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Vibrationally-Fluidized Granular Flows: Impact and Bulk
Velocity Measurements Compared with Discrete Element
and Continuum Models
by
Kamyar Hashemnia
A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy
Mechanical and Industrial Engineering University of Toronto
© Copyright by Kamyar Hashemnia 2015
ii
Vibrationally-Fluidized Granular Flows: Impact and Bulk Velocity
Measurements Compared with Discrete Element and Continuum
Models
Kamyar Hashemnia
Doctor of Philosophy
Mechanical and Industrial Engineering
University of Toronto
2015
Abstract
A new laser displacement probe was developed to measure the impact velocities of particles
within vibrationally-fluidized beds. The sensor output was also used to measure bulk flow
velocity along the probe window and to provide a measure of the media packing. The
displacement signals from the laser sensors were analyzed to obtain the probability distribution
functions of the impact velocity of the particles. The impact velocity was affected by the
orientation of the laser probe relative to the bulk flow velocity, and the density and elastic
properties of the granular media. The impact velocities of the particles were largely independent
of their bulk flow speed and packing density.
Both the local impact and bulk flow velocities within a tub vibratory finisher were predicted
using discrete element modelling (DEM) and compared to the measured values for spherical steel
media. It was observed that the impact and bulk flow velocities were relatively insensitive to
uncertainties in the contact coefficients of friction and restitution. It was concluded that the
iii
predicted impact and bulk flow velocities were dependent on the number of layers in the model.
Consequently, the final DE model mimicked the key aspects of the experimental setup, including
the submerged laser sensor. The DE method predictions of both impact velocity and bulk flow
velocity were in reasonable agreement with the experimental measurements, with maximum
differences of 20% and 30%, respectively.
Discrete element modeling of granular flows is effective, but requires large numerical models.
In an effort to reduce computational effort, this work presents a finite element (FE) continuum
model of a vibrationally-fluidized granular flow. The constitutive equations governing the
continuum model were calibrated using the discrete element method (DEM). The bulk flow
behavior of the equivalent continuum media was then studied using both Lagrangian and
Eulerian FE formulations. The bulk flow velocities predicted by the Lagrangian approach were
in good agreement with those obtained using DEM simulations over a wide range of tub wall
amplitudes. The local impact velocity distribution predicted by the DEM was also compared to
the continuum model using the shear rate as a measure of the granular temperature.
iv
Acknowledgments
I would like to express my great appreciation to my supervisor, Professor Jan K. Spelt, for
providing me with his continuous guidance, enthusiasm and encouragement to assist me in
conducting a successful research.
I would like to thank my Ph.D. committee, Professor Markus Bussmann and Professor Nasser
Ashgriz for their valuable comments and suggestions offered during my annual exams and
reports. I also appreciate the thesis reviews and helpful comments provided by Professors Carl
Wassgren and Murray Grabinsky. I would like to acknowledge the Department of Mechanical
and Industrial Engineering of the University of Toronto for preparing a wonderful atmosphere to
pursue my Ph.D. I am also grateful of the financial support from Natural Sciences and
Engineering Research Council (NSERC) of Canada for providing the necessary means to fund
this project.
I would like to extend my acknowledge to the other members of the Materials and Process
Mechanics Laboratory, Amirhossein Mohajerani, Reza Haj Mohammad Jafar, Hooman Nouraei,
Amir Nourani, Saeid Akbari, Kavin Kowsari, Neda Tamannaee for not only their technical
assistance but also for creating a very good atmosphere to work and share ideas. I would like to
appreciate Amir Nourani for his support in learning Abaqus during the final year of my PhD. I
should also appreciate all my valuable friends who supported me during my Ph.D. experience,
especially: Mohammad Hossein Amini, Mohammad Ali Amini, Roshanak Banan, Ali Ebrahimi,
Javad Esmaeilpanah, Pakeeza Hafeez, Mohammad Reza Kholghy, Masih Mahmoodi, Esmaeil
Safaei, Ashkan Aryaee, Saber Jafarpur and Azadeh Hushmandi.
I would also like to thank my father and mother for their help and support as well as my sister
and brother. Their unconditional support and love has been extremely helpful during each stage
of my life. If they had not supported me emotionally and spiritually, I know that I could not have
reached any of the achievements in my life.
v
Table of Contents
Acknowledgments .......................................................................................................................... iv
Table of Contents ............................................................................................................................ v
List of Tables ................................................................................................................................. ix
List of Figures ................................................................................................................................ xi
List of Appendices ..................................................................................................................... xviii
Chapter 1 ......................................................................................................................................... 1
1 Introduction ................................................................................................................................ 1
1.1 Overview ............................................................................................................................. 1
1.2 Literature review ................................................................................................................. 2
1.3 Objectives ........................................................................................................................... 5
1.3.1 Objective 1 .............................................................................................................. 5
1.3.2 Objective 2 .............................................................................................................. 6
1.3.3 Objective 3 .............................................................................................................. 6
1.4 Thesis outline ...................................................................................................................... 6
1.5 References ........................................................................................................................... 7
Chapter 2 ....................................................................................................................................... 10
2 Development of a Laser Displacement Probe to Measure Particle Impact Velocities in
Vibrationally-Fluidized Granular Flows .................................................................................. 10
2.1 Introduction ....................................................................................................................... 10
2.2 Experiments ...................................................................................................................... 12
2.2.1 Measurement approach ......................................................................................... 12
2.2.2 Tub motion characterization ................................................................................. 14
2.2.3 Displacement sensor characterization ................................................................... 17
vi
2.2.4 Test procedures in the vibratory finisher .............................................................. 18
2.3 Signal analysis .................................................................................................................. 24
2.3.1 Ball motion in the vibratory finisher and the laser sensor displacement signals .. 24
2.3.2 Impact velocity calculation ................................................................................... 30
2.3.3 Double impact hypothesis ..................................................................................... 31
2.3.4 Error due to tangential bulk flow velocity ............................................................ 33
2.4 Probe evaluation in the tub finisher .................................................................................. 33
2.5 Conclusions ....................................................................................................................... 43
2.6 References ......................................................................................................................... 44
Chapter 3 ....................................................................................................................................... 47
3 Particle Impact Velocities in a Vibrationally-Fluidized Granular Flow: Measurements and
Discrete Element Predictions ................................................................................................... 47
3.1 Introduction ....................................................................................................................... 47
3.2 Discrete element modeling ............................................................................................... 49
3.2.1 Contact models and properties .............................................................................. 49
3.2.2 Reduced shear modulus ........................................................................................ 54
3.3 Sensitivity of DEM impact velocities to contact properties ............................................. 58
3.4 Measurement of contact properties ................................................................................... 61
3.5 Results and discussion ...................................................................................................... 66
3.5.1 Effect of model width ........................................................................................... 66
3.5.2 Comparisons with immersed laser-probe velocity measurements ........................ 70
3.6 Conclusions ....................................................................................................................... 75
3.7 References ......................................................................................................................... 76
Chapter 4 ....................................................................................................................................... 79
4 Finite Element Continuum Modeling of Vibrationally-Fluidized Granular Flows ................. 79
4.1 Introduction ....................................................................................................................... 80
vii
4.2 Continuum constitutive equations ..................................................................................... 84
4.3 Determining the continuum model parameters ................................................................. 88
4.3.1 Coefficient of internal friction and volume fraction ............................................. 88
4.3.2 Media equivalent Young's modulus and media-wall effective contact stiffness .. 92
4.3.3 Media-wall equivalent coefficient of friction ....................................................... 97
4.4 Finite element implementation of the continuum model ................................................ 102
4.4.1 Lagrangian FE continuum model ........................................................................ 102
4.4.2 Eulerian FE continuum model ............................................................................ 107
4.5 Sensitivity to uncertainty in continuum model parameters ............................................. 110
4.6 Comparison of FEA and DEM bulk flow velocities ....................................................... 112
4.7 Comparison of FEA and DEM local impact velocities .................................................. 115
4.8 Conclusions ..................................................................................................................... 118
4.9 References ....................................................................................................................... 118
Chapter 5 ..................................................................................................................................... 122
5 Conclusions ............................................................................................................................ 122
5.1 Experimental measurements ........................................................................................... 122
5.2 Discrete element modeling ............................................................................................. 123
5.3 Continuum modeling ...................................................................................................... 124
6 Future work ............................................................................................................................ 126
Appendix A: Analytical sensitivity study in Chapter 3 .............................................................. 127
Appendix B: MATLAB Codes of Chapter 2 .............................................................................. 132
Appendix B.1. Fast Fourier transformation (FFT) of the tub motion based on the
measurements .................................................................................................................. 132
Appendix B.2. Acceleration, velocity and position of the vibratory finisher center of
gravity ............................................................................................................................. 133
Appendix B.3. Path of the top points of the vibratory finisher (A and B in Fig. 2.2a) .......... 136
viii
Appendix B.4. The impact velocity calculation based on the laser displacement sensor
measurements .................................................................................................................. 137
Appendix C: MATLAB Codes of Chapter 3 .............................................................................. 139
Appendix C.1. The calculation of the in-plane components of the impact velocity based on
DEM simulation results .................................................................................................. 139
Appendix C.2. The calculation of the out- of-plane component of the impact velocity
based on DEM simulation results ................................................................................... 140
Appendix C.3. The calculation of coefficient of friction between spherical steel balls
based on the linear tribometer measurements ................................................................. 141
Appendix D: MATLAB Codes of Chapter 4 .............................................................................. 143
Appendix D.1. The calculation of local stress tensor, pressure and the coefficient of
internal friction based on the DEM data to calibrate the equivalent properties needed
in the continuum model .................................................................................................. 143
Appendix D.2. The calculation of local shear rate based on the DEM data to calibrate the
equivalent properties needed in the continuum model ................................................... 145
ix
List of Tables
Table 2.1: Physical and mechanical properties of steel and porcelain media (±
indicates standard deviation, 10 measurements) 15
Table 2.2: Bulk flow velocity parallel to window, particle passage frequency, packing
parameter and impact velocity of the steel and porcelain media for locations H, M and
L in different directions (number of data points in brackets). 95% confidence
intervals on the mean values based on student t distribution. 35
Table 3.1: Material properties used in the DEM simulations. 52
Table 3.2: Measured tub center of gravity vibration components used in DEM. 52
Table 3.3: Change in the x and z components of the impact velocity (denoted IV) and
bulk flow velocity (denoted BFV) at the points H and RU (Fig. 3.2) due to a shear
modulus reduction of 1, 2 and 3 orders of magnitude (denoted G-1, G-2, G-3,
respectively) and their corresponding errors (Er). Velocities in mm/s. Root mean
square error (RMS Er) is for all components and locations of either IV or BFV. Time
to run DEM simulation given in last row. 57
Table 3.4: Percentage uncertainty in the linear and angular rebound velocities due to
10% uncertainty in each of the contact parameters. 59
Table 3.5: Percentage uncertainty (U) in the single-layer DEM predictions of the
impact and bulk flow velocities due 10% uncertainty in the contact parameters
normalized by the following mean velocities from the DEM: IVx =75 mm/s, IVz =95
mm/s, BFVx =14 mm/s, BFVz =20 mm/s. p-p particle-particle, p-w particle-wall.
Location H of Fig. 3.2. 60
Table 3.6: The coefficient of friction (µ) between the steel ball of finishing media and
the sample of the polyurethane tub wall as a function of sliding speed. 61
x
Table 3.7: The measured coefficients of friction and restitution for different pairs of
materials (mean ±95% confidence interval). 65
Table 3.8: The average slope and ±95% confidence intervals of the free surface in
models with various numbers of particle layers in the width direction, y, between the
glass partitions. 67
Table 3.9: Comparison of the predicted and measured impact velocities (±95%
confidence intervals based on at least 200 data points from the DEM and at least 1,000
laser-probe velocity measurements; mm/s) and bulk flow velocities (mean value;
mm/s) at locations H and M in different directions (Figs. 3.2 and 3.8). The bracketed
bulk flow velocity at location M-rightwards refers to the DEM absolute oscillating
velocity. 73
Table 4.1: Material properties used in the DEM simulations [19]. 89
Table 4.2: Measured tub center of gravity vibration components used in DEM [19]. 89
Table 4.3: Material and contact properties used in the FE simulations. The media
effective Young’s modulus, and the contact parameters were evaluated over the
indicated ranges. The values of the media Young's modulus and the media-wall
contact stiffness in parentheses were calculated at the mean volume fraction φmean; i.e.
the best estimate. 105
Table 4.4: Sensitivity of the Lagrangian FE model mean bulk flow velocity (MFV)
and free-surface angle to changes in the effective media-wall contact stiffness, media
equivalent Young's modulus, and media-wall coefficient of friction. 112
xi
List of Figures
Fig. 1.1: Photo of the tub vibratory finisher 2
Fig. 2.1: Schematic of laser displacement sensor triangulation principle. 14
Fig. 2.2: Schematic of the tub vibratory finisher showing the bulk flow circulation
direction: (a) side view, (b) top view. The probe is shown at location H. The curved
arrows show the flow streamlines around the elliptical tube. Dimensions are in
millimeters and are drawn to scale. 16
Fig. 2.3: Displacement of a steel ball falling onto a glass plate acquired by the laser
displacement sensor (LK-G 82). The time step of data acquisition was 50 s (20 kHz
sampling rate). The arrows show the first, second and third impact points. 18
Fig. 2.4: (a) Schematic of the test apparatus used to measure the impact velocity of the
balls in the vertical direction in the vibratory finisher. Sensor, tub and balls are drawn
to scale. Note that the steel rod supporting the laser sensor was mounted directly to a
brick wall and was thus isolated from vibration. (b) enlarged portion of (a). (c)
enlarged view of the probe used to measure the impact velocity in the horizontal
(upstream or downstream) direction. 21
Fig. 2.5: The vibration of the window in Fig. 2.4c and the simultaneous vibration of
the shielding tube. The solid line represents the window motion while the dotted line
represents the shielding tube motion. 22
Fig. 2.6: Schematic of the vibrating tub showing the bulk flow pattern and the three
locations of the probe as it was when measuring toward the right. The x-y distances
correspond to the fixed locations in the flow used in all measurement directions.
Dimensions are in millimeters. 23
Fig. 2.7: (a) Inadequate laser beam reflection from the ball surfaces with excessively
steep slopes; (b) Laser beam reflecting from a non-diametral plane; (c) The range for
the laser sensor to accurately detect the position of the ball in a diametral plane (θ1 ≠ 25
xii
θ2).
Fig. 2.8: (a) Displacement signal (normal to the glass window) generated by the
passage of five balls parallel to the glass window in the rightward direction at location
H. Arrows show readings when laser passed between balls. (b) a magnified portion of
(a) corresponding to the passage of a single ball; (c) a magnified portion of (b); (d) a
magnified portion of Fig. 2.3 which shows the drop test results. The vertical arrows in
(c) and (d) show the points corresponding to contact between the window and ball. 27
Fig. 2.9: Chord length a used to calculate the bulk flow velocity of the balls passing
the sensor in a diametral plane. 28
Fig. 2.10: Predicted positions of the window (dashed line) and ball showing first and
second collisions giving rise to the apparent rounding of the impact peaks. Modeled
using data from location H in the steel media in the rightward direction (Fig. 2.6). 32
Fig. 2.11: Schematic of a ball having both normal (impact) and tangential (bulk flow)
velocity components relative to the window. 33
Fig. 2.12: Raw displacement signals of balls moving in the vibratory finisher at
location H: (a) rightward, (b) downward, (c) leftward (Fig. 2.6). The signal from the
window is shown in the narrow band at the top of each graph. 38
Fig. 2.13: The probability density distributions of the impact velocity of steel balls in
the vibratory finisher (a) at location H in the three orthogonal directions and (b) at
locations M and L in the directions indicated (Fig. 2.6). The types of distributions are
shown in the figure. 39
Fig. 2.14: The probability density distributions of the impact velocity of porcelain
balls in the vibratory finisher at locations H and M in the directions shown (Fig. 2.6).
All the distributions are Gaussian. 40
Fig. 2.15: The mean values of bulk flow velocity, impact velocity and packing
parameter (normalized by the largest values for each media) in different locations and 42
xiii
directions (Fig. 2.6) with: (a) steel, and (b) porcelain media.
Fig. 3.1: (a) Photo of the tub vibratory finisher including the laser probe. Schematic
of the tub vibratory finisher showing the bulk flow circulation direction (drawn to
scale): (b) side view, (c) plan view. The probe is shown at location H. The curved
arrows show the flow streamlines around the elliptical outer tube of the laser velocity
probe. Dimensions in mm [9]. 53
Fig. 3.2: Snapshot of a single particle-layer simulation and the three measurement
points located at (x,z): RU (200, 300), H (100, 300), M (0, 280) (mm). At each
location, the small and large boxes were for impact velocity and bulk flow velocity
calculation, respectively. Figure drawn to scale. Straight line approximation to free-
surface slope established at steady state. Arrow shows average bulk flow direction. 56
Fig. 3.3: Schematic showing the lower steel ball sliding against the upper steel ball
that was constrained in the y-direction to measure vertical displacement depth only. T
is the measured horizontal force. y=0 taken to be the minimum measured depth at the
apex of the upper ball. 62
Fig. 3.4: Tribometer measurements of the horizontal sliding force vs. (a) time and (b)
angle (θ) for steel balls moving against each other. The solid straight lines in (a) and
(b) show the analytical prediction. 63
Fig. 3.5: The drop test apparatus used to measure the coefficient of restitution of: (a)
the steel ball colliding with the polyurethane surface, (b) two colliding balls (side and
plan views). 65
Fig. 3.6: Predicted impact velocities in: (a) horizontal (x) direction, (b) vertical (z)
direction, and (c) transverse (y) direction, in the DEM simulations with different
numbers of layers. 69
Fig. 3.7: Predicted bulk flow velocities in (a) horizontal (x) direction and (b) vertical
(z) direction, in the DEM simulations with different numbers of layers. 70
xiv
Fig. 3.8: Snapshot of the simulation constructed using the plane of symmetry through
the elliptical laser velocity probe used in [9] located at point M (as if measuring
velocities of particles approaching from the right). (a) side view, (b) plan view, (c)
side view of probe and the measurement bins on the probe surface oriented with the
laser window to the right. The large bin with solid boundaries and the small bin with
dotted boundaries were used to obtain the impact and bulk flow velocities,
respectively. (d) Side view of a portion of the probe surface showing the particle bulk
flow with the measurement bin in the downward directions. In (c) and (d), large
arrows show the bulk flow direction, dashed smaller ones represent the laser beam. 71
Fig. 3.9: Probability density distributions of impact velocities predicted by the DEM
and measured in the experiments of ref. [9]. All the distributions were fitted with log-
normal functions. Locations defined in Figs. 3.2 and 3.8. 74
Fig. 4.1: (a) Photograph of the tub vibratory finisher including the laser probe used to
measure the media impact velocities in ref. [9], and two glass partitions within the tub.
(b) Schematic of the tub vibratory finisher showing the bulk flow circulation direction
from the side view. Dimensions in mm [19]. 83
Fig. 4.2: Variation of (a) the equivalent coefficient of internal friction [29] and (b) the
volume fraction with the inertial number seen in simple shear test [28]. 87
Fig. 4.3: The average streamlines of particles moving in a counter-clockwise flow in
the first layer of the 4-layer DE model during a 1 s interval. 89
Fig. 4.4: Measurement bin locations, H1, H2, L1 and L2 used to calculate the shear
rates and stress tensors from the DEM. Three in-plane bins at each location and a forth
bin behind the main bin. 91
Fig. 4.5: DEM predictions of the coefficient of internal friction as a function of: (a) the
inertial number, (b) the local pressure and (c) the shear rate; (d) Local volume fraction
vs. the inertial number. The solid and dashed lines are the average and the 95%
confidence bounds. 92
xv
Fig. 4.6: The equivalent Young's modulus of the vibrationally-fludized bulk flow, Eb,
vs. the solid fraction, φ, based on Eqs. (4.9-4.11) and the DEM. 95
Fig. 4.7: Seven measurement bins used to study the media solid fraction variation with
distance from the wall when: (a) wall moves toward media leading to maximum
compression, (b) wall moves away from media leading to maximum decompression of
the media and some separation between the media and the wall. 96
Fig. 4.8: Average solid fraction, φ, variation with distance from the wall using data
from the measurement bins of Fig. 4.7. 96
Fig. 4.9: Eight measurement locations used to determine average shear and normal
forces acting on the media along the tub wall. 97
Fig. 4.10: Probability density distributions of the ratio of the shear to normal forces
applied to the media by the wall (Ft/Fn) at the 8 locations of Fig. 4.9 for: (a) all data
(columns ordered from left to right as positions 1-8), and (b) for Ft/Fn <0.8. Based on
all particle-wall impact events recorded by the DEM during 47 vibration cycles 99
Fig. 4.11: Shear force vs. normal force at location 2 (Fig. 4.9) for all wall impacts over
47 vibration cycles. The best-fit lines represent the mean effective coefficient of
friction (µmw=0.25) with the correlation coefficient, and the measured COF for sliding
of a single steel ball against the polyurethane wall material (µmw=1.8). The line for
µmw=0.7 is also shown since it will be used in the continuum modelling. 100
Fig. 4.12: The average effective coefficient of friction between the fluidized media
and the tub wall at the 8 locations shown in Fig. 4.9 based on all impacts over 47
vibration cycles for the 4-layer and 8-layer DEM. The horizontal lines show the
average and 95% confidence bounds. 101
Fig. 4.13: Grand average shear force acting on the media by the wall over each 1/12 of
a cycle over 235 vibration cycles. Shear forces recorded over the high wall (HW;
locations 1-3), low wall (LW; location 4-6), and bottom wall (BW; locations 7-8) as
shown in Fig. 4.9. Positive shear force produced a counter-clockwise torque on the 102
xvi
media.
Fig. 4.14: The granular media domain meshed using Lagrangian elements: Free
triangular elements of sizes (a) 2 cm and (b) 1 cm; free quadrilateral elements of sizes
(c) 2 cm and (d) 1 cm; tub wall meshed using free triangular elements of size 2 cm. 104
Fig. 4.15: The bulk flow velocity distribution obtained from the Lagrangian analysis:
Free triangular elements of sizes (a) 2 cm and (b) 1 cm; free quad elements of sizes (c)
2 cm and (d) 1 cm. The simulation parameters: K=30.2 MPa/m, Eb=257 kPa and
µmw=0.3. Nodal velocity scale shown in lower right. Some arrows omitted for clarity.
Mean bulk flow velocity (MFV) across the dark horizontal lines. 106
Fig. 4.16: Material deformations in the Lagrangian and Eulerian analyses [45]. 108
Fig. 4.17: The media meshed by Eulerian elements of sizes (a) 1 cm and (b) 0.5 cm.
The tub wall was meshed with Lagrangian elements. 109
Fig. 4.18: Bulk flow velocity distributions obtained from the Eulerian analysis for
rectangular elements of sizes: (a) 1 cm and (b) 0.5 cm. Nodal velocity scale shown in
upper left. Some arrows omitted for clarity. 110
Fig. 4.19: (a) mean bulk flow velocity and (b) inclination of free surface as a function
of tub amplitude expressed as the fraction of the actual tub amplitude, A. Predictions
of the DEM and of the Lagrangian FE continuum model that used the optimized
media-wall effective coefficient of friction, µmw =0.7. FE predictions with µmw=0.5
included for comparison. 114
Fig. 4.20: Distribution of bulk flow velocity perpendicular to the horizontal line
defined in Fig. 4.15 in the 4-layer DE and the Lagrangian FE using the optimized
µmw=0.7 (K=30.2 MPa/m, Eb=295 kPa). Zero distance corresponds to the center of
circulation and 160 mm is at the tub wall. 115
Fig. 4.21: (a) Average particle impact velocity, Vimp, and its components vs. the shear
rate, , at points H1, L1, L2, H2 (ordered from left to right) in the 4-layer DEM (Fig. 117
xvii
4.4). (b) Shear rate in FE and DE simulations at the same points.
Fig. 3-A.1: Collision of two disks having arbitrary initial linear and angular velocities:
(a) velocities after collision, and (b) impulses during impact. 127
xviii
List of Appendices
Appendix A: Analytical sensitivity study in chapter 3 127
Appendix B: MATLAB Codes of Chapter 2 132
Appendix C: MATLAB Codes of Chapter 3 139
Appendix D: MATLAB Codes of Chapter 4 143
1
Chapter 1
1 Introduction
1.1 Overview
Vibratory finishing is widely used to deburr, polish, burnish, harden and clean metal, ceramic
and plastic parts. In a tub vibratory finisher, a container filled with granular media is oscillated
by an eccentric rotating shaft so that it develops a vibrationally-fluidized circulatory bulk flow of
the media that is largely two-dimensional (Fig. 1.1). The media have both a large-scale bulk
flow velocity and a much smaller-scale local impact velocity. Therefore, work-pieces that are
entrained in the flowing media are subjected to the repetitive, high-frequency impacts of the
surrounding particles. The granular media can be made of many materials of various shapes
such as spheres, cylinders or pyramids, with either a smooth or abrasive texture.
Within a vibratory finisher, the erosive wear and plastic deformation of workpieces are largely
affected by the velocity, frequency, and direction of the impacts with particles. High impact
velocities can cause fracture and fragmentation while low impact velocities can make the process
less efficient. The impact velocity is also closely related to the breakage of granular materials in
many processes such as drug tablet coating within rotating drums, bulk materials handling, food
processing, and particle attrition in vibratory finishing. Large particle impact velocities may also
result in erosion of machine components in processes such as vibratory sieving and mixing.
