 May 2013 Frame 001 20 May 2013 Frame 001 20 May 2013

Bo=20

Bo=8

Bo=2

Frame 001 10 Jun 2013

Frame 001 10 Jun 2013 Frame 001 10 Jun 2013

Frame 001 10 Jun 2013 Frame 001 10 Jun 2013

Bo=0.02

Frame 001 13 Oct 2013

Frame 001 13 Oct 2013

Frame 001 13 Oct 2013 Frame 001 13 Oct 2013

Frame 001 13 Oct 2013

Frame 001 13 Oct 2013

Frame 001 13 Oct 2013

82 Rev 19 rev 39 rev


4-108

In order to obtain the time-averaged longitudinal velocity and porosity

distribution in the mixer, the mixer space is divided into cubic cells of size of

40x40x50mm3, and the particle velocities and porosity of each cell are averaged over a

time interval of 5s. Figure 4.14shows the average velocity and porosity in a

longitudinal section between x=-40 and 40 mm at different Bo numbers. The results

show that the axial flow deteriorated with an increase of particle cohesion as seen from

the reduction of arrow lengths. Figure 4.14also shows the average porosities of the cells

by colour. Porosity increased with an increase of Bond number as seen from the

increase of orange colour areas. At high cohesion, particles cannot become free to be

engrained in mixing due to strong cohesion between particles. The average velocities of

the particles in cells decreased with increase of Bo number. Quantification of velocities

in the tangential, radial and horizontal directions is needed to further understand the

particle behaviour.

Figure 4.14Average velocity and porosity of particles at t=25-30s, in a longitudinal

segment between x=-20 to 20 and y= -255 to 255.

4.5.2.4 Cohesion effects on the radial, tangential and horizontal velocity

To obtain the average velocity components of particles for each cell, velocities

of particles in each cell are averaged over a 90s time interval. The probability density

distributions of time-averaged velocity components are shown in Figure 14 (a). It shows

B0=0.02

Frame 001 13 Jun 2013

B0=2

Frame 001 13 Jun 2013

Frame 001 13 Jun 2013

B0=8

Frame 001 13 Jun 2013

B0=20

Z

Y

0 200 400 600 800

-200

0

200

400

600

Porosity

0.95

0.9

0.85

0.8

0.75

0.7

0.65

0.6

0.55

0.5

0.45

Frame 001 13 Jun 2013

V


4-109

that the number of cells with larger tangential velocity has increased somewhat with an

increase in the Bo number, which does not lead to improved mixing behavior . The

stronger tangential motion is created by the impeller with increasing Bo number. Figure

4.15 shows that the time-averaged tangential velocity increases with cohesion or Bo

number. On the other hand, the majority of particles receive radial velocities in the cases

of Bo=0.02 and 2 as seen from the wider spread of the distributions. However, the

distributions of radial velocity of particles are sharper around an average value of zero

for Bo=8 and 20 compared to the cases of Bo=0.02 and 2 as in Figure 4.15 (b), implying

that most particles are not moving radially (both ways) with stronger particle cohesion.

Figure 4.15(c) shows the axial velocity distributions. Similar to radial velocity

component, average horizontal (axial) velocity vary significantly with the cohesion,

with most particle showing zero axial velocity with increasing Bo number. Figure 4.16

shows that the average values of both the radial and axial velocity components are close

to zero.

0

0.005

0.01

0.015

0.02

0.025

0.03

-80 -40 0 40 80

Bo=0.02Bo=2Bo=8Bo=20

Prob

abili

ty D

ensi

ty

Average Tangential Velocity (m/s)

0

2

4

6

8

10

-0.9 -0.45 0 0.45 0.9


Prob

abili

ty D

ensi

ty

Average Radial Velocity (m/s)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-5 -2.5 0 2.5 5


Prob

abili

ty D

ensi

ty

Average Axial Velocity (m/s) (a) (b) (c)

Figure 4.15 Probability density distributions of velocity components: (a) tangential, (b) radial and (c) axial velocity

-2.0

0.0

2.0

4.0

6.0

8.0

10.0

0 5 10 15 20 25

VrVtVz

Mea

n R

adia

l, Ta

ngen

tial V

eloc

ity a

nd V

z

Bo Number

Figure 4.16 Time averaged mean velocity components as a function of Bo number: radial velocity Vr, tangential velocity Vt and axial velocity Vz.


4-110

4.5.2.5 The stress analysis of particles with different cohesion

The mixer space is divided into cubic cells of the size 40x40x50mm3 as before,

and the contact stresses of particles in the cells are averaged over a time interval of 10s

at steady state. The equations used for the evaluation of stresses are shown below.

jiji

C

ij tTnRRV

nNnRV

)(121

12 1R

(4.5)

where the summations are over the C contacts in the volume V, R1 and R2 are the radii

of the two spheres in contact, N and T are the magnitudes of the normal and tangential

contact forces for the contact orientation defined by the unit vector normal to the contact

plane ni and ti defines the unit vector parallel to the contact plane. The time and cell

averaged contact stresses of particles are analysed using probability distributions to

interpret the cohesion effects on the stresses in particles forces. Figure 4.17 shows the

stress components in three perpendicular planes, r, and z, in r, z and directions as

identified by rr, r, rz, z and zz, where the first suffix denotes the plane and the

second the direction. Particles received larger radial stresses rr with the increase of Bo

number in the Bo number range of 0.02 to 8, with particles in some cells receiving

radial stresses over 140Pa. However, with the increase of Bo number to 20, all stress

components have become reduced to less than 60 Pa. For this case mixing also has

deteriorated. Generally, the radial stresses rr is the largest, while the axial stress is the

next larger one, both of which being needed to move the particles in tangential and axial

directions. Shear stresses, r, rz, z are responsible for breaking the cohesive bonding

between particles. The particles with lower cohesion have larger radial and axial

stresses, which lead to stronger axial and radial velocities, lower coordination number

contributing to increased particle convective and angular motion and mixing in the

ribbon mixer.


4-111

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-20 0 20 40 60 80 100

rr

r

rz

z

zz

Prob

abili

ty D

ensi

ty

Stress (Pa)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 20 40 60 80 100 120 140

rr

r

rz

z

zz

Prob

abili

ty D

ensi

ty

Stress (Pa) (a) Bo=0.02 (b) Bo= 2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-20 0 20 40 60 80 100 120 140

rr

r

rz

z

zz

Prob

abili

ty D

ensi

ty

Stress (Pa)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-20 0 20 40 60 80 100 120 140

rr

r

rz

z

zzPr

obab

ility

Den

sity

Stress (Pa)

(c) Bo=8 (d) Bo=20

Figure 4.17 Probability density distributions of average normal stress in the vessel at

steady-state for different Bo numbers.

4.6 Conclusions

The effect of cohesion on particle mixing behaviour was investigated using DEM. Two

systems, namely micro and macro systems were tested for this purpose by creating

geometrically and dynamically similar conditions. To establish dynamic similarity

between the two systems, Froude and Bond numbers were each made identical in the

two systems. In addition, in the macro system, particle density was made smaller to

make the particles lighter, while in the micro-system, particles were made heavier by


4-112

choosing a larger density to account for the differences in momentum effects of the two

systems. The following conclusions are made regarding the two systems.

Under geometric and dynamic similar conditions, the macro and micro systems

displayed similarity in the mixing behaviour, velocity field and coordination

number variation with time when the particles in the two systems are mostly

non-cohesive. With the increase of cohesion, the mixing behaviour of macro-

system becomes increasingly slower compared to the micro system although Bo

numbers are identical. The reason for this discrepancy was considered to be due

to selecting the shaft speed based on Fr number matching, which resulted in a

high shaft speed. Identifying the fact that the shaft speed is the shear rate, led to

the conclusion that the cohesive bonds is easily broken at high shaft speeds of

the micro-system, which may have caused the faster mixing rate compared to the

macro system at higher cohesion. Thus it is not possible to find fixed operational

conditions for the two systems, which will make the two systems dynamically

similar in a wide range of cohesion. Although more work is necessary to

establish the connection between the two systems, the macro-system has the

advantage that computations can be done faster. More work is necessary to

establish the relationship between axial mixing and several other variables such

as particle size, density, cohesion and shaft speed.

Using the macro-system, the effect of particle cohesion on mixing behaviour was

investigated. The followings are the main findings of the effects of cohesion.

At a fixed Bo number, the radial side-by-side particle layout has a faster mixing

rate than the axial side-by-side particle layout. Use of heavier particles led to a

delay in the mixing rate, but better mixture quality at the steady-state. It can be

also deduced from the results that heaver particles are conveyed axially at a

slower rate, which leads to slower mixing.

Using lighter large particles, it was possible to obtain cohesive effects in the

macro-system. The mixing rate and uniformity of mixing are deteriorated with

an increase of cohesion. Overall coordination number increased with an increase

in cohesion. That is,particles showed higher tangential velocity with an increase

in cohesion. Conversely, horizontal or axial velocity increased when Bo number


4-113

is reduced. Radial stress is larger than the axial stress and increased with the

cohesion for moderately cohesive mixtures. However, with the increase of Bo

number to 20, all stress components have become reduced to less than 60 Pa.

The study demonstrated that cohesive effects of particles on mixing in a ribbon

mixer can be investigated using lighter large particles in a large-scale system.

Further work is needed to establish the particle size effect on convective and

diffusive mixing in the mixer.

Nomenclature D Vessel diameter, (m) d Particle diameter, (mm) E Young’s modulus, N/m2 Fc,ij Contact force vector between i and j, N Fd,ij Damping force vector between i and j, N Fr Froude number, (-) Fv, ij Cohesive force between particles i and j

zf Average vertical force on one type of particles (N) t

zf Time average of the instantaneous vertical force zf in time t(N) g Acceleration due to gravity, (m/s2) h Gap between two interacting particles, (m) Ha Hamaker constant, J Hap Hamaker constant of particle material, J Haw Hamaker constant of wall material, J Ii Moment of inertia of particle i, (kg m2) ki Number of particles in contact with particle i M Particle-scale mixing index defined in Eq.(3) , (-) Mij Vector of rolling friction torque on particle i, (Dury, Ristow et al.) Mp Predicted mixing index at steady-state, (-) n Actual particle number fraction of the two types of particles, (-) Ni Number of particles in the immediate neighbourhood of particle i Nl

Number of large particles Np Total number of particles of one type in the mixture Ns

Number of small particles Nt The number of sample points of instantaneous average values in time t p Number ratio of the target type particles to all the particles, (-) pi Particle fraction of a target type particle in the neighborhood of particle i, (-)

tp Average value of pi at time t for the entire mixture, (-) R Equivalent radius, m S0 Standard deviation of pi at fully-segregated state, (-) SR Standard deviation of fully-mixed state for uniform-sized particles of particle fraction of p, (-)


4-114

St Standard deviation of pi with respect to tp at time t, (-) Tij Vector of rolling friction torque on particle i, N m Vb Blade speed, (m/s) Vi Velocity of particle i, m/s Z Height from vessel base (mm)

Greek letters

α Volume fraction

β Size ratio ls dd / γ Density ratio ls / Average particle-scale sample size for the mixture θ Radial section measured horizontally from mid-plane of the blade

ik Number of particles in contact with particle i Mass diffusivity Density of particles of a uniform system, ( 3kgm )

l Density of large particle, ( 3kgm )

s Density of small particle, ( 3kgm ) Shaft rotational speed, (rad/s) i Angular velocity vector of particle i, (rad/s)

CHAPTER 5 Mixing Performance of Ribbon Mixers: Effect of Operational Parameters

5-115

Chapter 5

Mixing Performance of Ribbon Mixers:

Effect of Operational Parameters


5-116

5.1 Introduction



a wide variety of high quality products is produced. The selection of a mixer for a

mixing operation depends on the product uniformity required and many other factors

(Poux et al., 1991). Ribbon mixers are considered to be suitable for mixing of dry

powders as well as free-flowing granular material (Poux et al., 1991). It is also reported

that a ribbon mixer can produce an improved homogeneity in powder mixing because of

large shear stresses in the mixer as well as that it can handle mixing of different size

particles (Muzzio et al., 2008). Therefore, a ribbon mixer with a horizontal shaft will be

the focus of study here.

The impeller speed of the mixer is a parameter that can affect the performance of

the mixer, with regard to mixture quality and stresses on particles. The mixing quality

decreased with an increased blade speed in continues ribbon mixer (Sanoh et al., 1974).

The power consumption increased with an increase of blade speed in ribbon mixers

(Masiuk, 1987). The energy can be saved up to 60% with a constant impeller rotational

speed in a ribbon mixer (Dieulot et al., 2002). The blending endpoint was never reached

at high impeller rotational speeds (Côté and Abatzoglou, 2006). It is reported that a

high impeller speed can have a negative effect on the mixture uniformity. Although a

low speed can reduce shear stresses on particles, it may result in a poor homogeneity for

cohesive mixtures (Muzzio et al., 2008).

The fill level of the blender is the next most important operational parameter. It is

reported that for the entire batch, the mixing never reached ‘end-point’ when the blender

is filled to the shaft filling level. The mixing reached the end-point of the overall end-

batch with the inner shafts fill level at a blade speed of 20rpm (Côté and Abatzoglou

2006). The power consumption per rotational speed increases with an expansion of

loading ratio of the dry sand and water to capacity of the ribbon mixer (Masiuk, 1987).

Muzzio et al. experimentally investigated the mixing performance of a ribbon

mixer(Muzzio et al., 2008) with regard to effect of loading (layering and off-center spot

loading method), and found that the layering method provides faster mixing and better

homogeneity. Fill level had a significant effect in the case of 3-spokes 2-bladed ribbon


5-117

impeller, with a lower fill level resulting in better homogeneity. However, the fill level

did not have a significant effect in the case of 5-spokes 2-bladed ribbon impeller.

In terms of the effect of impeller speed and fill level of the other mixers, research

shows that the axial diffusion coefficient increased with an increase of blade speed,

however it decreases with an increase of the fill level (Martins et al., 2013). Recent

studies of the effects of fill level and blade speed in a mixer with a single or multiple

blades provide some relevance to the present study (Laurent and Bridgwater, 2002;

Laurent and Cleary, 2012; Wachs et al., 2012). Deteriorated mixing states were obtained

with high blade speeds for mixing of free-flowing particles in a cylindrical mixer

(Chandratilleke et al., 2010). However, the speed effect on the homogeneity and flow

pattern of the solid have not been fully clarified from microdynamic aspects in these

studies.

Research on a complicated mixer such as the ribbon mixer are few, and the

effects of the impeller speed, fill level and blade number on non-cohesive and cohesion

particle mixtures on the mixer performance have not been clearly established. The

objective of this study is thus, to investigate the effect of the impeller speed and fill

level on the mixing behaviours of cohesive as well as non--cohesive particle mixtures

for two different ribbon impellers, by using simulations based on the discrete-element-

method (Cundall and Strack, 1979).

This chapter is organized as follows. Section 5.2 discusses the simulation

method and conditions in section 5.3, which is followed by the results and discussions

section, where in Section 5.4, effect of impeller speed is investigated for a two-blade

impeller for both non-cohesive and cohesive particle mixtures. The blade motion and

mixing dynamics introduced in section 5.4.1. In Section 5.4.2, the effect of blade speed

is discussed focusing on both non-cohesive and cohesive particles using a two-bladed

impeller. Next, the effect of filling level is investigated by comparing the performance

of the two-bladed impeller with that of a four-bladed impeller at different fill levels and

different cohesive particles at the fixed shaft speed in section 5.4.3 . In section 5.4.4 the

mixing performance compared both in cylindrical mixer ad ribbon mixer. Finally,

summarizes the conclusions in section 5.5 of the chapter.


5-118

5.2 Methods to be Evaluated

The DEM model used here is essentially the same as that previously developed

and validated by Zhou et al. (Zhou et al., 2004; Zhou et al., 1999).

The model uses two momentum conservation equations to describe the translational and

rotational motion of particle i in a system at time t subjected to the gravity g and

interactions with the neighboring particles, blade and walls:

ii N

jijv

k

jijdijci

ii m

dtdm

1,

1,, FFFgV

(5.1)

and

ik

jijij

ii d

dI1t

MT (5.2)

where mi, Ii, Viand i are the mass, moment of inertia, translational and rotational

velocities of the particle respectively; k is the number of particles that are in contact

with particle i, Fc represents the elastic contact force which is the summation of the

normal and tangential forces.Fd represents the damping force, which is the summation

of the normal and tangential damping force respectively at the contact point with

particle j; and T and Mare the torque and rolling friction torque on particle i due to

particle j. Expressions for the forces and torque in Eqs. (5.1) and (5.2) are given in

Table 5.1.

Table 5.1 Formulae for contact and non-contact forces and torques

Force Equation

Particle-particle

forces


22

2/32

ssssnijs


max,max,

ijjijiji


RRhRRA nF ˆ)422()22(

)(646 2222

33

iji

ivdwi hRhh

RA nF ˆ)2(

23 222

3




5-119

5.3 Simulation Conditions

The impeller geometries of the ribbon mixers used are shown in Figure 5.1. The

impeller in Figure 5.1(a) has two helical blades starting from each end, but spiraling in

opposite directions, and this mixer is called a two-bladed mixer here. The impeller in

Figure 5.1(b) has two more inner blades, and this mixer is called a four-bladed mixer.

The shaft of each impeller is placed horizontally along the axis of a horizontal

cylindrical vessel, the dimensions of which and other input values for DEM simulations

are shown in Table 5.2. Simulations are conducted for the cases of cohesive and non-

cohesive particles for investigating the effect of impeller speeds. Bond numbers of the

particles used in the study of fill level effect are shown in Table.3. Bond number is

defined as the ratio of van der Waals force to particle weight. The mixture quality or

mixing index is quantified by a particle scale mixing index based on coordination

number(Chandratilleke et al., 2012), and also by a macroscopic index, both indexes

being based on Lacey index (Chandratilleke et al., 2010). In both these methods, all the

particles of the mixtures are considered, and the effect due to the sample size not being

constant is also considered (Chandratilleke et al., 2010). To analyze the mixing

performance of the mixers, particles of equal numbers are deposited in the side-by-side

arrangement, and the impeller is rotated at a constant speed.

Table 5.2. Simulation Input variables and their values

Input variable Value

Vessel diameter Vessel length Blade pitch Blade width Shaft diameter Rake angle Blade gap Base Particle number Particle dia, d Shaft speed,ω Fill level Young’s modulus, E Poisson’s ratio , ms(P-P) (P-W) mR(P-P) (P-W)

216 (mm) 970 (mm) 484 (mm) 70 (mm) 31.75 (mm) 45°, 135° 8.5 (mm) 18,000 15 mm 20, 50, 100, 200 r pm 16%, 26%, 46%,56% 1108 N/m2 0.29 0.3 0.002


5-120

(a) (b)

Figure 5.1. Geometry of ribbon impellers: (a), 2-bladed impeller, and (b), 4-bladed impeller.

Table 5.3.Bond number for a single contact at different values of Hamaker constant.

Bond number (P-P)

Bond number (P-W)

Hamaker Constant, Hap

(J)

Hamaker Constant Hap-w (J)

2 5.54E-16 8.76E-18 0.2 5.54E-17 2.77E-18 0.02 5.54E-18 8.76E-19

45°

S

D

width


5-121


Below the effects of impeller speed and fill level on particle mixing behaviour are

investigated. The effects are evaluated in terms of the mixing performance using a

mixing index, changes in the flow pattern and contact forces. Finally, the effect of

impeller configuration is investigated by comparing the performance of the two-bladed

impeller with that of a four-bladed impeller at different fill levels and a fixed shaft

speed.

5.4.1 Blade Motion and Mixing Dynamics

Before investigating the effects of shaft speed and fill level, we first examine the

blade and particle motions for one typical particle mixture of the study in the case of the

two blade impeller. Figure 5.2 shows the blade motion and mixing behavior of uniform

particles, deposited initially in the side-by-side arrangement in the axial direction, in

about one revolution of the shaft. If we follow the snapshots in the order of time starting

from t=1.7s, we can see that when one blade (say blade 1) advances towards the vessel

centre from one end of the vessel, the blade from the other end (say blade 2), having

advanced ahead of blade 1 disappears at the vessel centre, the two blades advancing and

disappearing alternately. The reason for the blades to be visible is that there is a void

space behind each of the advancing blades as observed previously in the case of flat

blades (Zhou, 2004). When a blade advances, particles in contact with it are conveyed

towards the vessel centre as can be seen from the interface of the two types of particles

becoming aligned with the shape of the blade. At the same time, the particles in contact

with the blades are rotated in the direction of shaft rotation (clockwise), the result of

which being the formation of two waves above the shaft, one from each blade as seen in

the snapshot for example at t=1.7s. It leads to mixing of particles at the interface of two

types of particles. Further, axial transportation of particles results in flow of particles

over the blade, as seen for example at t=2.3s and 1.4s when red particles flow over the

blade that is in contact with the blue particles. Thus, particle mixing will eventually

occur in all three directions: circumferential, axial and radial. This is the general particle

mixing behavior of this mixer based on visual observations of the animations of particle

motion, and flow patterns which are further clarified by investigating the velocity fields

below. Note also that although the blades have opposite spiraling directions, particles in

either half of the vessel are rotated in the same direction, which is the direction of the

shaft rotation.