2
Fig. 1.1: Photo of a tub vibratory finisher
1.2 Literature review
The flow of granular materials has been studied in non-fluidized beds such as in hopper
discharge [1, 2], rotating drums, conveyers, chutes and mixers [3, 4], and in fluidized beds such
as in vibratory finishers and vibrating sieves [5-8]. A granular media becomes fluidized as the
vertical acceleration amplitude exceeds gravity, resulting in a marked decrease in the contact
pressure [8].
Measurement of local quantities like the impact velocity between vibrating particles in a
vibrationally-fluidized granular media is challenging, because of the relatively small scale of the
motion and difficulties in designing probes capable of measuring local quantities without
disturbing the media. A number of purely experimental studies have been done on the local
behavior of granular media [9-11]. Yabuki et al. [11] used a 3 mm diameter “floating” pad
3
connected to four resistive force sensors to measure both normal and tangential impact forces.
They found that forces normal to the sensor were approximately 10 times greater than the
tangential forces in both the wet and dry conditions. In the wet condition, a small percentage of
the media had a tangential to normal force ratio approximately equal to the measured dynamic
coefficient of friction between media, indicating that sliding had occurred. The force signals also
showed that the packing of the media in the vibrating bowl finisher was relatively loose, with
many gaps that were akin to bubbles.
The variation of the impact forces in steel and porcelain media in a tub vibratory finisher was
investigated in [10, 11] using a unidirectional high-speed impact force sensor. Local impact
velocities were inferred from impact forces, but this indirect measurement of impact velocity was
dependent on a system calibration and required a relatively large sensor that disrupted the bulk
flow in the finisher [10]. These impact force measurements were sensitive to the stiffness of the
sensor, and hence were specific to the experimental setup. This points to a key advantage of
measuring impact velocities directly; the impact velocities of particles will be unaffected by the
sensor compliance. Comparing the results of investigations in tub and bowl-type finishers, and
in finishers operating with different amplitudes [10-12], it can be concluded that the major
characteristics of the local behavior of the particles are largely independent of the type of
vibratory finisher producing the granular flow. For example, the vibrationally-fluidized flows
tend to be loosely packed and collisions with workpiece surfaces are dominated by the normal
component of impact.
Most research in the field of flowing granular media has focused on the bulk flow of granular
media inside different machines such as hopper discharge [1, 2], rotating drums [3, 4], and
vibratory finishers [7, 8]. The discrete element method (DEM) is a numerical technique that is
often used to solve problems involving transient dynamics of systems comprising a large number
of moving bodies that interact with each other. Generally, discrete element modeling (DEM)
simulations [13, 14] give reasonable predictions of bulk flow velocity and volume fraction in
both fluidized and non-fluidized flows [15-17]. For instance, the behavior of the convection
cells produced by a spherical media within a vertically vibrated container has been studied using
DEM and the results were validated by the experimental data [16]. As another example, the bulk
flow velocity distribution of glass beads in a vertical axis mixer with rotating flat blades was
4
obtained in DEM simulations and validated by the positron emission particle tracking (PEPT)
measurements [17].
The local behavior of non-fluidized granular flows has also been investigated experimentally and
numerically [2, 18-20], while some studies have examined vibrationally-fluidized flows [10-12,
21-23]. For instance, DEM was used to obtain the planar distributions of the collision velocity
and frequency of the granular media inside a horizontal rotating drum [2]. In the same geometry,
DEM was used to predict the effects of particle stiffness, size and coefficient of restitution on the
solid fraction, collision frequency, and impact velocity inside the granular media [18]. A
vibrated bubbling fluidized bed was simulated using DEM to give the instantaneous particle
velocity and local volume fraction [23]. However, only in ref. [19] were DEM predictions of
local behavior compared with experimental measurements. In that case, the local volume
fraction, granular temperature and bulk flow velocity in the surface layer of a shear flow were
measured using digital particle tracking velocimetry and compared with DEM predictions [19].
Previous efforts to model granular flows using a continuum approach have focused on the bulk
flow behavior of the non-fluidized granular media such as in the flow down inclined chutes [24-
27], plane shear flow [28, 29], flow in rotating drums [27, 28], flow in annular shear cells [24,
27, 28], and silo discharge flows [25, 27, 28]. In these cases of non-fluidized flows, constitutive
equations were defined to describe the bulk flow of the granular materials under quasi-static and
moderate flow (liquid-like) regions [24, 26, 30]. To the author’s knowledge, no papers have
been published on continuum modelling of vibrationally-fluidized granular flows. The
continuum models that have been used for quasi-static and moderate flow include different
elasto-plastic formulations of the equivalent continuum media [24, 30, 32], and visco-plastic
formulations considering only the plastic behavior of the equivalent media under time-varying
shear deformation [26-27]. In many cases, DEM was used to obtain the equivalent properties
needed to model the granular flows as a continuum [25, 27-29, 31, 32]. For example, the
equivalent stress tensor, pressure and shear rate at different points of a flowing granular bed were
obtained using DEM, and were then used to determine the equivalent continuum media elastic
and plastic properties [25, 30, 31].
Generally, the average streamlines, and hence the bulk flow behavior, determined through these
continuum simulations have been in fairly good agreement with the predictions of discrete
5
element modeling [24, 27, 30]. For example, Kamrin [24] proposed an elasto-plastic constitutive
law for use in a Lagrangian finite element model of granular flows in inclined chutes, rectangular
silos and annular Couette cells. The predicted flow fields were compared with those calculated
using DEM. Andrade et al. [30, 31] used the same material law to model the static 3D
compression of sand particles. Forterre and Pouliquen [27] used a visco-plastic constitutive law
with the fixed-grid finite difference method, which is equivalent to an Eulerian mesh to simulate
granular flows in the geometries of [24] and in rotating drums. They compared their results with
experimental measurements and DEM predictions of bulk flow [27]. The Lagrangian
formulation has been used in most of the papers that modelled granular flows using the finite
element method [24, 32].
1.3 Objectives
The first two objectives of the research were to measure the impact velocities of particles in the
granular flow produced by a tub vibratory finisher, and to compare these with the impact
velocities and flow characteristics predicted by numerical simulations using the discrete element
method (DEM). It was expected that many of these research results would be useful in modeling
the local impact conditions found in other applications of granular flow. The third objective was
to investigate whether continuum models might offer a simpler means of predicting bulk flow
and impact velocities.
1.3.1 Objective 1
It was of interest to develop a new probe based on a high-speed laser displacement sensor to
measure directly the surface-normal impact velocities of particles in vibrationally-fluidized
granular flows. The probe was demonstrated in a tub vibratory finisher where it was used to
make measurements in various locations and directions within flows of steel and porcelain
media.
6
1.3.2 Objective 2
The main goal was to compare the bulk flow and impact velocities obtained from a three-
dimensional discrete element model with those measured by using the setup including the
submerged laser velocity probe used in the earlier experiments.
1.3.3 Objective 3
The goal was to compare the bulk flow velocities in a vibrationally-fluidized granular flow
modeled as a visco-plastic media using different finite element formulations with those obtained
in DEM simulations and experimental measurements. The equivalent material properties of the
granular media had to be estimated using grain-scale 3D DEM simulations prior to modeling the
media as a continuum.
1.4 Thesis outline
The first Chapter of this thesis gives an introduction to vibrationally-fluidized granular flows and
how to study their bulk and local behavior experimentally and numerically. Chapter 2 covers the
experimental procedure using an impact velocity probe, the corresponding measurements and the
analysis of the results. This work has been published as:
K. Hashemnia, A. Mohajerani, J. K. Spelt, Development of a laser displacement probe to
measure particle impact velocities in vibrationally fluidized granular flows, Powder Technology,
235 (2013) 940-952
7
In Chapter 3, the vibrationally-fluidized bed of a tub vibratory finisher was modeled using the
discrete element method and the resulting bulk flow and impact velocities of the particles at
different points in the granular flow were compared with the experimental measurements. This
study has been published as:
K. Hashemnia, J. K. Spelt, Particle impact velocities in a vibrationally fluidized granular flow:
measurements and discrete element predictions, Chemical Engineering Science, 109 (2014) 123–
135.
Chapter 4 presents a study of how the vibrationally-fluidized bed can be modeled as a continuum
using different finite element methods, and compares the results with DEM simulations. The
following article has been submitted on this part of the project:
K. Hashemnia, J. K. Spelt, Finite Element Continuum Modeling of Vibrationally-Fluidized
Granular Flows, Chemical Engineering Science, 2014, submitted.
Finally, Chapter 5 reviews the overall conclusions of the Ph.D. research.
1.5 References
[1] W. R. Ketterhagen, J. S. Curtis, C. R. Wassgren, A. Kong, P. J. Narayan, B. C. Hancock,
Granular segregation in discharging cylindrical hoppers: A discrete element and experimental
study, Chem. Eng. Sci. 62 (2007) 6423 – 6439.
[2] W. R. Ketterhagen, J. S. Curtis, C. R. Wassgren, B. C. Hancock, Predicting the flow mode
from hoppers using the discrete element method, Powder Technol. 195 (2009) 1–10.
[3] R.Y. Yang, R.P. Zou, A.B. Yu, Microdynamic analysis of particle flow in a horizontal
rotating drum, Powder Technol. 130 (2003) 138–146.
[4] V. Jasti, C. F. Higgs, Experimental study of granular flows in a rough annular shear cell,
Phys. Rev. E 78 (2008) 041306-1-8.
[5] N.G. Deen, M. Van Sint Annaland, M.A. Van der Hoef, J. A. M. Kuipers, Review of discrete
particle modeling of fluidized beds, Chem. Eng. Sci. 62 (2007) 28-44.
8
[6] X. Z. An, C.X. Li, R.Y. Yang, R.P. Zou, A.B. Yu, Experimental study of the packing of
mono-sized spheres subjected to one-dimensional vibration, Powder Technol. 196 (2009) 50–
55.
[7] S.E. Naeini, J.K. Spelt, Two-dimensional discrete element modeling of spherical steel media
in a vibrating bed, Powder Technol. 195 (2009) 83-90.
[8] S.E. Naeini, J.K. Spelt, Development of single-cell bulk circulation in granular media in a
vibrating bed, Powder Technol. 211 (2011) 176-186.
[9] C. Fan, X. T. Bi, J. R. Grace, A. Goto, Grid zone performance of a fluidized bed through
analysis of local solids holdup signals, Powder Technol. 219 (2012) 37–44.
[10] D. Ciampini, M. Papini, J. K. Spelt, Impact velocity measurement of media in a vibratory
finisher, J. Mater. Process Tech. 183(2007) 347-357.
[11] A. Yabuki, M. R. Baghbanan, J. K. Spelt, Contact forces and mechanisms in a vibratory
finisher, Wear 252 (2002) 635-643.
[12] S. Wang, R. S. Timsit, and J. K. Spelt, Experimental Investigation of Vibratory Finishing of
Aluminum, Wear 243 (2000) 147-156.
[13] A. Munjiza, The combined finite discrete element method, John Wiley & Sons, Ltd,
Chichester, West Sussex, England, 2004.
[14] H.P. Zhu, Z.Y. Zhou, R.Y. Yang, A.B. Yu, Discrete particle simulation of particulate
systems: A review of major applications and findings, Chem. Eng, Sci. 63 (2008) 5728-
5770.
[15] C.H. Tai, S.S. Hsiau, C.A. Kruelle, Density segregation in a vertically vibrated granular bed,
Powder Technol. 204 (2010) 255-262.
[16] M. Majid, P. Walzel, Convection and segregation in vertically vibrated granular beds,
Powder Technol. 192 (2009) 311-317.
[17] R. L. Stewart, J. Bridgwater, Y. C. Zhou, A.B.Yu, Simulated and measured flow of granules
in a bladed mixer - A detailed comparison, Chem. Eng. Sci56 (2001) 5457-5471.
[18] B. Freireich, J. Litster, C. Wassgren, Using the discrete element method to predict collision-
scale behavior: A sensitivity analysis, Chem. Eng. Sci. 64 (2009) 3407- 3416.
[19] J.J. McCarthy, V. Jasti, M. Marinack, C.F. Higgs, Quantitative validation of the discrete
element method using an annular shear cell, Powder Technol. 203 (2010) 70-77.
9
[20] W. R. Ketterhagen, J. S. Curtis, C. R. Wassgren, Stress results from two-dimensional
granular shear flow simulations using various collision models, Phys. Rev. E 71 (2005)
061307-1-11.
[21] K. Hashemnia, A. Mohajerani, J. K. Spelt, Development of a laser displacement probe to
measure particle impact velocities in vibrationally fluidized granular flows, Powder
Technol., 235 (2013) 940-952.
[22] K. Hashemnia, J. K. Spelt, Particle impact velocities in a vibrationally fluidized granular
flow: Measurements and discrete element predictions, Chem. Eng. Sci., 109 (2014) 123–
135.
[23] L. Xiang, W. Shuyan, L. Huilin, L. Goudong, C. Juhui, L. Yikun, Numerical simulation of
particle motion in vibrated fluidized beds, Powder Technol. 197 (2010) 25–35.
[24] K. Kamrin, Nonlinear elasto-plastic model for dense granular flow, Int. J. Plasticity, 26
(2010) 167–188.
[25] C. H. Rycroft, K. Kamrin, M. Z. Bazant, Assessing continuum postulates in simulations of
granular flow, J. Mech. Phys. Solids, 57 (2009) 828–839.
[26] P. Jop, Y. Forterre, O. Pouliquen, A constitutive law for dense granular flows, Nature, 441
(2006) 727-730.
[27] Y. Forterre, O. Pouliquen, Flows of dense granular media, Annu. Rev. Fluid Mech. (2008)
1-24.
[28] G. D. R Midi, On dense granular flows, Eur. Phys. J. E (2004) 341-365.
[29] F. Cruz, S. Emam, M. Prochnow, J. N. Roux, and F. Chevoir, Rheophysics of dense
granular materials: Discrete simulation of plane shear flows, Phys. Rev. E 72 (2005)
021309-1-17.
[30] J. E. Andrade, Q. Chen, P. H. Le, C. F. Avila, T. M. Evans, On the rheology of dilative
granular media: Bridging solid- and fluid-like behavior, J. Mech. Phys. Solids 60 (2012)
1122–1136.
[31] J. E. Andrade, C. F. Avila, S. A. Hall, N. Lenoir, G. Viggiani, Multiscale modeling and
characterization of granular matter: From grain kinematics to continuum mechanics, J.
Mech. Phys. Solids 59 (2011) 237–250.
[32] J. E. Andrade, X.Tu, Multiscale framework for behavior prediction in granular media,
Mech. Mater. 41 (2009) 652–669.
10
Chapter 2
2 Development of a Laser Displacement Probe to Measure Particle Impact Velocities in Vibrationally-Fluidized Granular Flows
2.1 Introduction
Vibratory finishing is widely used to deburr, polish, burnish, harden and clean metal, ceramic
and plastic parts. In a tub vibratory finisher, a container filled with granular media is oscillated
by an eccentric rotating shaft so that it develops a vibrationally-fluidized circulatory bulk flow of
the media that is largely two-dimensional. The media have both a large-scale bulk flow velocity
and a much smaller-scale local impact velocity. Therefore, work-pieces that are entrained in the
flowing media are subjected to the repetitive, high-frequency impacts of the surrounding
particles. The granular media can be made of many materials of various shapes such as spheres,
cylinders or pyramids, with either a smooth or abrasive texture.
Granular flow has been studied in non-fluidized beds such as in hopper discharge, conveyers,
chutes and mixers [1-8], and in fluidized beds such as in vibratory finishers, vibrating sieves and
rotating drums [3, 4, 9-14]. A granular media becomes fluidized as the vertical acceleration
amplitude exceeds gravity, resulting in a marked decrease in the contact pressure and constraint
provided by surrounding media [12, 14]. In addition to bed vibration, fluidization can result
from interstitial gas flow and from particle-wall interactions resulting from high-speed chute
flow [15].
Within a vibratory finisher, the erosive wear and plastic deformation of workpieces are largely
affected by the velocity, frequency, and direction of the impacts with particles. High impact
velocities can cause fracture and fragmentation while low impact velocities can make the process
less efficient. The impact velocity is also closely related to the breakage of granular materials in
11
many processes such as drug tablet coating within rotating drums, bulk materials handling, food
processing, and particle attrition in vibratory finishing [16]. Large particle impact velocities may
also result in erosion of machine components in processes such as vibratory sieving and mixing
[16].
Measurement of local quantities like the impact velocity between vibrating particles in a
vibrationally-fluidized granular media is challenging, because of the relatively small scale of the
motion and difficulties in designing probes capable of measuring local quantities without
disturbing the media. Therefore, most existing research in the field of flowing granular media
has focused on the bulk flow of granular media inside different machines [1, 2, 5-9, 11-14, 17-
19]. The discrete element method (DEM) is a numerical technique that is often used to solve
problems involving transient dynamics of systems comprising a large number of moving bodies
that interact with each other. DEM simulations have been used to study the bulk flow behavior
of granular media and the numerical results have been verified experimentally [7, 12].
Generally, these investigations have shown that discrete element simulations give reasonable
predictions of bulk flow velocity and volume fraction in both fluidized and non-fluidized flows.
Only a few studies have focused on the local behavior of the media in granular flows [7, 14, 20-
22, 23, 24] and some of these have investigated vibrationally-fluidized flows [14, 23]. For
example, DEM was used to obtain the planar distributions of the collision velocity and the
collision frequency of the granular media inside a horizontal rotating drum [7]. This
configuration was also examined in [20], using DEM to predict the effects of particle stiffness,
size and coefficient of restitution on the solid fraction, collision frequency, and impact velocity
inside the granular media. In most of these studies, the predicted collision scale quantities such
as impact velocity, collision frequency or impact energy, were not validated experimentally [7,
20, 22, 23]. Therefore, it is not known if the commonly used approaches and parameters in DEM
simulations yield correct predictions of the impact velocity. This remains as a significant
limitation since the impact forces that govern erosion, wear and fracture in granular flows are
fundamentally linked to the particle impact velocity and kinetic energy [20].
A number of purely experimental studies have been done on the local behavior of granular media
[24- 26]. Yabuki et al. [26] used a 3 mm diameter “floating”padconnected to four resistive
force sensors to measure both normal and tangential impact forces. They found that forces
12
normal to the sensor were approximately 10 times greater than the tangential forces in both the
wet and dry conditions. In the wet condition, a small percentage of the media had a tangential to
normal force ratio approximately equal to the measured dynamic coefficient of friction between
media, indicating that sliding had occurred. The force signals also showed that the packing of the
media in the vibrating bowl finisher was relatively loose, with many gaps that were akin to
bubbles. The variation of the impact forces in steel and porcelain media in a tub vibratory
finisher was investigated in [25, 26] using a unidirectional high-speed impact force sensor.
Local impact velocities were inferred from impact forces, but this indirect measurement of
impact velocity was dependent on a system calibration and required a relatively large sensor that
disrupted the bulk flow in the finisher [25]. These impact force measurements were sensitive to
the stiffness of the sensor, and hence were specific to the experimental setup. This points to a
key advantage of measuring impact velocities directly; the impact velocities of particles will be
unaffected by the sensor compliance.
Comparing the results of investigations in tub and bowl-type finishers, and in finishers operating
with different amplitudes [25-27], it can be concluded that the major characteristics of the local
behavior of the particles are largely independent of the type of vibratory finisher producing the
granular flow. For example, the vibrationally-fluidized flows tend to be loosely packed and
collisions with workpiece surfaces are dominated by the normal component of impact.
The main objective of the present investigation was to develop a new probe based on a high-
speed laser displacement sensor to measure directly the surface-normal impact velocities of
particles in vibrationally-fluidized granular flows. The probe was demonstrated in a tub
vibratory finisher where it was used to make measurements in various locations and directions
within flows of steel and porcelain media.
2.2 Experiments
2.2.1 Measurement approach
13
There are different methods to measure the particles velocity in the fluidized beds, such as: laser-
Doppler velocimetry, photographic and video techniques, optical fiber probes and laser
displacement sensors. Photographic methods have been used to measure bulk flow properties,
but the limited spatial resolution of these approaches makes them unsuited to the measurement of
local impact velocities [11, 27]. Laser-Doppler velocimetry is restricted to the low solid
concentrations (loose media). Optical fiber probes have been used to measure bulk flow
velocities and the void fraction of media passing transversely across the end of the sensor [28].
Some designs can measure displacements along the optical probe axis using a correlation with
the amount of reflected light, but such devices become inaccurate when the light reflected by the
particle varies because of a significant transverse velocity across the probe tip. Therefore, laser
displacement sensors were chosen to measure the impact velocity of particles inside a relatively
packed granular flow produced in the tub vibratory finisher.
Laser displacement sensors measure the distance to an object using triangulation as illustrated in
Fig. 2.1. Laser light reflected from the object is concentrated on a linearized charge-coupled
device (LI-CCD) such that its position on the LI-CCD depends on the distance to the object.
Two laser sensors with different working distances were used: LK-G 82 with a working distance
of 80±15 mm and a resolution of 0.2 µm, and LK-G 157 with a working distance of 150±40 mm
and a resolution of 0.5 µm (Keyence Inc.). Both sensors had a red semiconductor laser light
source, but the LK-G 82 had a round beam diameter of 70 µm while the LK-G 157 had an
elliptical beam with minor diameter of 120 µm and a major diameter of 1700 µm. The data
acquisition rate of these laser sensors was adjustable from 1-20 kHz [29].
14
Fig. 2.1: Schematic of laser displacement sensor triangulation principle.
2.2.2 Tub motion characterization
The laser probe developed in the present work was tested in a tub vibratory finisher (Burr-Bench
2016, Brandon Industries, TX) shown in Fig. 2.2a. Two types of granular media were used: 91
kg of steel balls (ABCO, Abbott Ball Company Inc., West Hartford, CT, USA) and 41 kg of
porcelain balls (MicrobriteTM
, Abrasive Finishing Inc., Chelsea, MI, USA). Additional mass was
added in the latter case to maintain the same tub dynamics. Table 2.1 shows the density,
Young's modulus, mass and diameter (mean ± standard deviation) of each particle in the steel
and porcelain media. The Young’s modulus and Poisson’s ratio of the porcelain balls were
measured ultrasonically on a disk (3 mm thick, 5 mm diameter) ground from a single ball. An
ultrasonic pulser-receiver (5072PR, Olympus Corp.) was used to measure the velocities of
longitudinal and shear waves to determine these elastic properties [30].
15
Table 2.1: Physical and mechanical properties of steel and porcelain media (± indicates standard
deviation, 10 measurements)
Material Young's
Modulus (GPa)
Poisson's
Ratio
Density
(g/cm3)
Mass (g)
[25]
Diameter
(mm) [25]
Steel 207 0.27 7.8 1.03±0.002 6.3±0.005
Porcelain 71.4 0.23 2.4 0.29±0.020 6.1±0.180
The granular media was confined to a compartment made of two 12 mm thick glass plates, 21 cm
apart, installed rigidly in the central portion of the tub (Fig. 2.2b). A tri-axial accelerometer and
two uni-axial accelerometers were attached to the top of the tub at points A and B (Fig. 2.2a) in
order to measure the horizontal and vertical acceleration components. The accelerometer signals
were filtered to eliminate high-frequency noise, and the dominant mode of vibration at 47 Hz
was determined using a fast Fourier transform. The velocity and displacement of points A and B
were then calculated using symbolic integration of the equations of motion in MATLAB. The
angular velocity, Ω, and the angular acceleration, α, that characterized the tub rocking motion
were then determined using:
B A AB ABa a r r (2.1)
B A ABv v r (2.2)
where rAB is the position vector from A to B. Equations (2.3) and (2.4) were then used to find
the amplitudes of displacement, velocity and acceleration at other points (P) on the tub wall.
P A AP APa a r r (2.3)
P A APv v r (2.4)
The accuracy of these procedures was assessed by comparing the measured vertical displacement
at point A and the calculated vertical displacement at point C (middle of tub floor) in an empty
tub with direct laser measurements at these two points. It was found that the tub displacement
amplitudes calculated from the tub accelerations measured by the accelerometers (0.86 mm at A
and 0.64 mm at C) were 10% deviated from the laser measurements, probably due to the noise in
the accelerometer readings since the errors did not display a consistent bias.
16
Fig. 2.2: Schematic of the tub vibratory finisher showing the bulk flow circulation direction: (a)
side view, (b) top view. The probe is shown at location H. The curved arrows show the flow
streamlines around the elliptical tube. Dimensions are in millimeters and are drawn to scale.
(a)
(b)
17
2.2.3 Displacement sensor characterization
The accuracy of the impact velocity measurement using the laser displacement sensor was
assessed by dropping a spherical steel ball of the type used in the finisher from an electromagnet
held various heights above a glass plate. The laser displacement sensor was positioned vertically
under the glass plate such that the laser beam reflected from the midpoint of theball’sbottom
surface, andtheballremainedwithinthesensor’sworkingdistance during its fall. To simulate
the oblique impacts of the balls in the vibratory finisher as they vibrate against a target while
moving with the bulk flow velocity, the glass plate was tilted between 0º and 16º (the maximum
impact angle corresponding to the largest bulk flow velocity), to examine impacts of different
incidence angles while keeping the incident laser beam normal to the glass.
Figure 2.3 shows the experimental results for the displacement and velocity of a 6.3 mm
diameter smooth steel ball falling onto the glass plate from a height of 12.7 mm and bouncing
five times. A quadratic relation was fitted to the approach portion of the first impact of Fig. 1.3
in order to determine the impact velocity from the first derivative. For a predicted impact
velocity of 500.0 mm/s (maximum uncertainty of 2.0 mm/s or 0.4%), the average measured
value was 496.9 mm/s (standard deviation of ±0.3 mm/s based on 6 measurements); i.e. a 0.6%
error. The coefficient of restitution corresponding to the ratio of the measured rebound and
incident velocities was 0.755±0.004. The surface-normal component of the impact velocity for
an oblique impact with a small inclination angle of the glass plate (less than 16) was predicted
with less than 10% error using the laser sensor.
The apparatus was also used to measure the coefficient of restitution of the porcelain balls
colliding with a glass plate. In this case, tweezers were used to release a porcelain ball from a
platform at a prescribed height. The coefficient of restitution was then calculated as the ratio of
the measured rebound and incident velocities to be 0.84±0.01 (mean ± standard deviation based
on 6 measurements).
18
Fig. 2.3: Displacement of a steel ball falling onto a glass plate acquired by the laser displacement
sensor (LK-G 82). The time step of data acquisition was 50 s (20 kHz sampling rate). The
arrows show the first, second and third impact points.
2.2.4 Test procedures in the vibratory finisher
In order to minimize the effects of vibration, the laser sensor was attached to an adjacent brick
wall using a horizontal steel beam connected to a vertical 30 mm diameter steel rod (Fig. 2.4a).
The laser was then directed down an elliptical aluminum shielding tube (70 mm major axis, 25
mm minor axis) that was immersed in the flowing media and supported by a 20×20 mm steel box
section that was also attached to the brick wall. The major axis was parallel to the bulk flow to
minimize disturbance to the flow. The laser sensor had no contact with the elliptical tube or the
media, and thus it was effectively stationary during the operation of the vibratory finisher. A
very small vibration of the sensor was due to acoustic wave propagation, but this found to be
negligible; i.e. less than 0.2 m. A glass window at the bottom of the tube was used to measure
impact velocities against the glass of upward flowing media. Two 12×40 mm glass windows at
opposite ends of the major axis of the cross-section were used to measure impact velocities in the
horizontal direction by redirecting the laser with a 45º polished steel mirror (Fig. 2.4c). To
19
minimize its vibration, the mirror was attached directly to the laser sensor using a rod passing
through the elliptical pipe without contact so that it was isolated from the vibration of the media.
As with the laser sensor, a negligible vibration was due to acoustic waves (measured to be less
than 0.5 m.)
Although the shielding elliptical pipe was supported by a steel bar connected to the brick wall,
some vibration of the three glass windows embedded in its wall was inevitable due to its
exposure to the granular flow. Although this window vibration did not influence the incident and
reflected laser light from the media, it was of interest to quantify the amplitude of this vibration
relative to the motion of the vibrating media. The vibration of the glass window could not be
recorded directly using the second sensor, since the window was covered by the media in the tub.
Therefore, a correlation was established between the window vibrations and the vibration of the
shielding tube to which they were bonded. To measure the vibration of each of the sideways
facing windows, the second laser sensor (LK-G 82) was positioned outside the media and
reflected from a point on the shielding tube in the horizontal direction (Fig. 2.4c). To measure
the vibration of the bottom window used to measure vertical impact velocities, the second laser
was reflected from a steel beam clamped to the shielding tube (Fig. 2.4b). A piece of tape was
put on each window to acquire the window vibration when the tub was operating using the
primary laser, and simultaneously the vibration of the shielding tube was recorded using the
second laser. A correlation was then established between the vibration amplitudes of the
windows and the output of the second laser. Figure 1.5 illustrates this for the window of Fig.
2.4c used to measure the particle horizontal impact velocities.
20
(b)
(a)
21
Fig. 2.4: (a) Schematic of the test apparatus used to measure the impact velocity of the balls in
the vertical direction in the vibratory finisher. Sensor, tub and balls are drawn to scale. Note that
the steel rod supporting the laser sensor was mounted directly to a brick wall and was thus
isolated from vibration. (b) enlarged portion of (a). (c) enlarged view of the probe used to
measure the impact velocity in the horizontal (upstream or downstream) direction.
(c)
22
10 10.02 10.04 10.06 10.08 10.1 10.12 10.14 10.16 10.18 10.2 10.22 10.24
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
Time (s)
Positio
n (
mm
)
Fig. 2.5: The vibration of the window in Fig. 2.4c and the simultaneous vibration of the shielding
tube. The solid line represents the window motion while the dotted line represents the shielding
tube motion.
The impact velocity in different orientations was measured at three locations in the tub. Each
location was 5 cm below the free surface of the flowing media which had the curved shape
shown in Fig. 2.6. The position of the shielding tube was adjusted slightly each time so that the
measurement locations in the flow were fixed regardless of the window being used. Location H
corresponded to one of the locations used in the same tub in ref. [25], allowing direct comparison
with the data obtained with the impact load cell. The tub as well as the type and amount of steel
and porcelain media were the same as in ref. [25], ensuring that the flow conditions were
identical to those in [25].
The vibrating tub finisher produced a single-cell circulation as illustrated in Fig. 2.6 because of a
resultant upward shear force between the media and the right wall of the tub [12]. This also
produced a free surface that hadapronouncedslopefromthe“highwall”tothe“lowwall”,and
a bulk flow near the free surface that accelerated as the media cascaded downward to the low
wall.
23
The impact velocities of the steel balls were measured with the laser sensor pointing in various
directions depending on the location: At location H, measurements were made in three
directions: horizontal upstream facing the bulk flow, horizontal downstream, and vertically
facing downward. At location M, the impact velocity was measured in the horizontal direction
facing upstream and in the downward vertical direction. At location L, the impact velocity was
measured in the horizontal direction facing upstream. In the porcelain media, impact velocities
were measured at locations H and M in the horizontal direction facing upstream toward the
oncoming flow and in the vertical direction facing downward. Two additional sets of
experiments were performed in the porcelain media at locations 2 cm above and 2 cm below
location M in order to assess the sensitivity of the impact velocity to depth.
Prior to each experiment, the relevant window was cleaned with acetone to minimize laser
reflection from the glass. The steady-state tub vibration was then measured using the
accelerometers described previously, and the laser sensor signals were acquired for 33 s.
Fig. 2.6: Schematic of the vibrating tub showing the bulk flow pattern and the three locations of
the probe as it was when measuring toward the right. The x-y distances correspond to the fixed
locations in the flow used in all measurement directions. Dimensions are in millimeters.
24
The porcelain particles were much lighter than the steel particles. Therefore, in the experiments
with the porcelain media, in order to maintain the same dynamics of the tub while preserving the
bed depth used with the steel media, additional 50 kg steel media was added equally to the
empty compartments located at either end of the tub on the outside of the glass partitions.
Displacement measurements confirmed that the vibration amplitude of the tub differed by less
than 5% between the experiments done with the steel and porcelain media.
2.3 Signal analysis
2.3.1 Ball motion in the vibratory finisher and the laser sensor displacement signals
Two components of particle motion were evident: a cyclic impact against the glass window at
the tub vibration frequency of 47 Hz, and bulk flow parallel to the glass window (Fig. 2.7a). The
angles shown in the Figs. 2.7b and 2.7c, illustrate the range through which the laser sensor can
accurately read the position of the points on the surface of a ball in the plane of the laser beams.
This range was greatest when the nominal plane of the sensor was on a diametral plane of the
ball (i.e. plane containing the ball axis) and became smaller as the path of the ball shifts further
from the plane of the sensor. In any event, Fig. 2-7c illustrates that the laser beam reflected
adequately over an asymmetric range of angles in the diametral plane (i.e. θ1 and θ2 were
approximately 80º and 65º, respectively).
The vibrations of the balls and the shielding tube were recorded simultaneously using two laser
sensors as described previously. Figures 2.8a-c shows the typical displacement signal obtained
from the motion of the steel balls in the finisher with a data acquisition rate of 2 kHz. Using the
correlation mentioned in the Section 2.4, the window vibration was calculated from the measured
shielding tube vibration. For example, Fig. 2.8a shows the window vibrations and the
simultaneous signals from 5 balls passing before the sensor. Figure 2.8b shows these signals for
the passage of a single ball, and illustrates that the window and the passing balls vibrated in
25
phase at 47 Hz. The displacement signal contained some erroneous readings of very small
displacements (indicated by arrows in Fig. 2.8a) generated when the laser beam passed through
the empty space between two adjacent balls, or when the laser was reflected from ball surfaces
with excessively steep slopes. In the former case, the sensor recorded the laser beam reflecting
from either the mirror surface or the glass window.
Laser
sensor
Emitter
Receiver
Direction of the
bulk flow
Balls path
Inappropriate
reflection
Window
Fig. 2.7: (a) Inadequate laser beam reflection from the ball surfaces with excessively steep
slopes; (b) Laser beam reflecting from a non-diametral plane; (c) The range for the laser sensor
to accurately detect the position of the ball in a diametral plane (θ1 ≠ θ2).
(a) (b)
(c)
26
29.4 29.5 29.6 29.7 29.8 29.9
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
Time (s)
Positio
n (
mm
)
(b)
(a)
27
29.48 29.485 29.49 29.495 29.5 29.505 29.51 29.515 29.52 29.525 29.53-0.9
-0.7
-0.5
-0.3
-0.1
0.1
0.3
0.5
Time (s)
Positio
n (
mm
)
0.04 0.06 0.08 0.1 0.12 0.14 0.16
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
Time (s)
Positio
n (
mm
)
Fig. 2.8: (a) Displacement signal (normal to the glass window) generated by the passage of five
balls parallel to the glass window in the rightward direction at location H shown with dotted
lines. Arrows show readings when laser passed between balls. (b) a magnified portion of (a)
corresponding to the passage of a single ball; (c) a magnified portion of (b); (d) a magnified
portion of Fig. 2.3 which shows the drop test results. The vertical arrows in (c) and (d) show the
points corresponding to contact between the window and ball. The solid lines show the window
position.
(c)
(d)
Δt
0.15Δt
0.3Δt
28
The large-scale sinusoidal variation of the average displacement reflects the curvature of the
balls and their bulk flow past the sensor (Fig. 2.8a). If the nearest approach of the ball was
coincident with the displacement of the window, the ball was passing before the sensor on a
diametral plane as shown in Fig. 2.8b. Figure 2.8c shows a portion of the displacement signal
corresponding to a portion of the ball surface that is not itself making contact with the window
(e.g. location B in Fig. 2.7b or a point between 1 and 2 in the diametral plane of Fig. 2.7c).
The bulk flow velocity was then be estimated by dividing the chord length a by the
corresponding time interval while the ball passed in front of the sensor along a diametral plane
(Fig. 2.9). The chord length a was calculated from
2 22 ( ( ) )a R R h (2.5)
where R is the ball radius, and h was chosen to be 0.5 mm in the calculations (i.e. h was the
maximum difference between the ball and window positions); therefore, a would be 3.1 mm for
the steel balls and 3.35 mm for the porcelain balls.
Fig. 2.9: Chord length a used to calculate the bulk flow velocity of the balls passing the sensor in
a diametral plane.
The bulk flow velocity at location H in the horizontal direction facing upstream determined from
this method was 13±1.0 mm/s (average ± 95% confidence interval based on 20 measurements).
Although it was not possible to visually record the bulk flow exactly at location H, video images
29
of the media flowing past one of the glass partitions in the finisher at the same x-y location (Fig.
2.2) were used to estimate the value. A video image with a 40×40 mm field of view was
replayed frame-by-frame to track individual balls over a distance of 20 mm. This gave a bulk
flow velocity adjacent to the glass partition of 15±1.0 mm/s (average ± 95% confidence interval,
20 measurements). While viewing the free surface (Fig. 2.2b), the average bulk flow velocity
midway between the glass partitions was measured similarly to be 85% of that at the glass
partitions (64 mm/s in the middle and 75 mm/s at the glass partitions). If it is assumed that this
correlation between bulk flow velocities in the middle of the flow and at the glass partitions was
also valid at location H, the actual bulk flow velocity at location H was 13 mm/s, in very good
agreement with the value from the laser sensor.
Another property of the granular flow that was obtained from the displacement signals was the
“particle passage frequency” defined as the number of balls passing in front of the sensor per unit
time. In this case, all balls passing within 0.5 mm of the diametral plane were considered (as
judged by the measured distance of nearest approach to the window). This provided a measure
that was a function of both the bulk flow velocity and the degree of packing in the granular flow.
The ascending line in each oscillation of Fig. 2.8b corresponded to the approach of a ball to the
surface of the window (i.e. the impact velocity). As can be seen in Fig. 2.8c, the displacement
curves were not as sharp near the impact points as they were in the drop tests (Fig. 2.3). The
main reason for this difference was that the window vibrated in-phase with the balls (Fig. 2.8b)
and with an amplitude that was 30-50% of the ball amplitude; therefore, after a ball collided with
the moving window, it rebounded from the window, but continued to move toward the sensor
with a reduced speed. The ball was then accelerated in the opposite direction by the window
moving outward once again (Fig. 2.11). This is discussed further in Section 3.3. Another factor
that can contribute to the rounding of the impact peaks in Fig. 2.8c was the tangential velocity of
the ball parallel to the window, which was about 17% of the average normal impact velocity.
This caused the laser reflection point to shift on the ball surface creating an error in the actual
displacement attributed to the surface-normal impact velocity; however, as explained below, this
error was negligible (less than 1%) in the present experiments.
30
2.3.2 Impact velocity calculation
Impact velocities normal to the sensor window were determined by calculating the slopes of the
straight line segments of the ascending (approach) portions of the curves shown in Fig. 2.8c prior
to contact. The following procedure was used to consistently identify the data points that were
fitted to a straight line to determine the impact velocity: 1. The minimum and maximum points
were used to calculate the half-period, Δt and the midpoint (Fig. 2.8c). 2. The selected data points
were in the range from 0.30Δt to the left of the midpoint to 0.15Δt to the right of the midpoint.
This criterion ensured that the ball had not yet contacted the window as discussed further below.
A MATLAB code was written to identify the correct displacement signals and perform these
calculations.
The calculated impact velocities were relatively insensitive to the exact range of the data used on
the approach portion of the ball displacement signal. For example, if the data points were
selected to be those in the interval 0.25Δt extending from the point 0.05Δt to the right of the
minimum, the difference in the calculated impact velocity was less than 3%. Although impact
velocities could also be calculated for balls passing a certain distance off the diametral plane, in
the present work it was convenient to use only the signals used in bulk flow velocity calculations
for impact velocity calculation; i.e. the signals in which the difference between the point of
closest approach and the window was less than 0.1 mm (10% of the vibration amplitude of the
tub center of gravity (1.1 mm)).
The impact velocities calculated as described above were always greater than the maximum
window velocity which was determined at each location and direction using the dominant
frequency of 47 Hz and the recorded amplitude of the sinusoidal motion. The raw displacement
data were smoothed to remove erroneous readings using the 5-point central moving average
method prior to calculating the impact velocities (Fig. 2.8a).
31
2.3.3 Double impact hypothesis
The double impact hypothesis was used above to explain the relatively blunt impact peaks in the
displacement signals. The displacement of the window (y1) and ball (y2) can be modeled as (Fig.
2.8c)
1 0 1 sin( )y y A wt
2 2 sin( )y A wt
(2.6)
(2.7)
where A1 and A2 are the corresponding vibration amplitudes and y0 is the initial distance between
the ball and the target. In Fig. 2.10 the ball and the window follow the solid and dashed lines,
respectively. Solving Eqs. (2.6) and (2.7) simultaneously, the time corresponding to the first
collision is
1 01
2 1
1sin ( )imp
yt
w A A
(2.8)
where w is the vibration frequency in rad/s. Then the target and the ball velocities immediately
prior to the first impact are, respectively
arg 1 1cos( )t et impV A w wt
2 1cos( )ball impV A w wt
(2.9)
(2.10)
The coefficient of restitution e is defined as
arg
arg
ball t et
t et ball
V Ve
V V
(2.11)
where the primed parameters correspond to the velocities after the impact and unprimed ones are
those before the collision. Since the mass of the window/shielding tube is much larger than that
of the ball, conservation of momentum indicates that the window velocity remains approximately
unchanged, so that the ball velocity after impact is
32
arg 1 2 1( ) cos( )ball t et impV V e A A w wt (2.12)
As shown in Fig. 2.10, the ball can be assumed to move with a new constant speed after the first
impact until the second impact occurs when the window moves outward again. The time of
impact, and the target and ball velocities prior to and after the second collision can be determined
using Eqs. (2.8) - (2.12). Figure 2.10 was drawn assuming e=0.75, A1 =0.15 mm, A2 =0.30 mm,
y0 =0.12 mm, and w=295.1 rad/s, and it is found that the second impact occurs 3 ms after the
first one. This is a relatively short time compared with the period of oscillation of the ball
(τ=1/47s=21ms).
0 1 2 3 4 5 6 7 8 9 100
0.05
0.1
0.15
0.2
0.25
Time (ms)
Ball
and targ
et positio
ns (
mm
)
Fig. 2.10: Predicted positions of the window (dashed line) and ball showing first and second
collisions giving rise to the apparent rounding of the impact peaks. Modeled using data from
location H in the steel media in the rightward direction (Fig. 2.6).
33
2.3.4 Error due to tangential bulk flow velocity
The bulk flow of the balls past the window changed the point of reflection of the laser and
caused an apparent change in the distance between the ball and the window, as illustrated in Fig.
2.11. Assuming an average value of the maximum bulk flow velocity parallel to the window of
νt=20.5 mm/s, the largest error this produced in the normal (impact) velocity measurement was
less than 1% and hence could be neglected.
Fig. 2.11: Schematic of a ball having both normal (impact) and tangential (bulk flow) velocity
components relative to the window.
2.4 Probe evaluation in the tub finisher
The probe was tested in the tub vibratory finisher using both the steel and porcelain spherical
media while maintaining the same tub vibration amplitude. Table 2.2 shows the component of
the bulk flow velocity parallel to the sensor window and the particle passage frequency of the
steel and porcelain media at locations H, M and L and in the different directions. The average
bulk flow velocity in the steel media was approximately 40% larger than that in the porcelain
34
media. Within each media, the bulk flow velocity along the window was approximately the
same in all cases, ranging from 16 to 20 mm/s in the steel media, except for location H rightward
where it was only 13 mm/s, and from 8.7 to 13 mm/s in the porcelain media.
The particle passage frequency showed a much greater variation, from 0.22 to 0.73 s-1
in the steel
media, and from 0.16 to 0.49 s-1
in the porcelain media. Since the bulk flow velocity was
approximately the same at all locations in the steel media except H rightward, and in all locations
in the porcelain media except M downward, the variation in particle passage frequency was
mostly attributable to differences in the packing density of the media. This was examined by
definingadimensionless“packingparameter”as theproductof theparticlepassage frequency
and the ball diameter divided by the bulk flow velocity, in order to eliminate the effect of the
bulk flow velocity. Table 2.2 shows that the packing parameter was maximum at location M
downward in both media, indicating that the media had the largest packing density at this
location and direction. In locations such as H leftward, this quantity was a minimum, probably
as a consequence of the wake formed behind the shielding tube. However, there was no
consistent relationship between the magnitude of the packing parameter and the flow direction.
For example, at location H it was essentially the same in the rightward and downward directions,
but at location M it was significantly greater in the downward direction. There was also no
consistent correlation between the relative packing and the type of media; i.e. in some locations
and directions the porcelain media were more highly packed, but not in others.
35
Table 2.2: Bulk flow velocity parallel to window, particle passage frequency, packing parameter
and impact velocity of the steel and porcelain media for locations H, M and L in different
directions (number of data points in brackets). 95% confidence intervals on the mean values
based on student t distribution.
Location and
orientation of laser
Bulk flow
velocity (mm/s)
Particle passage
frequency (s-1
)
Packing
parameter
Impact velocity
(mm/s)
Steel balls
H rightward 13±1.0 (20) 0.28±0.05 (12) 0.139±0.0179 74±1.1 (1,032)
H downward 16±1.3 (20) 0.33±0.04 (14) 0.132±0.0136 80±1.3 (1,068)
H leftward 20±1.5 (20) 0.22±0.05 (15) 0.0693±0.0116 61±1.0 (621)
M rightward 19±2.0 (20) 0.35±0.06 (29) 0.113±0.0177 91±1.2 (1,254)
M downward 20±2.1 (20) 0.73±0.05 (32) 0.233±0.0200 110±1.6 (1,618)
L rightward 19±1.7 (20) 0.63±0.07 (19) 0.208±0.0219 72±1.8 (939)
Porcelain balls
H rightward 10±1.4 (17) 0.16±0.02 (25) 0.0976±0.0133 85±0.8 (2,786)
H downward 8.7±1.2 (17) 0.22±0.04 (20) 0.146±0.0262 140±1.3 (1,665)
M rightward 10±1.1 (17) 0.22±0.02 (24) 0.134±0.0138 67±1.3 (1,525)
M rightward
(2 cm above)
32±1.5 (559)
M rightward
(2 cm below)
94±1.4 (1,849)
M downward 13±1.0 (17) 0.49±0.03 (25) 0.232±0.0162 120±1.2 (1,946)
Figures 2.12a and 2.13a show the displacement signal and the probability density distribution of
the normal impact velocity of the balls in the vibratory finisher at location H in the rightward
direction. Figure 2.13a also shows the best-fit log-normal probability distribution. Figure 2.13a
was calculated from the slopes of more than 1,032 displacement signals corresponding to the
passage of 120 balls using 15 recordings of 33 s duration. The impact velocity distribution had a
mean value of 74 mm/s with more than 90% of the impact velocities in the range of 50-100
mm/s. In comparison, more than 80% of the impact velocities measured at the same location and
in the same direction in ref. [25] using an impact force sensor were in the range of 0-20 mm/s.
There are several reasons for this difference, although the most significant is probably the linear
correlation that was assumed to exist between measured impact force and the impact velocity
over the range 0-1,000 mm/s [25]. Since the impact force varied nonlinearly with the impact
velocity [4], the slope of a line fitted to the lower impact velocity range 0-100 mm/s was
36
observed to be twice the slope of the line corresponding to the impact velocity over the range 0-
1,000 mm/s [4, 25]. Therefore, the correlation in [25] underestimated the impact velocities since
most of them were below 100 mm/s. Another reason for the difference in impact velocities was
that “non-impact”contactevents(e.g.aballthatdwelledoverthewindowwithoutvibrating)that
were excluded in the present measurements were assigned an effective impact velocity in [25]
based on their quasi-static contact force. Thirdly, although the location and measurement
direction were identical, the impact probe used in ref. [25] was significantly larger than the
present probe (twice the volume and twice the projected area normal to the average bulk flow).
This may have altered the local conditions at the probe surface.
Figures 2.12b and 2.12c show the displacement signals of the steel balls moving in the vibratory
finisher at location H in the downward and leftward and directions, respectively. The best-fit
impact velocity probability distributions corresponding to the different directions at location H in
the steel media were quite different in both their breadth and their mean, as seen in Fig. 2.13a.
Figure 2.13b shows that this was also true for the impact velocity distributions for the rightward
and downward directions at point M and the rightward direction at point L in the steel media.
As at location H rightward, a log-normal probability density distribution of impact velocities was
fitted to the steel media data at location H downward and location M rightward (Fig. 2.13). In
contrast, the distributions of impact velocities at locations M downward and L rightward were
found to fit better to a Gaussian curve (Fig. 2.13).
The probability density distribution of steel media impact velocities at location H leftward was
skewed more than could be fitted using a log normal distribution, similar to the distributions
found in [25]; therefore, a generalized extreme value distribution [31] was used to fit this impact
velocity distribution (Fig. 2.13a).
Freireich et al. [20] used the discrete element method to study the effects of different parameters
on the impact velocity of 3 mm diameter glass beads moving in a rotating drum. They also
found that the probability density distribution of impact velocity was skewed with an elongated
tail to the rights, similar to the log-normal and generalized distributions used with the present
steel media.
37
In contrast to the steel results, a Gaussian probability density distribution was found to fit best to
the impact velocities at locations H and M in the porcelain media, as shown in Fig. 2.14.
(a)
(b)
38
Fig. 2.12: Raw displacement signals of balls moving in the vibratory finisher at location H: (a)
rightward, (b) downward, (c) leftward (Fig. 2.6). The signal from the window is shown in the
narrow band at the top of each graph.
(c)
39
0 20 40 60 80 100 120 140 160 180 200 2200
0.05
0.1
0.15
0.2
0.25
Impact velocity (mm/s)
Pro
babili
ty d
ensity d
istr
ibution
0 20 40 60 80 100 120 140 160 180 200 2200
0.02
0.04
0.06
0.08
0.1
0.12
Impact velocity (mm/s)
Pro
babili
ty d
ensity d
istr
ibution
Fig. 2.13: The probability density distributions of the impact velocity of steel balls in the
vibratory finisher (a) at location H in the three orthogonal directions and (b) at locations M and L
in the directions indicated (Fig. 2.6). The types of distributions are shown in the figure.