5-122

t=1.7s t=2.1s

t=1.8s t=2.0s

t=1.9s t=2.3s

t=2.0s t=2.4s

Figure 5.2 Snapshots depicting blade motion, axial transport and circumferential motion

of particles in two-bladed mixer at 25% fill level (Bo=0.2).

5.4.2 Effect of impeller speed

5.4.2.1 Mixing performance

Figures 5.3(a) and (b) show instantaneous particle-scale mixing index M, as a function

of the number of revolutions for cohesive (Bo number=0.2) and non-cohesive particles,

respectively at different impeller speeds. M is the average for the particle bed. The

figures show that impeller speed affects the mixing characteristics significantly,

especially when the particles are non-cohesive. We define a mixing rate here as the

gradient of the mixing curve initially, using following first order equation:

M=1-(1-M0) e-kt (5.3)

Here, k (s-1) is the mixing rate.The mixing rate generally increases with the

impeller speed in both cases up to 100 rpm, but a further speed increase results in a


5-123

decrease in the mixing rate; note that such a trend has been observed experimentally in a

previous study on ribbon mixers, which reports that the mixing operations should be

carried out preferably in the impeller speed range of about 50 to 70 rpm (Sanoh et al.,

1974). Also affected by the speed in the present work is the steady-state mixing index of

non-cohesive particles, but no such effect is observed for cohesive particles. This will

be discussed in the flow pattern section 3.1.2.

The macroscopic index (Chandratilleke et al., 2010) is also now used to analyze

the results for cohesive mixing shown in Figure 5.3(a), which will enable us to compare

the two indexes. Figure 5.4 shows the results of the analysis. Clearly, there are

differences in the dependence of the rate of mixing on the impeller speed when

evaluated by macroscopic and particle scale indexes. Figure 5.4 shows that 200 rpm

produces a faster mixing rate than 100 rpm, which is opposite in trend to that in Figure

5. 3(a). Also, there is less sensitivity in the results for 20 to 100 rpm. It should be stated

here that the particle-scale index evaluates the mixture quality at the particle-scale,

which is much smaller in scale than the sample size used for the macroscopic index.

Thus, in the first place, the macroscopic index cannot guarantee mixing at a scale

smaller than the sample size. It should be noted that mixing state in a smaller scale

cannot be obtained by simply reducing the sample size of macroscopic index. This is

because, it can be shown that the smaller the sample size, the lower the macroscopic

index even for the well-mixed state of a mixture, when analyzed by the macroscopic

index using samples of equal size.

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Partt

ical

scal

e m

ixin

g in

dex,

M

Revolutions

20 rpm

50 rpm

100 rpm

200 rpm

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Parti

cle

scal

e m

ixin

g in

dex,

M

Revolutions

20 rpm

50 rpm

100 rpm

200 rpm

(a) Cohesive mixture (Bo=0.2) (b) Non-cohesive mixture

Figure 5.3 Effect of impeller speed: Comparison of mixing behaviours of cohesive and non-cohesive particles at 25% fill-level.


5-124

Although not shown here, the macroscopic index for non-cohesive particle mixing

at 200 rpm reaches unity in about 33.5 revolutions. Conversely, Figure 5.3(b) shows

that the particle scale index for non-cohesive mixing does not reach unity at 200 rpm

within 100 revolutions tested. Thus, the mixing state reaching unity in the macro scale

does not necessarily mean that mixing at the particle scale will also be well-mixed.

Even in the case of a cylindrical mixer, mixing state did not reach the fully-mixed state

at high shaft speeds (Chandratilleke et al, 2009). In the discussions to follow, we thus

use particle-scale index as the preferred index to maintain the consistency and avoid

discrepancies due to sample size and number.

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Mac

rosc

opic

mix

ing

inde

x

Revolutions

200 rpm

20, 50, 100 rpm

Figure 5.4 Effect of impeller speed on cohesive mixing: Macroscopic index

(Bo=0.2 and 25% fill-level)

5.4.2.2 Velocity and Flow Pattern

The investigating flow pattern in the mixer is essential because it enables us to

identify mixing mechanisms of the mixer. Flow patterns in the mixer are complicated

because of the unsteady nature of the flow and complex impeller configuration as seen

from Figure 5.1 and 5.2. To analyze the particle flow patterns in the mixer, velocity

fields are examined below, first in the central segment of the vessel between the axial

positions, z=350mm and 450 mm, z being measured from one end of the vessel.

Figure 5.5 shows the velocity fields in the cross-section of the mixer vessel at

impeller speeds of 50, 100 and 200 rpm for both cohesive and non-cohesive particles.

The snapshots were taken at t=25s, and the velocity vectors of the particles of the two

types deposited side-by-side initially are shown coloured red and blue. In the non-

cohesive mixtures, one can observe local re-circulations, which may not be so effective

in the mixing because the particles are deposited axially in side-by-side arrangement


5-125

initially. By contrast, cohesive particles tend to move circumferentially almost in full

circle even at the speed of 50 rpm, which may involve mixing in the axial direction as

well.

Figure 5.5 Velocity fields in vessel central segment (between z=350 and 450mm)

for non-cohesive and cohesive particles mixing at impeller speeds of 50, 100, and

200 rpm from the top to bottom, respectively, at t=25 s.

It would be appropriate to compare the flow patterns of the two sides of the

vessel at this stage because the blades have opposite spiral directions (Figure 5.1(a)) .

Figure 5.6 compares the velocity field in the section between z=350 and 450mm to that

between z= 485 and 600mm. It can be seen that the particles are rotating in the direction

of the shaft rotation despite the fact that the two blades have opposite spiral directions.

This observation has an impact on the circumferential velocity component of particles

as will be discussed later. It can be also noticed that the dominant particle colour is

different on both sides: blue on the left and red on the right.

Cohesive

Material

0.5

Frame 001 01 Feb 2012

Material

0.5

Frame 001 01 Feb 2012

Material

0.5

Frame 001 01 Feb 2012

Non-cohesive

Material

0.5

1

Frame 001 09 May 2012

Material

0.5

1

Frame 001 09 May 2012

Material

0.5

1

Frame 001 09 May 2012

50rpm 100rpm 200rpm

Material

0.5

Frame 001 01 Feb 2012

Material

0.5

1

Frame 001 09 May 2012

20rpm


5-126

(a) z=350 and 450mm (b) z= 485 and 600mm

Figure 5.6 Comparison of velocity fields in the two halves of the 2-bladed mixer

(at 50 rpm).

Figure 5.7 shows the velocity fields in the longitudinal section of the vessel

between x=-20 and +20mm, where x is horizontal distance from the shaft-axis

perpendicular to the shaft. It can be seen that particles circulate mostly in the lower part

of the mixer in the non-cohesive particle mixture for speeds less than 200 rpm. However,

cohesive particles occupy a large area in this longitudinal section, and thus there is more

chance of mixing. This observation is supported by a comparison of Figure 5.3(a) with

3(b), which shows that cohesive particles have a much faster mixing rate than the non-

cohesive ones, below 100 rpm.

(a)Non-cohesive

(b) Cohesive

50rpm 100rpm 200rpm

Figure 5.7 Velocity field in a longitudinal section between x=-20 and 20mm at impeller

speeds of 50, 100, and 200 rpm from top to bottom, respectively at time t=25s, x being

measured horizontally from the shaft-axis perpendicular to it.

Material

0.5

Frame 001 01 Feb 2012

Frame 001 01 Oct 2013

Z

Y

0 200 400 600 800-400

-200

0

200

400

Material

0.5

Frame 001 01 Feb 2012

Z

Y

0 200 400 600 800-400

-200

0

200

400

Material

0.5

Frame 001 01 Feb 2012

Z

Y

0 200 400 600 800-400

-200

0

200

400

Material

0.5

Frame 001 01 Feb 2012

Material

0.5

1

Frame 001 09 May 2012

Material

0.5

1

Frame 001 09 May 2012

Material

0.5

1

Frame 001 09 May 2012


5-127

With a further increase in the impeller speed, the red and blue particles seem to be

more separated in the longitudinal as well as in the cross-sections for non-cohesive

particles than for the cohesive ones. This observation is supported by the particle-scale

mixing curves of Figures 5.3(a) and (b) for 200 rpm.

To quantify the shaft speed effect on the velocity field, the probability density

functions of radial, tangential and axial velocity components of the particles are

considered as shown in Figures 5.8 and 5.9 for non-cohesive and cohesive mixtures,

respectively.

0.0

0.2

0.4

0.6

0.8

1.0

-4 -2 0 2 4

Prob

abili

ty D

ensi

ty

Radial Velocity(m/s)

50 rpm

100 rpm

20rpm200 rpm

0.0

0.5

1.0

1.5

-2 0 2 4 6 8 10

Prob

abili

ty D

ensi

ty

Tangential Velocity(m/s)

50rpm

100rpm

20rpm

200 rpm

(a) (b)

0.0

0.5

1.0

1.5

2.0

-3 -2 -1 0 1 2 3 4

Prob

abili

ty D

ensi

ty

Axial Velocity(m/s)

50 rpm

100 rpm

20rpm

200 rpm

-1

0

1

2

3

4

Vr_non_cohVt_non_cohVz_Non_coh

40 80 120 160 200

Tim

e av

erag

ed m

ean

radi

al v

eloc

ity (m

/s)

Blade speed (rpm)

(c) (d)

Figure 5.8 Probability density distributions of velocity components at different

blade speeds for non-cohesive particles at steady state.

From Figures 5.8 and 5.9, it is clearly seen that the tangential velocity

distributions each have a large spread than the distributions of the other two components

for both mixtures. Therefore, particles generally have a larger tangential velocity


5-128

compared to axial or radial components at a given impeller speed. Figures 5.8(d) and

5.9(d) show that the mean of the tangential particle velocities increases with the shaft

speed, the increase being sharp in the case of non-cohesive mixture. It should be noted

that the mean value of the tangential velocity distribution can not be zero as it is for the

other two velocity components. Figures 5.8(a) and 5.8(c) show that the spread of the

distributions for radial and axial velocity components increase with the shaft speed,

which means more particles are gaining higher axial and radial velocities with the shaft

speed increase, although their average values are close to zero as seen from Figures

5.8(d) due to the mixing vessel being a closed one. Similar arguments apply for the

radial, and axial velocity distributions of Figures 5.9(a) and 5.9(c) for cohesive particles.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

-3 -2 -1 0 1 2 3 4

Prob

abili

ty D

ensi

ty

Radial Velocity(m/s)

50 rpm

100 rpm

20rpm

200 rpm

0.0

0.5

1.0

1.5

2.0

-2 0 2 4 6 8 10

Prob

abili

ty D

ensi

ty

Tangential Velocity(m/s)

50rpm

100rpm

20rpm

200 rpm

(a) (b)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

-4 -3 -2 -1 0 1 2 3 4

Prob

abili

ty D

ensi

ty

Axial Velocity(m/s)

50 rpm

100 rpm

20rpm

200 rpm

-1

0

1

2

3

4

VtVzVr

40 80 120 160 200

Tim

e av

erag

ed m

ean

radi

al v

eloc

ity (m

/s)

Blade speed (rpm)

(c) (d)

Figures 5.9 Probability density distributions of velocity components at different

shaft speeds for cohesive particles (Bo=0.2) at steady state.

The mixing quality at 200rpm in the case of non-cohesive particles (Figure 5. 3(b))

shows a reduction at the steady-state, which is probably because of the increased


5-129

tangential velocity of particles delaying or preventing the mixing of particles in axial

direction. This will be especially true in the present case where the particles have been

initially deposited in the side-by-side arrangement in the axial direction.

5.4.2.3 Contact forces

The contact forces between particles in the ribbon mixer were investigated at

different impeller speeds for the non-cohesive and cohesive particle mixtures as shown

in Figure 5.10. Figure 5.10 shows an instantaneous network of contact forces among

the particles, with the colors indicating the ranges of magnitudes of the forces and red

being the strongest force (see Figure 5.10 caption for details). The diagrams show that

the contact forces between particles are stronger in the cohesive mixture compared to

the non-cohesive one for impeller speeds less than 100 rpm. In this speed range, the

magnitude of the contact forces increased with the impeller speed in both mixtures.

Figure 5.10 Force-network diagrams at different impeller speeds for cohesive and non-cohesive particles in the vessel segment between the axial positions, z= 350 and 450 mm at t=37-39 s; impeller speeds are 50, 100, and 200 rpm from the top to bottom, respectively; and colours and corresponding ranges of contact forces f are: blue, f < 0.235 N; green, 0.235 < f < 0.936 N; light green, 0.936< f < 1.17 N; and red, 1.17 < f <1.62N

However, at speeds more than 100 rpm, the cohesive mixture shows a reduction

in the strength of the contact forces, which is also true compared to the non-cohesive

mixture at the same speed. As seen from the figure, the contact force network is more

uniformly distributed in the cohesive mixture at 200rpm than in the non-cohesive one,

leading to lesser contact forces in the branches of the force network. Better spreading of

Non-cohesive

Cohesive

50rpm 100rpm 200rpm 20rpm


5-130

particles in the cohesive mixture is probably the reason for its better mixing at 200 rpm

as seen from Figures 5.3(a) and (b).

To be quantitative, Figure 5.11 shows the probability density for normal force,

shear force and axial force in the shaft speed range 20-100rpm. The normal force is

found to be varying between -0.5 and 4 N. The particles received higher normal force

than tangential and axial forces. The increased number of particles receives a higher

normal force with an increment of shaft speed. Figure 5.11(a) shows the normal contact

forces received by the majority of particles are similar at 50rpm and 100rpm blade

speeds which can be confirmed from the force-network diagrams. The increment of

normal contact forces can be observed from Figure 5.11(c), which shows the mean

values of the probability density curve as a function of shaft speed.

0

0.2

0.4

0.6

0.8

1

10-4

10-3

10-2

10-1

100

101

20rpm50rpm100rpm200rpm

Cum

ulat

ive

Prob

abili

ty

Normal Force(N)

0

0.2

0.4

0.6

0.8

1

10-5

10-4

10-3

10-2

10-1

100

101


Cum

ulat

ive

Prob

abili

ty

Shear Force(N) (a)

(b)

0.2

0.4

0.6

0.8

1

40 80 120 160 200

FnFt

Mea

n va

lues

of d

istri

butio

ns (N

)

Shaft speed (rpm) (c)

Figure 5.11 Probability density distributions of force components for non-cohesive

mixture at steady state.


5-131

The diagrams that correspond with Figure 5.11 are shown in Figure 5.12 for the

cohesive case (Bo=0.2). In general, the effect of shaft speed is not clear at this cohesion.

However, it can be seen that the curves for both normal and shear forces have moved to

larger force range compared to the non-cohesive case. The mean values of the

distributions are shown in Figure 5.12(c), which shows that both the normal force and

shear force are larger compared to the non-cohesive case at any shaft speed. It also

shows that the mean value does not change with the shaft speed. Further, the difference

between the two forces has increased in this case.

0

0.2

0.4

0.6

0.8

1

10-1

100

101


Cum

ulat

ive

Prob

abili

ty

Normal Force(N)

0

0.2

0.4

0.6

0.8

1

10-2

10-1

100

101


Cum

ulat

ive

Prob

abili

ty

Shear Force(N) (a) (b)

0.4

0.8

1.2

1.6

2

40 80 120 160 200

Fn

Ft

Mea

n va

lues

of d

istri

butio

ns (N

)

Shaft speed (rpm) (c)

Figure 5.12 Mean values of cumulative probability distributions of the force

components for cohesive mixture at steady state


5-132

5.4.3 Effect of fill level

The effect of fill level on particle mixing is investigated at different cohesion

levels using the particle-scale mixing index and mixing rate for the two bladed and four

bladed mixers. The shaft speed is chosen as 100 rpm in both cases, in order to compare

with the 2-bladed mixer. The velocity fields and flow patterns in the two and four

bladed mixers are also investigated for non-cohesive and cohesive mixtures. The effects

on velocities and forces are quantified using probability distribution curves in the case

of the 2-bladed mixer for the non-cohesive mixture.

5.4.3.1 Mixing Index and mixing rate

To investigate the effect of fill level on mixing, fill level is varied from 16% to

56% using mixtures of different cohesion at a shaft speed of 100 rpm. The mixing rate

is calculated using the following first order Eq (5.3). Bond number (Bo) which is the

van der Waals force scaled by the particle weight is used here for characterizing

differences in cohesion.

Figure 5.13 shows in each diagram the effect of fill level on the mixing

performance of the two-bladed mixer for a mixture of given cohesion, with Bo number

(or the mixture cohesion) being increased from 0 to 2 in diagrams (a) to (c) in that order.

Figure 5.14 shows the corresponding results for a four-bladed mixer. With an increase

in the fill level and Bo number, mixing takes increasingly longer times to reach the

steady state for both mixers. The effect of fill level is much more pronounced with a

two-bladed mixer than with a four-bladed mixer. On the other hand, the two bladed

mixer shows even a deterioration in the steady-state mixture quality for Bo=2, which is

not the case for the four-bladed mixer. Thus, it can be concluded, the four-bladed mixer

is a better mixer than the two-bladed one, especially for mixing of cohesive particles.


5-133

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Fill level=16%

Fill level=26%

Fill level=46%

Fill level=56%

Part

tical

Sca

le M

ixin

g In

dex

Revolutions 0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Fill level=16%

Fill level=26%

Fill level=46%

Fill level=46%Part

tical

Sca

le M

ixin

g In

dex

Revolutions 0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Fill level=16%

Fill level =26%

Fill level=46%

Fill level=56%

Part

tical

Sca

le M

ixin

g In

dex

Revolutions

(a) Bo=0 (b) Bo=0.2 (d) Bo=2

Figure 5.13 Effect of fill level on the mixing performance of a 2-bladed mixer at

different Bo numbers (shaft speed = 100 rpm).

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Fill level=16%Fill level=26%Fill level=46%Fill level=56%Pa

rttic

al S

cale

Mix

ing

Inde

x

Revolutions 0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Fill level=16%Fill level=26%Fill level=46%Fill level=56%Pa

rttic

al S

cale

Mix

ing

Inde

x

Revolutions 0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Fill level=16%Fill level=26%Fill level=46%Fill level=56%

Partt

ical

Sca

le M

ixin

g In

dex

Revolutions

(a) Bo=0 (b) Bo=0.2 (c) Bo=2

Figure 5.14 Effect of fill level on the mixing performance of a 4-bladed mixer at

different Bo numbers. (shaft speed = 100 rpm).

Figure 5.15 shows the mixing rate k as a function of the fill level for the two

mixers, the parameter in the diagrams being the Bo number. For both mixers, k

decreases with an increase in the fill level. Similarly k decreases with an increase in Bo

number for both mixers. A comparison of Figures 5.15(b) and 5.15(a) shows that k is

larger for the four bladed mixer for any Bo numbers considered. It can be also observed

that the differences in k due to changes in Bo number become reduced when the fill

level increases. Particularly, in the case of the four-bladed mixer, it disappears for fill

levels more than 45%.


5-134

0

0.05

0.10

0.15

0.20

0.25

10 20 30 40 50 60

Non-cohesive Bo=0.02Bo=0.2Bo=2

Mix

ing

Rat

e, k

(1/s

ec)

Fill level (%)

0

0.05

0.10

0.15

0.20

0.25

10 20 30 40 50 60

Non-cohesive Bo=0.02Bo=0.2Bo=2

Mix

ing

Rat

e, k

(1/s

ec)

Fill level (%)

(a) Two-bladed mixer (b) Four-bladed mixer

Figure 5.15 Mixing rates k for the two-bladed and four-bladed mixers (shaft speed

= 100 rpm)

5.4.3.2 Velocity field and quantification of velocity

The velocity field, and time and cell averaged velocities are produced here to

understand the particle flow behaviour in micro-dynamic aspects for both the non-

cohesive and cohesive mixtures at different fill levels. The probability density

distributions of particle velocities are used to quantify the effects of fill level on the

velocity field.

Figure 5.16 shows the particle flow patterns as investigated from the velocity

fields at different fill levels for the non-cohesive and cohesive mixtures in the two

bladed and four bladed mixers at the 80th revolutions. It can be seen that the flow

patterns as similar to those in a rotating drum in the case of the non-cohesive mixture at

low fill levels. For example, one can see cascading, recirculation and sliding-like

motions in the non-cohesive mixture both for the two bladed and four bladed mixers. In

the case of non-cohesive mixture, the two bladed mixer shows recirculation motion on

the right lower part in the mixer at 26% fill level; similarly, the four bladed mixer shows

such motion at 16% and 26% fill levels. The cascading motion is seen in the mixture

with 26% and 56% fill levels in the non-cohesive mixture for two-bladed as well as the

four-bladed mixers.


5-135

Due to the cohesion between the particles, the particles tend to move together

enhancing the circumferential flow in the case of the cohesive mixture in the 4-bladed

mixer.

Figure 5.16 Velocity fields at different fill levels in the two bladed and four

bladed mixers at the 80th revolution for the two-bladed and four-bladed mixers in

cases of non-cohesive and cohesive mixtures (shaft speed = 100 rpm).