(b)
(a)
Rightward-L-Steel
(Gaussian)
Downward-M-steel
(Gaussian)
Rightward-M-Steel
(Lognormal)
Rightward-H-steel (Lognormal)
Leftward-H-steel (Generalized extreme value)
Downward-H-steel (Lognormal)
40
0 20 40 60 80 100 120 140 160 180 200 2200
0.02
0.04
0.06
0.08
0.1
Impact velocity (mm/s)
Pro
babili
ty d
ensity d
istr
ibution
Fig. 2.14: The probability density distributions of the impact velocity of porcelain balls in the
vibratory finisher at locations H and M in the directions shown (Fig. 2.6). All the distributions
are Gaussian.
The mean values of impact velocities in the steel and porcelain media along with the 95%
confidence intervals are listed in Table 2.2. As the sample sizes were sufficiently large (typically
N>30), the mean values of the impact velocities were normally distributed, regardless of the
underlying population distribution [32]; therefore, the 95% confidence intervals for the means
were calculated using the t distribution.
Table 1.2 shows that the packing parameter and the average impact velocity at location H were
smaller in the leftward direction than in the rightward direction. This can be interpreted as a
result of the wake produced behind the shielding tube.
In the porcelain media, it was observed that the impact velocity in the downward direction (Fig.
2.14 and Table 2.2) and the corresponding packing parameter (Table 2.2) were higher than those
in the rightward direction at location H and M, similar to what was observed in the steel media,
except at location H where the steel packing parameter was approximately the same in the
downward and rightward directions.
Rightward-H-Porcelain
Rightward-M-Porcelain
Downward-M-Porcelain
Downward-H-Porcelain
41
Comparing the values of the rightward impact velocities at location M in the porcelain media
with those at points 2 cm above and below it (Table 2.2), it was concluded that impact velocity
of the particles is strongly dependent on depth, increasing sharply with distance from the free
surface.
Figures 2.13, 2.14, and Table 2.2 show that the average impact velocities in the porcelain media
were almost 15% greater than those in the steel media. There are two possible reasons for this
observation: Firstly, the density and mass of the porcelain balls was lower than that of steel balls;
therefore, for the same vibrational kinetic energy transferred from the wall, the porcelain balls
had a higher velocity than the steel balls. Secondly, based on the measurements of Section 2.3, it
may be assumed that the coefficients of restitution of impacts of porcelain balls with each other
and with the tub walls were larger than those corresponding to the steel balls; therefore, less
kinetic energy was dissipated during the collisions, thereby increasing the impact velocities of
the porcelain balls compared with the steel balls.
Freireich et al. [20] reported that the predicted mean impact velocity increased and the
probability density distribution of the impact velocity becomes less skewed as the coefficient of
restitution increased. This is consistent with the present experimental results where it was
observed that the impact velocity of the porcelain balls (with larger coefficient of restitution)
fitted better to a normal distribution (symmetric) rather than the lognormal distribution that was
used to fit the impact velocity of the steel balls.
Figure 2.15 shows the normalized mean values of bulk flow velocity, impact velocity and
packing parameter in the different locations and directions for the steel and porcelain media. It is
evident that all three parameters varied considerably in the various locations and directions, and
that there was no simple correlation between them. Moreover, the relative magnitudes of the
impact velocity, packing density and bulk flow velocity, and their dependence on location and
direction, were often different in the steel and porcelain media.
42
Fig. 2.15: The mean values of bulk flow velocity, impact velocity and packing parameter
(normalized by the largest values for each media) in different locations and directions (Fig. 2.6)
with: (a) steel, and (b) porcelain media.
(b)
(a)
43
2.5 Conclusions
A laser displacement sensor was used to construct a probe that can be immersed in a granular
flow to measure local impact velocities, bulk flow velocities and a measure of particle packing.
The accuracy of impact velocity measurements was verified using drop tests with the same
granular media. The sensor output was modeled to interpret laser reflections from media passing
before the sensor in different trajectories and to distinguish the media vibrational impact velocity
from the oscillations of the sensor window.
The sensor was demonstrated in a tub vibratory finisher with two types of media – steel and
porcelain spheres. In this case, the average bulk flow velocity parallel to the sensor window in
the steel media was almost 40% larger than that in the porcelain media, and both varied
appreciably over the different measurement locations within a single media. Similarly, the
impact velocity and particle packing varied considerably among the various locations and in
different directions. Although it was concluded that there was no direct proportionality between
the impact velocity of particles, the bulk flow velocity, and the packing density of the media, the
smallest packing density and impact velocity occurred in the wake formed behind the sensor
shielding tube. These two parameters were larger when the sensor was facing the average
moving bulk flow. It was also observed that the impact velocity of the particles was strongly
dependent on depth in the flow, increasing sharply with distance from the free surface. The
average impact velocities in the steel media were approximately 15% smaller than those in the
porcelain media due to differences in the density and coefficient of restitution. In addition to
providing a means of quantifying the impact energy distribution within a vibratory bed, the
experimental results can be used to validate the predictions of numerical simulations such as
discrete element modeling (DEM). Such models can help to understand the complex patterns of
behavior that have been observed within these vibrationally fluidized beds. The laser
displacement probe and the current procedures are directly applicable to the measurement of the
local impact velocities of non-spherical particles. The measurement of the bulk flow velocity in
the case of irregular particle shapes would be more difficult, since the variation in the average
displacement signal would be more irregular.
44
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[14] C. H. Tai, S.S. Hsiau, Dynamic behaviors of powders in a vibrating bed, Powder Technol.
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[15] J. K. Spelt, C. E. Brennen, and R. H. Sabersky, Heat Transfer to Flowing Granular Material,
Int. J. Heat Mass Transf., 25 (6) (1982) 791-796.
[16]EDEMsoftware’swebsite,http://www.dem-solutions.com/academic, 2011.
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[17] C. S. Campbell, Granular material flows–An overview, Powder Technol. 162 (2006) 208–
229.
[18] H.P. Zhu, Z.Y. Zhou, R.Y. Yang, A.B. Yu, Discrete particle simulation of particulate
systems: A review of major applications and findings, Chem. Eng, Sci. 63 (2008) 5728-5770.
[19] A. Wu, Y. Sun, Granular Dynamic Theory and Its Applications, first ed., Metallurgical
Industry Press, Beijing and Springer-Verlag GmbH Berlin Heidelberg, 2008.
[20] B. Freireich, J. Litster, C. Wassgren, Using the discrete element method to predict collision-
scale behavior: A sensitivity analysis, Chem. Eng. Sci. 64 (2009) 3407- 3416.
[21] J.J. McCarthy, V. Jasti, M. Marinack, C.F. Higgs, Quantitative validation of the discrete
element method using an annular shear cell, Powder Technol. 203 (2010) 70–77.
[22] W. R. Ketterhagen, J. S. Curtis, C. R. Wassgren, Stress results from two-dimensional
granular shear flow simulations using various collision models, Phys. Rev. E 71 (2005) 061307-
1-11.
[23] L. McElroy, J. Bao, C.T. Jayasundara , R.Y. Yang, A.B. Yu, A soft-sensor approach to
impact intensity prediction in stirred mills guided by DEM models, Powder Technol. 219 (2012)
151–157.
[24] C. Fan, X. T. Bi, J. R. Grace, A. Goto, Grid zone performance of a fluidized bed through
analysis of local solids holdup signals, Powder Technol. 219 (2012) 37–44.
[25] D. Ciampini, M. Papini, J. K. Spelt, Impact velocity measurement of media in a vibratory
finisher, J. Mater. Process Tech. 183(2007) 347-357.
[26] A. Yabuki, M. R. Baghbanan, J. K. Spelt, Contact forces and mechanisms in a vibratory
finisher, Wear 252 (2002) 635-643.
[27] S. Wang, R. S. Timsit, and J. K. Spelt, Experimental Investigation of Vibratory Finishing of
Aluminum, Wear 243 (2000) 147-156.
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and Validation, AIChE Journal 49 (2003) 1405-1420.
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47
Chapter 3
3 Particle Impact Velocities in a Vibrationally-Fluidized Granular Flow: Measurements and Discrete Element Predictions
3.1 Introduction
The flow of granular materials has been studied in non-fluidized beds such as in hopper
discharge [1,2], rotating drums, conveyers, chutes and mixers [3,4], and in fluidized beds such
as in vibratory finishers and vibrating sieves [5-8]. A granular media becomes fluidized as the
vertical acceleration amplitude exceeds gravity, resulting in a marked decrease in the contact
pressure [8]. Vibratory finishing is widely used to polish, burnish, harden, and clean metal,
ceramic and plastic parts. In a tub vibratory finisher, the two-dimensional vibration of the walls
produces a vibrationally-fluidized circulatory bulk flow of the media. The media have both a
large-scale bulk flow velocity resulting from the resultant shear force with the walls, and a local
impact velocity during each vibration cycle [9,10].
The erosive wear and plastic deformation of a workpiece within a vibratory finisher are
determined mainly by the velocity, frequency, and direction of the impacts of the granular
finishing media [11,12]. Moreover, these impacts can lead to breakage of the finishing media.
In processes such as drug tablet coating within rotating drums, and bulk materials handling, the
impact of the granular products can contribute to their fragmentation. The machine components
may also be eroded in processes such as vibratory sieving and mixing because of large particle
impact velocities [12].
Most research in the field of flowing granular media has focused on the bulk flow of granular
media inside different machines such as hopper discharge [1,2], rotating drums [3,4], and
vibratory finishers [7,8] . Generally, discrete element modeling (DEM) simulations [14, 15] give
48
reasonable predictions of bulk flow velocity and volume fraction in both fluidized and non-
fluidized flows [16-18]. For instance, the behavior of the convection cells produced by a
spherical media within a vertically vibrated container has been studied using DEM and the
results were validated by the experimental data [16]. As another example, the bulk flow velocity
distribution of glass beads in a vertical axis mixer with rotating flat blades was obtained in DEM
simulations and validated by the positron emission particle tracking (PEPT) measurements [17].
Relatively few studies have dealt with the local behavior of the media in granular flows [16, 19-
23], and some of these studies have investigated vibrationally-fluidized flows [9, 11, 16, 23].
For instance, DEM was used to obtain the planar distributions of the collision velocity and
frequency of the granular media inside a horizontal rotating drum [3]. In the same geometry,
DEM was used to predict the effects of particle stiffness, size and coefficient of restitution on the
solid fraction, collision frequency, and impact velocity inside the granular media [18]. A
vibrated bubbling fluidized bed was simulated using DEM to give the instantaneous particle
velocity and local volume fraction [11]. However, only in ref. [19] were DEM predictions of
local behavior compared with experimental measurements. In that case, the local volume
fraction, granular temperature and bulk flow velocity in the surface layer of a shear flow were
measured using digital particle tracking velocimetry and compared with DEM predictions [19].
Among the three different contact models used in these DEM simulations, only the large plastic
dissipation model predicted both the measured local volume fraction and bulk flow velocity with
reasonable accuracy (maximum error of 35%); however, the predictions of granular temperature
had an error of approximately 120% [19]. A key objective of the present study was to compare
DEM predictions with local impact velocity measurements made previously in the tub vibratory
finisher using the submerged high-speed laser displacement probe of Fig. 1b, c [9].
Although it has been shown that collision-level quantities are relatively insensitive to the choice
of the contact model used in DEM [23], it is not known how sensitive the particle impact
velocities are to uncertainties in the contact coefficients used in these models, such as the
coefficients of restitution and friction corresponding to the particle-particle and the particle-wall
interactions. Therefore, it was of interest to begin the present work with an analytical and
numerical study of the sensitivities of the predicted bulk flow and impact velocities to these
contact coefficients used in the DEM. This was then used to guide the accuracy requirements for
the measurement of these coefficients for the spherical steel media used in the urethane-lined tub
49
finisher with glass end walls. This was then followed by investigations of the relationship
between the DEM accuracy and the number of particle layers modelled between the glass end
walls of the tub finisher, and the use of a reduced shear modulus in order to decrease the model
run time. Finally, the DE model was expanded to mimic the complete experimental setup
including the submerged laser velocity probe used in the earlier experimental study of the bulk
flow and impact velocities [9, 10].
3.2 Discrete element modeling
3.2.1 Contact models and properties
In DEM, the contact forces between interacting particles and between the particles and the
container boundaries are calculated from the magnitude of the overlaps between the bodies in the
normal and tangential directions. The contact force models that can be used in DEM simulations
are defined using different assumptions about the elasticity of the contacting particles [24].
Some of the models are elastic such as the linear-spring and Hertz models, while some are visco-
plastic like the linear-spring/dashpot (LSD), Kuwabara-Kono and Lee-Hermann models, and
some are elasto-plastic such as the Hertz-Mindlin, Thornton and Walton- Braun models [25, 26].
Most of these models are non-linear as they were defined based on the Hertz theory. In the
present simulations, since the amount of plastic deformation of the steel spheres was marginal,
the Hertz-Mindlin contact model was selected. In this model, both elastic and plastic
deformations of particles are calculated in the normal and tangential directions by considering
the coefficients of restitution, friction, and rolling resistance [12]. The normal elastic force, e
nF ,
was calculated using the common Hertz formulas (Eq. 3.1) [24].
3
* * 24
3
e
n nF E R (3.1a)
*
1 1 1
i jR R R (3.1b)
50
22
*
111 ji
i j
vv
E E E
(3.1c)
where E* and R
* are the equivalent Young's modulus and the equivalent radius, respectively, and
the indices i and j indicate the particles in contact. δn is the normal deformation (overlap)
between the particles. The normal plastic (damping) force, d
nF , is determined from the
knowledge of the normal overlap and the normal coefficient of restitution, e, and the relative
normal velocity between particles before impact, rel
nv (Eq. 3.2) [23]. The coefficient of
restitution is defined as the ratio of the relative normal velocity just after collision to that just
prior to the collision (Eq. 3.2e) [23].
*52
6
d rel
n n nF S m v (3.2a)
*
1 1 1
i jm m m (3.2b)
* *2n nS E R (3.2c)
2 2
ln
e
ln e
(3.2d)
2 1
1 2
n n
n n
v ve
V V
(3.2e)
Here, m*
is the equivalent mass of the contacting particles, β is a function of the normal
coefficient of restitution, and Sn is called the normal stiffness. Vn and vn are the normal velocities
before and after the collision, respectively. The tangential elastic force, e
tF , is calculated using
tangential overlap, δt, and tangential stiffness, St (Eq. 3.3) [23]. The tangential overlap is defined
as the relative displacement of the particles during impact in the tangential direction, having
subtracted the rolling contribution [23]. G*=E
*/2(1+v)
is the equivalent shear modulus of the
contacting bodies.
e
t t tF S (3.3a)
* *2t tS G R (3.3b)
51
The tangential plastic (damping) force, d
tF , is determined using the tangential overlap, the
tangential coefficient of restitution, et, and the relative tangential velocity between particles
before impact, rel
tv (Eq. 3.4) [23].
*52
6
d rel
t t t tF S m v (3.4a)
2 2
ln
tt
t
e
ln e
(3.4b)
2 1
1 2
t tt
t t
v ve
V V
(3.4c)
The tangential force is constrained by the Coulomb friction force fs= µsFn such that:
e d
t t t sF F F f , where µs is the coefficient of static friction.
The present DEM simulations were made using EDEM 2.5 (DEM Solutions Inc. 2012) and its
standard library of contact models [12]. The shear modulus, Poisson's ratio and density of the
steel balls, polyurethane wall of the tub and the glass end partitions were input to the software,
along with the particle-particle and particle-wall coefficients of restitution, friction and rolling
resistance, which were measured as described in Section 3.3. The glass partition at either end of
the tub was assumed to be frictionless. Table 3.1 gives the material properties used in the DEM.
The tub geometry was defined (Fig. 3.1a) and its motion was determined using the accelerometer
and laser displacement measurements described in ref. [9]. The tub had sinusoidal translations in
the x and z directions of Fig. 3.1b and one sinusoidal rotation in the plane of Fig. 3.1b, all at 47
Hz. Table 3.2 gives the measured amplitudes of these vibration components at the tub centre of
gravity and the phase difference between the vertical and horizontal vibrations. The submerged
velocity probe shown in Fig. 3.1b, c could be oriented to the right, left or downward so that the
laser sensor measured particle impact velocities in these three directions.
52
Table 3.1: Material properties used in the DEM simulations.
Material Shear modulus (GPa) Poisson's ratio Density (kg/m3)
Steel 76 0.29 7800
Polyurethane 0.0086 0.50 1200
Glass 26 0.23 2500
Table 3.2: Measured tub center of gravity vibration components used in DEM.
Horizontal (x)
displacement
amplitude (mm)
Vertical (z)
displacement
amplitude (mm)
Phase difference
between
translations (deg.)
Angular
displacement
amplitude (deg.)
Vibration
frequency
(Hz)
0.85 0.66 103 0.244 47
(a)
53
Fig. 3.1: (a) Photo of the tub vibratory finisher including the laser probe. Schematic of the tub
vibratory finisher showing the bulk flow circulation direction (drawn to scale): (b) side view, (c)
plan view. The probe is shown at location H. The curved arrows show the flow streamlines
around the elliptical outer tube of the laser velocity probe. Dimensions in mm [9].
(b)
(c)
Z
54
The duration of an elastic impact between two bodies, Ti, was calculated based on the Hertz
theory from Eq. (3.5),
2
2
1
* 5
* *2.87i rel
n
mT
R E v
(3.5)
At least six time steps should be included during an impact in order to accurately solve the
equations of motion of the particles in contact [12]. To make the solution stable, the simulation
time step was set to be 20% of the Rayleigh critical time step, ΔtR, defined as the time taken for a
shear wave to propagate through a solid particle. The Rayleigh critical time step is a function of
the particle diameter d, density ρ, shear modulus G and Poisson's ratio ν, and it was calculated
from Eq. (3.6) to be 3.43 µs [12].
/ 2 / / 0.1631 0.8766Rt d G v
2 1
EG
v
(3.6)
The simulation time step was then calculated to be 0.686 µs for the 6.3 mm diameter steel balls
used in the experiments.
3.2.2 Reduced shear modulus
By decreasing the shear modulus used for the steel balls, the simulation time step could be
increased so that the total simulation run time was decreased, as indicated by Eq. (3.6). Previous
studies have used a reduced shear modulus in DEM to determine bulk-flow fields [26], but it was
important in the present case to study the effect of the shear modulus reduction on the impact
velocities and forces. Considering the conservation of momentum of two particles during
impact, the particle velocity change is proportional to the product of the normal impact force (Fn)
and the impact time duration (Δt) (Eq. (3.7)). ΔVn and m are, respectively, the change in the
55
normal velocity and the mass of a single particle in a binary impact. Fn is calculated using Eq.
(3.8) knowing the relative normal velocity of particles before impact, rel
nv [23].
, Δ 0.2nn n n R
F tV v V t t
m
(3.7)
0.4
*0.6 1.2 0.4 1.220.1431 ( ) ( )
1
rel rel
n n n
G dF m V A G V
v
(3.8)
The coefficient A is constant, since all other quantities except rel
nV and G are constant. Equation
(3.9) gives the particle velocity change after one time step as derived by combining Eqs. (3.6-
3.8):
0.1
0.21 0.1102n n nv V VG
(3.9)
The parameter
0.1
G
was varied from 0.199 to 0.316 as the shear modulus was reduced by one
order of magnitude. Considering the approach velocities to be equal to the maximum values
observed in the experiments (0.5 m/s) [9], the relative error in the normal component of the
particle rebound velocity (vn) introduced by using the reduced modulus instead of the actual one
was less than 2%.
Equations (3.8) and (3.9) give the normal component of the impact force calculated using either
the actual shear modulus (Ga) or the reduced one (Gr), Fna and Fnr, respectively.
1.20.4 1.2 0.4 1.022na a na a nrF A G v A G V
0.4 1.20.4 1.2 1 10 1.044nr r nr a nrF A G v A G V
(3.10)
Fnr was found to be 2.5 times smaller than Fna due to the shear modulus reduction.
Therefore, although, the contact forces were decreased significantly as the shear modulus was
artificially reduced, the impact velocities, which were the main results to be obtained from the
present DEM, remained almost unchanged.
56
In order to confirm that the impact velocities calculated using a reduced shear modulus for the
steel media were acceptably accurate, DEM simulations were run for the single particle-layer
model of Fig. 3.2 using the actual shear modulus and the shear modulus reduced by 1, 2 and 3
orders of magnitudes. The material properties and the tub motion given in Tables 3.1 and 3.2,
and the contact parameters provided in Table 3.7 (Section 3.4), were used in these simulations.
The model had the same bed depth as in the vibratory tub experiments of [9], with 2,500
particles introduced to the DEM at 100,000 particles/s at the beginning of the simulation. The
tub was then held stationary for 1 s for the particles to settle before the sinusoidal vibrations were
applied to the tub for 19 s.
Fig. 3.2: Snapshot of a single particle-layer simulation and the three measurement points located
at (x,z): RU (200, 300), H (100, 300), M (0, 280) (mm). At each location, the small and large
boxes were for impact velocity and bulk flow velocity calculation, respectively. Figure drawn to
scale. Straight line approximation to free-surface slope established at steady state. Arrow shows
average bulk flow direction.
The average impact and bulk flow velocities predicted by the DEM at points H and RU of Fig.
3.2 along with the corresponding total simulation run time are presented in Table 3.3. It was
observed that the calculated impact velocities changed less than 2% as the shear modulus was
H RU
M
Surface slope line
H2
z x
57
decreased from its true value by a factor of 10 (Er-1), while the predicted bulk flow velocity
increased by 6.6%; however, the run time was reduced appreciably from 24 h to 10 h. These
errors increased quickly as the shear modulus was further reduced by factors of 100 (Er-2) and
1,000 (Er-3), making these additional decreases in run time impossible. Therefore, it was
concluded that the shear modulus could be safely reduced by one order of magnitude without
excessively compromising the accuracy of the predictions of either the impact or bulk flow
velocities. The simulation time step corresponding to this reduced shear modulus was 2.17 µs,
which was 9 times smaller than the elastic impact duration T, thereby satisfying the stability
condition mentioned in [12]. The DEM data acquisition rate was 94 Hz, which was twice the
frequency of the tub vibration 47 Hz.
Table 3.3: Change in the x and z components of the impact velocity (denoted IV) and bulk flow
velocity (denoted BFV) at the points H and RU (Fig. 2.2) due to a shear modulus reduction of 1,
2 and 3 orders of magnitude (denoted G-1, G-2, G-3, respectively) and their corresponding errors
(Er). Velocities in mm/s. Root mean square error (RMS Er) is for all components and locations
of either IV or BFV. Time to run DEM simulation given in last row.
Actual
G
Reduced
G-1
Er-1
(%)
Reduced
G-2
Er-2
(%)
Reduced
G-3
Er-3
(%)
H-IVx 74.7 74.1 -0.8 62.7 -16 60.4 -19
H-IVz 90.9 91.2 0.3 90.5 -0.4 81.3 -11
RU-IVx 147.8 144.5 -2.2 136.7 -7.5 118.3 -20
RU-IVz 140.6 138.1 -1.8 126 -10 108.8 -23
RMS Er %
1.3
8.6
18
H-BFVx 12.8 12.8 0.0 13.4 4.7 4.2 -67
H-BFVz 18.8 16.2 -14 16.4 -13 6.6 -65
RU-BFVx 4.8 5.4 12 5.1 6.3 0.1 -98
RU-BFVz 19.3 19.3 0.0 16.1 -17 12.1 -37
RMS Er % 6.6 10 67
Run time (h) 24 10 5 1
58
3.3 Sensitivity of DEM impact velocities to contact properties
The dependence of the bulk flow velocity and particle impact forces on coefficients of friction in
vibrationally-fluidized beds has been demonstrated in several earlier studies. Decreasing the
coefficient of friction between spherical steel and glass media, by applying a thin film of water,
lowered the bulk flow velocity significantly in the same tub finisher as in the present work [8].
The bulk flow velocity also decreased when the coefficient of friction between the particles and
the tub walls was reduced by applying a smooth sheet of polytetrafluoroethylene [8]. This is
consistent with the observation that the velocity of cylindrical work-pieces entrained in a bowl-
type finisher decreased when the media were made slightly wet [27]. Measurements of
individual particle impact forces in a bowl-type vibratory finisher showed that reducing the
coefficient of friction by means of a thin film of water resulted in lower impact forces, implying
that the particle impact velocities were also reduced [28]. Therefore, in order to establish the
accuracy of the DEM in predicting impact and bulk flow velocities, it was necessary to assess the
sensitivity of the predictions to uncertainties in the contact coefficients used in the DEM
simulations. This could then be used to guide the accuracy required in the experimental
measurement of the coefficients.
To begin, a pair of simple two-body collisions was analyzed: (1) the collision of two moving
disks, and (2) the collision of a moving disk with a moving wall. The procedure described in
Appendix 2-A was used to determine the sensitivity of the rebound velocities (normal, vn,
tangential, vt, and angular, ω) to the contact parameters. Based on the measurements described
in Section 3.4, the coefficient of restitution, e, was investigated over the range 0.4-0.95, the
coefficient of friction, µ, between 0.4 and 2.0, and coefficient of rolling resistance, µr, was varied
from 0.01 to 0.09, in both particle-particle (p-p) and particle-wall (p-w) interactions.
Table 3.4 gives the average percentage uncertainties in the rebound velocities due to 10%
uncertainty in each of the contact parameters calculated using the equations in Appendix 3-A.
Each uncertainty was normalized by the average rebound velocity calculated using the mean
values of the three contact parameters (e=0.675, µ=1.2 and µr=0.05). These average velocities
used in each normalization are given in Appendix 3-A.
59
Table 3.4: Percentage uncertainty in the linear and angular rebound velocities due to 10%
uncertainty in each of the contact parameters.
vn (case 1) vt (cases 1) ω (case 1) vn (case 2) vt (cases 2) ω (case 2)
e 10 4 4 7 4 4
µ 0 10 10 0 9 10
µr 0 0 0.2 0 0.8 0.4
It is concluded that the normal rebound velocity, vn was only dependent on the coefficient of
restitution. The tangential velocity, vt and the angular velocity, ω were affected most by the
coefficient of friction. The angular velocity and the tangential velocity were weakly dependent
on the coefficient of rolling resistance. Overall, Table 3.4 illustrates that the uncertainties in the
rebound velocity components were relatively insensitive to uncertainties in the contact
parameters, being at most directly proportional to such uncertainty. It was also confirmed that
the rebound velocities varied directly with each of the three contact parameters (e.g. they
increased as each of the contact parameters increased), consistent with the observations of [27]
and [28].
These conclusions regarding the sensitivity of the rebound velocities to the coefficients of
restitution and friction were confirmed using the same single-layered DE model of the tub
described in Fig. 3.2. This model used the material properties given in Table 3.1 and the tub
motion defined in Tables 3.2. Design of experiments (DOE; Taguchi method) was used to
reduce the number of required runs of the model to 25 in order to investigate the relative
sensitivity of the velocity predictions to the 6 contact parameters under consideration [29]. Five
values were selected for each coefficient within the same ranges assumed in the analytical study
(i.e. e: 0.4, 0.6, 0.8, 0.9, 0.95; µ: 0.4, 0.8, 1.2, 1.6, 2.0, and µr: 0.01, 0.03, 0.05, 0.07, 0.09). For
any particular combination of contact parameters within the interaction table, the average impact
and bulk flow velocities were calculated at points H and RU (Fig. 3.2) in the horizontal and
vertical directions using the method described in Section 3.2. Then, 25 combinations of the
contact parameters and the values corresponding to one of the velocity components were used to
calculate the sensitivities.
60
Table 3.5 shows the uncertainties of the predicted impact and bulk flow velocities due to 10%
uncertainty in the contact parameters at location H. The uncertainty of each velocity component
was normalized by its mean value calculated from the DEM results (Table 3.5). In order to
combine the uncertainties in the x and z velocity components, the root mean square (RMS) values
of the uncertainties are also shown in Table 3.5. The coefficient of rolling resistance was not
included in this study because it does not contribute significantly to energy dissipation, and
Table 3.4 suggested its effect would be negligible as it is much smaller than the coefficient of
friction (less than 5%) [17]. It is seen that the impact velocities predicted by the single-layer DE
model were quite insensitive to uncertainty in the contact parameters, being considerably smaller
than the specified 10% uncertainty in each contact parameter. This agrees with the conclusions
of Table 3.4 from the analytical study. The relatively small changes that did occur in the
predicted particle impact and bulk flow velocities varied directly with the coefficients of friction
and the coefficients of restitution of the particle-particle and particle-wall interactions. This is
consistent with the observations of refs. [8, 28, 29]. Similar results were obtained at location RU
of Fig. 3.2. As with the results of Table 3.4, it was confirmed that the impact velocities varied
directly with the contact parameters. This was also true for the bulk flow velocities, in agreement
with the experimental observations of [27].
Table 3.5: Percentage uncertainty (U) in the single-layer DEM predictions of the impact and bulk
flow velocities due 10% uncertainty in the contact parameters normalized by the following mean
velocities from the DEM: IVx =75 mm/s, IVz =95 mm/s, BFVx =14 mm/s, BFVz =20 mm/s. p-p
particle-particle, p-w particle-wall. Location H of Fig. 3.2.
Impact velocity Bulk flow velocity
Contact
parameters Ux (%) Uz (%)
RMS
U(%) Ux (%) Uz (%)
RMS
U(%)
e (p-p) 0.6 0.4 0.5 5 3 4
µ (p-p) 0.2 0.3 0.3 1 4 3
e (p-w) 1 0.2 1 1 2 2
µ (p-w) 2 0.4 1 0.9 1 1
61
3.4 Measurement of contact properties
A linear tribometer (Nanovea Inc. 2012 [30]) was used to measure the coefficient of friction as a
function of sliding speed between a steel ball and the tub polyurethane wall, and between two
steel balls. In the first test, a 6.3 mm diameter ball of the steel media was fixed in the tribometer
as the stationary element against a flat sheet cut from the polyurethane tub wall that reciprocated
with an amplitude of 1 mm (2 mm peak-to-peak, sinusoidal oscillation). A weight of 1 N was
applied to the ball holder and a variety of reciprocating frequencies were tested to obtain a range
of sliding speeds up to 31 mm/s. The estimated maximum sliding speed at the wall in the
vibratory finisher was previously estimated to be 50 mm/s using data from a submerged high-
speed laser displacement probe [9, 10]. Each measurement involved 15 cycles and each
experiment was repeated 5 times on a different surface each time. Before each trial, the
contacting surfaces were wiped with a tissue soaked in ethanol. Table 3.6 shows the average and
maximum sliding speeds during the sinusoidal oscillation of the friction force corresponding to
each frequency, and the average and maximum µ during the cycle. The variation in µ with
sliding speed was negligible; therefore, the spindle speed was adjusted to be 5 rpm for the rest of
the experiments. This confirmed that a single value of µ could be used to model all impacts with
the tub walls.
Table 3.6: The coefficient of friction (µ) between the steel ball of finishing media and the sample
of the polyurethane tub wall as a function of sliding speed.
Spindle
speed (rpm)
Mean sliding
speed (mm/s)
Maximum sliding
speed (mm/s)
µ mean±95% confidence
interval (maximum value)
5 0.330 0.520 1.73±0.12 (1.88)
20 1.31 2.08 1.63±0.02 (1.65)
50 3.33 5.22 1.86±0.09 (1.94)
80 5.32 8.36 1.76±0.14 (1.90)
100 6.67 10.5 2.01±0.07 (2.07)
300 20.0 31.4 1.84±0.06 (1.91)
Average 1.80±0.06
62
The measurement of the coefficient of friction between two steel balls was complicated by the
curved contacting surfaces during sliding. It was obtained by fitting a theoretical sliding force
versus time curve to the measured curve. One ball was fixed while the other was glued to the
stage reciprocating at the mean speed of 0.167 mm/s (spindle speed of 5 rpm). As in the
previous tests, 1 N was applied to the fixed ball. The upper ball vertical displacement, y, (Fig.
3.3) was measured using the tribometer depth sensor and the corresponding horizontal
displacement, x, and the inclination angle, θ, between the tangent to the contact point and the X-
axis, were calculated using Eqs. (3.11) and (3.12), respectively. The ball radius, R, was 3.15 mm
and the maximum depth at the end of each cycle was 30 µm giving a total arc length amplitude
of 0.5 mm.
22x R R y (3.11)
arccosR y
R
(3.12)
Fig. 3.3: Schematic showing the lower steel ball sliding against the upper steel ball that was
constrained in the y-direction to measure vertical displacement depth only. T is the measured
horizontal force. y=0 taken to be the minimum measured depth at the apex of the upper ball.
63
Fig. 3.4: Tribometer measurements of the horizontal sliding force vs. (a) time and (b) angle (θ)
for steel balls moving against each other. The solid straight lines in (a) and (b) show the
analytical prediction.
The surface normal force, N, and the tangential friction force, f, were calculated using Eq. (3.13)
from the applied vertical force, W= 1 N, and the measured horizontal force, T. A horizontal
force signal acquired by the tribometer is shown in Fig. 3.4a. The sloped portions of the force
(a)
(b)
(b)
(a)
64
signal corresponded to sliding while the vertical lines corresponded to the changes in the force
direction. The small fluctuations observed in the sloped portions were due to the surface
roughness, and were considered to be straight lines. Assuming that the upper ball moves to the
right relative to the lower ball as is shown in Fig. 3.3:
T fcos Nsin
W fsin Ncos (3.13)
considering θ to be positive; therefore, when the upper ball was to the left of the lower ball, θ
was negative value in Eq. (3.16). The coefficient of friction f
N , was expressed as a
function of T, W and θ as in Eq. (3.14). Rather than obtain µ directly from Eq. (3.14) using the
measured T as a function of time or θ, it was determined using a best fit of the predicted function
T(θ) given by Eq. (3.15) to the measured function, treating µ as an adjustable parameter (Fig.
3.4b).
Tcos Wsin
Tsin Wcos
(3.14)
cos sin
cos sinT W
(3.15)
The coefficients of restitution of a steel ball colliding with the polyurethane and of two
impacting steel balls were measured using the apparatus of Fig. 3.5. A solenoid actuator was
used to hold a steel ball above the urethane target (Fig. 2.5a) while a high-speed laser
displacement sensor (Keyence, Osaka, Japan, LK-G157) with an accuracy of 0.5 µm was used to
measure the impact and rebound velocities as in [9]. The ball height was adjusted such that the
incident speed was 0.5 m/s as in ref. [9]. Two laser sensors (LK-G82 and LK-G157 with
accuracies of 0.2 µm and 0.5 µm, respectively) were used to measure the incident and rebound
velocities of colliding steel balls as shown in Fig. 3.5b. One of the balls was left stationary on
the two glass guide rods while the other ball was released from rest through an inclined tube and
hit the stationary ball with an incident speed of 0.5 m/s. Equation (3.2e) and the conservation of
linear momentum were used to calculate the coefficient of restitution of the colliding steel balls.
Since the glass guide rods were very smooth, the falling ball tended mostly to slide into contact
65
with the second ball. The possible influence of rolling was assessed by repeating the
measurement with the glass rods covered with tape so that the ball rolled. The coefficient of
restitution was unchanged. Table 3.7 shows the mean and 95% confidence intervals of the
coefficients of friction and restitution corresponding to the different material pairs for the 5
repeat experiments in each case. The friction results were consistent with the observation that
the coefficient of friction decreases as the hardness of the materials increases, because of reduced
plowing [31]. As well, since the structural damping ratio (proportional to the collisional energy
dissipation [32]) of polyurethane is about 3 times larger than that of steel [32], steel balls
colliding with each other should indeed have a larger coefficient of restitution than a steel ball
colliding with a polyurethane surface.
Fig. 3.5: The drop test apparatus used to measure the coefficient of restitution of: (a) the steel
ball colliding with the polyurethane surface, (b) two colliding balls (side and plan views).
Table 3.7: The measured coefficients of friction and restitution for different pairs of materials
(mean ±95% confidence interval).
Materials e µ
Steel-Polyurethane (p-w) 0.662±0.004 1.81±0.170
Steel-Steel (p-p) 0.851±0.030 0.616±0.026
(b) (a)
66
3.5 Results and discussion
3.5.1 Effect of model width
The present tub vibratory finisher produced a mostly two-dimensional oscillation and bulk flow
that was bounded at each end by a glass partition (Fig. 3.1). One of the objectives of the present
study was to assess the relation between the predicted DEM impact velocities and the width of
the model; i.e. the number of particle layers between the glass walls. In a series of models using
the material properties, tub motion and contact parameters provided in Tables 3.1, 3.2 and 3.7,
the glass partitions were separated by 1d, 2d, 4d, 8d and 12d where d is the particle diameter and
the instantaneous particle velocities (impact velocities) were acquired at each data time step in
two measurement bins, 9×9×6.3 mm (1.5d×1.5d×d) in the x, z and y dimensions, respectively),
located half way from the partitions at points H and M (Fig. 3.2). In the 1d simulation, a
clearance of 0.1 mm was added between the glass and the particles. These locations
corresponded to the locations of the laser probe used in [9] to measure the impact velocities in
this same tub finisher. The size of the measurement bin had an insignificant effect on the impact
velocities recorded from the model, with a difference of less than 5% when the bin size was
increased to 3d×3d×d.
As seen in Fig. 3.2, the free surface of the vibrationally-fluidized bed was sloped at steady state
as a result of the bed expansion and bulk flow. The establishment of a constant free surface
slope was used to assess whether the DEM had reached steady-state, and to compare the flows
predicted by the models of different width in the y-direction. Table 3.8 shows the average and
95% confidence interval for slopes calculated every second from 10-20 s after the start of the
simulation. Since the maximum variation in the slope angle over this time interval was just 10%,
it was decided that these simulations had attained steady state, and that DEM data collection
could begin 9 s after the start of tub vibration for each of the models of different width.
Table 3.8 also shows that the mean slopes corresponding to the single layer and double layer
models were significantly greater than those of the 4, 8, 12 and 16-layer models. The main
reason for this was that the particles in the 1d and 2d models were prevented from moving freely
in the y-direction (perpendicular to the simulated glass partitions). Even though the tub
67
vibrations and the dominant bulk flow were in the x-z plane, this constraint in the y-direction
produced considerable changes in the bulk flow and solid fraction compared with the wider
simulations in which the diffusion of particles between the layers was less restricted. This
constraint effect appeared to vanish for models of 4 layers and greater, as the predicted free-
surface slopes were nearly the same (Table 3.8).
Table 3.8: The average slope and ±95% confidence intervals of the free surface in models with
various numbers of particle layers in the width direction, y, between the glass partitions.
1 layer 2 layers 4 layers 8 layers 12 layers 16 layers
21±1.6 21±2.2 16±0.9 15±0.9 16±0.9 15±0.9
The mean impact velocity was calculated as the average of the absolute values of the
instantaneous velocities of the particles within the measurement bin acquired during 10 s of
simulation, beginning 9 s after the tub started to move. Extremely small particle velocities (less
than 1 mm/s or about 2% of the average impact velocity [9]) were excluded since they were
unreliable and would have no influence on the surface finishing process.
Figure 3.6 shows the mean impact velocities of particles at points H, M and H2 (Fig. 3.2) in the x
(horizontal), z (vertical), and y (transverse) directions for different simulations with various
layers of particles in the transverse (y) direction. As the number of particle layers increased from
1-16, the average impact velocities in the three directions tended to become relatively constant.
This was expected as a consequence of the diminishing relative influence of the container side
walls on the particle behavior in the central part of the flow field. Since the tub motion was in
the x-z plane, the x and z impact velocities were much larger than those in the transverse y-
direction. The average impact velocities in the x-direction (Fig. 3.6a) decreased sharply from 1-4
layers at locations H and M, but not at the deeper location H2, which was 8 cm below point H.
There it was constant for all of the layer models, and significantly greater than at the two
shallower locations. In contrast, the impact velocity in the z-direction (Fig. 3.6b) displayed only
a small decrease with increasing numbers of layers, and the behavior was essentially the same at
all three locations. In the y-direction, the impact velocity increased sharply from 1-4 layers, but
then became approximately constant for the 8, 12 and 16-layer models (Fig. 3.6c). This result
68
was a logical consequence of the reduced spatial confinement in the y direction as the number of
layers increased. However, the magnitude of the impact velocity in the y-direction depended on
the location in the flow, being appreciably greater at the deeper location H2, just as it was for the
x-component of impact velocity. The reason for this is unclear, and illustrates the variability of
the local impact conditions from point to point in the flow. This variability was also evident in
the experimental measurements of impact velocity that will be compared with DEM predictions
in the next section. These large variations in local impact velocity highlight the need for
accurate DE models capable of predicting the distribution of local impact conditions within a
vibratory finisher in order to understand how the degree of surface finishing changes within the
flow.
The average bulk flow velocity was obtained from the DEM using one of two procedures that
were shown to give equivalent results. The first measured the transit time for 20 particles across
a 20×20×6.3 mm (3d×3d×d) bin located adjacent to one of the glass partitions in either the H or
M locations in order to be consistent with the procedure that was used in [9] for the experimental
measurements of the bulk flow velocity using video recordings through the glass partitions. It is
noted that locating the bin half way between the glass partitions yielded the same results in all of
the simulations. The second procedure used the average of the instantaneous velocities of 60
particles inside the bins after assigning them a positive and negative sense so that the net bulk
flow was calculated. The average bulk flow velocities of the two approaches agreed to within
5%, and so the second procedure was adopted because it did not require the visual tracking of
particles entering and exiting the bins.
Figure 3.7a shows that the bulk flow velocity in the x-direction varied with location as expected
from the overall circulation in the tub, indicated by the arrow in Fig. 3.2; i.e. the largest x-
component at location M where the bulk flow is mostly parallel to the free surface, and the
smallest component at the location H2 where the flow is mostly vertical. The z-direction bulk
flow velocities of Fig. 3.7b show the expected corresponding trends in the vertical direction, with
the largest bulk flow velocity at location H2 and the smallest at location M. In all cases, the bulk
flow velocity became approximately constant beyond about 8 layers. The average bulk flow
velocity in the transverse (y) direction was zero for all of these simulations, consistent with the
symmetry of the model.
69
Fig. 3.6: Predicted impact velocities in: (a) horizontal (x) direction, (b) vertical (z) direction, and
(c) transverse (y) direction, in the DEM simulations with different numbers of layers.
(a)
(c)
(b)
70
Fig. 3.7: Predicted bulk flow velocities in (a) horizontal (x) direction and (b) vertical (z)
direction, in the DEM simulations with different numbers of layers.
3.5.2 Comparisons with immersed laser-probe velocity measurements
The dependence of the model results on the number of simulated layers, and the potential
influence of the immersed laser probe on the bulk flow and impact velocities, motivated the DE
modelling of the actual experimental setup of ref. [9], including the immersed laser probe as
shown in Fig. 3.8. In order to decrease the simulation time, only half of the geometry in the y-
direction was modeled (105 mm; i.e. 16 particle diameters), using a smooth, frictionless
symmetry boundary made of the same material as the particles [33]. The model was, however,
insensitive to this friction coefficient, and essentially the same results were obtained if the
symmetry boundary was assigned the coefficient of friction measured for particle-particle
contact (Table 3.6).
(a) (b)
71
Fig. 3.8: Snapshot of the simulation constructed using the plane of symmetry through the
elliptical laser velocity probe used in [9] located at point M (as if measuring velocities of
particles approaching from the right). (a) side view, (b) plan view, (c) side view of probe and the
measurement bins on the probe surface oriented with the laser window to the right. The large bin
with solid boundaries and the small bin with dotted boundaries were used to obtain the impact
and bulk flow velocities, respectively. (d) Side view of a portion of the probe surface showing
the particle bulk flow with the measurement bin in the downward directions. In (c) and (d), large
arrows show the bulk flow direction, dashed smaller ones represent the laser beam.
Five models were constructed to mimic the experimental measurement locations and orientations
used in ref. [9] where the probe was located at either position H or M (Fig. 3.2) and then
positioned to record particle velocities normal to the plane of the laser window in the leftward,
(b)
(d)
Y
x
z x
(a)
(c)
72
rightward and downward directions; i.e. H-rightward, leftward, downward; M-rightward and
downward. These models used the material properties, tub motion and contact parameters given
in Tables 3.1, 3.2 and 3.7, respectively. In each simulation, the impact velocities of the particles
colliding with the probe, normal to the laser window of the probe, in a d×3d×d bin adjacent to
the probe with its 3d side parallel to the window (Fig. 3.8c) were acquired in 10 s and then
averaged. The mean bulk flow velocity in the DEM was calculated as the average velocity
(considering both positive and negative signs) of particles moving in a d×d×d bin adjacent to the
probe and parallel to its surface (Fig. 3.8c). The bulk flow velocity measurement bin was
smaller to have only one particle in the bin at each time step. This was consistent with the
experimental measurements where the bulk flow velocity was calculated considering only the
laser reflections from the particle layer immediately adjacent to the probe [9].
In the experiments of ref. [9], although the laser was stationary, the external protective shell and
glass laser window of the submerged probe experienced some vibration with a velocity between
20 to 45 mm/s, depending on the location in the flow [9]. Consequently, the average impact
velocity was determined considering only those particles with velocities normal to the window
were greater than that of the shell of the probe [9]. In the DEM simulations, the probe was fixed,
but the same velocity filter was applied to the model data to be consistent with the experimental
procedure; i.e. only surface-normal particle velocities greater than that of the probe.
Table 2.9 shows that the predicted impact and bulk flow velocities for the five DEM simulations
agreed quite well with the measured values from ref. [9]. At point H, the DEM impact velocities
overestimated the measured values by between 10% and 19%, while at point M the impact
velocities were underestimated by less than 7%. One cause for these differences was the
sensitivity of the impact velocities measured in [9] to the laser probe depth relative to the free
surface of the flow. For example, shifting the probe 1 cm upward produced approximately a
25% change in the impact velocity, while moving it down 1 cm, changed it by about 15%.
Fig. 3.9 shows the log normal probability density distributions of the impact velocities calculated
using the DEM results and the experimental data of [9]. It is seen that the numerical and
experimental distributions were very similar in most of their main features such as their mean
values and skewness. However, the DEM probability distributions were wider than those
determined experimentally (Fig. 3.9).
73
Table 3.9: Comparison of the predicted and measured impact velocities (±95% confidence
intervals based on at least 200 data points from the DEM and at least 1,000 laser-probe velocity
measurements; mm/s) and bulk flow velocities (mean value; mm/s) at locations H and M in
different directions (Figs. 3.2 and 3.8). The bracketed bulk flow velocity at location M-
rightwards refers to the DEM absolute oscillating velocity.
Measurement
locations and
directions
Impact
velocity
(DEM)
Impact
velocity (Expt
[9])
Diff
(%)
Bulk flow
velocity
(DEM)
Bulk flow
velocity (Expt
[9])
Diff (%)
H-rightwards Vx =86±4.5 Vx=74±1.1 +16 Vz =12 Vz =13 -7.7
H-downwards Vz =95±4.6 Vz=80±1.3 +19 Vx=15 Vx=16 -6.3
H-leftwards Vx=70±5.4 Vx=61±1.0 +10 Vz =15 Vz =20 -25
M-rightwards Vx=85±3.8 Vx=91±1.2 -6.6 Vz =5 (18) Vz =19 -74%
(-5.5%)
M-downwards Vz =110±7.7 Vz =110±1.6 0 Vx=14 Vx=20 -30%
74
0 20 40 60 80 100 120 140 160 180 2000
0.05
0.1
0.15
0.2
Impact velocity (mm/s)
Pro
ba
bili
ty d
en
sity d
istr
ibu
tio
n
H-rightward-DEM
H-rightward-exp
H-leftward-DEM
H-leftward-exp
M-rightward-DEM
M-rightward-exp
Fig. 3.9: Probability density distributions of impact velocities predicted by the DEM and
measured in the experiments of ref. [9]. All the distributions were fitted with log-normal
functions. Locations defined in Figs. 3.2 and 3.8.
The predicted mean bulk flow velocities at point H in different directions and at point M-
downwards in the x-direction were reasonably close to the measured values, underestimating
them by between 6% and 30%. However, the bulk flow velocity in the z-direction at point M-
rightwards was apparently underestimated by 74%. In reality, this DEM prediction was probably
more accurate than the measurement due to a limitation of the laser sensor velocity data which
did not distinguish the direction of ball travel parallel to the laser window. In other words, all of
the velocities from the laser displacement signals in ref. [9] were assumed to be in the same bulk
flow direction, which was correct in most cases. However, in locations where the bulk flow was
weak and the balls were moving in more random patterns, this assumption would lead to an
artificially high apparent bulk flow velocity, since the sign of ball velocity was not taken into
account. This was indeed the case at location M-rightward, where the particles oscillated back
and forth in the z-direction as the flow collided with the upstream side of the elliptical shell of
75
the laser probe. Therefore, the laser probe likely overestimated the bulk flow velocity in this
case. A better simulation of these erroneous laser measurements at location M-rightwards could
be obtained by taking the average of the DEM absolute oscillating velocities. Doing this, the
average absolute oscillating z-direction velocity predicted by the DEM for the M-rightward
experiment was 18 mm/s, as shown in brackets in Table 3.9. This agreed well with the laser
velocity measurement of 19 mm/s [9]. However, the absolute oscillating velocities at point H in
different directions and at point M-downwards in the x-direction were the same as the bulk flow
calculated using signed velocities, and so the differences between the measured and predicted
bulk flow velocities in these cases was due to other causes.
3.6 Conclusions
The impact and bulk flow velocities in a tub vibratory finisher predicted by discrete element
modeling (DEM) were compared to the measured values for steel granular media. This was the
first time measurements of local impact velocities have been compared with DEM predictions in
a vibrationally-fluidized granular flow. The sensitivity of the predicted local impact velocities
and the bulk flow velocities to the DEM contact parameters (coefficients of friction, restitution
and rolling resistance) was investigated both analytically and numerically. It was found that the
predictions of both the bulk flow and impact velocities were relatively insensitive to
uncertainties in these impact coefficients.
The vibration of the tub finisher walls was essentially two-dimensional, with negligible motion
in the transverse direction. The effect of the number of DE particle layers in the transverse
direction was investigated to determine the minimum number required for accurate predictions of
the particle impact and bulk flow velocities. It was concluded that accurate velocities could only
be predicted using at least 4-12 layers. Fewer layers created too much spatial constraint to allow
realistic particle motion. Moreover, a reduced shear modulus could be used to decrease the
model run time without significantly affecting the accuracy of the predicted impact and bulk
flow velocities.
76
The complete DE model simulated the actual setup used in an earlier experimental study that
measured the bulk flow and local impact velocities in the tub vibratory finisher. The coefficients
of friction and restitution between the steel balls, and between the balls and the tub walls, were
measured in order to obtain the most accurate DE simulations. The predicted impact velocities in
several directions and at two locations in the tub displayed the same log-normal probability
density distributions as did the previously measured impact velocity distributions. Moreover, the
predicted mean impact velocities were in good agreement with the experimental measurements,
with a maximum error of 19%. The DE predictions of the mean bulk flow velocities were also
consistent with the experimental measurements, having a maximum error of 30%.