In order to obtain the time-averaged longitudinal velocity distribution in the

mixers, the mixer space is divided into cubic cells of the size 40x40x40mm3, and the

velocities of particles in each cell are averaged over a time interval of 2s. Figure 5.17

shows the average velocity and porosity in a longitudinal section between x=-40 and 40

mm for the non-cohesive mixture in the 2-bladed mixer averaged over 2s up to the

30 rev. It shows that a circulating flow is established from either end of the vessel

towards the centre of the vessel and back towards the vessel ends; although not shown

here, similar flow patterns are observed in the case of 4-blade mixer. However, such a

flow pattern will not improve axial mixing in the present case where a side-by-side

Fill level=16% Fill level=26% Fill level=46% Fill level=56%

2blade-non-cohesive

4blade cohesive

Frame 001 20 Jul 2013 Frame 001 20 Jul 2013 Frame 001 20 Jul 2013 Frame 001 20 Jul 2013


2blade -cohesive


Frame 001 20 Jul 2013 Frame 001 20 Jul 2013

4blade non-cohesive



5-136

initial arrangement has been used. To improve mixing rate, it is thus more appropriate

to deposit particles in top-bottom arrangement or by the layering method as reported

elsewhere (Muzzio et al., 2008). Porosity over the impeller seems to decrease with an

increase in the fill level, which is true for both types of mixers.

Fill level=16% Fill level=26%

Fill level=46% Fill level=56%

Figure 5.17 Average velocity and porosity of particles in a longitudinal section

between x=-40 and 40 mm at the 30 revolution for non-cohesive mixture in the 2-

bladed mixer, x being measured at right angle to the shaft from the shaft axis

(shaft speed = 100 rpm).

To quantify the fill level effect on particle velocities, probability density

distributions of instantaneous velocity components are considered in Figures 5.18 and

5.19 for non-cohesive mixtures in the 2-bladed and 4-bladed mixers, respectively, as

was done in Figure 5.9. Generally, Vz and Vr have their distributions centred around 0,

although there is slight deviation from it due to the velocity being instantaneous.

Particularly, Vz shows a shift towards large positive values in Figure 5.18(c). Its time-

average is not so close to zero over 2s time interval for some fill levels, which indicates

that the traversing of particles from one end to the other may be occurring in a much

Frame 001 29 Jul 2013

Frame 001 29 Jul 2013


Porosity

0.95

0.9

0.85

0.8

0.75

0.7

0.65

0.6

0.55

0.5

0.45

Frame 001 29 Jul 2013


5-137

longer period of time. Vt shows a slight shifting of its mean value in the positive

direction with an increase in the fill level, which means more particles are moving

circumferentially at a higher speed. The time-averaged Vt also show a slight increase

with the fill level. Vr is more or less centered around 0 as expected, and its time-

averaged value is mostly 0.

In the case of the 4-bladed mixer (Figure 5.19), Vt shifts towards higher

velocities with an increase in the fill level as seen from its distribution curve (Figure

5.19 (b)). This effect is clearly seen from the time-averaged Vt shown in Figure 5.19(d).

The time-averaged Vz and Vr are zero at higher fill levels, but show non-zero values at

lower fill levels. The reason could be that particles take longer time to traverse from one

end to the other of the vessel at low fill levels.

0

0.2

0.4

0.6

0.8

1

1.2

-3 -1.5 0 1.5 3

16%26%46%56%

Prob

abili

ty D

ensi

ty

Radial Velocity (m/s)

0

0.1

0.2

0.3

0.4

0.5

0.6

-2.25 0 2.25 4.5

16%26%46%56%

Prob

abili

ty D

ensi

ty

Circumferencial Velocity (m/s) (a) Radial velocity Vr (b) Circumferential velocity Vt

0

0.2

0.4

0.6

0.8

1

-2 -1 0 1 2

16%26%46%56%

Prob

abili

ty D

ensi

ty

Axial Velocity (m/s)

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

10 20 30 40 50 60

VrVtVz

Tim

e-av

erag

ed m

ean

velo

city

(m/s

)

Fill level (%) (c) Axial velocity Vz (d) Time-averaged mean velocities

Figure 5.18 Probability distributions of velocity components and time-averaged mean

velocities at different fill levels for the non-cohesive mixtures in 2-bladed mixer at 100

rpm shaft speed.


5-138

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-3 -1.5 0 1.5 3

16%26%46%56%

Prob

abili

ty D

ensi

ty


0

0.1

0.2

0.3

0.4

0.5

-2.25 0 2.25 4.5

16%26%46%56%

Prob

abili

ty D

ensi

ty

Circumferencial Velocity (m/s) (a) Radial velocity Vr (b) Circumferential velocity Vt

0

0.1

0.2

0.3

0.4

0.5

0.6

-2 0 2 4

16%26%46%56%

Prob

abili

ty D

ensi

ty


-0.2

0

0.2

0.4

0.6

0.8

1

16% 26% 46% 56%

Vr

Vt

Vz

Tim

e-av

erag

ed m

ean

velo

citie

s (m

/s)

Fill level (%) (c) Axial velocity Vz (d) Time-averaged mean velocities

Figure 5.19 Probability distributions of velocity components and time-averaged

mean velocities at different fill levels for non-cohesive mixtures in the 4-bladed

mixer at 100 rpm shaft speed.

5.4.3.3 Contact forces

The effect of the fill level on the contact forces are analyzed here using

probability density distributions of contact forces in the 2-bladed and 4-bladed mixers

for mixing of non-cohesive particles as done previously in the shaft-speed section. The

results are shown in Figures 5.20, with the left-column showing those for the 2-bladed

mixer and the right column showing those for the 4-bladed one, respectively.


5-139

0

0.2

0.4

0.6

0.8

1

10-5

10-4

10-3

10-2

10-1

100

101

16%26%46%56%

Cum

ulat

ive

Prob

abili

ty

Normal Force(N)

0

0.2

0.4

0.6

0.8

1

10-5

10-4

10-3

10-2

10-1

100

101

16%26%46%56%

Cum

ulat

ive

Prob

abili

ty

Normal Force(N) (a1) Normal forces: 2-bladed (b1) Normal forces: 4-bladed

0

0.2

0.4

0.6

0.8

1

10-5

10-4

10-3

10-2

10-1

100

101

16%26%46%56%

Cum

ulat

ive

Prob

abili

ty

Shear Force(N)

0

0.2

0.4

0.6

0.8

1

10-4

10-3

10-2

10-1

100

101

16%26%46%56%

Cum

ulat

ive

Prob

abili

ty

Shear Force(N) (a2) Shear forces: 2-bladed (b2) Shear forces: 4-bladed

0

0.05

0.1

0.15

0.2

16% 26% 46% 56%

FnFt

Mea

n Fo

rce

of C

PD (N

)

Fill level (%)

0

0.05

0.1

0.15

0.2

20 25 30 35 40 45 50 55

FnFt

Mea

n Fo

rce

of C

PD (N

)

Fill level (%) (a3) Mean values of (a1) and (a2): (b3) Mean values of (a1) and (a2):

2-bladed. 4-bladed.

Figure 5.20 Probability distributions of contact force components of particles of

the non-cohesive mixture at different fill levels for the 2-bladed and 4-bladed

mixers at 100 rpm shaft speed.


5-140

The figure shows that the normal forces are slightly increased with the fill level

in the case of 2-bladed mixer as seen by the slight right-shift in the probability

distribution curves. Conversely, fill level does not affect the normal forces in the case of

4-bladed mixer. The average values shown in Figures 5.20(a3) and (b3) confirm this

observation. As for the shear or tangential contact forces, the fill level has the effect of

increasing the shear force in the case of 2-bladed mixer, but not as much in the case of

4-bladed mixer as seen from a comparison of Figures 5.20(a2) and 5.20(b2), In fact, for

both the 2-bladed and 4-bladed mixers, the contact forces are at about the same level.

With its ability to mix cohesive particles (up to Bo=2) without deterioration in the

steady-state mixture quality, the 4-bladed mixer can thus be considered as a better

choice to mix cohesive particles.

5.4.4 Comparison of mixer performances: Ribbon versus cylindrical mixers

It is also worthwhile to make a statement about the level of normal and shear

forces that particles experienced in the ribbon mixers in comparison with the cylindrical

mixers (Chandratilleke et al., 2009). In a 2-bladed cylindrical mixer with a vertical shaft,

the normal force on the average is about 0.022N at a shaft speed of 100 rpm, which is

about 1/5th of the present normal forces. Shear forces in the cylindrical mixer is about

0.005N at 100 rpm, which is about 1/10th of the present shear forces. However, the

particle number being 18,000 in the present case in comparison to 16,000 of the

cylindrical mixer, one can expect some increase in the contact forces in the ribbon

mixer. However, the effect of this difference is not clear at this stage, but should be

clarified in a future work.

In addition, if we consider the mixing characteristics of Figures 5.4, which is the

macroscopic mixing index for the 2-bladed mixer at 25% fill-level, the fully mixed state

is reached after about 60 revolutions at 100 rpm for particles laid down initially in the

side-by-side configuration in the axial direction. On the other hand, for 16,000 non-

cohesive particles laid down initially in side-by-side configuration in a cylindrical mixer,

the fully mixed state is reached in just about 6 revolutions (Chandratilleke et al., 2009).

Such a difference may have been caused by the size difference of the mixers; the

cylindrical mixer is of 250mm diameter while the length of the ribbon mixer vessel is

970mm. The ratio of vessel diameter to particle size is 50 for the cylindrical mixer and


5-141

that of the vessel length to particle size is 64 for the ribbon mixer. Therefore, as a matter

of fact, mixing in the case of ribbon mixer will be delayed in comparison to the

cylindrical mixer. The strengths of the two mixers need to be thus properly assessed in a

future work.

5.5 Conclusions

The mixing performances of ribbon mixers were investigated using the discrete

element method. The two operational parameters, the shaft speed and fill level were

varied to investigate their effects on the mixing performance. In order to establish the

effect of the shaft speed, a 2-bladed ribbon mixer was used at a fixed fill-level of 25%.

Next, having determined a suitable shaft speed, the fill-level was varied over a wide

range. In the study, both non-cohesive and cohesive mixtures were also used to

highlight how the particle cohesion can influence the effects of the two operational

parameters on the mixing performance. Further, performances of two ribbon impellers

were also compared based on fill-level effects at different particle cohesion levels. A

comparative discussion was also made of ribbon mixers and a vertically-shafted

cylindrical mixer. The following conclusions were drawn.

Shaft speed affects the mixing performance of a 2-bladed ribbon mixer significantly,

especially when the mixture is non-cohesive. A shaft speed of 100 rpm was found

to be suitable for both non-cohesive and cohesive mixtures as it provided the fastest

mixing rate in the range of speeds studied. At speeds exceeding 100 rpm, not only

the mixing rate, but also the steady-state mixing quality deteriorated for non-

cohesive mixtures, although it was not the case with cohesive mixtures.

With an increase in the shaft speed, the circumferential velocity of particles

increased, but the average axial and radial velocity components remained close to

zero, although their absolute values increased with the shaft speed, all of which

contributing to mixing improvement with shaft speed increment.

Shaft speed increased the contact forces of particles in non-cohesive mixtures.

Conversely, the shaft speed did not affect the contact forces in cohesive mixtures.

However, contact forces in cohesive mixtures were larger than those in non-

cohesive mixtures.

The fill level affected the mixing rate, particle flow and particle velocities

significantly in the non-cohesive and cohesive mixtures in both the 2-bladed and 4-


5-142

bladed ribbon mixers. On the other hand, fill level did not affect the contact forces

in both mixers for non-cohesive mixtures. The mixing rate deteriorated with an

increase in the fill level and cohesion in both the mixers. The particle flow

gradually changed from a recirculation flow to a sliding flow to a cascading flow

with an increase in the fill level in non-cohesive mixtures for both mixers tested.

The 4-bladed mixer is more capable of handling mixing at high fill levels,

especially in the case of cohesive mixtures without a sacrifice in the mixture quality

at 100 rpm.

Overall, the ribbon mixers, especially the 4-bladed ones are promising for mixing

of cohesive particles. However, proper comparison work is necessary to highlight its

advantages over the simple vertically-shafted cylindrical mixers, which may provide

faster mixing with lower stresses on the particles to be mixed.

Nomenclature

D Vessel diameter, (m)

d Particle diameter, (mm)

E Young’s modulus, N/m2

Fc,ij Contact force vector between i and j, N

Fd,ij Damping force vector between i and j, N

Fv, ij Cohesive force between particles i and j

g Acceleration due to gravity, (m/s2)

h Gap between interacting two particles, (m)

Ha Hamaker constant, J

Hap Hamaker constant of particle material, J

Haw Hamaker constant of wall material, J

Ii Moment of inertia of particle i, (kg m2)

ki Number of particles in contact with particle i

M Particle-scale mixing index defined in Eq.(3) , (-)

Mij Vector of rolling friction torque on particle i, (Dury et al.)

Mp Predicted mixing index at steady-state, (-)

n Actual particle number fraction of the two types of particles, (-)

Ni Number of particles in the immediate neighbourhood of particle i


5-143

Nl Number of large particles

Np Total number of particles of one type in the mixture

Ns Number of small particles

Nt The number of sample points of instantaneous average values in time t

p Number ratio of the target type particles to all the particles, (-)

pi Particle fraction of a target type particle in the neighborhood of particle i, (-)

tp Average value of pi at time t for the entire mixture, (-)

R Equivalent radius, m

S0 Standard deviation of pi at fully-segregated state, (-)

SR Standard deviation of fully-mixed state for uniform-sized particles of particle

fraction of p, (-)

St Standard deviation of pi with respect to tp at time t, (-)

Tij Vector of rolling friction torque on particle i, N m

Vb Blade speed, (m/s)

Vi Velocity of particle i, m/s

Z Height from vessel base (mm)

Greek letters

α Volume fraction

Average particle-scale sample size for the mixture

θ Radial section measured horizontally from mid-plane of the blade

ik Number of particles in contact with particle i

Density of particles of a uniform system, ( 3kgm )

Shaft rotational speed, (rad/s)

i Angular velocity vector of particle i, (rad/s)

CHAPTER 6 DEM Simulation of Powder Mixing in a Ribbon Mixer: Effects of Impeller Geometry

6-144

Chapter 6

DEM Simulation of Powder Mixing in a Ribbon Mixer: Effects of

Impeller Geometry


6-145

6.1 Introduction

A ribbon mixer is known to possess a combination of mixing mechanisms such as

convection, diffusion and shear when mixing granular matter. Ribbon mixers are widely used

in practice because they are capable of providing high speed convective mixing, especially in

the circumferencial direction. The efficiency of such operations and quality of the resulting

mixtures will depend on the impeller geometry. Thus, it is important to understand the effects

of the impeller geometry and use the right geometry for the product to be manufactured based

on industry requirements, which can be different from one industry to another, some examples

of different industries being pharmaceuticals, food, cosmetics and powder metallurgy

industries.

Much of the work on the effect of ribbon geometry on mixing has been reported for

liquid mixing (Masiuk and Lacki, 1993). For example, Masiuk et al (1992) and Muzzio, Lusa

et al. (2008) reported that helix pitch and width have a perceptible influence on the mixing

time and power consumption for mixing of liquids. Masiuk et al. (1993) also reported that the

shape of the ribbon agitator has a significant influence on the energy required for mixing of

liquids. Takahashi et al. (1988) reported that even though the primary circulation patterns are

approximately the same, the impeller geometry can strongly affect the secondary circulation

flows. The following parametric effects have been reported for liquid mixing with a helical

blade. When the pitch of the helical blade is decreased with the wall clearance fixed, the shear

rate becomes higher (Delaplace et al., 2000). It is also reported that doubling the width of a

helical ribbon blade increased the axial forces by 40-50% while the power consumption

remained practically constant (Bortnikov et al., 1973). Further, doubling the helical ribbon

width has resulted in an increase of 10% in the power required for mixing of liquids

( ). A helical ribbon mixer with a greater clearance has a lower

power consumption, but it takes a longer time for the homogenization. Conversely, an agitator

with a smaller clearance has exhibited a greater efficiency (Rieger et al., 1986). Thus, the

design of a blade has a significant impact on the homogenization of mixtures. However, there

is not much information available in the literature regarding this effect for granular mixing

(Muzzio et al., 2008). Further, the studies of this effect on the mixing of cohesive particles in

a horizontal ribbon mixer are rare to the best of our knowledge.


6-146

Muzzio et al. (2008) experimentally investigated the mixing performance of a ribbon

mixer with regard to the effect of loading (layering and off-center spot loading method). They

reported that layering method results in faster mixing and better homogeneity.They also found

that the fill level has a significant effect on the mixing performance of a 3 spokes 2-bladed

ribbon impeller, but not so much on a 5 spokes 2-bladed ribbon mixer.

The objective of this study is to investigate the effect of blade geometry on the mixing

performance of a horizontal ribbon mixer. The discrete elmement method (Cundall and Strack,

1979) is an effective way of acquiring such information because it can be difficult and/or

expensive to use traditional experimental approaches in such a study as has beed reported

elsewhere (Cleary and Sinnott 2008). Therefore, the effect of the impeller geometry of a

ribbon mixer is investigated here by using DEM and the mixing behaviour analyzed by a

particle-scale mixing index (PSMI). Here, the impeller geometry is changed by varying the

blade number, blade angle, blade width, blade pitch and blade clearance, and the

performacnce of the impellers are evaluated with different cohesive mixtures at a fixed fill

level of 35%.

6.2 Numerical Method

The DEM model used here is based on an extension to the original DEM model

proposed by Cundall and Strack (Cundall and Strack, 1979) to account for the rolling friction

of particles, and is essentially the same as that previously developed and validated by Zhou et

al. (Zhou et al., 1999; Zhou et al., 2004). The model uses two momentum conservation

equations to describe the translational and rotational motion of particle i in a system at time t

subjected to the gravity g and interactions with the neighboring particles, blade and walls:

ii N

jijv

k

jijdijci

ii m

dtdm

1,

1,, FFFgV

(6.1)

ik

jijij

ii d

dI1t

MT (6.2)


6-147

Here mi, Ii, Viand i are the mass, moment of inertia, translational and rotational

velocities of particle i respectively;kiis the number of particles that are in contact with particle

i;Fc,ij represents the elastic contact force, which is the summation of the normal and tangential

forcesrespectively at the contact point with particle j;Fd,ijrepresents the damping force, which

is the summation of the normal and tangential damping force respectively at the contact point

with particle j;and Tij and Mijare the torque and rolling friction torque on particle i due to

particle j respectively. Expressions for the forces as shown in the Table6.1.

Table 6.1. Equations used to calculate forces in the DEM simulations

Force Equation

Particle-particle forces 13, 18-20


22

2/32

ssssnijs


max,max,

ij

jijiji


RRhRRA nF ˆ)422()22(

)(646 2222

33

iji

ivdwi hRhh

RA nF ˆ)2(

23 222

3



6.3 Mixing quantification

The index is based on the use of the coordination number to define the particle

fraction piof one type of particles (or target type particles) at particle i in a binary mixture, and

the variance St2of pi is obtained relative to the instantaneous mean value of pi for the mixture.

Next, St2is used in the following formula of L ce ’ m x g ex , M to obtain the

instantaneous mixing index for the mixture:

220

220

R

t

SSSSM

(6.3)


6-148

where, ppS 120 and /12 ppSR are the variances of the fully-segregated

and fully-mixed mixed states respectively, with p and ƞ representing the particle number ratio

of the target type particles to the total particle number in the binary mixture and average

particle-scale sample size for the mixture, respectively. The average sample size is determined

from the average value of the coordination number for the mixture. In determining St2, one

has to use a particle-contact condition, which is taken as an inter-particle gap size of 5% of

the small particle diameter, to be consistent with our previous work.

Mixing rate

The mixing rate is calculated using following first order equation:

M=1-(1-M0) e-kt (6.4)

Here, k (s-1) is the mixing rate.

6.4 Simulation conditions and procedures

The mixer used in this study is a horizontal ribbon mixer, and the ribbon impeller

designs tested are shown in Figure 6.11. Two types of impellers are investigated here. In the

first type, the impeller has only outer two blades (see the top row in Figure 6.1, the base case

being the leftmost one). This impeller is termed the type-1 impeller. In the second type, one

inner set of two blades are also used (see bottom row in Figure 6.1, the base case being the

leftmost one again). This impeller is termed here the type-2 impeller.

To investigate its mixing performance, an impeller is placed axially in a horizontal

cylindrical vessel, whose the major dimensions are as listed in Table 6.3. Here, 27 numerical

experiments are conducted to investigate effects of the following four variables related to the

impeller geometry, on mixture quality: blade number, blade angle, blade pitch, and blade

clearance. For mixing studies, two types of particles (different only in color) are laid in the

mixer in the side-by-side arrangement in the axial direction. After the particles have settled

down, the impeller is rotated from the stationary state at a constant speed of 100 rpm.


6-149

The study is carried out in two parts. In the first part, effects of geometry of type-1

impeller on cohesive particle mixing are investigated. The geometric parameters considered

are blade pitch, width and clearance of blades with the vessel wall. In the second part,

performances of type-1 impellers are compared with those of type-2 impellers using a non-

cohesive particle mixture. Note that when the blade pitch is reduced in the base case geometry

of each type of impellers, blades are needed to be added to fill up the vacant space between

the two end-blades of the impeller. Therefore, the pitch reduction is equivalent to an increase

in the blade number for both types of impellers. The cohesion between the objects in the

system is modelled by the van der Waals force. The values of Hamaker constant, Ha used for

particle-to-particle and particle-to-wall contacts are shown in the Table 6.2. Hamaker constant,

Hap-w for particle-wall contacts is obtained from wp HaHa , where Hap and Haw are Hamaker

constants for particle and wall materials, respectively (Kruusing, 2008).