Therefore, it was concluded that DEM can be used to give reasonably accurate predictions of
both the local impact velocities and the bulk flow of particles in vibrationally-fluidized beds.
This will be useful in predicting the impact energy and force of the media on workpiece surfaces,
and hence the resulting wear and surface deformation during vibratory finishing.
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in a vibrating bed, Powder Technol. 195 (2009) 83-90.
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[21] L. McElroy, J. Bao, C.T. Jayasundara , R.Y. Yang, A.B. Yu, A soft-sensor approach to
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properties during normal impacts, Powder Technol. 154 (2005) 99 –109.
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79
Chapter 4
4 Finite Element Continuum Modeling of Vibrationally-Fluidized Granular Flows
Nomenclature
Dij Rate of strain tensor Vs Speed of sound in air
dev
ijD Deviator rate of strain tensor Kb Media bulk modulus
σij Stress tensor ν Poisson's ratio
dev
ij Deviator stress tensor Eb Media equivalent Young's
modulus
σ1 Maximum principal stress Ew Wall Young's modulus
σ3 Minimum principal stress Lb Effective thickness of the media
Shear rate Lw Wall thickness
η Shear viscosity Kmedia Media stiffness
µ Coefficient of internal
friction Kwall Wall stiffness
I Inertial number K Effective contact stiffness
ρs Solid phase density Ft/Fn Shear to normal force ratio
d Particle diameter µmw
Media-wall coefficient of
friction
p Local pressure u'i, v
'i, w
'i Particle velocity fluctuations
Δ Dilatation ui, vi, wi Particle instantaneous velocity
( , )k l
iF Contact force vector ū, ῡ, w Particle average velocity
( , )k l
jx Relative center-to-center
distance of particles l Mean free path
80
vi Particle velocity vector e Coefficient of restitution
Mk Mass of particle k fi Body force vector
V Measurement bin volume N Number of particles in a bin
Lij Velocity gradient tensor ui Displacement vector
φ Volume fraction T Granular temperature
Ud Sound wave propagation
speed V Average bulk flow velocity
ρb Bulk density Vimp Average local impact velocity
ρg Gas phase density
4.1 Introduction
The flow of granular materials has been studied in non-fluidized beds such as in hopper
discharge [1, 2], rotating drums, conveyers, chutes and mixers [3, 4], and in fluidized beds such
as in vibratory finishers and vibrating sieves [5-8]. Vibratory finishing is widely used to polish,
burnish, harden, and clean metal, ceramic and plastic parts. In a tub vibratory finisher, the two-
dimensional vibration of the walls produces a vibrationally-fluidized circulatory bulk flow of the
media (Fig. 4.1). The granular media becomes fluidized as the vertical acceleration amplitude
exceeds that of gravity, resulting in a marked decrease in the contact pressure [8]. The media
have both a large-scale bulk flow velocity resulting from the resultant shear force with the walls,
and a local impact velocity during each vibration cycle [9].
The erosive wear and plastic deformation of a workpiece within a vibratory finisher are
determined mainly by the velocity, frequency, and direction of the impacts of the granular
finishing media [10-12]. Moreover, these impacts can lead to breakage of the finishing media.
In processes such as drug tablet coating within rotating drums, and bulk materials handling, the
impact of the granular products can contribute to their fragmentation [12].
Generally, discrete element modeling (DEM) simulations [14, 15] give reasonable predictions of
bulk flow velocity and volume fraction in both fluidized and non-fluidized granular flows [16-
18]. The local behavior of the non-fluidized granular flows has also been investigated
81
experimentally and numerically [2, 21-23], while some studies have examined vibrationally-
fluidized flows [9-12, 19, 20]. Recently, it has been shown that DEM predictions of both bulk
flow and local impact velocities in a tub vibratory finisher agreed well with measurements made
using a high-speed laser displacement probe immersed in the flow [19].
Previous efforts to model granular flows using a continuum approach have focused on the bulk
flow behavior of the non-fluidized granular media such as in flow down inclined chutes [24, 26,
30], plane shear flow [28, 29], flow in rotating drums [27, 28], flow in annular shear cells [24,
27, 28], and silo discharge flows [25, 27, 28]. In these cases of non-fluidized flows, constitutive
equations were defined to describe the bulk flow of the granular materials under quasi-static and
moderate flow (liquid-like) regions [24, 26, 30]. To the author’s knowledge, no papers have
been published on continuum modelling of vibrationally-fluidized granular flows. The
continuum models that have been used for quasi-static and moderate flow include different
elasto-plastic formulations of the equivalent continuum media [24, 30, 32], and visco-plastic
formulations considering only the plastic behavior of the equivalent media under time-varying
shear deformation [26-27]. In many cases, DEM was used to obtain the equivalent properties
needed to model the granular flows as a continuum [25, 27-29, 31, 32]. For example, the
equivalent stress tensor, pressure and shear rate at different points of a flowing granular bed were
obtained using DEM, and were then used to determine the equivalent continuum media elastic
and plastic properties [25, 30, 31]. Generally, the average streamlines, and hence the bulk flow
behavior, determined through these continuum simulations have been in fairly good agreement
with the predictions of discrete element modeling [24, 27, 30]. For example, Kamrin [24]
proposed an elasto-plastic constitutive law for use in a Lagrangian finite element model of
granular flows in inclined chutes, rectangular silos and annular Couette cells. The predicted flow
fields were compared with those calculated using DEM. Andrade et. al [30, 31] used the same
material law to model the static 3D compression of sand particles. Forterre and Pouliquen [27]
used a visco-plastic constitutive law with the fixed-grid finite difference method, which is
equivalent to an Eulerian mesh to simulate granular flows in the geometries of [24] and in
rotating drums. They compared their results with experimental measurements and DEM
predictions of bulk flow [27]. The Lagrangian formulation has been used in most of the papers
that modelled granular flows using the finite element method [24, 32].
82
The present work modelled the tub vibratory finisher shown in Fig. 4.1a filled with 6.3 mm
diameter steel balls. Its motion was determined using the accelerometer and laser displacement
measurements as described in ref. [9]. The tub had sinusoidal translations in x and z directions
and sinusoidal rotation in the plane of Fig. 4.1b, all at 47 Hz. The equivalent material properties
of the granular media needed in modeling the media as a continuum were estimated using grain-
scale 3D DEM simulations.
The vibrationally-fluidized granular flow was then modeled as a continuum visco-plastic media
using both Lagrangian and Eulerian finite element formulations [34]. The bulk flow velocities
were compared with those obtained in DEM simulations and with experimental measurements
made in the same tub finisher [9, 19].
83
Fig. 4.1: (a) Photograph of the tub vibratory finisher including the laser probe used to measure
the media impact velocities in ref. [9], and two glass partitions within the tub. (b) Schematic of
the tub vibratory finisher showing the bulk flow circulation direction from the side view.
Dimensions in mm [19].
(b)
Z
(a)
84
4.2 Continuum constitutive equations
Since the flow in the present vibrationally-fluidized bed was in the moderate regime as will be
discussed below, the visco-plastic model of ref. [24] was used, as given by Eq. (4.1). This visco-
plastic constitutive equation expresses the stress tensor, σij, in terms of the rate of strain tensor,
Dij, and the pressure, p, which is equal to the mean value of the diagonal components of the
stress tensor (Eq. (4.2a)) [24-26]. The bulk modulus of elasticity, Kb, was obtained by dividing p
by one third of the dilatation, Δ, which is the trace of the strain tensor (Eq. (4.2b)) [35]. Kb is
also related to the Young's modulus, Eb, and Poisson's ratio, v, of the continuum material
representing the granular media [35]. The shear rate, , is defined as the norm of the deviator
strain rate tensor dev
ijD (Eq. (4.3)), and the shear viscosity, η, defined in Eq. (4.4), is a function
of the pressure, the shear rate, and the internal coefficient of friction, µ, which was defined as the
ratio of the maximum shear stress to the mean normal stress (Eq. (4.5a)), where σ1 and σ3 are the
maximum and minimum principal stresses, respectively [25][24, 25]. It is noted that µ could
also be calculated using the Drucker-Prager criterion (Eq. (4.5b)) where dev
ij is the norm of the
deviator stress tensor.
The inertial number, I, defined in Eq. (4.6) captures the effects of the shear rate and the
hydrostatic pressure, and the media properties ρs and d, the particle density and diameter,
respectively. The inertial number reflects the relative magnitudes of inertial forces (dependant
on the shear rate) and confining forces (dependant on the pressure) [28, 29]. It can also be
interpreted as the ratio of two time scales: (i) a microscopic time scale of particle
rearrangements,/ s
d
p , which represents the time taken for a particle to move a distance d
under the pressure p, and (ii) a macroscopic time scale, 1/ , corresponding to the mean shear
deformation [27, 28]. Small values of I correspond to a quasi-static regime where particle
inertial forces are small compared to confining forces, whereas large values of I correspond to
rapid flows with high shear rates. Therefore, the coefficient of internal friction, µ, is a function
of I as illustrated in Fig. 4.2a [24, 26, 29]. The data presented of Fig. 4.2 were obtained from a
85
2D shear test in which the particles between two parallel planes are confined by a normal force
and sheared by moving one of the planes [29].
( , ) dev
ij ij ij ij ijp p p D (4.1)
1
3iip (4.2a)
1; 3(1 2 )
3b b bp K K v E (4.2b)
3 3
1 1
1
2
dev dev dev
ij ij ij
i j
D D D
(4.3)
( , )p
p
(4.4)
1 3
1 3
(4.5a)
2
dev
ij
p
,
3 3
1 1
1
2
dev dev dev
ij ij ij
i j
(4.5b)
/ s
dI
p
(4.6)
In the quasi-static and moderate flow regimes up to I=0.01, the coefficient of internal friction, μ,
and the volume fraction are often modeled as being independent of I [24], as shown in Fig. 4.2a.
As will be explained in Section 4.3, DE modelling of the present vibrationally-fluidized granular
media in the tub finisher (Fig. 4.1) revealed that I was smaller than 0.012 throughout the flow.
This also permitted the assumption of a constant volume fraction, since Fig. 4.2b shows that φ is
approximately constant if I is less than about 0.1.
Two stress components contribute to the stress tensor σij in granular flows, as shown by Eq.
(4.7a): the collisional stress due to contact forces and the kinetic (or streaming) stress due to
momentum transfer by particle diffusion [24, 36]. The kinetic stress tensor is produced by the
transfer of momentum by particle diffusion through the bulk material, and is appreciable in
highly agitated flows with relatively large mean free paths [36]. Equation (4.7a) expresses these
stresses in terms of ( , )k l
iF and ( , )k l
jx which are i-component of the contact force vector and j-
86
component of the center-to-center vector corresponding to the contact of particle k and its
neighbor particle l. vi, Mk and V are, respectively, the velocity i-component, the mass of particle
k, and the volume of the measurement bin. In most quasi-static and moderate granular flows, the
kinetic stress is much smaller than the collisional stress since the momentum of most of the
particles is transferred by particle impact rather than the particle diffusion. Therefore, most
previous studies have ignored the second term of Eq. (4.7a), thereby reducing computational
time [24, 25]. The same assumption was made in the current study leading to (Eq. (4.7b)). For
highly agitated flows, this assumption cannot be made and the kinetic stress becomes dominant.
( , ) ( , ) ( ) ( )
1 1 1
1( ) ( )
n m nk l k l k k
ij i j k i j
k l k
F x M v vV
(4.7a)
( , ) ( , )
1 1
1( )
n mk l k l
ij i j
k l
F xV
(4.7b)
( , ) ( , )
1 1
1( ( ))
n mk l k l
ij i j
k l
sym F xV
(4.7c)
The stress tensors calculated using Eq. (4.7b) were asymmetric because, in granular flows, local
body moments are produced everywhere in the media due to the frictional forces acting between
the contacting particles [36-39]. Therefore, only the symmetric part of the stress tensor was
considered when determining the principal stresses and the coefficient of internal friction µ (Eq.
(4.7c)) in the vibrationally-fluidized granular flow under investigation.
87
Fig. 4.2: Variation of (a) the equivalent coefficient of internal friction [29] and (b) the volume
fraction with the inertial number seen in simple 2D shear test [28].
The stress tensor from Eq. (4.1) was then used to define the equation of motion of the equivalent
continuum material as shown in Eq. (4.8), where fi and ui are the body force and displacement
vectors, respectively, ρb is the media equivalent bulk density, and xj and t are the position vector
and time, respectively [35]. The media equivalent bulk density is the product of the particle
density and the average volume fraction, φ which was defined as the total volume of particles in
a measurement bin divided by the volume of the bin. As discussed above, φ, is constant except
in the collisional regime where it decreases due to highly energetic impacts (Fig. 4.2b) [25][24,
28]. Equation (4.8) was then solved using the finite element method.
2
2
ij ii b
j
uf
x t
(4.8)
(a) (b)
µ
φ
I
I
Collisional
Quasi-static Moderate
88
4.3 Determining the continuum model parameters
4.3.1 Coefficient of internal friction and volume fraction
A three-dimensional DE model of the vibratory finisher with the glass partitions separated by a
distance equal to 4 particle diameters was used to obtain the equivalent continuum media
properties used in the constitutive model (Eq. (4.1)).
The present DEM simulations were made using EDEM 2.5 (DEM Solutions Inc. 2014) and its
standard library of contact models [12] following the procedure used in the ref. [19] The DEM
input parameters including the coefficients of friction and restitution, and the tub vibration
amplitudes were the same as those measured and used in [19], and are listed in Tables 4.1 and
4.2. The only departure from the procedures used in ref. [19] was that the actual shear modulus
of the steel balls was used in the present simulations in order to predict accurate impact forces.
In contrast, it was possible to use a reduced ball shear modulus in [19] to decrease the
computational effort since only the impact velocities were of interest. The appropriate simulation
time step was calculated to be 0.686 µs for the 6.3 mm diameter steel balls [19]. An 8-layer DE
model was also simulated to assure that the results were not dependent on the distance between
the glass partitions. Figure 4.3 shows the granular flow streamlines (trajectories) of the particles
moving in the first layer of the 4-layer DE model during a 1 s interval. The shear rate and the
stress tensor were calculated at four points in the flow (H1, H2, L1 and L2 in Fig. 4.4) to
determine the average equivalent continuum properties. At each of these points a 3d×3d×1.5d
measurement bin was defined at the mid-plane between the glass partitions to measure the stress
tensor and consequently, the equivalent coefficient of internal friction (Eq. (4.5)) and the local
pressure (Eq. (4.2)).
89
Fig. 4.3: The average streamlines of particles moving in a counter-clockwise flow in the first
layer of the 4-layer DE model during a 1 s interval.
Table 4.1: Material properties used in the DEM simulations [19].
Material Shear modulus (GPa) Poisson's ratio Density (kg/m3)
Steel 76 0.29 7800
Polyurethane 0.0086 0.50 1200
Glass 26 0.23 2500
Table 4.2: Measured tub center of gravity vibration components used in DEM [19].
Horizontal (x)
displacement
amplitude (mm)
Vertical (z)
displacement
amplitude (mm)
Phase difference
between x and z
translations (deg.)
Angular
displacement
amplitude (deg.)
Vibration
frequency
(Hz)
0.85 0.66 103 0.244 47
z
x
90
The stress tensor in the granular media was calculated according to Eq. (4.7b) using a MATLAB
code that processed the normal and tangential impact forces acquired during 10 s of the DEM
simulation. The local pressure, p, and the coefficient of internal friction, μ, were then calculated
using Eqs. (4.2) and (4.5), respectively. The independence of these values from the
measurement bin size was assessed by changing the side dimension to 4d and 5d at the
measurement point L1 (Fig. 4.4). Since the coefficient of internal friction calculated using
different bin sizes remained unchanged, the 3d bin was used in all measurements. Varying the
number of particle layers between the glass partitions from 4 to 8 did not change the stress
tensors, as expected from the results of ref. [19].
In order to calculate the shear rate, three more measurement bins of the same size were defined
adjacent to the original bin and were located on the right side, top side and back side of the
points H1, L1 and L2 and on the left side, bottom side and back side of the point H2 (Fig. 4.4).
The instantaneous particle velocities were acquired in the four bins in three orthogonal directions
(x ,y ,z) and then used as an input to another MATLAB code to calculate the velocity gradient
tensor L. The ij-component of the velocity gradient tensor was defined as the variation in the
mean bulk flow velocity in direction i from the main bin to its adjacent bins divided by the
distance between the bin centers (equal to the size of the bin) along direction j (Eq. (4.9a)). The
mean bulk flow velocity was calculated as the average of the instantaneous velocity (considering
the resultant of velocities with positive and negative signs) of the particles moving in the main
measurement bin during 10 s [19]. The strain rate tensor, D, is the symmetric part of the velocity
gradient tensor, L, (Eq. (4.9b)), and the shear rate was dev
ijD (Eq. (4.3)). The inertial number
was then calculated using Eq. (4.6) taking the steel particle density and diameter as ρs=7800
kg/m3 and d=6.3 mm, respectively.
( ) ( )i j j i j
ij
j
V x x V xL
x
(4.9a)
2
ij ji
ij
L LD
(4.9b)
Figure 4.5a shows that the coefficient of internal friction did not vary significantly with the
inertial number at the four measurement points; i.e. =0.12±0.01 as I varied from 0.005 to 0.012.
The coefficient of internal friction also did not change significantly with either the local pressure
91
over the range of 50-170 kPa, or with the shear rate over range 2.5-8 1/s (Figs. 4.5b and 4.5c).
Therefore, a single constant value of =0.12 was used in the continuum model.
The volume fraction was calculated in the 4d measurement bins, and was also found to be
independent of bin size. As with , Fig. 4.5d shows that φ was essentially independent of I,
being the same at the four measurement locations (φ=0.57-0.61). Therefore, the volume fraction
was assumed to be constant, equal to 0.6.
Fig. 4.4: Measurement bin locations, H1, H2, L1 and L2 used to calculate the shear rates and
stress tensors from the DEM. Three in-plane bins at each location and a fourth bin behind the
main bin.
H1
L1
L2
H2
z
x
92
Fig. 4.5: DEM predictions of the coefficient of internal friction as a function of: (a) the inertial
number, (b) the local pressure and (c) the shear rate; (d) Local volume fraction vs. the inertial
number. The solid and dashed lines are the average and the 95% confidence bounds.
4.3.2 Media equivalent Young's modulus and media-wall effective contact stiffness
It was necessary to determine the equivalent Young's modulus of the bulk media, Eb, to use in the
continuum model as shown in Eqs. (4.1) and (4.2b). The equivalent Young's modulus, Eb, of
quasi-static granular flows has been conventionally determined in tri-axial compression tests
either experimentally or numerically using DEM [25, 30-32]. It is also possible to use the
(a) (b)
(c) (d)
93
Mindlin theory to derive equations for sound wave propagation speed in the granular media
assuming that the particles are always in contact and that force chains remain largely unchanged
[30]. Therefore, for the present vibrationally-fluidized beds, these approaches give inaccurate
values of the equivalent Young's modulus since the force chains change at every time step [30].
An alternate approach was therefore employed, using the sound wave propagation speed, Ud, as
calculated from Eq. (4.10) using the density, stiffness and volume fractions of the solid and fluid
phases in the vibrationally-fluidized bed [25, 40-42].
;(1 ) (1 )
g g
d s s g s
b s
U V V
(4.10)
bd
b
KpU
(4.11)
where φ is the solid fraction, ρb is the bulk density calculated from ρb= φ ρs+(1- φ) ρg, and ρs and
ρg are the solid and gas phase densities, respectively. Vs is the speed of sound in air at standard
pressure and temperature; 330 m/s. In deriving this equation, the mixture was assumed to be
compressible, pseudo-homogeneous, and without relative motion between the phases [41]. The
latter assumption was mostly valid in the present vibrationally-fluidized bed since the interstitial
air moved with the particles. As the air density is much smaller than that of the steel particles, ρb
was approximated by φρs in Eq. (4.10). The speed of sound in any media can also be obtained
using the Newton-Laplace equation (Eq. (4.11)) and depends on its bulk modulus of elasticity,
Kb, and the bulk density[25] [41, 42]. Knowing φ and hence ρb from the DEM simulations, Kb in
the vibrationally-fluidized state was determined by equating Eqs. (4.10) and (4.11). Finally,
assuming the Poisson's ratio to be ν=0.25, which is a typical value for granular flows [32], the
equivalent Young's modulus was obtained from:
3(1 2 )
bb
KE
v
(4.12)
Figure 4.6 shows these predictions of the equivalent Young's modulus, Eb, for the present
vibrationally-fluidized media as a function of the solid fraction, φ, which varied with distance
from the wall. For the present tub finisher, the maximum Eb=498 kPa corresponded to the
maximum solid fraction φ=0.60 inside the media far from the wall, while the minimum value
94
Eb=257 kPa corresponded to φ=0.23 obtained near the wall through the procedure described
below.
The effective normal contact stiffness between the vibrationally-fluidized media and the
tub wall, which was needed to apply the appropriate boundary condition to the media-wall
interface in the continuum model, was defined as the slope of the pressure-overlap curve; i.e. the
amount of pressure needed to cause an overlap of unit distance of the contacting surfaces.
Although the relationship might be slightly nonlinear [24], it was approximated as linear so that
the contact stiffness, K, between the media and the wall was given by Eq. (4.13) as
1 1 1
media wallK K K (4.13)
where Kmedia=Eb/Lb and Kwall=Ew/Lw where Ew is the Young's modulus of the polyurethane wall,
and Lb and Lw are, respectively, the effective thickness of the media in the vicinity of the wall,
and the wall thickness. Since the modulus of the polyurethane wall was so much larger than that
of the bulk fluidized media (i.e. Ew=26 MPa, Eb=498 kPa) the effective contact stiffness could be
approximated as K=Eb/Lb. The parameter Lb represents a form of boundary layer thickness, over
which media-wall contact produces significant changes in the solid fraction, and hence the bulk
modulus and contact stiffness. In order to estimate the media effective thickness, Lb, the
variation of the media solid fraction normal to the wall near location H2 (Fig. 4.4) was obtained
using 7 measurement bins of size d×12d×16d (7×76×100 mm) in the 12-layer DEM simulation
(Fig. 4.7). The solid fraction variation was evaluated under both maximum compression (Fig.
4.7a) and maximum decompression (Fig. 7b) corresponding to the wall at its maximum
vibrational displacement to the left and right, respectively, in Fig. 4.7. Figure 4.8 shows how the
solid fraction varied with distance from the wall of Fig. 4.7 using 10 s of DEM simulation to
obtain average solid fraction values over many instances of maximum compression and
decompression. As expected, the largest variation in the solid fraction occurred near the wall
within the first few particle diameters. There was also only a relatively small difference between
the curves as maximum compression and decompression. The media effective thickness was
defined initially as the distance from the wall where the solid fraction changed less than 10%
from bin to bin, which was Lb=24 mm in Fig. 4.8, yielding Kb=15 MPa/m. The sensitivity of the
95
continuum model to this choice of the effective media thickness at the wall will be examined
below.
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65250
300
350
400
450
500
Solid fraction
Eq
uiv
ale
nt
Yo
un
g's
mo
du
lus
(k
Pa
)
Fig. 4.6: The equivalent Young's modulus of the vibrationally-fludized bulk flow, Eb, vs. the
solid fraction, φ, based on Eqs. (9-11) and the DEM.
96
Fig. 4.7: Seven measurement bins used to study the media solid fraction variation with distance
from the wall when: (a) wall moves toward media leading to maximum compression, (b) wall
moves away from media leading to maximum decompression of the media and some separation
between the media and the wall.
Fig. 4.8: Average solid fraction, φ, variation with distance from the wall using data from the
measurement bins of Fig. 4.7.
(a) (b)
HW HW
97
4.3.3 Media-wall equivalent coefficient of friction
The effective coefficient of friction between the wall and the vibrationally-fluidized media, mw,
which was needed to apply the appropriate boundary condition to the media-wall interface in the
continuum model, was determined using the three-dimensional DEM model with 8 particle
layers between the glass partitions as used in ref. [19]. The shear forces (parallel to wall) and
normal forces acting on the media were determined during 1 s of simulated operation (equivalent
to 47 tub vibration cycles) at each of 8 locations along the tub wall as shown in Fig. 4.9, using
4d×6d×4d measurement bins. The appropriate simulation time step was calculated to be 0.686 µs
for the 6.3 mm diameter steel balls [19].
Fig. 4.9: Eight measurement locations used to determine average shear and normal forces acting
on the media along the tub wall.
z
x
1
2
3
4
5
6
7
8
HW
LW
BW
98
Figure 4.10 shows the probability density distributions of the shear to normal force ratio of all
particle-wall contacts, Ft/Fn, at each measurement location for 47 tub vibration cycles. The
maximum value of the predicted Ft/Fn in the distribution was 1.8, which corresponded to the
coefficient of friction that was measured for the 6.3 mm diameter steel balls sliding against the
urethane tub wall in ref. [19] and was used as an input to the DEM. It was observed that most of
particle collisions with the tub wall involved rolling rather than sliding since the most probable
values of Ft/Fn were smaller than 0.7 (Fig. 4.10a). The sliding impact of particles without rolling
is evident as a small peak at the right end of the probability distribution (Fig. 4.10a). Figure
4.10b shows that the probability density distributions of Ft/Fn at the different measurement
points were quite similar. This is a useful observation, because it means that a single relation can
represent the effective friction behaviour on the tub wall.