A total of 18,000 uniform particles of 15 mm dia. are used in the study, the particle

properties used being as follows: You g’ mo u u =1108N/m2 Po o ’ r t o=0.29,

damping coefficient=0.3, static sliding friction coefficient=0.3 and rolling friction

coefficient= 0.002. The time step is chosen as 5.8x10-6s.

Table 6.2 Particle material properties

Hamaker Constant, Hap (J)

Hamaker Constant Hap-w

(J)

Hamaker Constant Haw(J)

5.54E-16 8.76E-18 1.4E-19 5.54E-17 2.77E-18

5.54E-18 8.76E-19

Table 6.3. Geometry parameters of ribbon mixer


6-150

Input variable Value

Vessel diameter 432 (mm)

Vessel length 970 (mm)

Blade pitch ratio S/D 1.12, 0.75, 0.56

Blade width W 10, 20, 30, 40, 60 (mm)

Shaft diameter 31.75 (mm)

Rake angle 45 , 135 , 50 , 80

Blade Clearance C 8.5 ,15,22, 27 (mm)

Particle number 18000

Particle dia., d 15 mm

Shaft speed, co 100 rpm

You g’ mod. E 1 108 N/m2

Po o ’ ratio, ν 0.29

ms (P-P) (P-W) 0.3

mR(P-P) (P-W) 0.002

Time step 5.8E-06

S/D1.123 S/D0.748 S/D0.561width

45°

S

D

Inner and outer blade

45°S

D

width

S/D=1.123 S/D=0.748 S/D=0.561

Figure 6.1 Impeller configurations: Top-row, type-1 impeller and bottom-row,

type-2 impeller.

S/D=1.123 S/D=0.748

S/D=0.561


6-151

6.5 Results and discussions

Below, the results of the numerical experiments are divided into two parts. In the first

part, the mixing performance of a mixer with a type-1 impeller (see Figure 6.1) is investigated

by varying the mpe er’ blade number (or pitch), as well as blade width, blade angle and

clearance of the blades with the vessel wall. In the second part, the performaces of type-1

impellers are compared with those of the type-2 (see Figure 6.1).

6.5.1 Geometrical effects of type-1 ribbon impellers

The geometric effects such as blade number (or pitch ratio), blade width, blade clearance

and blade angle on mixing behaviour in a ribbon mixer with a type-1impeller is investigated

here using DEM method. Such effects are investigated at different cohesion with the objective

of exploring the merits of ribbon mixers for powder mixing. The effects of the blade number

(or pitch ratio) and blade width on particle mixing behavior are examined in terms of a

particle-scale mixing index and mixing rate, coordination number, velocities and forces. The

shaft speed is chosen as 100rpm. The mixing rate, k is calculated using Eq (6.4)

6.5.1.1 Blade number or blade pitch effect

To investigate the effect of number of blades added to the base-case ribbon geometry

(leftmost type-1 impeller in Figure 6.1), the blade number is increased from 2 to 4 as shown in

the top row of Figure. 6.1. To add the new blades into the impeller, the blade pitch S is

reduced. Thus, the increase of blade number corresponds to a decrease in the blade pitch. The

pitch ratio S/D of the impellers are shown below each diagram, where D is the cylindrical

vessel diameter. With the reduction of the blade pitch, the length of the blade measured along

the spiral will increase.

6.5.1.1.1 Mixing index and mixing rate

Figure 6.2 shows the mixing characteric of each impeller of type 1 as a function of shaft

revolutions at different Hap values. The results show that mixing gets delayed with an increase

of cohesion. Mixing rate of the mixing curves are shown in Fig. 6.3 as a function of either S/D

or blade number. Mixing curve for the ribbon impeller with S/D=0.75 (or 3-blades) has a


6-152

higher mixing rate and the steady-state reached earlier compared to other impellers as shown

in Figs. 6.2 and 6.3, for all Hap values tested. The mixing rate k deteriorated with an increase

of cohesion for 2 and 4-bladed type 1 impellers as shown in Figs. 6.3.

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

S/D=1.12S/D=0.75S/D=0.56

Partt

ical

scal

e m

ixin

g in

dex,

M

Revolutions

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

S/D=1.12S/D=0.75S/D=0.56

Part

tical

sca

le m

ixin

g in

dex,

M

Revolutions

(a) Hap = 5.54 x 10-18 J (Bo=0.02) (b) Hap = 5.54 x 10-17 J (Bo=0.2)

Figure 6.2 Effect of blade number or pitch ratio on mixing at different Hap for type-

1 impellers

The reason of particles mixing in the mixer with pitch ratio 0.75 shows a high mixing

rate and high uniformity probably is that the blade arrangement improves particle axial

movement, mixing easily reached the pick value of the mixing index. However because of

enhanced particle axial movement, the particle flow divided into two parts perform

segregation around 10 revolutions.

Figure 6.3 (a) shows that the mixing rate, as a function of pitch ratio increases and then

decreases with an increase of pitch ratio up to S/D=0.75 and with a decrease of cohesion of

the mixture. It is investigated that the particles in the ribbon impeller with pitch ratio

S/D=0.75 show similar mixing rate in the mixture with cohesion range from Hap = 5.54 x 10-

18 J to Hap = 5.54 x 10-17 J. Figure 6.4 (b) shows the mixing rate as a function of the Ha

constants. It is clearly seen that the mixing rate decrease with an increase of Ha constant.


6-153

0

0.05

0.1

0.15

0.2

0.5 0.6 0.7 0.8 0.9 1 1.1

Mix

ing

Rat

e, k

(1/s

ec)

Pitch ratio S/D

Hp= 5.54x10-18

Hp= 5.54x10-17

0

0.1

0.2

2 3 4

Mix

ing

Rat

e, k

(1/s

ec)

Blade Number

Hp= 5.54x10-18

Hp= 5.54x10-17

Figure 6.3 Dependence of mixing rate on pitch ratio or blade number at different particle

cohesion.

6.5.1.1.2 Coordination number

Overall coordination number gives an indication of expansion of the particle bed

within the vessel. Figure 6.4 shows the total (overall) coordination numbers as a function of

Ha and Pitch ratio S/D . The results show the total coordination number increases in the

mixer with pitch ratio 0.56 and 1.12, it means that the particles compacted in middle or two

halves and increased total coordination number, thus the mixing deteriorated which is can be

confirmed in mixing curve as shown in Figure 6.4. This is also can be investigated from

snapshots of particle flow when the mixer with the pitch ratio 1.12 at different cohesion at

40th rev as shown in Figure 6.12. The uniformity of mixing deteriorated when the mixture

cohesion increased as shown in Figure 6.5 (a) and (b) . Figure 6.4 (b) shows the total

coordination number decreases up to pitch ratio 0.9, and then increases while the pitch ratio

incremented. The Total coordination number increased when the cohesion decreases with the

pitch ratio range between 0.62 and 1.08. It implies that the geometric dimension with pitch

ratio in this range, the total coordination number decreases with an increase of cohesion of the

mixture due to the porous structure enhanced with an increase of cohesion, thus there are less

contact between particles and no particle compact in the mixer , produce premium particle

mixing.


6-154

2.6

2.8

3.0

3.2

3.4

3.6

3.8

10-18

10-17

S/D=1.12

S/D=0.75

S/D=0.56

Tota

l Coo

rdin

atio

n N

umbe

r

Hp(J)2.0

2.5

3.0

3.5

4.0

4.5

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

Hp = 5.54x10-18

Hp = 5.54x10-17

Tota

l Coo

rdin

atio

n N

umbe

r

Pitch ratio S/D

(a) (b)

Figure 6.4 Comparison of total coordination number: (a) Total coordination number

as a function of Hap; (b) Total coordination number as a function of pitch ratio S/D

(a) Hap = 5.54 x 10-18 J (b) Hap = 5.54 x 10-17

Figure 6.5 Snapshots of particle flow in the mixer for 1.12 pitch ratio at different

cohesion (at 40th revolution.)

(a) Hap = 5.54 x 10-18 J (b) Hap = 5.54 x 10-17

Figure 6.6 Snapshots of the particle flow in the mixer at pitch ratio 0.75 at different

cohesion at 40th revolution


6-155

Figure 6.6 shows the snapshots of particle flow in the mixer with pitch ratio 0.75 at

different cohesion. The uniformity of mixing deteriorated with an increase of cohesion of the

mixture. This is concurring with the Figure 6.4 that the cohesion leads the porosity increase

in the mixture, therefore the coordination number decreases with an increase of the cohesion.

6.5.1.1.3 Flow pattern

To investigate the particle flow at different cohesion and different pitch ratios, the

time and cell averaged paricle velocities are obtained. Note that the mixer space is divided

into cubic cells of the size 40×40×40 mm3, and the velocities of the particles in each cell are

averaged over a time interval of 2s. Figure 6.7 shows the average velocity of the particles in

a longitudinal section between x=-40 and 40mm for different pitch ratio and different

cohesion. Figure 6.7 shows the average velocity Vz, Vy and |V| decreases with an increase of

particle Ha constant in the mixer with a different pitch ratio respectively. However the

concentration of average velocity can be seen in the upper part of the vessel as a red color

when cohesion increased with a higher pitch ratio.

S/D=1.12 S/D=0.75 S/D=0.56

|V|

14

13

12

11

10

9

8

7

6

5

4

3

2

1

Frame 001 13 Jul 2013

Frame 001 13 Jul 2013 Frame 001 13 Jul 2013 Frame 001 13 Jul 2013


Hap = 5.54 x 10-18

Hap = 5.54 x 10-17

Figure 6.7 The averaged velocity in the mixer at a different pitch ratio S/D and Hap;

x =-40–40, y= -260– 260, z= 0 –1000, rev 30th

The particle average velocity shows that the particle movement is symmetric, and the

high velocity located upper and middle part of the vessel. The most particle mixing

y Vz

V


6-156

movement happened in the middle section of the ribbon with pitch ratio 1.123. The particles

flow or average of Vz and Vy can be seen from very right to the left part of the vessel with

pitch ratio 0.75. The red color located upper and very right part of the vessel represents the

higher average velocity of particles due to pushed by the blades which is tilted to the left axial

direction of the impeller. This blade arrangement and dimensions of the mixer perform higher

mixing rate and homogeneity of the mixture due to particle flow enhanced by the axial and

radial movement of the particles which is initially loaded side by side.

The particle flow in the ribbon mixer with pitch ratio 0.56 as follows: the particle flow

moving from middle to the right and left part of the vessel, show less homogeneity due to the

division of the particle flow into two parts.

6.5.1.1.4 Probability density of velocity and forces

To quantify the pitch ratio effect on particle velocity and force, probability density

distribution of the instantaneous velocity and cumulative probability of instantaneous force

are considered in the mixer with different pitch ratio in Figure 6.8 and Figure 6.9 respectively.

The range of the velocity between -1.26 m/s and 3.25 m/s. The increased number of particles

receives an incremented of radial velocity in the ribbon impeller as shown Figure 6.8 (a). The

radial velocities have not affected by the pitch ratio significantly, however, the axial velocity

of particles increased with an increment of pitch ratio. Overall, mean tangential velocity

higher than the axial and radial velocity.

Figures 6.9 (a) and (b) show respectively the cumulative probability distribution of

normal and tangential forces at 30rev in the mixer with different pitch ratio. The probability

distribution curve of the normal and tangential force each shift to slightly high forces value in

the entire force range, when the pitch ratio increased. It indicates that particles experience

larger inter-particle forces when the pitch ratio increased.


6-157

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-1.262 0 1.262 2.525

S/D=1.12S/D=0.75S/D=0.56

Prob

abili

ty D

ensi

ty


0

0.1

0.2

0.3

0.4

0.5

0.6

-1.625 0 1.625 3.25

S/D=1.12S/D=0.75S/D=0.56

Prob

abili

ty D

ensi

ty

Tangential Velocity (m/s)

(a) (b)

0

0.2

0.4

0.6

0.8

-3 -1.5 0 1.5 3

S/D=1.12S/D=0.75S/D=0.56

Prob

abili

ty D

ensi

ty


-0.05

0

0.05

0.1

0.15

0.2

0.25

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

VrVtVz

Mea

n V

eloc

ities

of P

roba

bilit

y D

ensi

ty

S/D

(c) (d)

Figure 6.8 Probability density function velocities of particles in the mixer at different

pitch ratio S/D rev 30 , Hap = 5.54 x 10-18 J


6-158

0

0.2

0.4

0.6

0.8

1

10-3

10-2

10-1

100

S/D=1.12

S/D=0.75

S/D=0.56

Cum

ulat

ive

Prob

abili

ty

Normal Force(N)

0

0.2

0.4

0.6

0.8

1

10-5

10-4

10-3

10-2

10-1

100

S/D=1.12

S/D=0.75

S/D=0.56

Cum

ulat

ive

Prob

abili

ty

Shear Force(N)

(a) (b)

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

Fn

Ft

Mea

n Fo

rces

of C

umul

ativ

e Pr

obab

ility

S/D

(c)

Figure 6.9 The Probability density function of forces of particles in the mixer at

different pitch ratio S/D, rev 30, Hap = 5.54 x 10-18 J


6-159

6.5.1.2 Effect of blade width at different Ha

Effect of blade width is investigated using mixing index, mixing rate, flow pattern and

particle velocity and contact forces at different cohesion for the base case of type-1 impeller

(see Figure 6.1).

6.5.1.2.1 Mixing index and mixing rate

Figure 6.10 shows that the mixing performance of a type-1 impeller depends on the

blade width at a given Hap value. A larger blade width casuses a cohesive mixture to reach the

steady state quicker. However, such differences disappear with an incrase in the cohesiveness

of the mixture.

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Partt

ical

scal

e m

ixin

g in

dex,

M

Revolutions

W=10 mm

W=30 mm

W=20 mm

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Partt

ical

scal

e m

ixin

g in

dex,

M

Revolutions

W=10 mm

W=30 mm

W=20 mm

(a) Hap = 5.54 x 10-18 J (b) Hap = 5.54 x 10-17 J

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Partt

ical

scal

e m

ixin

g in

dex,

M

Revolutions

W=10 mm

W=30 mm

W=20 mm

(c) Hap = 5.54 x 10-16 J

Figure 6.10 Effect of blade width at different Hap for type-1 impeller


6-160

Above trends are also clear from the diagrams of mixing rates shown in Fig. 6.18.

Figure 6.11(a) shows the mixing rate k as a function of Hap at fixed blades widths W, and Fig.

6.18(b) shows that k as a function of W at fixed Hap. The two figures show that k is

deteriorated with an increase in Hap at a fixed W. Conversely, k increases with an increase in

W at fixed Hap. k can be represented by the equation, k=c1-c2log10 (Hap), where c1 and c2 are

linear functions of W which can be found from the results in Fig. 6.18(b).

0.0

0.0

0.0

0.1

0.1

0.1

10-18 10-17 10-16 10-15

Hap(J)

Mix

ing

Rat

e, k

(1/s

)

k=-c1-c2 log10

(Hap)

W=10

W=20

W=30

0.0

0.0

0.0

0.1

0.1

0.1

5 10 15 20 25 30 35Width, W

Hp = 5.54x10-18

Hp = 5.54x10-17

Hp = 5.54x10-16 M

ixin

g R

ate,

k (1

/s)

(a) (b)

Figure 6.11 Effect of blade width on mixing rate for a type-1 impeller: (a) mixing rate

as a function of Hap; and (b) mixing rate as a function of blade width W.

Below, the effect of the blade width is investigated by examining the particle flow in

the mixer.

6.5.1.2.2 Particle flow

The particle flow is investigated by obtaining the time averaged velocity in the mixer.

To do so, the mixer space is divided into cubic cells of the size 40x40x40 mm3, and the

velocities of particles in each cell are averaged over a time interval of 2s (or 3.3 rev.). Figure

6.12 shows the time and cell averaged particle velocities in the longitudinal section of the

mixer obtained at the 30th shaft revolution. Particle motion in the axial direction is important

for the mixing process because the particles are initially deposited side-by-side arrangement


6-161

in the axial diection. The colour shows the magnitude of the particle velocity. Generally,

particles from both ends are transported towards the middle of the vessel, where the mixing

happens. At a given cohesion, particle velocity in the longitudinal section increases with an

increase in the blade width. However, at a given blade width, with an increase in the particle

cohesion, velocity in the longitudinal section decreases sharply, which is consistent with the

reduction in mixing rate with the cohesion. Also, the regions where the magnitude of velocity

is high gradually gets shifted to the ends of the vessel with the increase of cohsesion. At high

cohesion and large blade width, the high velocity region mostly is located at the vessel ends,

and mixing becomes deteriorated as a result.

The effect on the velocity field in the particle bed is quantified below using the

probability density distributions of particle velocity.

W=10 W=20 W=30


|V|

15

14

13

12

11

10

9

8

7

6

5

4

3

2

1

Frame 001 15 Jul 2013



Hap = 5.54 x 10-18

Hap = 5.54 x 10-17

Hap = 5.54 x 10-16

Figure 6.12 Time and cell averaged particle velocities in the mixer at different W and

Hap in the longitudinal section, where x values range between -40 and 40 (at 30th rev.)


6-162

6.5.1.2.3 Velocity and force analysis

To quantify the blade width effect on particle velocities and contact forces, probability

density distributions of instantaneous velocity and force components are considered in Figures

6.13 and 6.14, respectively. The distributions of radial velocity Vr are shown in Figure

6.13(a). The mean values of Vr are seen to be close to zero as expected because the vessel is a

closed system. To show the effect of W on Vr, the mean of the absolute Vr values are shown

in Figure 6.13(d) as a function of W. It shows that radial particle velocities increase with the

blade width. Similarly, the tangential velocity Vt of the particles increase when the blade

width increases as shown in Figures 6.13(b) and (d). It implies that the number of particles

with a negative Vt decreases with an increase in W, which is clear from the right-shift in the

distribution curves in Figure 6.13(b). Thus, there is a reduction in the recirculation flows in

the vessel cross-sections (or in x-y plane) with an increase in the blade width. Axial velocity

distributions are shown in Fig. 6.13(c), and their mean values are closed to zero as expected

because of the symmetry in the closed vessel. However, if we consider the mean of the

absolute Vz values, it increases with an increase in W as shown in Figure 6.13 (d). This can be

also observed from the flow patterns in Figure 6.12, whic shows that particles are gaining

higher axial velocities with an increase in the blade width. Thus, all velocity components are

increased when the blade width is increased.

The effect of the blade width on contact forces are analyzed below using cumulative

probability distributions of contact forces for a cohesive mixture (Bo=0.02) in the case of a

type-1 mixer. Figure 6.14(a) shows the cumulative probability distributions of normal contact

forces and those of shear contact forces in Figure 6.14(b). The figures show that the

distributions curves shift to the high force range in both cases with an increase in the impeller

blade width. Figure 6.14 shows that both normal and shear contact forces decrease slightly at

W=20mm, but increase again for W>20mm. Thus, W=20mm appears to be an appropriate

value as the blade width in order to reduce particle stresses. However, if one wants a higher

mixing rate, W=30mm is appropriate because the contact forces are then the same as at

W=10mm.


6-163

0

0.2

0.4

0.6

0.8

-1.375 0 1.375 2.75

W=10W=20W=30

Prob

abili

ty D

ensi

ty

Radial Velocity (m/s)0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-1.625 0 1.625 3.25

W=10W=20W=30

Prob

abili

ty D

ensi

ty


(a) (b)

0

0.2

0.4

0.6

0.8

-2.5 -1.25 0 1.25 2.5

W=10W=20W=30

Prob

abili

ty D

ensi

ty


0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

5 10 15 20 25 30 35

Mea

n of

Vt , (

m/s

)

Mea

ns o

f abs

olut

e va

lues

of V

r and

Vz, (

m/s

)

Blade width, W (mm)

Vt

Vz

Vr

(c) (d)

Figure 6.13 Probability density distributions of particle velocity components in the

mixer at different blade widths W at Hap = 5.54 x 10-18 J (Bo=0.02) and at the 30th rev

for type-1 mixer.


6-164

0

0.2

0.4

0.6

0.8

1

10-3

10-2

10-1

100

W=10

W=20

W=30C

umul

ativ

e Pr

obab

ility

Normal Force (N)

0

0.2

0.4

0.6

0.8

1

10-4

10-3

10-2

10-1

100

W=10W=20W=30

Cum

ulat

ive

Prob

abili

ty

Shear Force(N)

(a) (b)

0

0.02

0.04

0.06

0.08

0.1

5 10 15 20 25 30 35

FnFtM

ean

Con

tact

For

ces (

N)

Blade Width (mm)

(c)

Figure 6.14 Cumulative probability density distributions of velocities of particles in

the mixer at different blade width W, Hap = 5.54 x 10-18 J and rev 30 for type-1 mixer.


6-165

6.5.1.3 Effect of blade clearance at different Ha

The effect of the clearance between the impeller blades and vessel wall on the mixing

behavior of particles are studied here. To do so, mixing index, mixing rate, flow pattern, and

quantification of velocities and forces of particles are considered as in other sections.

6.5.1.3.1 Mixing quantification

The particle scale mixing index as a function of a time diagram obtained by an

increase of blade clearance and particle cohesion as shown in the Figures 6.15 (a), (b)and (c).

The results show that the higher mixing rate and homogeneity can be obtained with a small

and high of clearance. The mixing in the horizontal ribbon mixer with the medium clearance

needed a the longest time to reach the steady state mixing as shown in the Figure 6.15. The

mixing index shows that the poor mixing rate and homogeneity are obtained in the mixture in

ribbon with 15mm clearance, which is the same as the particle diameter as shown in Figure

6.15. Related to the effect of the clearance, in the casse of the helical ribbon mixer, it is

reported that the mixer with smaller clearance has a grater efficient, and the mixer with a large

greater clearance has a lower power consumption (Rieger et al., 1986).

The mixing rate is calculated using first order equation Eq (6.4). Figure 6.16 (a)

shows that the mixing rate, k as a function of the impeller clearance C at different cohesion. k

decreases and increases when the particle with lower cohesion. The impeller clearance

slightly affects on the mixing rate when the particles have a higher cohesion. Figure 6.16(b)

shows the mixing rate, k as a function of Ha ad it shows that the mixing rate decreased

linearly in the impeller with clearance 8.5mm and 27mm. Mixing rate does not change with

an increase of Ha for the clearances 15mm and 22mm.


6-166

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Partt

ical

scal

e m

ixin

g in

dex,

M

Revolutions

C=8.5 mmC=27.0mm

C=22.0mm

C=15.0mm

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Partt

ical

scal

e m

ixin

g in

dex,

M

Revolutions

C=8.5 mm

C=27.0mm

C=22.0mm

C=15.0mm

(a) Hap = 5.54 x 10-18 J (b) Hap = 5.54 x 10-17 J

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Partt

ical

scal

e m

ixin

g in

dex,

M

Revolutions

C=8.5 mm

C=27.0mm

C=22.0mm

C=15.0mm

(c) Hap = 5.54 x 10-16 J

Figure 6.15 The effect of blade clearance at different cohesion (type-1 impeller)


6-167

0.00

0.02

0.04

0.06

0.08

0.10

5 10 15 20 25 30

Hap = 5.54x10-18

Hap = 5.54x10-17

Hap= 5.54x10-16

Impeller Clearance, C

Mix

ing

Rat

e, k

(1/s

ec)

0.00

0.02

0.04

0.06

0.08

10-18 10-17 10-16 10-15

C=8.5C=15C=22C=27

Hap(J)

Mix

ing

Rat

e, k

(1/s

ec)

(a) (b)

Figure 6.16 Comparison of mixing rate: (a) Mixing rate as a function of Hap;

(b) Mixing rate as a function of clearance C

6.5.1.3.2 Spatial- average velocity

In order to obtain the time-averaged longitudinal velocity distribution in the mixers,

the mixer space is divided into cubic cells of the size 40x40x40mm3, and the velocities of

particles in each cell are averaged over a time interval of 2s. Figure 6.17 shows that the

average velocity of the mixture in the ribbon mixer with different blade clearance. It is

investigated from a cross section snapshot of the ribbon that the more particles pushed by the

blade or the particle flow when the clearance either small (8.5mm) or greater (27mm), thus

the particles receive reduced velocity, with a medium the blade clearance 15mm and 22mm.

The particle average velocities decrease significantly with the increase of particle’s Ha

constant. The particles received higher average Vx at the lower part of the ribbon mixer with

lower clearance in the snapshot longitudinal section of the ribbon mixer. Particle axial

movement was enhanced in the ribbon impeller with clearance 8.5mm and 27mm. The reason

for the particles receive higher the mixing rate in the mixer with clearance 8.5 mm probably is

that the particles pushed by the blade at the lower part of the vessel enable particles receive

higher Vx, thus the circumferential and axial movement of the particles were enhanced, and

the mixing rate and homogeneity of the mixture were improved. The particles receive higher

Vz and Vy, therefore the axial flow and homogeneity of the particles were enhanced in the

mixer with clearance 27mm, which is can be confirmed by mixing index in Figure 6.15.


6-168

C=8.5

C=15

C=22

C=27

Frame 001 16 Jul 2013

Vx

1

0

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

Frame 001 16 Jul 2013

Vx

1

0

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

Frame 001 16 Jul 2013

Vx

1

0

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

Frame 001 16 Jul 2013

Vx

1

0

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

Frame 001 16 Jul 2013

Vx

1

0

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

Frame 001 16 Jul 2013

Vx

1

0

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

Frame 001 16 Jul 2013

Vx

1

0

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

Frame 001 16 Jul 2013

Vx

1

0

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

Frame 001 16 Jul 2013

Vx

1

0

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

Frame 001 16 Jul 2013

Vx

1

0

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

Frame 001 16 Jul 2013

Vx

1

0

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

Frame 001 16 Jul 2013

Vx

1

0

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

Frame 001 16 Jul 2013

yz

x

Hap = 5.54 x 10-17 J Hap = 5.54 x 10-16 JHap = 5.54 x 10-18 J

Figure 6.17 The averaged velocity of particles in the mixer at different blade width

W and Hap ; x =-40-- 40, y= -260– 260 z= 0 --1000 , 30th rev


6-169

6.5.1.3.3 Velocity and force components

To understand the effect of blade clearance on the particles velocities and forces, the

probability density distribution of instantaneous velocity and force components are considered

here. Figure 6.18 shows the increased number of particles received higher radial, tangential

and axial velocities in the mixer with clearance 8.5mm and 27mm. That is can be confirmed

from Figure 6.17 that the particle axial and radial flow were enhanced in the impeller with

clearance 8.5mm and 27mm. The effect of blade clearance on the contact forces are analyzed

using cumulative probability distributions of contact forces in type 1 impeller. Figure 6.19

shows that the particles encounter higher normal and shear forces in the mixer with clearance

27mm and 8.5mm. The cumulative probability curves of the particles shift left when the

mixer clearance 15mm and 22mm. It implies that the particles receive the smaller normal and

shear forces. These results can be supported by the Figure 6.15 and 6.18(c).


6-170

0

0.5

1

1.5

-0.817 0 0.817 1.634

C=8.5C=15C=22C=27

Prob

abili

ty D

ensi

ty

Radial Velocity (m/s)0

0.2

0.4

0.6

0.8

1

-1.75 0 1.75 3.5

C=8.5C=15C=22C=27

Prob

abili

ty D

ensi

ty


(a) (b)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-1.5 0 1.5 3

C=8.5C=15C=22C=27

Prob

abili

ty D

ensi

ty


(c)

Figure 6.18 The Probability density function of velocities of particles in the mixer

with different blade clearance C, Hap = 5.54 x 10-18 J, rev 30


6-171

0

0.2

0.4

0.6

0.8

1

10-4

10-3

10-2

10-1

100

C=8.5C=15C=22C=27

Cum

ulat

ive

Prob

abili

ty

Normal Force(N)

0

0.2

0.4

0.6

0.8

1

10-4

10-3

10-2

10-1

100

C=8.5C=15C=22C=27

Cum

ulat

ive

Prob

abili

ty

Shear Force(N)

(a) (b)

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

5 10 15 20 25 30

Fn

Ft

Mea

n Fo

rces

of C

umul

ativ

e Pr

obab

ility

Blade Clearance(mm)

(c)

Figure 6.19 Probability density function of forces of particles in the mixer at different

blade clearances C, Hap = 5.54 x 10-18 J, rev 30th

The effect of clearance, width, pitch ratio and Ha on mixing index at 30th rev

combined into one mixing curve. Figure 6.20 shows that the mixing performance at different

cohesion of the particles is similar.


6-172

0.0

0.2

0.4

0.6

0.8

1.0

C=15 C=27 W=20 S/D=0.56 S/D=1.12M

ixin

g In

dex

at 3

0 re

v

Clearance _Width_Pitch ratio

Hp = 5.54x10-18

Hp = 5.54x10-17

Hp = 5.54x10-16

Hp = 2.22x10-15

Figure 6.20 Mixing index at 30rev as a function of blade clearance, width, pitch ratio

and Hap


6-173

6.4.1.4 Effect of blade angle

The effects of outer and inner blade angle are investigated using inner and outer

horizontal ribbon blender mixer by means of DEM method. The Particle Scale mixing Index

shows that the inner angles have not any effect on the homogeneity of the cohesive particle

mixture. However, the outer ribbon angles have a slight influence on the uniformity of the

cohesive particle mixture in ribbon mixer. The mixing rate increased with an increase of the

inner blade angle from 45 to 70 as shown in the Figure 6.21.

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80

=45?=50?

=80o

Parti

cle

Scal

e M

ixin

g In

dex

Revolutions

0

0.02

0.04

0.06

0.08

0.1

40 50 60 70 80 90

Mix

ing

Rat

e, k

(1/s

ec)

Outer blade angle, ` (a) (b)

Figure 6.21 Effect of outer rake angle (inner blade angle is fixed at 45°):

(a) Mixing index variation with shaft revolutions; (b) Mixing rate k as a function of

outer blade angle


6-174

6.5.2 Comparison of type 1 and type 2 impellers

To investigate the effect of blade number (or pitch ratio) on homogeneity and mixing

rate, two ribbon designs (type-1 and 2 impellers) are chosen as shown in Figure 6.1. In the

study, non-cohesive particles are used, and the effect is analyzed based on mixing index,

mixing rate and velocity fields. To demonstrate the blade number effect (or pitch ratio), the

performances of the two types of impellers are compared below at fixed pitch ratios (or blade

numbers) and blade widths in the sections 6.4.2.1 and 6.4.2.2, respectively.

6.5.2.1 Blade number (or pitch ratio) effect

Figures 6.22(a) and (b) show the instantaneous particle-scale mixing index M, as a

function of shaft revolutions for different pitch ratios at a fixed blade width of 20mm. The

figures show that the pitch ratio affects the mixing performance of both impellers significantly.

Mixing rate k obtained from Eq. (6.4) is shown in Figure 6.22(c) for both impellers. At a

given pitch ratio, type-2 impeller shows faster mixing compared to the type-1 impeller except

at 1.12 pitch ratio. Generally, both impellers show a peak mixing rate at certain pitch ratio, the

peak being higher for the 4-blade impeller. The peak occurs at about 0.75 patio for this

impeller. The high mixing rate of 4-blade impeller is understood from the fact that the steady

state is reached in about 10 shaft revolutions, compared to 30 revolutions of the type-1

impeller at 0.75 pitch ratio (see Figsures 6.22(a) and (b)). The particle mixing is significantly

delayed with a pitch ratio of 0.56, especially in the case of the type-1 impeller. However, the

mixtures reach the well-mixed state within 100 revolutions with both impellers at higher pitch

ratios such as 0.748 and 1.123, but delaying further at lower pitch ratios such as 0.561.

In fact, Figure 6.22(c) also shows the effect of blade number at a given pitch ratio for a

fixed blade width. Increasing the blade number from 2 to 4 at a fixed pitch ratio and width

results in an increase in the mixing rate as seen from the diagram for over most of the range of

the pitch ratio tested. Having 4 blades is most efficient for the mixing operation when the

pitch ratio is about 0.75. The reason for this increase with 4-blades will be investigated in the

section of the velocity field below.


6-175

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Partt

ical

Sca

le M

ixin

g In

dex

Revolutions

S/D=0.561

S/D=0.748

S/D=1.123

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Partt

ical

Sca

le M

ixin

g In

dex

Revolutions

S/D=0.561

S/D=0.748S/D=1.123

(a) Type-2 impeller (b) Type-1 impeller

0

0.2

0.4

0.6

0.8

1

1.2

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

Mix

ing

Rate

, k (1

/sec

)

Pitch ratio, S/D

Type-1 impellers

Type-2 impellers

(c) Mixing rate

Figure 6.22 Effect of blade number (or pitch ratio S/D) on mixing performance for

non-cohesive mixtures. (a) type 2 impeller, (b) type 1 impeller (c) mixing rate k as a

function of pitch ratio.

To understand the pitch ratio effect on the particle flow, the time and cell averaged

particle velocities are shown in Figure 6.23 in the central longitudinal planes for the two

bladed mixer at the 38th revolution (or at t=13s). In order to obtain the time-averaged

longitudinal velocity distribution in the mixer, the mixer space is divided into cubic cells of


6-176

size 40x40x40mm3, and the velocities of particles in each cell are averaged over a time

interval of 2s. Figure 6.23 shows that the particle flow is significantly affected by the pitch

ratio S/D: the mixing behavior for S/D=1. 123 is symmetric with respect to the mid-plane

perpendicular to the shaft, with particles moving towards the vessel center from both ends,

producing steady mixing as shown Figure 6.23(a). The particles are pushed towards the center

by the blades from the left and right of the mixer for S/D=0.748, increased the particle axial

movement and produced higher mixing rate as shown in Figure 6.23(b). The particle flow in

the mixer with pitch ratio 0.56 (Figure 6.23 (c)) shows that the particles pushed primarily to

move in the left and right halves of the mixer, therefore the flow divided into two parts

decreased homogeneity of the mixture, due to the blade arranged backward tilted from middle

to two sides of the mixer. Porosity over impeller seems to increase with a decrease in pitch

ratio.

Frame 001 11 Mar 2013 Frame 001 11 Mar 2013

S/D=0.561S/D=0.748S/D=1.123

Frame 001 11 Mar 2013

(a) (b) (c)

Figure 6.23 Average velocity and porosity of the particle mixture, snap shot at

revolution at 38, t=13s,X =-20-20, at ZY plane in type-1 impeller

For comparison the flow in type-1 ribbon mixer and in the type-2 impeller, the particle

flow in four blade mixer being investigated. The results show that the flow pattern is very

similar to the flow pattern in the two bladed impellers. Snapshots of the particles in Figure

6.24 show that there is particle recirculation movement in the lower part of the vessel with

pitch ratio 0.561 which has contributed to the division of particle flow. This phenomena

decreased with an increase of the pitch ratio that the particles pushed by blade located in the

very right easily moved to left producing higher mixing rate.


6-177

S/D=0.748

S/D=0.561

S/D=1.123

F ram e 0 0 1 2 8 Aug 2 0 1 2

F ram e 0 0 1 2 8 Aug 2 0 1 2

F ram e 0 0 1 2 8 Aug 2 0 1 2

Frame 001 11 Mar 2013

Frame 001 11 Mar 2013

Frame 001 11 Mar 2013

Central cross-section Z=400-450 mm Central longitudinal section(width=40mm) Snapshots at the 38th revolution (t=13s)

Figure 6.24 Velocity fields and snapshots of the mixing states at steady-state for the

type-2 impellers of different pitch ratios, blue and red represent the two types of

particles initially laid in the side-by-side arrangement

6.5.2.2 Blade width

The effect of blade width on the uniformity of particle mixing is investigated for the

two type of impellers below: the blade width is varied as 20, 40 and 60 mm. Figure 6.25

shows the particle scale mixing index M, as a function of revolutions at different blade width

in two and four bladed mixer. The particle mixing homogeneity and mixing rate are higher in

type 2 mixer than type 1 impeller. The particle mixing index diagram show that the mixing

rate and uniformity of particle mixing in ribbon mixer with the width 40mm are higher than

that in the particle mixing in the ribbon with width 20 and 60mm as shown in Figure 6.25 (a)

and (b). The particles in the impeller with blade width 60mm perform higher mixing rate but

lower steady state mixing index compared to the mixing behavior in the mixer with blade

width 20mm in the type 1 mixer. The mixing rate k calculated using Eq (6.4). Figure 6.25 (c)

shows the mixing rate, k as a function of blade width show that the mixing rate in type 2

impeller higher than that in the two blade mixer.


6-178

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Partt

ical

Sca

le M

ixin

g In

dex

Revolution

W=60

W=40

W=20

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Partt

ical

Sca

le M

ixin

g In

dex

Revolutions

W60

W40

W20

0

0.02

0.04

0.06

0.08

0.1

10 20 30 40 50 60 70

Mix

ing

Rat

e, k

(1/s

ec)

Width (mm)

2 blade impeller

4 blade impeller

(a) type-2 (b) type-1 (c) Mixing rate

Figure 6.25 Effect of blade width on mixing behaviour at a pitch rratio of 1.12.

The average velocity and porosity counter in the two blade ribbon show that the

average velocity toward the middle section of the ribbon and the recirculation happened in the

bottom part of the mix; the upper part of the mixer has large porosity where less particle areas

in the ribbon mixer with 3 different widths as shown Figure 6.26(a). However the magnitude

of the vector in the ribbon width 40mm width slightly higher compare to the other ribbon.

The high porosity regions, red color, located at the right and left lower part of the ribbon

mixer with width 60mm compare to the other two ribbon mixer.

Frame 001 07 Mar 2013 Frame 001 07 Mar 2013 Frame 001 07 Mar 2013

W40W20 W67

Porosity

0.95

0.9

0.85

0.8

0.75

0.7

0.65

0.6

0.55

0.5

Frame 001 07 Mar 2013

Frame 001 06 Mar 2013 Frame 001 06 Mar 2013 Frame 001 06 Mar 2013

Revolution 80 , t=25s, X =-20-20

W=20 W=40 W=60

Figure 6.26 Mixing behaviour for different blade widths for type-1 impeller design. Top:

Steady-state velocity fields and spatial distribution of void fraction in the longitudinal central

plane; and bottom: snapshot of particles at 80threvolution (t=25s) in the longitudinal central

plane, with blue and red representing two types of particles.

The real mixing can be seen Figure 6.26 (b) red and blue represents of two types of

particles. It is shown that there is significant white space right and left part of the mixer with

60 mm wide, slightly less in the ribbon mixer with 40mm wide as shown in Fig.6.26(b). The

(

a)

(

b)


6-179

results show that the mixing rate and homogeneity of the mixture in ribbon with 40 and 20mm

width significantly higher than that in the mixer with a width 60mm which is explained in the

particle mixing index in the Figure 6.25.

Figure 6.27 shows the velocity fields in the type-2 ribbon mixer that the particle

cascaded and then pushed by the blade in axial direction of the middle part of the vessel,

enhanced axial movement of particles when the blade width W=40mm which is promoting

uniformity and mixing rate of the particles. The particle circumferential flow of the local area

increased with an increase of blade width, which is delaying the mixing rate.

Frame 001 29 Aug 2012

W=40

W=20

W=60

Frame 001 29 Aug 2012

Frame 001 29 Aug 2012

Frame 001 29 Aug 2012

F ram e 0 0 1 2 9 Aug 2 0 1 2

F ram e 0 0 1 2 9 Aug 2 0 1 2

Central cross-section Z=400-450mm Central longitudinal section(width=40mm) Snapshots at the 40th revolution (t=13.6s) Figure 6.27 Velocity fields and snapshots of the mixing states at steady-state for the

type-2 impellers of W=20, 40 and 60 mm: blue and red represent the two types of

particles initially laid in side-by-side arrangement.

The effect of pitch ratio and a blade width of the type-1 and type-2 ribbon on the

mixing index, mixing rate and particles flow show that the blade number affects the mixing

behavior significantly. Type-2 ribbon impeller is a more efficient mixer than the type-1 mixer.


6-180

6.6 Conclusions

The effects of blade width, pitch angle, pitch ratio, clearance of the impeller and the

number of blades on cohesive particle mixing in a ribbon mixer are investigated here for two

impeller configurations, i. e., a 2-bladed and 4-bladed ribbon impellers by means of DEM.

Increased blade number enhances the mixing homogeneity, mixing rate, axial flow and

tangential flow. Type 2 mixer shows higher mixing capabilities compared to the type-

1 mixer.

Pitch ratio affects on the flow pattern of the mixture significantly. The convective and

circumferential flow are enhanced by particles with higher axial, tangential and radial

velocities when the pitch ratio moderate or high. Particles primarily move in the right

and left halves of the mixer. Particles encounter higher axial, radial forces, because of

there are more blades in the impeller with S/D=0.56 compare to the other impeller.

Blade width affects the mixing rate, velocity and force of particles. The mixing rate

increases with an increase of blade width. The wider blade width enables particles

have higher velocities in the axial and, tangential directions and higher radial force.

Blade clearance affects the mixing rate significantly. The impeller with larger

clearance enables particles to get involved in circumferential flow pushing particles to

the outer periphery of the vessel, and receiving higher axial, tangential and radial

velocities and forces, thus increasing the mixing rate. The mixing rate deteriorated

when the impeller clearance C is =15mm which is same as particle diameter. In this

case, particles probably cannot pass through the blade clearance and to involved the

circumferential flow reducing the mixing rate.