99
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Shear to normal force ratio Ft/Fn
Pro
bab
ilit
y d
en
sit
y d
istr
ibu
tio
n
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Shear to normal force ratio Ft/Fn
Pro
bab
ilit
y d
en
sit
y d
istr
ibu
tio
n
Fig. 4.10: Probability density distributions of the ratio of the shear to normal forces applied to
the media by the wall (Ft/Fn) at the 8 locations of Fig. 4.9 for: (a) all data (columns ordered from
left to right as positions 1-8), and (b) for Ft/Fn <0.8. Based on all particle-wall impact events
recorded by the DEM during 47 vibration cycles
Figure 4.11 shows how the shear force varied with the normal force considering all impact
events over 47 vibration cycles at location 2 in Fig. 4.9. The slope of the best-fit line was 0.25
(a)
(b)
100
with a coefficient of determination R2=0.7. This represents the average effective coefficient of
friction between the flowing steel balls and the tub wall at location 2 (µmw). As expected this
was much smaller than the measured coefficient of friction in sliding (µmw=1.80), since rolling
was prominent during contact. This is consistent with observations of Yabuki et al. [28] who
found that abrasive ceramic media did not slide on an aluminum surface in a vibratory finisher,
resulting in an average effective coefficient of friction of about 0.10, compared with a measured
value for sliding of 0.35-0.40 [28]. The average effective coefficient of friction, µmw, at the eight
locations on the tub wall varied from 0.25 to 0.35, giving an overall mean and standard deviation
of 0.28 and 0.02, respectively, as shown in Fig. 4.12. Figure 4.12 also shows the results from a
DEM simulation that used only 4 particle layers between the glass partitions, illustrating that the
predicted shear and tangential forces were independent of the number of layers.
Fig. 4.11: Shear force vs. normal force at location 2 (Fig. 4.9) for all wall impacts over 47
vibration cycles. The best-fit lines represent the mean effective coefficient of friction (µmw=0.25)
with the correlation coefficient, and the measured COF for sliding of a single steel ball against
the polyurethane wall material (µmw=1.8). The line for µmw=0.7 is also shown since it will be
used in the continuum modelling.
101
Fig. 4.12: The average effective coefficient of friction between the fluidized media and the tub
wall at the 8 locations shown in Fig. 4.9 based on all impacts over 47 vibration cycles for the 4-
layer and 8-layer DEM. The horizontal lines show the average and 95% confidence bounds.
It is noted that the average shear force recorded in these DEM simulations explain the origin of
the bulk flow of the granuar media. Figure 4.13 shows how the average normal and shear forces
applied to the media by the high wall (HW), low wall (LW) and bottom wall (BW) (Fig. 4.9)
changed during a single cycle (average forces over each interval of 1/12 of a cycle from 5 s (235
vibration cycles) of DEM simulation using the 8-layer model). For example, the first point in
Fig. 4.14 represents the average of the impact forces acquired in the first time interval of 1/12 of
a cycle. In Fig. 4.13, shear force acting on the media is positive if it was applied as a counter-
clockwise torque (Fig. 4.13), thereby creating the observed bulk flow in Fig. 4.3.
102
Fig. 4.13: Grand average shear force acting on the media by the wall over each 1/12 of a cycle
over 235 vibration cycles. Shear forces recorded over the high wall (HW; locations 1-3), low
wall (LW; location 4-6), and bottom wall (BW; locations 7-8) as shown in Fig. 4.9. Positive
shear force produced a counter-clockwise torque on the media.
4.4 Finite element implementation of the continuum model
Both Lagrangian and Eulerian finite element formulations of the continuum model were
investigated using the ABAQUS 6.11 software package (Dassault Systèmes Corp. 2014) [34].
4.4.1 Lagrangian FE continuum model
The explicit plane strain Lagrangian element was used to mesh the 2D domain of the tub
vibratory finisher using two mesh densities and either free triangular elements or free
quadrilateral elements as shown in Fig. 4.14. These element sizes of 2 cm and 1 cm were,
respectively, 5.0% and 2.5% of the domain characteristic length of 40 cm. The mesh was
103
adapted using 3 to 5 sweeps per increment to correct for excessive mesh distortion [34]. The
mesh aspect ratio was restricted to be larger than 0.1.
The interaction between the media and the tub wall was defined as a surface-to-surface contact
[34]. The average effective coefficient of friction and normal contact stiffness were determined
from the DEM as in Sections 4.3.3 and 4.3.2, respectively. The displacement and rotation
components of the tub wall harmonic motion were the same as those used in the DEM (taken
from accelerometer measurements made in ref. [9]). In most previous studies of continuum
modeling of non-fluidized granular flows, the simple no-slip boundary condition was assumed at
the wall surface [25][24, 25, 27, 30]. However, the nature of the wall-media interaction in the
present vibrationally-fluidized bed could not be assumed as being either slip or no-slip, since it
was a combination of rolling and sliding of the media on the wall as described in Section 4.3.3.
Therefore, an effective media-wall coefficient of friction that accounted for media rolling and
sliding, and a normal contact stiffness were considered to apply the appropriate boundary
condition to the wall-media interface. These parameters as well as the media equivalent density,
Young's modulus, and coefficient of internal friction were determined as in Section 4.3, and are
listed in Table 4.3. The only body force exerted on the media was gravity.
104
Fig. 4.14: The granular media domain meshed using Lagrangian elements: Free triangular
elements of sizes (a) 2 cm and (b) 1 cm; free quadrilateral elements of sizes (c) 2 cm and (d) 1
cm; tub wall meshed using free triangular elements of size 2 cm.
(a) (b)
(c) (d)
105
Table 4.3: Material and contact properties used in the FE simulations. The media effective
Young’smodulus, and the contact parameterswere evaluated over the indicated ranges. The
values of the media Young's modulus and the media-wall contact stiffness in parentheses were
calculated at the mean volume fraction φmean; i.e. the best estimate.
Material Density, ρb
(kg/m3)
Young's modulus,
Eb (kPa)
Coefficient of
internal friction, µ
Media 4680 257-498
(340 at φmean) 0.12
Tub wall 1200 26000 N/A
Contact Media-wall contact stiffness, K
(MPa/m)
Coefficient of media-wall
friction, µmw
Media-tub wall interaction 15.0-30.2 (19.7 at φmean) 0.1-0.7 (mean=0.3)
As an example, Fig. 4.15 shows the bulk flow velocity distributions obtained from the
Lagrangian analysis using the different elements with the media equivalent Young's modulus
Eb=257 kPa, the effective normal contact stiffness K=30.2 MPa/m, and the media-wall
coefficient of friction µmw=0.3. It is seen that the element type and size did not affect the bulk
flow patterns significantly, and that in all cases the flow fields were qualitatively similar to that
of the equivalent DEM, shown in Fig. 4.3. The mean bulk flow velocity was determined by
averaging the nodal vertical velocities perpendicular to a horizontal line from the center of
circulation to the right wall as shown in Fig. 4.15. The average inclination of the free surface
was the slope of a straight line fitted to the free-surface of the media [19]. Although the flow
fields had the correct form, the mean bulk flow velocity obtained in this Lagrangian FE
continuum model was 4.5 mm/s compared to 14 mm/s from the DE simulation. Section 4.5
describes the introduction of an adjustable parameter to the continuum model in order to improve
the accuracy of the bulk flow velocity.
The present FE model was two-dimensional and did not allow for the out-of-plane particle
motion that can occur in the tub finisher and in the three-dimensional DEM. The DEM
simulations of ref. [19] showed that the average out-of-plane particle impact velocities can be
106
approximately 20% of the in-plane components. This might be one of the reasons for the
difference between the bulk flow velocities predicted by the FE and DE simulations.
Fig. 4.15: The bulk flow velocity distribution obtained from the Lagrangian analysis: Free
triangular elements of sizes (a) 2 cm and (b) 1 cm; free quad elements of sizes (c) 2 cm and (d) 1
cm. The simulation parameters: K=30.2 MPa/m, Eb=257 kPa and µmw=0.3. Nodal velocity scale
(a) (b)
(c) (d) 2 mm/s
MFV: 4.5 mm/s
Surface angle: 15
MFV: 4.5 mm/s
Surface angle: 15
MFV: 5 mm/s
Surface angle: 17
MFV: 5 mm/s
Surface angle: 17
107
shown in lower right. Some arrows omitted for clarity. Mean bulk flow velocity (MFV) across
the dark horizontal lines.
4.4.2 Eulerian FE continuum model
In an Eulerian analysis, the mesh is spatially fixed and the material can move through it as
demonstrated in Fig. 4.16 [45]. An Eulerian analysis is normally used when the material
deformation is relatively large and Lagrangian elements become excessively distorted. Although
remeshing can resolve this distortion, it is computationally expensive, and it was of interest to
investigate the accuracy of an Eulerian analysis, which is faster to execute.
In the present case, an Eulerian mesh was generated to contain the tub wall and ensure that the
contact between the media and the tub wall was enforced accurately and the media did not leave
the Eulerian domain (Fig. 4.17) [34]. A Lagrangian mesh was used for the tub wall. A single
layer of 2 cm thick 3D hexagonal elements, the only Eulerian element type available in
ABAQUS, was used for the Eulerian analysis. The elements were assumed to be rectangular in
order to decrease the computational time and to increase the accuracy of the results [34]. Two
element sizes were evaluated, 1 cm and 0.5 cm, which were respectively 2.5% and 1.25% of the
domain characteristic length (Fig. 4.17). Since the wall deformations were very much smaller
than those of the media, a rigid body constraint was applied to the Lagrangian wall domain to
greatly decrease the run time.
108
Fig. 4.16: Material deformations in the Lagrangian and Eulerian analyses [45].
The volume fraction tool in ABAQUS [34] was used to fill the Eulerian elements occupied by
the media at time zero. By refining the mesh, the calculation of the volume fraction used to
define the material initial location became more accurate, minimizing the artefact of media-wall
overlap [34]. The contact between the media and the tub wall was modelled using the same
effective coefficient of friction and normal contact stiffness considered in the Lagrangian
analysis (Table 4.3). The velocity component normal to the plane of the bulk flow was again
considered to be zero as the flow was two-dimensional.
Lagrangian analysis Eulerian analysis
109
Fig. 4.17: The media meshed by Eulerian elements of sizes (a) 1 cm and (b) 0.5 cm. The tub
wall was meshed with Lagrangian elements.
Figure 4.18 shows the Eulerian mesh occupied by the media and the bulk flow velocity
distributions using the two sizes of Eulerian rectangular elements and the same parameters as
used in the Lagrangian analysis (K=30.2 MPa/m, Eb=257 kPa and µmw=0.3). The global
circulation observed in Fig. 4.18 was not dependent on the mesh size and was similar to that for
the Lagrangian model (Fig. 4.15). However, the mean bulk flow velocity obtained from the
Eulerian analysis was 90 mm/s, much larger than that computed in the Lagrangian analysis (4.5
mm/s), and in the DEM simulation (14 mm/s). The reason for this much higher bulk flow
velocity is unclear. Consequently, only the Lagrangian FE formulation was used to perform the
investigations of Sections 4.5, 4.6 and 4.7.
(a) (b)
110
Fig. 4.18: Bulk flow velocity distributions obtained from the Eulerian analysis for rectangular
elements of sizes: (a) 1 cm and (b) 0.5 cm. Nodal velocity scale shown in upper left. Some
arrows omitted for clarity.
4.5 Sensitivity to uncertainty in continuum model parameters
As discussed in Sections 4.3.2 and 4.3.3, the DEM of the vibrationally-fluidized flow showed
that there was considerable uncertainty in the equivalent continuum model parameters in the
vicinity of the media-wall interface. In particular, the media-wall coefficient of friction varied
about µmw=0.3 (Fig. 4.11), while uncertainty in the effective thickness of the media-wall
interaction zone, Lb, resulted in uncertainty in the dependent parameters Eb, the media equivalent
Young's modulus, and K, the media-wall effective contact stiffness (Eq. (4.12) and Fig. 4.6). It
was therefore of interest to assess the sensitivity of the continuum model to these uncertainties.
This was done using the Lagrangian FE analysis since the Eulerian FE simulations predicted
velocities significantly larger than the typical velocities observed in the DE simulations. The
mean bulk flow velocity was chosen as the measure of the continuum model response to these
media-wall parameters. Three values of the effective contact stiffness (K=15.0 MPa/m
(corresponding to Lb=24 mm), 19.7 MPa/m (Lb=16 mm), 30.2 MPa/m (Lb=10 mm)) and two
(a) (b)
100 mm/s
111
values of the media Young's modulus (Eb=257 (corresponding to Lb=10 mm) kPa, 498 kPa
(Lb=24 mm)) were selected while the coefficient of friction was fixed at µmw=0.3 to study the
sensitivity of the bulk flow behavior to K and Eb (first six combinations in Table 4). Then, to
study the effect of the coefficient of media-wall friction on the bulk behavior, four values of µmw
(0.1, 0.3, 0.5, 0.7) were used with the media equivalent Young's modulus and the media-wall
effective contact stiffness fixed as K=30.2 MPa/m, Eb=257 kPa (last four combinations in Table
4). The media equivalent density, coefficient of internal friction, and Poisson's ratio were held
constant at the values of Table 3. The first six combinations in Table 4 show that the mean bulk
flow velocity increased marginally (from 3 to 5 mm/s) as the effective contact stiffness, K,
increased. Presumably this resulted from the larger coefficient of restitution that followed from
an increase in the contact stiffness. Similarly, the mean bulk flow velocity was insensitive to
changes in the media equivalent Young's modulus, Eb. Much larger changes in the mean bulk
flow velocities (2.5 to 14 mm/s) were seen as the media-wall coefficient of friction, µm-w, was
varied from 0.1 to 0.7 (last four combinations in Table 4). It is noted that these conclusions
pertain to the mean bulk flow velocity, and that different trends may have been observed with
other characteristics of the flow.
112
Table 4.4: Sensitivity of the Lagrangian FE model mean bulk flow velocity (MFV) and free-
surface angle to changes in the effective media-wall contact stiffness, media equivalent Young's
modulus, and media-wall coefficient of friction.
K (MPa/m), Eb (kPa), µmw
MFV
(mm/s)
Free-surface
angle ()
Eb=257 kPa, µm-w=0.3
K=15.0 MPa/m 3.0 8.0
K=19.7 MPa/m 3.5 11
K=30.2 MPa/m 4.0 15
Eb=498 kPa, µmw =0.3
K=15.0 MPa/m 3.0 10
K=19.7 MPa/m 4.0 12
K=30.2 MPa/m 5.0 15
K=30.2 MPa/m, Eb=257 kPa
µmw=0.1 2.5 11
µmw=0.3 4.5 15
µmw=0.5 7.5 20
µmw=0.7 13 20
4.6 Comparison of FEA and DEM bulk flow velocities
As discussed in Section 4.4, although the flow fields predicted by the FE continuum models were
similar to those of the DEM, there were considerable differences in the predicted mean bulk flow
velocities, particularly with the Eulerian formulation. This is not too surprising given the
uncertainties in the media-wall contact parameters discussed in Section 4.5. It was therefore of
interest to adjust one of these contact parameters to obtain the best fit with the DEM, and then to
assess whether the continuum model with this optimized parameter continued to provide
satisfactory agreement for other vibrationally-fluidized flows.
113
The results of the Section 4.5 showed that the Lagrangian FE continuum model was most
sensitive to the media-wall coefficient of friction; therefore, it was selected as the adjustable
parameter. Trials with a range of µmw within the plausible range shown in Fig. 4.11 indicated
that the best agreement with the mean bulk flow velocity, MFV, occurred with µmw=0.7. Figure
4.19a compares the predictions of the Lagrangian FE model and the DEM for the actual
measured tub vibration amplitude, A, and for amplitudes of 0.5A, 0.75A and 1.5A. It is seen that
there is good agreement with the MFV for the actual tub amplitude, and that this agreement
persists when the vibrationally-fluidized flow is changed significantly by altering the tub wall
amplitude; i.e. the changes in MFV were approximately in proportion to the changes in the wall
amplitude. Similarly, the free surface angle increased with increasing wall amplitude as shown
in Fig. 4.19b. The angle predicted by the FE model with the optimized µmw=0.7 was 4 larger
than that of the DEM for the actual tub amplitude, A, and this difference increased to 7 when the
amplitude was 1.5A.
Figure 4.20 compares the distribution of the component of the bulk flow velocity perpendicular
to a horizontal line from the center of rotation to the wall as shown in Fig. 4.15. The local bulk
flow velocities obtained from the DEM and the FE simulation with µmw=0.7 were in reasonable
overall agreement, with the under-prediction of the FE model growing near the tub wall. This
may have been due to the rapid change in the bulk flow density and other contact parameters
near the wall (Figs. 4.6 and 4.8).
Overall, it is concluded that using a media-wall coefficient of friction µmw=0.7, optimized using
the DEM for the actual tub wall amplitude, provided reasonably accurate continuum predictions
of bulk flow velocity and free-surface inclination over a wide range of other vibrationally-
fluidized flow fields created by varying the tub wall amplitude.
114
Fig. 4.19: (a) mean bulk flow velocity and (b) inclination of free surface as a function of tub
amplitude expressed as the fraction of the actual tub amplitude, A. Predictions of the DEM and
of the Lagrangian FE continuum model that used the optimized media-wall effective coefficient
of friction, µmw=0.7. FE predictions with µmw=0.5 included for comparison.
(b)
µmw=0.7
µmw=0.5
µmw=0.7
µmw=0.5
(a)
115
Fig. 4.20: Distribution of bulk flow velocity perpendicular to the horizontal line defined in Fig.
4.15 in the 4-layer DE and the Lagrangian FE using the optimized µmw=0.7 (K=30.2 MPa/m,
Eb=295 kPa). Zero distance corresponds to the center of circulation and 160 mm is at the tub
wall.
4.7 Comparison of FEA and DEM local impact velocities
The previous sections considered the DEM and the FE continuum model predictions of the bulk
flow velocity and the inclination of the free surface. It is also of great interest to consider
whether the continuum model can yield predictions of the local particle impact velocity, or at
least predictions of parameters related to this velocity, since the erosion and surface deformation
produced in vibratory finishing are proportional to the particle kinetic energy.
The granular temperature, T, of an assembly of N particles is a function of the fluctuations of the
velocity components (u'i, v
'i, w
'i) as shown in Eq. (4.14a) [15], where the fluctuations are the
difference between the instantaneous (ui, vi, wi) and the mean velocities ū, ῡ, w (Eq. (4.14b)).
The granular temperature is the also a function of the mean free path, l, the shear rate and the
coefficient of restitution e (Eq. (4.15a)) [46]. The mean free path is related to the volume
116
fraction through Eq. (4.15b). Considering the particles to be spherical, the mean free path would
be proportional to the particle diameter, d, divided by the volume fraction, φ.
2 2 2
1
1( )
3
N
i i i
i
T u v wN
(4.14a)
, ,w ,i i i i i iu u u v v v w w (4.14b)
2 2
2(1 )
c lT
e
(4.15a)
2
3
dl
(4.15b)
Combining Eqs. 4.14(a) and 4.14(b), it is seen that the square of the granular temperature, T2, is
proportional to the difference between the average of the instantaneous velocities squared,
2
1
/N
i
i
V N
, and the bulk flow velocity squared, 2V (Eq. (4.16a, b, c)). Since the granular
temperature is proportional to the shear rate squared through Eq. (4.15a), it is seen that a measure
of the local impact velocities, 2
1
/N
i
i
V N
, is related to the shear rate. Therefore, if the shear rate
follows the same trend in the DE and FE simulations, it can be inferred that the impact velocities
would follow the same trend.
2
2 2 2 21
1
1 1( ) ( ) ( )
3 3 3
N
iNi
i i i
i
V
T u u v v w w VN N
(4.16a)
2 2 2 2V u v w (4.16b)
2
21 3
N
i
i
V
T VN
(4.16c)
Figure 4.21a confirms this hypothesis by showing that the mean impact velocity and its
components increased with increasing shear rate calculated at the four locations shown in Fig.
4.4 in the 4-layer DE model. The mean impact velocities were obtained by averaging the
instantaneous particle velocities (the absolute values of the impact velocities) over 10 s, as
117
described in ref. [19]. Thus the shear rate is a possible proxy for the magnitude of the local
average impact velocity of the media.
The average shear rate was then determined in the Lagrangian FE model (K=30.2 MPa/m,
Eb=295 kPa, µmw=0.7) by following the same procedure described in Section 4.3.1 in the
calculation of the shear rate in the DEM. Figure 4.21b shows that the shear rate obtained from
the DE and FE simulations followed the same trend at the four assessment points in the flow
field. Therefore, it is concluded that the local shear rate predicted by the FE continuum model
correctly reflected the trend of the local impact velocity distribution in the present vibrationally-
fluidized granular flow. This implies that the FE continuum model may prove to be useful as a
means of predicting the impact velocity distribution within a vibratory finisher, and hence the
distribution of the impact energy imparted to workpiece surfaces as they move with the bulk
flow.
Fig. 4.21: (a) Average particle impact velocity, Vimp, and its components vs. the shear rate, , at
points H1, L1, L2, H2 (ordered from left to right) in the 4-layer DEM (Fig. 4.4). (b) Shear rate in
FE and DE simulations at the same points.
(a) H1 L1 L2 H2 (b)
118
4.8 Conclusions
A continuum finite element model was used successfully to simulate the flow behavior in a tub
vibratory finisher. The effective bulk and media-wall properties for the constitutive equation
governing the visco-plastic continuum model of the spherical, vibrationally-fluidized media were
found using a discrete element model. It was seen the continuum model could be simplified by
using constant, average values of the constitutive properties for the vibrationally-fluidized media
(density, modulus, internal friction), and for the media-wall interaction parameters (friction,
contact stiffness).
The bulk behavior of the vibrationally-fluidized granular flow was modelled using both a
Lagrangian and an Eulerian finite element implementation of the continuum model. The
predicted patterns of the bulk flow velocity distributions agreed with the DEM simulations and
with experimental observations. Good quantitative agreement was obtained by treating the
media-wall coefficient of friction as an adjustable parameter. The generality of this approach
was demonstrated by using this optimized value to predict the bulk flow velocities created by
different tub vibration amplitudes. It was also observed that the local impact velocities of the
vibrationally-fluidized media could be correlated with the local shear rate in the continuum
model through its relation to the granular temperature. It was concluded that the bulk and local
behavior of vibrationally-fluidized flows can be modelled using finite element implementations
of a visco-plastic continuum model. This can lead to significant reductions in computational
effort compared with discrete element modelling.
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122
Chapter 5
5 Conclusions
5.1 Experimental measurements
A laser displacement sensor was used to construct a probe that can be immersed in a granular
flow to measure local impact velocities, bulk flow velocities and a measure of particle packing.
The accuracy of impact velocity measurements was verified using drop tests with the same
granular media. The sensor output was modeled to interpret laser reflections from media passing
before the sensor in different trajectories and to distinguish the media vibrational impact velocity
from the oscillations of the sensor window. The main conclusions of the experimental
measurements were as follows:
The sensor was demonstrated in a tub vibratory finisher with two types of media-steel
and porcelain spheres. In this case, the average bulk flow velocity parallel to the sensor
window in the steel media was almost 40% larger than that in the porcelain media, and
both varied appreciably over the different measurement locations within a single media.
Similarly, the impact velocity and particle packing varied considerably among the various
locations and in different directions. Although it was concluded that there was no direct
proportionality between the impact velocity of particles, the bulk flow velocity, and the
packing density of the media, the smallest packing density and impact velocity occurred
in the wake formed behind the sensor shielding tube. These two parameters were larger
when the sensor was facing the average moving bulk flow.
It was also observed that the impact velocity of the particles was strongly dependent on
depth in the flow, increasing sharply with distance from the free surface.
The average impact velocities in the steel media were approximately 15% smaller than
those in the porcelain media due to differences in the density and coefficient of
restitution.
123
In addition to providing a means of quantifying the impact energy distribution within a
vibratory bed, the experimental results can be used to validate the predictions of
numerical simulations such as discrete element modeling (DEM). Such models can help
to understand the complex patterns of behavior that have been observed within these
vibrationally fluidized beds.
The laser displacement probe and the current procedures are directly applicable to the
measurement of the local impact velocities of non-spherical particles. The measurement
of the bulk flow velocity in the case of irregular particle shapes would be more difficult,
since the variation in the average displacement signal would be more irregular.
5.2 Discrete element modeling
The impact and bulk flow velocities in a tub vibratory finisher predicted by discrete element
modeling (DEM) were compared to the measured values for steel granular media. This was the
first time measurements of local impact velocities have been compared with DEM predictions in
a vibrationally-fluidized granular flow. The sensitivity of the predicted local impact velocities
and the bulk flow velocities to the DEM contact parameters (coefficients of friction, restitution
and rolling resistance) was investigated both analytically and numerically. The vibration of the
tub finisher walls was essentially two-dimensional, with negligible motion in the transverse
direction. The effect of the number of DE particle layers in the transverse direction was
investigated to determine the minimum number required for accurate predictions of the particle
impact and bulk flow velocities. Moreover, a reduced shear modulus could be used to decrease
the model run time without significantly affecting the accuracy of the predicted impact and bulk
flow velocities. The complete DE model simulated the actual setup used in an earlier
experimental study that measured the bulk flow and local impact velocities in the tub vibratory
finisher. The coefficients of friction and restitution between the steel balls, and between the balls
and the tub walls, were measured in order to obtain the most accurate DE simulations. The main
conclusions drawn from the discrete element simulations were:
124
It was found that the predictions of both the bulk flow and impact velocities were relatively
insensitive to uncertainties in the contact parameters.