Nomenclature

D Vessel diameter, (m)


E You g’ mo u u N/m2

Fc,ij Contact force vector between i and j, N


6-181

Fd,ij Damping force vector between i and j, N

Fv, ij Cohesive force between particles i and j


h Gap between two interacting particles, (m)

Ha Hamaker constant, J

Hap Hamaker constant of particle material, J

Haw Hamaker constant of wall material, J


k Mixing rate, (s-1)

ki Number of particles in contact with particle i


Mij Vector of rolling friction torque on particle i, (Dury et al.)


n Actual particle number fraction of the two types of particles, (-)

Ni Number of particles in the immediate neighbourhood of particle i

Nl Number of large particles

Np Total number of particles of one type in the mixture

Ns Number of small particles

Nt The number of sample points of instantaneous average values in time t




R Equivalent radius, m



fraction of p, (-)


Tij Vector of rolling friction torque on particle i, N m


Vi Velocity of particle i, m/s

Z Height from vessel base(mm)


6-182

Greek letters

α Volume fraction

Average particle-scale sample size for the mixture

θ Radial section measured horizontally from mid-plane of the blade


Density of particles of a uniform system, ( 3kgm )



CHAPTER 7 A Study of Binary Particle Mixtures in a Ribbon Mixer

7-183

Chapter 7

A Study of Binary Particle Mixtures in

a Ribbon Mixer


7-184

7.1 Introduction



a wide verity of high quality products are produced. The selection of a mixer for a

mixing operation depends on the product uniformity required and many other factors

including the ease with which it can be cleaned (Poux et al., 1991). One type of mixers

often used in mixing operations in pharmaceutical industry is the ribbon mixer, which

has a ribbon impeller with two or more helical blades starting from each end, but

spiraling in opposite directions towards the vessel centre. Thus, the blades on one end of

the vessel push particles axially towards the centre and those at the other end push in the

opposite direction towards the centre thus particles being mixed at the vessel centre. In

this process, material is stretched tangentially and conveyed to and fro axially by the

opposite blades, resulting in stretching and dispersion of material. The helical blades

makes cleaning the mixer difficult because of their complicated design; cleaning is

important when dealing with different mixture compositions.

The mixer can be used for mixing a wide range of materials, from dry powders

to pastes. In this type of mixers, radial mixing is good, but axial mixing is less so

(Metzger and Glasser, 2013). A ribbon mixer displays a combination of mixing

mechanisms such as convection, diffusion and shear when processing granular matter.

Therefore, ribbon mixers are considered to be suitable for mixing of dry powders as

well as free-flowing granular material (Poux, Fayolle et al. 1991). It is also reported that

a ribbon mixer can produce an improved homogeneity in powder mixing because of

large shear stresses in the mixer. Ribbon mixers have been examined in the literature in

the following aspects for powder mixing relationship between power consumption and

blade speed ((Masiuk, 1987); energy saving (Dieulot, Delaplace et al. 2002); mixture

uniformity (Côté and Abatzoglou 2006); loading method (Côté and Abatzoglou, 2006),

stresses, homogeneity and mixing performance in the case of cohesive mixtures, effect

of fill level on the different impellers (Muzzio, Llusa et al. 2008); and fill level (Cite

and Abatzoglou 2006).

It is also reported that a ribbon mixer can handle mixing of different sized

particles (Muzzio, Llusa et al. 2008). Further, a mixture of particles with large density


7-185

and size differences can also be mixed in a ribbon mixer (Poux et al., 1991). Mixture

quality is significantly affected by size differences of powder particles, and an increase

in either the size or density differences results in increased segregation tendencies (Fan

et al., 1975; Stephens and Bridgwater, 1978). The effect of size and density on particle

mixing has been widely investigated for different mixers such as rotating drums

(Alonso et al., 1991; Eskin and Kalman, 2000; Jayasundara et al., 2012; Liu et al., 2013;

Metcalfe and Shattuck, 1996; Xu et al., 2010), a cylindrical mixer (Chandratilleke et al.,

2011; Zhou et al., 2003), and vibrated granular systems (Rosato et al., 1987; Shinbrot

and Muzzio, 1998; Yang, 2006). An optimum combination of the size and density

differences is also shown to exist (Alonso et al., 1991; Metcalfe and Shattuck, 1996;

Nitin et al., 2005; Zhou et al., 2003).

An expression has been developed to predict the transition from mixing to

segregation due to percolation and buoyancy effects under different particle densities

and sizes and volume fractions (Alonso et al., 1991). However, research on a

complicated mixer such as the ribbon mixer are few, and the effects of particle size and

density on the mixer performance have not been clearly established. The objective of

this study is, thus, to investigate the effect of size and density differences of particles on

the mixing behaviours of non-cohesive binary particle mixtures in a ribbon mixer, by

using simulations based on the discrete-element-method (Cundall and Strack, 1979).

This chapter is organized as follows. Sections 7.2 and 7.3 discusses the

simulation method and conditions, which is followed by the results and discussions

section, where in Section 7.4.1, the effect of density ratio is examined when the size

ratio is fixed. In Section 7.4.2, the effect of size ratio is examined when density ratio is

fixed. Next, a prediction equation is developed for the steady-state mixing index

relating it to size and density differences. Finally Section 7.5 summarizes the

conclusions of the chapter.


7-186

7.2 Method of Analysis


and validated by Zhou et al. (Zhou et al., 2004; Zhou et al., 1999). The model uses two


particle i in a system at time t subjected to the gravity g and interactions with the

neighboring particles, blade and walls:

ik

jijdijci

ii m

dtdm

1,, FFgV (7.1)

and

ik

jijij

ii d

dI1t

MT (7.2)

where mi, Ii, Vi and i are the mass, moment of inertia, translational and

rotational velocities of the particle respectively; k is the number of particles in contact

with particle i, Fc, ij represents the contact force which is the summation of the normal

and tangential forces. Fd,ij represents the damping force, which is the summation of the

normal and tangential damping forces at the contact point with particle j. Tij and Mij are

the torque and rolling friction torque on particle i due to particle j. Expressions for the

forces and torques in Eqs. (7.1) and (7.2) are given in Table 7.1.

7.3 Simulation Conditions

The geometry of the ribbon impeller used here is shown in Figure 7.2. The

impeller has two helical blades starting from each end, but spiraling in opposite

directions. The impeller is placed horizontally in a horizontal cylindrical vessel so that

their axes coincide. The dimensions of the impeller, vessel and other input values for

DEM simulations are shown in Table 7.2. The total volume of particles used in the

ribbon mixer is fixed and chosen the same as in the case of the cylindrical mixer used in

Chapter 3. For a fill level of 30% and assuming a void fraction of 0.36, the mixer length

and diameter were chosen. The blade width, pitch, angle and clearance were selected


7-187

based on the optimum values and in the same scale ratio as in the base ribbon mixer of

Chapter 6 (see Table 6.1). Simulations are conducted for the cases of non-cohesive

particles for investigating the effect of particle size and density.


The mixture quality or mixing index M is quantified by a particle scale mixing

index based on coordination number (see Chapter 3, section 2) (Chandratilleke et al.,

2012). To analyze the mixing performance of the mixers, particles of equal numbers are

deposited in the side-by-side arrangement in the axial direction, and the impeller is

rotated at a constant speed of 100 rpm. The size ratio rs is defined as ls dd / and density

ratio rd as ls / , where s and l refer to small and large particles respectively. Volume

fraction xl is the ratio of volume of large particles to the total particle volume. The

density of large particles ρl is fixed at 6040 kg/m3 , the diameter of the large particles is

fixed at 8mm in the study.

Forces and torques


Normal contact force

Normal damping force


Tangential damping force

Rotational torque



7-188

Figure 7.1. Geometry of ribbon impeller

Table 7.2 Input variables and their values

Input variable Value Vessel diameter Vessel length Blade pitch Blade width Shaft diameter Rake angle Blade gap Base Particle number Particle dia.,d (large) Shaft speed,ω Fill level Young’s modulus, E Poisson’s ratio , s(P-P) (P-W) R(P-P) (P-W) Time step

120 (mm) 200 (mm) 100 (mm) 11 (mm) 8.5 (mm) 45°, 135° 2.36 (mm) 18,000 8 ( mm) 100 rpm 30% 1108 N/m2 0.29 0.3 0.002 (m) 3.41E-05 (s)


7-189

Table 7.3 Particle information for density effect cases

Table.7.4 Particle information for size effect cases

case DL DS NL NS ρL ρs rs rd Vol

1 8 4.56 810 4374 6040 3624 0.57 0.6 0.5

2 8 4.88 810 3569 6040 5436 0.61 0.9 0.5

3 8 4.8 810 3750 6040 1812 0.6 0.3 0.5

4 8 4.8 810 3750 6040 2416 0.6 0.4 0.5

5 8 4.8 810 3750 6040 4530 0.6 0.75 0.5

6 8 4.56 162 7873 6040 3624 0.57 0.6 0.1

7 8 4.88 162 6423 6040 5436 0.61 0.9 0.1

8 8 4.8 162 6750 6040 1812 0.6 0.3 0.1

9 8 4.8 162 6750 6040 2416 0.6 0.4 0.1

10 8 4.8 162 6750 6040 4530 0.6 0.75 0.1

11 8 4.56 1458 875 6040 3624 0.57 0.6 0.9

12 8 4.88 1458 714 6040 5436 0.61 0.9 0.9

13 8 4.8 1458 750 6040 1812 0.6 0.3 0.9

14 8 4.8 1458 750 6040 2416 0.6 0.4 0.9

15 8 4.8 1458 750 6040 4530 0.6 0.75 0.9

case DL DS NL NS ρL ρs rs rd Vol

1 8 6.4 810 1582 6040 1812 0.8 0.3 0.5

2 8 6.24 810 1707 6040 1812 0.78 0.3 0.5

3 8 7.6 810 945 6040 1812 0.95 0.3 0.5

4 8 4.8 810 3750 6040 1812 0.6 0.3 0.5

5 8 3.2 810 12656 6040 1812 0.4 0.3 0.5

6 8 2.4 810 30000 6040 1812 0.3 0.3 0.5

7 8 6.4 162 2848 6040 1812 0.8 0.3 0.1

8 8 6.24 162 3072 6040 1812 0.78 0.3 0.1

9 8 7.6 162 1701 6040 1812 0.95 0.3 0.1

10 8 4.8 162 6750 6040 1812 0.6 0.3 0.1

11 8 3.2 162 22781 6040 1812 0.4 0.3 0.1

12 8 2.4 162 54000 6040 1812 0.3 0.3 0.1

13 8 6.4 1458 316 6040 1812 0.8 0.3 0.9

14 8 6.24 1458 341 6040 1812 0.78 0.3 0.9

15 8 7.6 1458 189 6040 1812 0.95 0.3 0.9

16 8 4.8 1458 750 6040 1812 0.6 0.3 0.9

21 8 3.2 1458 2531 6040 1812 0.4 0.3 0.9

22 8 2.4 1458 6000 6040 1812 0.3 0.3 0.9


7-190


7.4.1 Effect of density on binary particle mixing

7.4.1.1 Mixing performances

The effect of density ratio on the mixing performance is investigated for

different volume fractions, in each case density being varied with the size ratio fixed at

0.66 and density of large particles ρl =6040 kg/m3; these cases are listed in Table 7.3.

Figure 7.2 shows the effect of density ratio on mixing in mixing index M versus time

diagrams. The particle scale mixing index M decreases initially and reaches steady state

value after 20 revolutions at volume fraction 0.1 as shown Figure 7.2 (a). The

uniformity of mixing decreases with an increase of density ratio at the volume fraction

0.1 because of the dominance of the size effect. For the volume fraction of 0.5, the

mixing rate and steady state mixing index increase slightly and decreases with an

increment of density ratio as shown in Figure 7.2(b). The mixing rate and homogeneity

of the particle mixing at steady state are nearly the same at different density ratios when

the volume fraction is 0.9 as shown in Figure 7.2(c).

0.9

0.92

0.94

0.96

0.98

1

0 10 20 30 40 50 60

rd=0.3

rd=0.4

rd=0.6

rd=0.9Pa

rticl

e Sc

ale

Mix

ing

Inde

x

Revolutions

0.96

0.965

0.97

0.975

0.98

0.985

0.99

0.995

1

0 20 40 60 80 100

rd=0.6

rd=0.9

rd=0.3

rd=0.4

Parti

cle

Scal

e M

ixin

g In

dex

Revolutions 0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 20 40 60 80 100

rd=0.6

rd=0.9

rd=0.3

rd=0.4

Parti

cle

Scal

e M

ixin

g In

dex

Revolutions

(a) (b) (c)

Figure 7.2 Particle scale mixing index as a function of revolutions at different

volume fractions when rs=0.66 : (a) xl=0.1; (b) xl=0.5; (c) xl=0.9

Figure 7.3 shows the particle scale mixing index M at steady-state (or the

mixture quality as will be called below) as a function of density ratio and volume

fraction. The mixing index M at the steady-state will be called as the mixture quality


7-191

below. A higher mixture quality is obtained for the volume fraction of 0.5 than for other

volume fractions, and the density ratio rd is seen to be having no significant effect on M

at this size ratio except at 0.1 volume fraction, where there is a slight effect. However, in

the case of a vertically-shafted cylindrical mixer, the mixture quality showed a gradual

increase with a decrease in rd (Chapter 3, Figure 3.2). Therefore, the effect of rd on

mixture quality is somewhat different here compared to a cylindrical mixer. Figure

7.3(b) shows that mixture quality displays a peak value with increasing volume fraction

as in the case of the cylindrical mixer.

0.88

0.9

0.92

0.94

0.96

0.98

1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

xl=0.1

xl=0.5

xl=0.9

Parti

cle

Scal

e M

ixin

g In

dex

rd

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

0 0.2 0.4 0.6 0.8 1

rd=0.3

rd=0.4

rd=0.6

rd=0.9

Parti

cle

Scal

e M

ixin

g In

dex

xl

Figure 7.3 Effects of rd and xl on M at rs=0.66: (a), effect of rd at different xl ; and (b), effect of xl at different rd

7.4.1.2 Velocity and force analysis

To understand the effects of binary particle mixing further, contact forces and

velocity field are examined at the t=50s when mixing is steady. Figure 7.4 shows that

the cumulative probability distributions of normal and tangential contact forces at

different rd values. It shows that both normal and tangential contact forces increase with

an increase in the density ratio when size ratio is fixed at 0.66, even if the effect of

density ratio on mixing index was not significant.


7-192

0

0.2

0.4

0.6

0.8

1

10-6

10-5

10-4

10-3

10-2

10-1

100

rs=0.6; r

d=0.6

rs=0.61; r

d=0.9

rs=0.6; r

d=0.3

rs=0.6; r

d=0.4

Cum

ulati

ve P

roba

bilit

y

Normal Force (N)

0

0.2

0.4

0.6

0.8

1

10-6

10-5

10-4

10-3

10-2

10-1

100

rs=0.6; r

d=0.6

rs=0.6; r

d=0.9

rs=0.6; r

d=0.3

rs=0.6; r

d=0.4

Cum

ulat

ive

Prob

abili

ty

Tangentail Force (N) (a) (b) Figure 7.4 Effects of rd on contact forces at rs=0.66: (a), normal force ; and (b),

tangential force for a volume faction of 0.5.

Figure 7.5 shows instantaneous filed of particle velocity in vertical section at

three different axial positions Z, measured from one end of the vessel. Note that the

velocity vectors of small particles are coloured red, while those of large particles are

coloured blue in diagrams. The particle velocity increased with a decrease of the

density ratio when size ratio fixed at rs=0.6 as seen from the increase of arrow lengths.

The reason probably is that the mass difference of particles increases and small particles

are driven by momentum of large particles, increasing the overall velocity of particles.

The particles are pushed towards the middle part of the vessel by the blades, and

therefore, a larger amount of particles can be seen in the middle sections from the

velocity field. Flow patterns are mostly identical in the corresponding sections even if

the density is changed suggesting that the homogeneity of a mixture is not much

affected by the density ratio when size ratio is fixed at 0.6 and volume fraction at 0.5 as

also can be confirmed from Figure 7.3 (a).


7-193

Figure 7.5 Effects of rd on velocity field at rs=0.6 at different cylindrical height

for a volume faction xl of 0.5

Figure 7.6 shows the distributions of particle velocity components for different

density ratios when the size ratio is fixed. The tangential velocity shows a slightly

increase in velocity when density ratio is 0.6 which is the maximum mixing index for a

volume fraction of 0.5 in Figure 7.2(b). The radial and tangential velocities are mostly

not affected by the density change.

Frame 001 08 Nov 2013

rs=0.6; rd=0.6

Z=90-130


Z=40-60 Z=14-160


rs=0.6; rd=0.9

Frame 001 08 Nov 2013

Frame 001 08 Nov 2013

rs=0.6; rd=0.4



7-194

0

1

2

3

4

5

6

7

8

-0.2 -0.1 0 0.1 0.2 0.3

rd=0.4; r

s=0.6

rd=0.6; r

s=0.6

rd=0.9; r

s=0.6

Prob

abili

ty d

ensi

ty fu

nctio

n

Radial Velocity 0

1

2

3

4

5

6

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

rd=0.4; r

s=0.6

rd=0.6; r

s=0.6

rd=0.9; r

s=0.6

Prob

abili

ty d

ensi

ty fu

nctio

n

Tangential Velocity 0

1

2

3

4

5

6

7

8

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

rd=0.4; r

s=0.6

rd=0.6; r

s=0.6

rd=0.9; r

s=0.6

Prob

abili

ty d

ensi

ty fu

nctio

n

Axial Velocity Figure 7.6 Instantaneous probability density distributions of particle velocity

components at different density ratios when size ratio is fixed at rs=0. 6

7.4.2 Effect of size ratio rs on particle mixing

7.4.2.1 Mixing Index

Figure 7.7 shows the effect of size ratio rs on mixing for different values of xl

with rd=0.33, the other parameters being given in Table 7.4. For a given volume

fraction, the mixing index increases to the steady-state value or remain unchanged.

Figure 7.8 (a) and (b) summarizes the steady-state values M as a function of rs at

different xl. Generally, mixture quality shows a peak value with respect to rs as in the

case of a vertically-shafted cylindrical mixer (Chapter 3, and Figure 3.3(b)). The peak

occurs at rs between 0.8 and 0.9, which is somewhat larger compared the cylinder’s

which occurs between 0.7 and 0.8. A higher mixture quality can be obtained when

xl=0.5, which is however is not the optimum value of xl as shown below.

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

rs=0.6

rs=0.78

rs=0.8

rs=0.95

Parti

cle

Scal

e M

ixin

g In

dex

Revolutions

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

rs=0.6

rs=0.78

rs=0.8

rs=0.95

Parti

cle

Scal

e M

ixin

g In

dex

Revolutions0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

rs=0.6

rs=0.78

rs=0.8

rs=0.95

Parti

cle

Scal

e M

ixin

g In

dex

Revolutions

(a) (b) (c)

Figure 7.7 Particle scale mixing index as a function of revolutions at different

values of rs with rd=0.33: (a) xl=0.1; (b) xl=0.5; and (c) xl=0.9


7-195

0.75

0.8

0.85

0.9

0.95

1

0.5 0.6 0.7 0.8 0.9 1

xl=0.1

xl=0.5

xl=0.9

Parti

cle

Scal

e M

ixin

g In

dex

rs

0.75

0.8

0.85

0.9

0.95

1

0 0.2 0.4 0.6 0.8 1

rs=0.6

rs=0.78

rs=0.8

rs=0.95

Parti

cle

Scal

e M

ixin

g In

dex

xl

(a) (b)

Figure 7.8 Effect of rs and xl on steady-state values of mixing index (or mixture

quality) for rd = 0.33 and l = 6040 kg/m3: (a) effect of rs at different xl; and (b)

representation of results in (a) as an effect of xl at different rs

Figure 7.8(b) shows the results of Fig. 7.8(a) as a function of xl with rs as the

parameter. The mixture quality shows a peak against xl as in the case of the cylindrical

mixer (Chapter 3, Fig 3.4). The peak occurs at xl=0.35 here, but in the cylindrical mixer

it occurs roughly at xl=0.55. The trends of the curves of mixture quality versus xl is

mostly the same at different size ratios. This result is similar to the trends for the

cylindrical mixer(Chapter3, Figures 3.3(b) and 3.4).


7-196

7.4.2.2 Force components and velocity fields

Figure 7.9 shows the effect of rs on particle contact forces when rd=0.33 at t=50s.

An increase in the particle size ratio will cause an increase in both the normal and

tangential forces of particles as seen from the figure.

0

0.2

0.4

0.6

0.8

1

10-6

10-5

10-4

10-3

10-2

10-1

100

rs=0.8; r

d=0.3

rs=0.95; r

d=0.3

rs=0.6; r

d=0.3

Cum

ulat

ive

Prob

abili

ty

Normal Force (N)

0

0.2

0.4

0.6

0.8

1

10-6

10-5

10-4

10-3

10-2

10-1

rs=0.8; r

d=0.3

rs=0.95; r

d=0.3

rs=0.6; r

d=0.3

Cum

ulat

ive

Prob

abili

ty

Tangentail Force (N)

(a) (b)

Figure 7.9 Effects of rs on contact forces when xl =0.9

To understand the effect of size ratio on the particle flow, the velocity field in

vertical sections perpendicular to the vessel axis are examined in Figure 7.10 as in the

case of density effect of Fig. 7.5. Figure 7.10 shows the instantaneous velocity fields in

sections at Z values indicated in the figure for different size ratios with density ratio

fixed at 0.33 and xl at 0.9. The blue colour represents large particles and red, the small

ones. Large particles are heavy and its density is kept fixed (see simulation conditions).

When the size ratio is decreased, arrows of different colours coexist in the sections.

However, it is difficult to conclude from such an observation that mixing will improve;

in fact, Fig. 7.8 shows that mixing is deteriorated with a reduction of size ratio at xl=0.9.


7-197

Figure 7.10 Effect of rs on velocity field when rd = 0.33 and xl =0.9

Figure 7.11 shows the size ratio effect on the velocity components. The

tangential velocity of the particles for rs=0.8 has slightly improved, and thus, the

circumferential flow is enhanced. For this case, a higher mixing index has been obtained

as shown in Figure 7.8. The size ratio has not affected the radial velocity significantly.

However, axial velocity has reduced with the increase of size ratio as seen from the

sharpening of the peak of the distribution curve in Figure 7.11(c).

rs=0.6; rd=0.3

Z=90-110Z=40-60 Z=140-160

rs=0.8; rd=0.3

rs=0.95; rd=0.3

Frame 001 08 Nov 2013 Frame 001 08 Nov 2013 Frame 001 08 Nov 2013




7-198

0

2

4

6

8

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

rs=0.6; r

d=0.3

rs=0.8; r

d=0.3

rs=0.95; r

d=0.3

Prob

abili

ty d

ensi

ty

Radial Velocity (m/s) 0

1

2

3

4

5

6

7

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

rs=0.6; r

d=0.3

rs=0.8; r

d=0.3

rs=0.95; r

d=0.3

Prob

abili

ty d

ensi

ty

Tangential Velocity (m/s) 0

4

8

12

16

-0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

rs=0.6; r

d=0.3

rs=0.8; r

d=0.3

rs=0.95; r

d=0.3

Prob

abili

ty d

ensi

ty


(a) (b) (c)

Figure 7.11 Probability density of velocity components at different size ratio

when rd = 0.33, xl =0.9 at t=50s

7.4.3 Prediction Equation

The above results show that size ratio, density ratio and volume fraction affect

the mixing behavior of binary mixture of the particles. Previously, a prediction equation

has been established for a vertically-shafted cylindrical mixer to account for the effects

of size, density and volume fractions on mixture quality. Here, it would be appropriate

to examine whether a correlation of the form for a vertical cylinder would be valid for

the ribbon mixer. If it exists, that would suggest that similar mixing mechanisms are in

operation in both mixers. A prediction equation of the form of Eq. 7-3 is obtained using

‘FindFit’ function of Mathematica software for the ribbon mixer. Simulation results and

prediction results from the equation have been compared and the error obtained as

shown in Table 7.5. The error is equal or less than 7% as shown in Table 7.5.

MI = 0.2981 + 1.605 rs - 1.074 rs2 + 0.2639 rd - 0.3854 xl

2 + 0.1535 rd xl - 0.259 rd2 + 0.2185 xl (7.3)

Contour plots have been draw in Figure 7.12 to illustrate the effects of the three

variables, size, density and volume fraction on the mixture quality MI using Eq (7.3) in

wide ranges of the three variables. The contour plot also shows the interchangeability

of size and density effects, and there is an optimum combinations of size and density

ratio for a given volume fraction which gives the highest mixture quality. The difference


7-199

between the correlations or predictive equations for the cylindrical mixer and ribbon

mixer is that the size and density interactions are significant in the cylindrical mixer,

which is not the case with the ribbon mixer tested. Conversely, the density and volume

fraction interactions are more significant in the case of the ribbon mixer.

Table 7.5 Comparison of mixing index of simulation and mixing index from equation

rs rd xl MI MI_Eq Error

0.6 0.3 0.1 0.956794 0.952921 0%

0.78 0.3 0.1 0.965042 0.975039 -1%

0.8 0.3 0.1 0.968078 0.973201 -1%

0.95 0.3 0.1 0.936 0.932026 0%

0.6 0.3 0.5 0.96735 0.966245 0%

0.78 0.3 0.5 0.985382 0.988363 0%

0.8 0.3 0.5 0.986659 0.986525 0%

0.95 0.3 0.5 0.973901 0.94535 3%

0.6 0.3 0.9 0.797539 0.856241 -6%

0.78 0.3 0.9 0.854902 0.878359 -2%

0.8 0.3 0.9 0.862373 0.876521 -1%

0.95 0.3 0.9 0.814755 0.835346 -2%

0.66 0.3 0.1 0.97497 0.968027 1%

0.66 0.4 0.1 0.97044 0.977822 -1%

0.66 0.6 0.1 0.97268 0.981872 -1%

0.66 0.9 0.1 0.96478 0.949097 2%

0.66 0.3 0.5 0.98901 0.981351 1%

0.66 0.4 0.5 0.99036 0.997286 -1%

0.66 0.6 0.5 0.99436 1.013616 -2%

0.66 0.9 0.5 0.98986 0.999261 -1%

0.66 0.3 0.9 0.93767 0.871347 7%

0.66 0.4 0.9 0.93243 0.893422 4%

0.66 0.6 0.9 0.93628 0.922032 1%

0.66 0.9 0.9 0.92238 0.926097 0%


7-200

x

l=0.1 x

l=0.5 x

l=0.9

Figure 7.12 Contour maps using the prediction equation, demonstrating the

equivalence of size and density effects at different volume fractions, xl of 0. 1, 0.5 and

0.9.

Figure 7.13 shows the comparison of the effects of size, density and volume

fraction on the particle mixing in the cylindrical and the ribbon mixer. The results show

that the trends of this effect are similar in both mixers. But the mixing indexes are

higher in the ribbon mixer compared to the cylindrical mixer. However, an exact

comparison has not been done here: for example, shaft speeds are different for the two

mixers, being 100 rpm and 20 rpm for the ribbon and cylindrical mixers respectively.

The important fact here is that mixing is less affected by the size and density effect in

the case of the ribbon mixer.

0.00

0.20

0.40

0.60

0.80

1.00

0.4 0.5 0.6 0.7 0.8 0.9 1

MI_cylinderEq_cylinderMI_ribbonEq_ribbon

Stea

dy-s

tate

mix

ing

inde

x, M

Size Ratio, rs, x

l=0.9,

Cylindrical mixer

Ribbon mixer

0.00

0.20

0.40

0.60

0.80

1.00

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

MI_cylinderEq_cylinderMI_ribbonEq_ribbon

Stea

dy-s

tate

mix

ing

inde

x

Density Ratio,rd , x

l=0.1

Cylindrical mixer

Ribbon mixer

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

MI_cylinderEq_cylinderMI_ribbonEq_ribbonPa

rticl

e -s

tate

mix

ing

inde

x

Volume fraction, xl

Ribbon mixer, rs=0.8

Cylindrical mixer, rs=0.875

(a) (b) (c) Figure 7.13 Comparison of the effects of rs, rd and xl on particle mixing behaviour in the cylindrical mixer and ribbon mixer; M is steady-state values of mixing index, solid line is representing prediction: (a) the effect of size ratio when xl =0.9 for cylindrical mixer rd=0.33; for ribbon mixer rd=0.22; (b) density effect when xl =0.1 for cylindrical mixer rs=0.6; for ribbon mixer rs=0.5; (c)volume fraction effect: for cylindrical mixer rs=0.875, rd=0.22; for ribbon mixer rs=0.8, rd=0.33.


7-201

7.5 Conclusions

The effects of size and density ratios of particles of binary mixtures were

investigated for a ribbon mixer at different volume fractions. There are similarities in

the effects of size ratio, density ratio and volume fraction in relation to those for a

vertically-shafted cylindrical mixer. For example, the mixing index showed a peak

value with respect to the size ratio and volume fraction. However, the effect of density

ratio on mixture quality is not significant in the case of the ribbon mixer. This is an

indication that the mixing mechanisms of the two systems are different.

The predictability of these effects is confirmed again for the ribbon mixer as for

a cylidrical mixer by obtaining a prediction equation for the ribbon mixer. The form of

the prediction equations are similar, but some differences exist. In the case of

cylindrical mixer, the size and density interaction is significant as observed from the

presence of rsrd term in the equation. On the other hand, for the ribbon mixer, density

and volume fraction interaction is significant as seen from the presence of rd xl term.

However, there is still some interation between size and density as both terms are

present in the correlation developed. The error between the DEM results and predictions

of the equation is 7% at the most. The equivalence of the size and density effects is

shown to exist by using the contour maps from the predictive equation. The maps show

that a maximum mixing index can be obtained using an optimum size and density

combinations at different volume fractions.

Nomenclature


dl Large particle diameter, (mm)

ds Small particle diameter, (mm)

E Young’s modulus, (N/m2)

Fc,ij Contact force vector between i and j, (N)

Fd,ij Damping force vector between i and j, (N)


7-202



ki Number of particles in contact with particle i, (-)


Mij Vector of rolling friction torque on particle, (Nm)


N Total number of particles in the mixture, (-)

Nt The number of sample/data points of instantaneous average values in time t, (-)




rs Size ratio ls dd / , (-)

rd Density ratio ls / , (-)



fraction of p, (-)


Tij Vector of rolling friction torque on particle i, (Nm)


Vi Velocity of particle i, (m/s)

xl Volume fraction, which is ratio of volume of large particles to total particle

volume (-)


Greek letters



Density of particles of a uniform system, (kg m-3)


s Density of small particle, (kg m-3)


7-203



CHAPTER 8 Radial Segregation of a Binary Mixture in a Rotating Drum

8-204

Chapter 8

Radial Segregation of a Binary Mixture in a Rotating Drum


8-205

8.1 Introduction

Segregation is one of the most intriguing properties of granular flows. It is well

known from experimental results that the perfect mixing cannot be obtained due to

differences in particles shape, surface roughness, size and density (Ahmad and Smalley,

1973; Aranson and Tsimring, 2006; Williams, 1976). The segregation is inevitable

when heavy particles of a binary mixture of light and heavy equal-size particles of a

binary mixture are located near the centre of mass of particles in the drum (Ristow,

1994). A two-dimensional experiment with a rotating drum using disks, shows that the

radial segregation occur in avalanches and continuous flow regimes in a half filled

rotating drum due to the size differences of the disks (Cantelaube and Bideau, 1995). It

is reported that smaller particles are dispersed into the centre of the mixture while the

larger ones dwell on the edge of the rotating drum (Clément et al., 1995). It is observed

that the percolation primarily occur in the rapid flow layer formed on bed surface

(Cantelaube et al., 1997). The size of core region increases with an increase of more

denser particles (Khakhar et al., 1997). The size segregation is counter-balanced by

density segregation when varying the density of small particles with the size ratio of a

binary particle mixture fixed (Dury and Ristow, 1999). It is reported that the segregation

of species of different size and surface properties would occur with the smallest and

roughest grains being found at the center of the drum (Makse, 1999). The mixing rate

increased, segregation deteriorated with a decrease of the size difference of particles

(Eskin and Kalman, 2000). A small difference in either the size or density leads to flow

induced segregation. The denser particles or smaller particles migrate towards the core

of the cylinders (Ottino and Khakhar, 2000). For a small size ratio, there is a

segregation of the large beads at the surface, but for large size ratios, the large beads

segregate inside the mixtures (Thomas, 2000). Ternary mixtures with different size and

density particles segregate due to size and density driving forces which may

complement or oppose each other (Hajra and Khakhar, 2011). Large particle size ratios

or density ratios lead to segregation, though segregation can be deteriorated due to

percolation effect in a ternary or multi sized system (Xu et al., 2010). The small

particles in ternary mixtures exhibit reverse segregation as in binary mixtures.

Segregation is nearly independent of the sizes of the medium size and large particles.

Ternary mixtures with different size and density particles segregate due to size and


8-206

density driving forces which may complement or oppose each other. (Hajra and

Khakhar, 2011). The radial segregation is driven by a density segregation flux, which

results in heavier particles tending to come to rest deeper in the bed, as a result of their

larger mass (Pereira et al., 2011). It is stated that a prediction equation to quantify the

combined effects of density and size differences would enable us to predict transition

from mixing to segregation by percolation due to size difference and buoyancy effect

induced by density differences (Alonso et al., 1991). The combined effects of

percolation and buoyancy enhanced segregation when smaller particles are heavier.

Otherwise, the peocolation and buoyancy effects negate each other and segregation is

reduced (Liu et al., 2013). The feasibility of prediction of segregation in rotating drum

still need to be further investigate.

Research on a relationship of the effects of the particle properties such as the

size and density differences of particles on the rotating drum performance have not been

clearly established. The objective of this study is thus, to investigate the effect of size

and density differences of particles on the binary mixing behaviours of non--cohesive

particle mixtures for rotating drum, by using simulations based on the discrete-element-

method (Cundall and Strack, 1979).

This chapter is organized as follows. Section 8.2 discusses the simulation

method, simulation conditions and quantification method, which is followed by the

results and discussions section, where in section 8.3.1, the effect of density ratio when

size ratio fixed. In Section 8.3.2, effect of size ratio is discussed with density ratio fixed.

Next, a prediction equation is developed to account for size and density effects on

mixture quality in section 8.3.4. Finally, Section 8.4 summarizes the conclusions of the

chapter.


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8.2 Analytical Methods

8.2.1 DEM Model


and validated by Zhou et al. (Zhou et al., 2004; Zhou et al., 1999). The model uses two


particle i in a system at time t subjected to the gravity g and interactions with the

neighboring particles, blade and walls:

ik

jijdijci

ii m

dtdm

1,, FFgV (8.1)

and

ik

jijij

ii d

dI1t

MT (8.2)

where mi, Ii, Viand i are the mass, moment of inertia, translational and

rotational velocities of the particle respectively; k is the number of particles that are in

contact with particle i, Fc represents the elastic contact force which is the summation of

the normal and tangential forces. Fd represents the damping force, which is the

summation of the normal and tangential damping force respectively at the contact point

with particle j; and T and M are the torque and rolling friction torque on particle i due to

particle j. Expressions for the forces and torques in Eqs. (8.1) and (8.2) are given in

Table 8.1.


8-208


Forces and torques


Normal contact force nF ˆ)2(

)1(32321

2, niijcn RE

Normal damping force n nVF ˆˆ)(

)1(23

21

212,

ijn

inijdn R

Emc

iijjijij RRVVV

i

i

RR

n ˆ


t

t

t

i

ijcnijct

F

23

max,

max,,,

,min11

tsF

Tangential damping force ijt

t

ttijcnsitijdt Fmc ,

21

max,

max,,,

1 6 VF

nst

)1(2)2(

max,

nnVV ˆˆ, ijijt

Rotational torque

ijdtijctiij ,, FFRT


ιijcnrij F ̂,M ι

ιi

ˆ

8.2.2 Simulation Conditions

The initial loading method is side-by-side axially as shown in Figure 8.1. Due to

the short axial length of the vessel sometimes particles may be mixed already, but we

are interested here on the steady-state mixture quality. Periodic boundary conditions

have been used in the axial direction of the drum. The key parameters used in DEM

simulations are shown in Table 8.2. Here, dS and ρS represent the diameter and density

of small particles, respectively, while dL and ρL denote the size and density of large

particles, respectively. The size ratio and density ratio are defined as rs=dS/dL and rd

=ρS/ρL , respectively. The small particles have dS=1.3mm and ρS=2500 kg/m3, those of

large particles were varied as shown in Table 8.3. The particle number for the

simulations is calculated based on the fixed volume fraction of 0.5 for each type of


8-209

particles. The drum’s fill level was fixed at 30%. The drum speed was chosen as 15rpm

to obtain the rolling mode of particle flow(Liu et al., 2013). The simulations were

conducted for the cases of non-cohesive particles for investigating the effect of size and

density. The simulation conditions are summarized in Table 8.2.

8.2.3 Segregation index

The mixture quality is quantified using Segregation index (Yamane, 2004).

20

2

I (8.3)

where, 02

is the sample variance of fully-segregated state and is the sample

variance of the mixture at time t, where pp 120 and p represents the particle

number ratio of the target particles to the total particle number in the binary mixture.

Here,

Ni

ii pp

N 1

2t

2 1 , where N is the total number of particles in the mixer, with

tp representing the average of pi at time t, pi being the fraction of the target type of

particles in each sample. Here, samples are taken at each particle to include only its

contacting particles (i.e. the immediate neighbourhood), and number fraction pi of the

target type of particles in each sample is found from pi=ni/Ni where ni and Ni are the

target type particles and total number of particles in a sample, respectively

(Chandratilleke et al., 2012). To find, tp , pi is averaged over the particle number N. In

determining 2, one has to use a particle-contact condition, which is taken as an inter-

particle gap size of 5% of the small particle diameter, to be consistent with our previous

work (Chandratilleke et al., 2012).


8-210

Table 8.2 Input variables and their values

Parameter Value

Number of Particles 7260 ~ 14000

Drum, D × L (mm) 104 × 7.8

Size of small particles, dS (mm) 1.3

Size ratio, dS/dL 0.3 ~ 0.86

Density of small particles, ρS (kg/m3)

2.5 × 103

Density ratio, ρS/ρL 0.25 ~ 4.0

Elastic modulus, E (Pa) 1.0 × 107

Poisson’s ratio, ν 0.29

Normal damping coefficient, (s-1) 1.0 × 10-6

Particle- particle sliding friction coefficient, µpp

0.4

Particle- wall sliding friction coefficient, µpw

0.5

Rolling friction coefficient, µr 0.001

Fill level, f (%) 30

Rotation speed, ω (rpm) 15

Figure 8.1. Initial loading pattern of particles in the rotating drum


8-211

Table 8.3 Particle information for size effect cases

Cases ρs/ρL ρL ρS ds/dL dL dS 1 0.615385 4062.5 2500 0.333333 3.9 1.3 2 0.64 3906.25 2500 0.363636 3.575 1.3 3 0.444444 5625 2500 0.432526 3.0056 1.3 4 0.5 5000 2500 0.410256 3.16875 1.3 5 0.333333 7500 2500 0.470588 2.7625 1.3 6 0.64 3906.25 2500 0.666667 1.95 1.3 7 0.592593 4218.75 2500 0.8 1.625 1.3 8 0.888889 2812.5 2500 0.714286 1.82 1.3 9 1.262626 1980 2500 0.666667 1.95 1.3 10 0.8 3125 2500 0.551724 2.35625 1.3 11 0.695652 3593.75 2500 0.444444 2.925 1.3 12 0.769231 3250 2500 0.363636 3.575 1.3 13 0.4 6250 2500 0.615385 2.1125 1.3 14 0.333333 7500 2500 0.571429 2.275 1.3 15 0.266667 9375 2500 0.588235 2.21 1.3 16 1.904762 1312.5 2500 0.8 1.625 1.3 17 1.454545 1718.75 2500 0.666667 1.95 1.3 18 1.257862 1987.5 2500 0.5 2.6 1.3 19 1.333333 1875 2500 0.4 3.25 1.3 20 1.454545 1718.75 2500 0.333333 3.9 1.3 21 0.4 6250 2500 0.856898 1.5171 1.3 22 0.333333 7500 2500 0.8 1.625 1.3 23 0.285714 8750 2500 0.761905 1.70625 1.3


8-212


The following effects are investigated below for binary mixtures using the

segregation index: the effect of size ratio when density ratio is fixed; effect of density

ratio when size ratio is fixed and combined effect of size and density ratios on flow

pattern and contact forces.

8.3.1 Effect of density ratio on binary particle mixing

8.3.1.1 Segregation Index

The effect of density ratio on mixing is investigated using the segregation index

with the density being varied while keeping ρS =2500 kg/m3 and size ratio at 0.66; these

cases are listed in Table 8.3. Figure 8.2 shows the effect of density ratio on segregation

index I as a function of time or drum revolutions. Segregation index decreases first and

then increases for rd= 0.64, while rd= 1.26 and rd= 1.46 show similar trends with a

steady increase after 2 rev as shown in Figure 8.2(a). Figure 8.2(b) shows the average

steady-state segregation index as a function of density ratio. It shows that the

segregation has slightly enhanced with an increase in density ratio for rd= 0.66. The

trend is the same as in other mixers, where mixing improved with a reduction of density

ratio.

Figure 8.3 shows the evolutions of radial segregation pattern investigated at

different density ratios when the size ratio is fixed at 0.66. Radially, a core of denser

particles (blue) is gradually developed over time due to the buoyancy effect (Zhou et al.,

2003) when small particles are more heavier as a result of an increase of density ratio.

Thus, it can be observed that segregation is enhanced with an increase of density ratio,

the small particles (blue) percolating to the core region. On the other hand, with a lower

density ratio of rd= 0.64, large particles became heavier, the small ones lighter, and the

percolation and buoyancy effects counterbalance each other producing better mixing at

t=35s. The segregation index for this case where rd= 0.64 and rs=0.66, shows a lower

value in Figure 8.2(a). These phenomena have been observed in a previous paper on

rotating drum as well (Liu, Yang et al. 2013)


8-213

0.00

0.20

0.40

0.60

0.80

1.00

0 5 10 15 20 25 30 35

0 1 2 3 4 5 6 7 8

rs =0.66; r

d =0.64

rs =0.66; r

d =1.26

rs =0.66; r

d =1.46

Segr

egat

ion

Inde

x

Time (s)

Drum revolutions

0.00

0.20

0.40

0.60

0.80

1.00

0.5 1 1.5

Steady-state values

Eq.Ave

rage

segr

egat

ion

inde

x

Density ratio, rd

(a) (b)

Figure 8.2 Effect of density ratio on mixing: (a) Segregation index as a function

of time and revolutions when rs=0.66; and (b) steady-state segregation index as

a function of density ratio.

Figure 8.3 Effect of rd on the segregation flow pattern at rs=0.66; blue represents small particles, and red the large ones.

t=5s

t=35s

rs =0.66; rd =0.64 rs =0.66; rd =1.26 rs =0.66; rd =1.46


8-214

8.3.1.2 Force analysis

To understand the role of density ratio on particle contact forces in the

segregation mechanism, the cumulative probability distributions of contact forces are

produced at t=30s, when the flow is steady. Figure 8.4 shows that the normal and

tangential forces increased with a decrease of density ratio while the size ratio is fixed at

0.66. This is an indication that particle interactions are becoming more dynamic. In the

case where rd= 0.64 and rs=0. 66, the particle rolling flow was enhanced with stronger

tangential forces, and higher normal contact forces of particles improved the buoyancy

and percolation effects, thus reducing the segregation. The buoyancy effect becomes

dominant as seen from the expulsion of large (red) particles to the periphery of the

mixture when rd= 1.26 and 1.46 in which case the normal and tangential force become

smaller and the density difference of particles is lower.

0.00

0.20

0.40

0.60

0.80

1.00

10-6

10-5

10-4

10-3

10-2

rs =0.66; r

d=0.64

rs=0.66; r

d =1.26

rs =0.66; r

d=1.46

Cum

ulat

ive

prob

abili

ty

Normal Force (N)

0.00

0.20

0.40

0.60

0.80

1.00

10-6

10-5

10-4

10-3

10-2

rs =0.66; r

d=0.64

rs=0.66; r

d =1.26

rs =0.66; r

d=1.46

Cum

ulat

ive

prob

abili

ty

Tangential Force (N) (a) (b)

Figure 8.4 Effects of rd on contact forces at rs=0.6: (a) normal force ; and (b) tangential

force


8-215

8.3.2 Effect of size ratio rs on particle mixing

8.3.2.1 Segregation patterns Figure 8.5 shows the effect of size ratio rs on the evolutions of radial segregation

pattern for rd=0.33, the other parameters being given in Table 2. The segregation index

steadily decreased to reach the steady state value as shown in Figure 8.5(a). Figure

8.5(b) summarizes average values I at steady-state as a function of rs. A higher

segregation can be seen for rs=0.47 as shown in Figure 8.5(a), and Figure 8.5(b) shows

that the average segregation index decreases with an increase in the size ratio.

0.00

0.20

0.40

0.60

0.80

1.00

0 5 10 15 20 25 30 35

0 1 2 3 4 5 6 7 8

rs =0.47; r

d = 0.33

rs = 0.57; r

d = 0.33

rs =0.8; r

d = 0.33

Segr

egat

ion

Inde

x

Time (s)

Drum revolutions

0.00

0.20

0.40

0.60

0.80

1.00

0.4 0.5 0.6 0.7 0.8 0.9

Average steady-state valueEqA

vera

ge S

egre

gatio

n In

dex

Size ratio, rs

(a) (b) Figure 8.5 Effect of size ratio: (a) Segregation index as a function of time and

revolutions when rd=0.33; and (b) Average segregation index as a function of size ratio;

the solid line is prediction value.


8-216

Figure 8.6 Effects of rs on the segregation flow pattern when rd = 0.33

Figure 8.6 shows segregation flow patterns with an increase of the size ratio. At

the initial stage of t=5s, the radial core is occupied by heavy large-particles, but it

becomes decayed at t=25~30s. When size ratio is increased, the mechanism of mixing

in which small particles percolate and large particles receive buoyancy forces, becomes

enhanced at the same time, and thus segregation index decreased with an increase in

size ratio as can be seen from Figure 8.5.

t=5s

t=25s

rs =0.47; rd = 0.33 rs = 0.57; rd = 0.33 rs =0.8; rd = 0.33


8-217

8.3.2.2 Force components

Figure 8.7 shows the effect of rs on particle contact forces for rd = 0.33 at t=30s

averaged over 1 s time interval. Stronger normal and tangential forces are seen in the

case of rs= 0.8, rd= 0.33, which shows the lower segregation index in the Figure 8.5(a)

and a lower segregation flow in the Figure 8.6. Particles encounter smaller normal and

tangential forces when the size ratio is decreased to 0.47 (Table 8.3 case 5).

0.00

0.20

0.40

0.60

0.80

1.00

10-6

10-5

10-4

10-3

10-2

rs =0.47; r

d = 0.33

rs = 0.57; r

d = 0.33

rs =0.8; r

d = 0.33

Cum

ulat

ive

prob

abili

ty

Normal Force (N)

0.00

0.20

0.40

0.60

0.80

1.00

10-6

10-5

10-4

10-3

10-2

rs =0.47; r

d = 0.33

rs = 0.57; r

d = 0.33

rs =0.8; r

d = 0.33

Cum

ulat

ive

prob

abili

ty

Tangential Force (N) (a) (b)

Figure 8.7 Effect of rs on contact forces: (a), normal force ; and (b), tangential force

8.3.3 Combined size and density effect

8.3.3.1 Size ratio r

s and density ratio r

d decrease at the same time

The evolution of the segregation pattern is investigated when the size ratio and

density ratio are both decreased at the same time. Figure 8.8 shows that the segregation

index decreased with a decrease of both the size ratio and density ratio at the same time.

Figure 8.9 shows the evolution of the segregation pattern when both the size ratio and

density ratio are reduced at the same time. Here, even if red particles are subjected to an

increase in the size and density at the same time, large particles being still lighter than

the small one, the red particles move to periphery under the buoyancy effect while small

particles percolate to the core region.


8-218

0.00

0.20

0.40

0.60

0.80

1.00

0 5 10 15 20 25 30 35

0 1 2 3 4 5 6 7 8

rs =0.8; r

d =1.9

rs =0.67; r

d =1.26

rs =0.5; r

d =1.25

Segr

egat

ion

Inde

x

Time (s)

Drum revolutions

Figure 8.8 Segregation index as a function of the time when

rs and rd are both decreased

Figure 8.9 Evolution of the segregation flow pattern when rs and rd decreased;

particle condition similar with Fig 8.3

rs =0.66; rd =1.26

t=20s

t=5s

rs =0.8; rd =1.9 rs =0.5; rd =1.25


8-219

8.3.3.2 Size ratio rs

is decreased and density ratio r

d increased

Figure 8.10 shows that the segregation index increased with a decrease of size

ratio rs , increase of density ratio rd. Figure 8.11 shows the flow patterns for the cases

shown in Figure 8.10.

0.00

0.20

0.40

0.60

0.80

1.00

0 5 10 15 20 25 30 35

0 1 2 3 4 5 6 7 8

rs=0.8; r

d=0.59

rs=0.6; r

d=0.64

rs=0.44; r

d=0.7

rs=0.36; r

d=0.77

rs=0.33; r

d=1.45

Segr

egat

ion

Inde

x

Time (s)

Drum revolutions

Figure 8.10 Segregation index as a function of time when rs decreases and rd

increases at the same time.

Figure 8.11 shows that the buoyancy effect becomes the dominant mechanism

when the size ratio decreases while the density ratio increases. It means that red

particles become larger but less dense when the size ratio decreases and density

decreases. The small particles become smaller and denser. Therefore small particles

percolate to the core of the mixture, while large particles float outside of the core

region.


8-220

Figure 8.11 Evolution of the segregation flow pattern when rs decreases and rd

increases simultaneously.

8.3.4 Prediction equation

It has been reported previously that it is possible to represent the effect of rs for a

binary system of rd=1 and effect of rd for rs=1 system by a single characteristic curve

using the particle weight ratio rd rs3 in the case of xl=0.5(Chandratilleke et al., 2012).

The predictability of the effect of size ratio, density ratio and volume fraction in a

cylindrical mixer and ribbon mixer has already been confirmed (see Chapters 3 and 7).

The predictability of these effects for the rotating drum also needs to be investigated.

The predictive relationship is shown by Eq.(8.4), which can predict the segregation

index for binary particles of different sizes and densities with an error up to 17%.

SI = -0.87 – 4.07 rs2 +5.07 rs+ 0.29 rd

2 – 0.197rd– 0.38rsrd (8.4)

t=20s

rs =0.8; rd = 0.59

t=5s

rs =0.44; rd = 0.77 rs =0.33 ; rd =1.45


8-221

Eq.(8.4) can predict in wide ranges of size, density and volume fraction their

effects on particle segregation in a rotating drum for the rolling mode as shown in

Figure 8.12. The contour plot also shows the size and density effects are

interchangeable, meaning that one combination of size and density ratios can produce

exactly the same mixing state as another combination of size and density ratios. The

map also shows that an optimum mixing state can be found at a certain size and density

combination.

Table 8.4 Comparison of segregation index and prediction

r s r d SI Eq error0.333333 0.615385 0.298667 0.278544 -2%

0.363636 0.64 0.409114 0.340029 -7%

0.432526 0.444444 0.558076 0.457736 -10%

0.410256 0.5 0.541743 0.420763 -12%

0.470588 0.333333 0.6 0.520793 -8%

0.666667 0.64 0.490458 0.532429 4%

0.8 0.592593 0.466598 0.386781 -8%

0.714286 0.888889 0.490458 0.490038 0%

0.666667 1.262626 0.616944 0.600574 -2%

0.551724 0.8 0.573861 0.550175 -2%

0.444444 0.695652 0.49627 0.465944 -3%

0.363636 0.769231 0.477428 0.350283 -13%

0.615385 0.4 0.573861 0.582431 1%

0.571429 0.333333 0.57 0.591761 2%

0.588235 0.266667 0.663577 0.61182 -5%

0.8 1.904762 0.558076 0.693651 14%

0.666667 1.454545 0.661729 0.6676 1%

0.5 1.257862 0.577386 0.625022 5%

0.4 1.333333 0.635729 0.563164 -7%

0.333333 1.454545 0.652079 0.518006 -13%

0.856898 0.4 0.151397 0.323224 17%

0.8 0.333333 0.5 0.446084 -5%

0.761905 0.285714 0.44671 0.51441 7%


8-222

Figure 8.12 Contour maps of segregation index using the prediction equation, to

demonstrate the effects of size and density on the mixing.

8.4 Conclusions

The effect of size and density ratios on radial segregation of binary mixtures

was investigated in a rotating drum by means of the discrete element method. Density

induced segregation occurs with an increase of density ratio, if the size ratio is fixed.

Then, small particles become denser and percolate to the core region of the mixture,

with the increase of the density ratio. The buoyancy and percolation effects compete

with each other when the size ratio is increased while density ratio is fixed, and the

segregation is reduced. When size and density ratios are reduced at the same time, larger

particles become lighter and buoyancy mechanism become dominant, and the large

particles moved outside the core region.

The effect of size and density can interact each other and decrease segregation.

These effects can be predicted. A correlation is established to predicted the segregation

index under different size and density ratios and volume fractions. The predictive

equation is a second order polynomial that is similar in form to those of the cylindrical

and ribbon mixers. It suggests that similar mixing mechanisms are in operation in all

three mixers.


8-223

Nomenclature


dl Large particle diameter, (mm)

ds Small particle diameter, (mm)

E Young’s modulus, (N/m2)

Fc,ij Contact force vector between i and j, (N)

Fd,ij Damping force vector between i and j, (N)



ki Number of particles in contact with particle i, (-)


Mij Vector of rolling friction torque on particle, (Nm)


N Total number of particles in the mixture, (-)

Nt The number of sample/data points of instantaneous average values in time t, (-)




rs Size ratio ls dd / , (-)

rd Density ratio ls / , (-)



fraction of p, (-)


Tij Vector of rolling friction torque on particle i, (Nm)


Vi Velocity of particle i, (m/s)

xl Volume fraction, which is ratio of volume of large particles to total particle

volume (-)



8-224

Greek letters



Density of particles of a uniform system, (kg m-3)


s Density of small particle, (kg m-3)



CHAPTER 9 Summary and Future Work

9-225

Chapter 9

Summary and Future work


9-226

The mixing behaviours of spherical particles were investigated in three types of mixers by

means of DEM simulations. The mixers tested are: a vertically-shafted cylindrical mixer, a

ribbon mixer and a rotating drum. The study focused on the effects of particle properties,

operational parameters and impeller and mixer geometries. The main conclusions are

summarized below for each of the mixers investigated.

Vertically-shafted cylindrical mixer:

Here, the objectives were the investigation of the effects of size and density ratios at different

volume fractions on mixing behaviour of binary particles at fixed operational conditions and

development of a correlation for predicting the size and density effects at different volume

fractions. The main conclusions are as follows.

The quality of a binary particle mixture varies depending on the particle size ratio,

density ratio and volume fraction of the particles. There is an interaction between each

pair of the variables, and therefore, the mixture quality can be improved by

appropriately selecting the variables. For example, mixture quality can be improved if

the density ratio is reduced with the size ratio and volume fraction fixed. Similarly, a

peak mixture quality is obtained for a given combination of size and density ratios

when the volume fraction is 0.55. Further, a size ratio of 0.76 produces a peak mixture

quality when the density ratio and volume fraction are fixed. A global peak could be

found in a large domain of size, density and volume fraction by developing a

correlation.

The mechanism of mixing improvement is related to the interplay between buoyancy

forces and particle weights in binary mixtures. If large particles are heavy enough to

counter the buoyancy forces on them, the larger particles can compete with lighter

small particles that sink to the vessel base, producing improved mixing.

A predictive equation could be developed to account for the effects of particle size,

density and volume fraction on mixture quality. Although limited in its capabilities at

present, with further work it should be possible to extend the predictive equations to

account for the effects of other variables such as particle number, mixer scale and

material properties. Thus, it may be possible to use a cylindrical mixer in studies of

particle mixing as a standard mixer for which results are predictable.


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Horizontal ribbon mixer:

Here, the objectives were the investigation of the effects of the impeller geometry,

operational variables and particle properties on particle mixing behaviour and development of

a correlation for predicting the size and density effects at different volume fractions. The

blade geometry was varied by changing the following parameters: blade pitch ratio, blade

width, blade clearance, blade angle and blade numbers. The operational variables are shaft

speed and filling level. Particles properties considered are particle size, density, volume

fraction and cohesiveness. The effects of geometrical parameters were examined in uniform

different cohesiveness mixture. The main conclusions are as follows.

The effect of cohesion on particle mixing was investigated in micro and macro

systems by creating geometrically and dynamically similar conditions. To create a

geometrical similarity between the two systems, the micro-system was linearly scaled

down to 1/150 of the size of the macro-system. To create the dynamic similarity,

Froude and Bond numbers were matched. For non-cohesive particles, several

variables showed similar trends: namely, mixing curves, velocity distributions and

contact forces. This was achieved by using lighter particles in the macro-system and

heavier ones in the micro-system. However, for cohesive particles, similarity in the

variables became poorer with an increase in cohesion, the possible reason for this

discrepancy being the Froude number matching, which results in a high shaft speed

and thus, a high shear rate in the micro-system. Thus, further work is required in this

regard. However, the macro-system showed generally expects trends for cohesive as

well as non-cohesive particle mixing, and thus, it was used for a parametric study, the

results of which are summarized below.

The mixing rate and homogeneity of cohesive particle mixtures deteriorated with an

increase in cohesion. The tangential velocity increased, and the radial and axial

velocities of particles decreased with an increase of cohesion. The radial stress was

higher than the axial stress and increased with an increase in cohesion.

The shaft speed showed a significant effect on the mixing behaviour of uniform

particles of cohesive and non-cohesive mixtures in a 2-bladed mixer. The mixing rate

and homogeneity of the cohesive and non-cohesive mixtures increased with an

increase of the shaft speed up to 100rpm due to increase of the circumferential

velocity of particles. The contact force of the particles increased with an increase of

shaft speed in non-cohesive mixtures, but no such effects on cohesive mixtures.


9-228

The fill level affected the mixing rate, particle flow and particle velocities

significantly both in cohesive and non-cohesive mixtures, in both the 2-blade and 4-

blade ribbon mixers. With an increase of the fill level and particle cohesion, the

mixing rate diminished, and a transition in the particle flow from a sliding flow to a

cascading flow was investigated. A 4- blade mixer was found to be more effective for

use at higher fill levels, especially in case of cohesive mixtures.

The mixture homogeneity, mixing rate, axial flow and tangential flow enhanced with

an increase of blade number in the ribbon mixer for 2-bladed and 4-bladed impellers.

The convective and circumferential flow increased with moderate to high pitch ratios

of the ribbon due to higher axial, tangential and radial velocities. A wider blade width

enabled the particles to receive higher axial, tangential and radial forces and velocities,

and thus the mixing rate increased. The particle mixing rate enhanced with a higher

blade clearance. The particles encounter higher axial, tangential and radial velocities

and forces with a larger blade width. The outer blade angle only slightly affected the

mixing rate.

Effects of size, density and volume fraction on binary particle mixtures were also

investigated for the ribbon mixer. There are similarities in these effects to those of the

cylindrical and ribbon mixers. For example, mixing index showed a peak each with

respect to changes in either the size ratio or volume fraction. The density effect is not

so significant on the particle mixing in the size range examined. A correlation has

been established for predicting the effects of size, density and volume fraction on the

behaviour of binary particle mixing. The form of the equation is quite similar to that

developed for a cylindrical mixer, which suggests that similar mixing mechanisms are

present in both mixers.

Rotating drum:

Here, the objectives were the investigation of the effects of size and density ratios at different

volume fractions on mixing behaviour of binary particles at fixed operational conditions in

the rolling mode and development of a correlation for predicting the size and density effects

at volume fraction 0.5. The main conclusions are as follows.

The segregation mechanism that size induced buoyancy competes with the density

induced segregation by percolation was observed in the rolling mode in the rotating

drum. The mechanism can be used to reduce the segregation of particles into a core


9-229

region and the region around it by making small particles lighter and large ones

heavier.

A correlation was developed to predict the size and density effects on mixing of

binary particles in a rotating drum for the rolling mode. The prediction equation

obtained was similar in the form to those developed previously for cylindrical and

ribbon mixers.

Similarities and differences of the mixers

The effects of size, density and volume fraction of the particle mixing behaviour were

similar in the three mixers, vertically-shafted cylindrical mixer, ribbon mixer and

rotating drum.

The prediction equations developed for the three mixers are similar in the form, which

is a second order of polynomial for three variables.

The generation of the driving forces and their competition with the particle weight are

the main mechanisms for the size and density effects in the cylindrical mixer. In a

ribbon mixer too, similar mechanisms can be present as suggested by the similarity in

the prediction equations for the two mixers. However, a ribbon mixer has a higher

shear rate compared to other two mixers as observed from its effectiveness in

cohesive mixing. In rotating drums, Buoyancy and percolation are considered to be

the mechanisms responsible mixing or segregation in the rotating drum, especially in

the rolling mode. These mechanisms are similar to those in other two mixers, but

their effectiveness may be overshadowed by other factors such as forced particle

convection and shearing by the blades.

Overall, the present work has used the discrete element method successfully in the study

of the effect of material properties of particles on particle mixing in different mixers; the

effects of operational and geometrical parameters of ribbon mixer on mixing have been

established; and predictability of particle mixing behaviours have been identified for

different mixers.

Future work

It is essential to further investigate the possibility of achieving dynamic similarity

between the micro and macro systems for mixtures of larger cohesion.


9-230

Scaling methods need to be investigated to take the full advantage of the macro

computational system, which produces computational results quicker.

Velocity field can be a useful method to investigate particle mixing behaviour

because the methods such as PEPT readily generate the velocity field even in an

opaque particle bed. Therefore, predicting the mixing behaviour based on the

velocity field is worthy of exploration for a ribbon mixer as well.

The diffusion, convection and breakage of particles due to strong blade motion in

ribbon mixer need to be investigated.

Different blade arrangements of the ribbon mixers produce different flow patterns,

and thus, further studies should be carried out to obtain such important

information useful for mixer designing.

Vessel shape of the ribbon mixer will effect on the mixture quality. The U shape

and other type of vessels need to be studied for providing an optimum vessel

design of ribbon mixers.

Comparison of the dynamics of the different industrial mixers is essential for

mixer designers.

231

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List of publications

Conference papers:

Musha, H., G. R. Chandratilleke, S. L. I. Chan, J. Bridgwater and A. B. Yu (2013). "Effects of size and density differences on mixing of binary mixtures of particles." AIP Conference Proceedings 1542(1): 739-742.

Musha, H., K. Dong, G. R. Chandratilleke, J. Bridgwater and A. B. Yu (2013). "Mixing behaviour of cohesive and non-cohesive particle mixtures in a ribbon mixer." AIP Conference Proceedings 1542(1): 731-734.

H. Musha+, K. J. Dong, G. R. Chandratilleke, S.L.I. Chan, and A.B. Yu, DEM, Simulation of Powder mixing in a Ribbon-mixer, DEM6 conference preceedings.

M. Halidan, G. R. Chandratilleke, S.L.I. Chan, A.B. Yu , and John Bridgwater, PREDICTION OF THE MIXING BEHAVIOUR OF BINARY MIXTURES OF PARTICLES IN A BLADED MIXER,(submitted for Journal publication).

Particle Mixing Study in Different Mixers - UNSWorks

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