It was concluded that accurate velocities could only be predicted using at least 4-12 layers.
Fewer layers created too much spatial constraint to allow realistic particle motion.
The predicted impact velocities in several directions and at two locations in the tub displayed
the same log-normal probability density distributions as did the previously measured impact
velocity distributions.
Moreover, the predicted average impact velocities were in good agreement with the
experimental measurements, with a maximum error of 19%. The DE predictions of the bulk
flow velocities were also consistent with the experimental measurements, having a maximum
error of 30%.
Therefore, it was concluded that DEM can be used to give reasonably accurate predictions of
both the local impact velocities and the bulk flow of particles in vibrationally-fluidized beds.
This will be useful in predicting the impact energy and force of the media on workpiece
surfaces, and hence the resulting wear and surface deformation during vibratory finishing.
5.3 Continuum modeling
A continuum finite element model was used successfully to simulate the flow behavior in a tub
vibratory finisher. The effective bulk and media-wall properties for the constitutive equation
governing the visco-plastic continuum model of the spherical, vibrationally-fluidized media were
found using a discrete element model. The bulk behavior of the vibrationally-fluidized granular
flow was modelled using both a Lagrangian and an Eulerian finite element implementation of the
continuum model. The main conclusions drawn from the finite element continuum modeling
were as follows:
125
It was seen the continuum model could be simplified by using constant, average values of
the constitutive properties for the vibrationally-fluidized media (density, modulus,
internal friction), and for the media-wall interaction parameters (friction, contact
stiffness).
The predicted patterns of the bulk flow velocity distributions agreed with the DEM
simulations and with experimental observations. Good quantitative agreement was
obtained by treating the media-wall coefficient of friction as an adjustable parameter.
The generality of this approach was demonstrated by using this optimized value to
predict the bulk flow velocities created by different tub vibration amplitudes.
It was also observed that the local impact velocities of the vibrationally-fluidized media
could be correlated with the local shear rate in the continuum model through its relation
to the granular temperature.
It was concluded that the bulk and local behavior of vibrationally-fluidized flows can be
modelled using finite element implementations of a visco-plastic continuum model. This
can lead to significant reductions in computational effort compared with discrete element
modelling.
126
6 Future work
The following topics may prove to be interesting areas for future research:
The experimental measurement of impact velocities using the laser displacement sensors
can be extended to the granular flows of non-spherical.
The continuum modeling of flows of non-spherical particles would be useful, although
the calibration of the equivalent bulk properties of non-spherical particles is expected to
be more challenging than spherical particles.
It would be of interest to apply the same procedure followed in the continuum modeling
of the granular flow produced in the vibratory finisher to fluidized beds in other
geometries, such as in a bowl vibratory finisher. It may also be possible to extend the
present single-phase continuum model to a two-phase model that could be applied to the
granular flows that are fluidized by a gas or liquid.
Using the concept of granular temperature to obtain an estimation of the average impact
velocities from the shear rates in the continuum modeling can be explored further.
127
Appendix A: Analytical sensitivity study in Chapter 3
This Appendix describes the derivation of the equations describing the sensitivity of particle-
particle and particle-wall rebound velocities to uncertainties in the collision coefficients. This
was then be used to determine the required measurement accuracy for these coefficients. The
linear and angular rebound velocities of disks were determined as functions of the contact
parameters using the equations of linear and angular momentum conservation [1]. The
coefficient of rolling resistance is defined as the ratio of the rolling resistance force to the normal
force (or the ratio of the rolling resistance torque to the product of the normal force and the
particle radius), due to the elastic deformation of the bodies in the contact region [2, 3][25].
Figure A.1 of case (1), is described by
1 1 2 2 1 1 2 2n n n nm v m v mV m V (A.1)
1 1 2 2 1 1 2 2t t t tm v m v mV m V (A.2)
V and v were designated for pre-collision and post-collision velocities, and n and t were
designated for normal and tangential directions, respectively. Conservation of angular
momentum gives
1 1 2 2 1 1 2 2J J J J (A.3)
where Ω and ω are the angular speeds before and after the collision, respectively. The normal
and tangential impulse are then defined as
1 1 1 2 2 2n n n n nP m v V m v V
1 1 1 1 1 1 1 2 2 2 2 2 2 2t t r t t tP m v V r m v r V r
(A.4)
The coefficient of rolling resistance ( rµ ) was calculated from Eqs. (A.6) and (A.7) [26].
1 1 1 2 2 2 1 1 1 1 1 1 2 2 2 2 2 2 2n n n n t t tm v V m v V m v r V r m v r V r (A.5)
128
1 1 1 1t rr P M J (A.6)
2 2 2 2t rr P M J (A.7)
Where µ=Pt/Pn is the coefficient of friction, and * r r nM µ R P is the rolling torque with R* being
the equivalent radius of curvature (Eq. 1b). Equations (A.1-A.7) can be written as Eq. (A.8) with
2
1 2 1 2 1 2, , 1/ 2m m m r r r J J mr .
1
1
2
2
1
2
1 0 1 0 0 0 1 0 1 0 0 0
0 1 0 1 0 0 0 1 0 1 0 0
1 0 1 0 0 0 0 0 0 0
1 1 1
0 2 0 2 0 2 0
0 0
n
t
n
t
r r
v
v
v e e
r r v
r r
µ µ r r µ
1
1
2
2
1
2
1
2
0 0
n
t
n
t
r r
V
V
V
r r V
r r
µ r r
(A.8)
Fig. A.1: Collision of two disks having arbitrary initial linear and angular velocities: (a)
velocities after collision, and (b) impulses during impact.
Equation (A.8) gives the rebound velocities of both disks as functions of their initial velocities
and the collision coefficients. The normal rebound velocity, v1n, the tangential rebound velocity,
v1t, and the angular rebound velocity, ω1, are written in Eq. (A.9) as functions of the contact
parameters and normal impact velocities prior to collision. The sensitivities of v1n, (Eq. (A.9a)),
(a) (b)
129
v1t, (Eq. (A.9b)), and ω1, (Eq. (A.9c)) to the collision coefficients were determined by
differentiating these velocities with respect to the coefficients as:
1 2 1 2 1
1 1
2 2n n n n nv V V e V V eC
12 1
1
2
nn n
vV V C
e
1 1 0,n n
r
v v
µ µ
(A.9a)
1 2 1
1(1 ) (1 )
2t n nv V V e e C
12 1
1
2
tn n
vV V C
e
12 1
11 1
2
tn n
ve V V e C
1 0,t
r
v
µ
(A.9b)
1 2 1
2 21(1 e) (1 e)
2
r r
n n
µ µV V C
r r
1
2 1
2 21
2
r r
n n
µ µV V C
e r r
1
2 1
2 2 2 21
2n n
e eV V C
r r
1
2 1
1 11
2n n
r
e eV V C
µ r r
(A.9c)
130
where 2 1
1
2n nC V V is half of the relative normal component of the impact velocity. The
mean values of the rebound velocity components were calculated using the mean values of the
contact parameters; i.e. e=0.675, µ=1.2 and µr=0.05 based on their ranges given in Section 3.3.
The disks were assumed to approach each other with the same speed and in opposite directions
so that the second term in the normal velocity expression in Eq. (A.9a) vanished.
1( ) 0.675C,n avev 1 ,nvC
e
(A.10a)
1 2.01C ,t avev 1 1.2 ,t
ave
vC
e
1 1.68 ,t
ave
vC
(A.10b)
1
4.
10,
Cave r
1 2.45 ,
ave
C
e r
1 3.35 ,
ave
C
r
1 1.73
r ave
C
µ r
(A.10c)
The normalized sensitivities of the rebound velocities to variations in the contact parameters
were then calculated using the derivatives in Eq. (A.9) with the mean values of the contact
parameters, and then dividing by the average rebound velocities of Eq. (A.10).
The same procedure was used for case (2), the collision between a disk and a moving wall. The
average wall velocity was assumed to be one third of the disk approach velocity, based on the
experimental data from the laser probe [4]. The sensitivities of the normal, tangential and angular
velocities of the moving disk in case (2) are shown in Eq. (A.11).
1( ) 1.85C,n avev 1 2nvC
e
(A.11a)
1 4.36C ,t avev 1 2.6 ,t
ave
vC
e
1 3.35 ,t
ave
vC
1 6.7 ,t
r
vC
µ
(A.11b)
1
8.
38,
Cave r
1 5 ,
ave
C
e r
1 6.7 ,
ave
C
r
1 6.7
r ave
C
µ r
(A.11c)
Equations A.9, A.10 and A.11 were used in Section 3.3 to calculate the percentage uncertainties
of the rebound velocities due to 10% uncertainty in the contact parameters.
131
References
[1] R.M. Brach, Friction, restitution and energy loss in planar impacts, J. Appl. Mech., 51
(1984) 164-170.
[2] J. Ai, J. F. Chen, J. M. Rotter, J. Y. Ooi, Assessment of rolling resistance models in discrete
element simulations, Powder Technol. 206 (2011) 269–282.
[3] T. N. Tang, Input parameters of discrete element methods, J. Eng. Mech., 132 (2006) 723-
729.
[4] K. Hashemnia, A. Mohajerani, J. K. Spelt, Development of a laser displacement probe to
measure particle impact velocities in vibrationally fluidized granular flows, Powder
Technol., 235 (2013) 940-952.
132
Appendix B: MATLAB Codes of Chapter 2
Appendix B.1. Fast Fourier transformation (FFT) of the tub motion based on the measurements
Inputs: The tub wall displacement signal measured by the laser sensor
Outputs: Dominant frequencies and their corresponding amplitudes
Fs = 10000; % Sampling frequency T = 1/Fs; % Sample time L = 65000; % Length of signal t = (0:L-1)*T; % Time vector % y[]; z=laser(:,2); figure; %plot(Fs*t,z)
NFFT = 2^nextpow2(L); % Next power of 2 from length of y y = fft(z,NFFT)/L; f = Fs/2*linspace(0,1,NFFT/2+1); a=2*real(y); b=-2*imag(y); S=2*abs(y); F=f';
% plot(f,2*abs(y(1:NFFT/2+1))) % % title('Single-Sided Amplitude Spectrum of y-displacement at point A') % xlabel('Frequency (Hz)') % ylabel('|Y(f)|')
% [z,p,k] = butter(6,500/5000,'high'); % [sos,g] = zp2sos(z,p,k); % Convert to SOS form % Hd = dfilt.df2tsos(sos,g); % Create a dfilt object % h = fvtool(Hd); % Plot magnitude response % set(h,'Analysis','freq') % Display frequency response
[B,A]=BUTTER(3,500/5000,'low'); Z=filter(B,A,z);
plot(t,z); hold on plot(t,Z,'red');
title('y-acceleration at point B filtered and unfiltered') xlabel('Time(s)') ylabel('noise')
Y=fft(Z,NFFT)/L;
133
figure plot(f,2*abs(y(1:NFFT/2+1))); hold on plot(f,2*abs(Y(1:NFFT/2+1)),'r');
title('Single-Sided Amplitude Spectrum of y-acceleration at point B') xlabel('Frequency (Hz)') ylabel('|Y(f)|')
Appendix B.2. Acceleration, velocity and position of the vibratory finisher center of gravity
Inputs: Acceleration of points A and B (Fig. 2.2a) measured by the accelerometers
Outputs: Acceleration, velocity and position of the vibratory finisher center of gravity
% Calculating angular velocity and angular acceleration of the tub with % respect to time and calculating the acceleration and vibration amplitude % of the tub center of gravity (including the media inside it) % angular velocity and acceleration are CCW (counter-clock-wise) by % default. x and y axis are positive Cartesian coordinates % tub vibration frequency: 47 Hz % t=[]; % time vector % AAp=[]; % acceleration of point A (in Volts), dimension n*2; first column
is % x-component and second column is y-component % ABp=[]; % acceleration of point B (in Volts), dimension n*2; first column
is % x-component and second column is y-component clc syms t % Fs=10000; % Sampling frequency % T=1/Fs; % Sample time % L=64000; % Length of signal % t=(0:L-1)*T; % Time vector % After fft analysis and knowing the coefficients
AAXF=-20.933*cos(2*pi*46.39*t)-
3.188*sin(2*pi*46.39*t)+10.522*cos(2*pi*139.3*t)+4.064*sin(2*pi*139.3*t)+10.2
24*cos(2*pi*325.2*t)-2.007*sin(2*pi*325.2*t)-0.167*cos(2*pi*371.7*t)-
14.052*sin(2*pi*371.7*t);
AAYF=-26.837*cos(2*pi*46.39*t)-61.254*sin(2*pi*46.39*t)-
8.391*cos(2*pi*139.3*t)-6.289*sin(2*pi*139.3*t)+7.328*cos(2*pi*185.9*t)-
16.361*sin(2*pi*185.9*t)-20.847*cos(2*pi*232.2*t)-6.868*sin(2*pi*232.2*t)-
19.459*cos(2*pi*325.2*t)-
3.416*sin(2*pi*325.2*t)+9.759*cos(2*pi*371.7*t)+8.150*sin(2*pi*371.7*t);
ABXF=-15.965*cos(2*pi*46.39*t)-
1.706*sin(2*pi*46.39*t)+2.880*cos(2*pi*418.1*t)+8.837*sin(2*pi*418.1*t);
134
ABYF=59.748*cos(2*pi*46.39*t)-
14.752*sin(2*pi*46.39*t)+5.646*cos(2*pi*92.93*t)
+6.508*sin(2*pi*92.93*t)+5.596*cos(2*pi*325.2*t)+7.035*sin(2*pi*325.2*t)-
4.364*cos(2*pi*418.1*t)+6.870*sin(2*pi*418.1*t);
rAB=.39; % (distance between A and B) theta=atan(19.5/25.4); % x and y components of the tub center of gravity r=.32; % distance between point A and center of gravity % n=length(t); % T1=T; % t1=(0:n-1)*T1; VAXF=int(AAXF,t);% velocity (m/s) of point A-x VAYF=int(AAYF,t);% velocity (m/s) of point A-y VBXF=int(ABXF,t);% velocity (m/s) of point B-x VBYF=int(ABYF,t);% velocity (m/s) of point B-y
XAXF=int(VAXF,t);% position (m) of point A-x XAYF=int(VAYF,t);% position (m) of point A-y XBXF=int(VBXF,t)+rAB;% position (m) of point B-x XBYF=int(VBYF,t);% position (m) of point B-y
% XBX(1,1)=rAB; % initial conditions; the rest are zero.
Lan=atan(XBYF-XAYF)/(XBXF-XAXF); % angle between rAB and x-axis
omegaF=(VBYF-VAYF)/(rAB*cos(Lan)); % angular velocity of the tub
alphaF=(ABYF-AAYF+rAB*(omegaF)^2*sin(Lan))/(rAB*cos(Lan));%angular
acceleration of the tub
% ACG=zeros(n,2); % VCG=zeros(n,2); % XCG=zeros(n,2);
XCGXF=XAXF+r*sin(theta+Lan); XCGYF=XAYF-r*cos(theta+Lan); VCGXF=VAXF+r*omegaF*cos(theta+Lan); VCGYF=VAYF+r*omegaF*sin(theta+Lan); ACGXF=AAXF-r*(omegaF)^2*sin(theta+Lan)+r*alphaF*cos(theta+Lan); ACGYF=AAYF+r*(omegaF)^2*cos(theta+Lan)+r*alphaF*sin(theta+Lan); % AACG=sqrt(ACGX^2+ACGY^2);
% They should be evaluated with respect to t first. Fs=1000; % Sampling frequency T=1/Fs; % Sample time n=1000; % Length of signal t=0; for i=1:n t=t+T; omega(i,1)=eval(omegaF); alpha(i,1)=eval(alphaF); ACGX(i,1)=eval(ACGXF); ACGY(i,1)=eval(ACGYF); VCGX(i,1)=eval(VCGXF); VCGY(i,1)=eval(VCGYF); XCGX(i,1)=eval(XCGXF);
135
XCGY(i,1)=eval(XCGYF); end figure; plot(t,omega); title('tub angular velocity vs. time') xlabel('Time(s)') ylabel('Omega(rad/s)') figure; plot(t,alpha); title('tub angular acceleration vs. time') xlabel('Time(s)') ylabel('Alpha(rad/s^2)') figure; plot(t,ACGX); title('x-acceleration of tub center of gravity vs. time') xlabel('Time(s)') ylabel('x-acceleration(m/s^2)') figure plot(t,ACGY); title('y-acceleration of tub center of gravity vs. time') xlabel('Time(s)') ylabel('y-acceleration(m/s^2)') figure plot(t,VCGX); title('x-velocity of tub center of gravity vs. time') xlabel('Time(s)') ylabel('x-velocity(m/s)') figure plot(t,VCGY); title('y-velocity of tub center of gravity vs. time') xlabel('Time(s)') ylabel('y-velocity(m/s)') figure plot(XCGX*1000,XCGY*1000); title('y-displacement of tub center of gravity vs. x-displacement') xlabel('x-displacement(mm)') ylabel('y-displacement(mm)') %fft analysis of omega, alpha, ACG(:,1), ACG(:,2), VCG(:,1), VCG(:,2) % Now, the amplitude of this acceleration (AACG) should be determined and % should be divided by square of 47 Hz=295 rad/s in order to obtain the
vibration amplitude
136
Appendix B.3. Path of the top points of the vibratory finisher (A and B in Fig. 2.2a)
Inputs: Acceleration of points A and B (Fig. 2.2a) measured by the accelerometers
Outputs: Path of points A and B
clc Fs=10000; % Sampling frequency T=1/Fs; % Sample time L=65000; % Length of signal t=(0:L-1)*T; % Time vector AA1=AAp*981; % acceleration (m/s^2) of point A AB1=ABp*981; % acceleration (m/s^2) of point B AA(:,1)=AA1(:,1)-14.01;%12.56; AA(:,2)=AA1(:,2)-13.98;%11.56; AB(:,1)=AB1(:,1)+8.02;%6.85; AB(:,2)=AB1(:,2)-15.03;%13.3; rAB=.39; % (distance between A and B) theta=atan(19.5/25.4); % x and y components of the tub center of gravity r=.32; % distance between point A and center of gravity % A=AB-AA; % acceleration difference between A and B n=length(t); T1=T; t1=(0:n-1)*T1; VA=zeros(n,2);% velocity (m/s) of point A VB=zeros(n,2);% velocity (m/s) of point B % numerical integration for determining velocity of points A and B for i=2:n VA(i,1)=(AA(i,1)+AA(i-1,1))/2*T1+VA(i-1,1); VA(i,2)=(AA(i,2)+AA(i-1,2))/2*T1+VA(i-1,2); % VB(i,1)=(AB(i,1)+AB(i-1,1))/2*T1+VB(i-1,1); % VB(i,2)=(AB(i,2)+AB(i-1,2))/2*T1+VB(i-1,2); end XA=zeros(n,2); XB=zeros(n,2); XB(1,1)=rAB; % numerical integration for determining position of points A and B for i=2:n XA(i,1)=(VA(i,1)+VA(i-1,1))/2*T1+XA(i-1,1); XA(i,2)=(VA(i,2)+VA(i-1,2))/2*T1+XA(i-1,2); % XB(i,1)=(VB(i,1)+VB(i-1,1))/2*T1+XB(i-1,1); % XB(i,2)=(VB(i,2)+VB(i-1,2))/2*T1+XB(i-1,2); end plot(t1,VA(:,1)); title('x-velocity of tub center of gravity vs. time') xlabel('Time(s)') ylabel('x-velocity(m/s)') figure plot(t1,VA(:,2)); title('y-velocity of tub center of gravity vs. time') xlabel('Time(s)') ylabel('y-velocity(m/s)') figure plot(XA(:,1)*1000,XA(:,2)*1000);
137
title('y-displacement of tub center of gravity vs. x-displacement') xlabel('x-displacement(mm)') ylabel('y-displacement(mm)')
Appendix B.4. The impact velocity calculation based on the laser displacement sensor measurements
Inputs: Displacement signals of the impact velocity probe window and the balls passing behind
the laser beam acquired by the laser sensors
Outputs: Impact velocity of the balls colliding the probe window
clc % y=[]; % ball motion after smoothing % z1=[]; % probe motion after smoothing t=.0005; %input('time increment'); n=65536; p=0; Z=0; % criteria should be changed for upstream 2 (very important) z=-1.446*z1+0.012; % glass target motion after smoothing emax=0.1; %0.12*GAmp % GAmp=0.86 mm is the amplitude of center of gravity (is
to be determined) taramp=0.12/2;% target motion amplitude(mm) depends on each signal pvdata=20; % pvdata is the number of data points from valley to peak (should
be read visually from the signal) min=0.2*pvdata; %should be specified max=0.6*pvdata; %should be specified for i=4:n-4 a1=(z(i-3,1)+z(i-2,1)+z(i-1,1))/3; a2=(z(i+3,1)+z(i+2,1)+z(i+1,1))/3; if z(i,1)>a1 && z(i,1)>a2 p=p+1; Z=Z+z(i,1); end end Zave=Z/p; % average of maximum z (target motion) peak values ll=-0.7; %should be specified visually hl=0.4; %should be specified visually j=0; a=0; c=0; v0=2*pi*47*taramp/2; % target reference velocity (1/3 (pi/6) of the data
points from the bottom points (valleys)) k=0; aa=0; bb=0; for i=30:n-4 a1=(y(i-3,1)+y(i-2,1)+y(i-1,1))/3; % can be changed to average of 2 a2=(y(i+3,1)+y(i+2,1)+y(i+1,1))/3; % can be changed to average of 2
138
if y(i,1)>a1 && y(i,1)>a2 % maximum points recognition e1=abs(y(i,1)-z(i,1)); if e1<emax % impact point recognition. e is in mm. difference between
peaks of ball and target signals; can be changed for m=i-(pvdata+5):i a11=(y(m-3,1)+y(m-2,1)+y(m-1,1))/3; a22=(y(m+3,1)+y(m+2,1)+y(m+1,1))/3; if y(m,1)<a11 && y(m,1)<a22 % minimum points recognition e2=z(m,1)-y(m,1); if e2<emax/2 % in order to avoid same motions of balls
and target break; else bb=1; end else aa=1; end if aa==1 || bb==1 if y(m,1)>ll && y(m,1)<hl if y(m+1,1)>y(m,1) j=j+1; if j>min && j<max if abs(y(m+1,1)-y(m,1))<0.2 % 0.2 maximum
acceptable distance between the data points in order to avoid the erroneous
data (jumps) a=a+y(m+1,1)-y(m,1); else break; end end else if j>max j=max; end c=j-min; if c>0.25*pvdata %should be specified v=a/(c*t); if v~=0 && v>v0 % k=k+1 % disp(c); V(i)=v; % disp((m-1)*t); end end end end end aa=0; bb=0; end a=0; j=0; end end end j=1; for i=1:length(V)
139
if V(i)~=0 Vf(j)=V(i); j=j+1; end end for i=1:65536 T(i,1)=(i-1)*.0005; end Vff=Vf'; S=Unique(Vff); plot(T,y); hold; plot(T,z,'r'); % figure; % hist(S,20);
Appendix C: MATLAB Codes of Chapter 3
Appendix C.1. The calculation of the in-plane components of the impact velocity based on DEM simulation results
Inputs: In-plane components of the particles instantaneous velocities acquired over 10 s of a
DEM simulation
Outputs: In-plane components of the mean impact velocity of the particles
% vnx=Input('particles x-relative normal '); % vnz=Input('particles z-relative normal '); % vx=Input('particles x-velocity '); % vz=Input('particles z-velocity ');
%V=zeros(n,4); V=[vnx vnz vx vz]
[n,m]=size(V); % V is the velocity matrix; 1st and 2nd columns: relative
normal x-velocity and z-velocity, 3rd and 4th columns: element A x-velocity
and z-velocity.
d=0.0063; % diameter
theta=zeros(n,1); vn=zeros(n,1); % absolute normal collision velocity (element A) vt=zeros(n,1); % absolute tangential collision velocity (element A)
for i=1:n a=((V(i,1))^2+(V(i,2))^2)^0.5; if a>0.001
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theta(i,1)=atan2(V(i,2),V(i,1)); vn(i,1)=V(i,3)*cos(theta(i,1))+V(i,4)*sin(theta(i,1)); vt(i,1)=-V(i,3)*sin(theta(i,1))+V(i,4)*cos(theta(i,1)); else V(i,3)=0; V(i,4)=0; end end
vn1=unique(vn); vt1=unique(vt);
vn2=mean(abs(vn1)); vt2=mean(abs(vt1));
vn2s=std(abs(vn1)); vt2s=std(abs(vt1));
k=0; for i=1:n if V(i,3)~=0 || V(i,4)~=0 k=k+1; Vx(k,1)=V(i,3); Vz(k,1)=V(i,4); end end
vx2=mean(abs(Vx)); vz2=mean(abs(Vz)); % vxBF=mean(Vx); % vzBF=mean(Vz); vx2s=std(abs(Vx)); vz2s=std(abs(Vz)); % vnx=Input('particles x-relative normal ');
Appendix C.2. The calculation of the out- of-plane component of the impact velocity based on DEM simulation results
Inputs: Out-of-plane component of the particles instantaneous velocities acquired over 10 s of a
DEM simulation
Outputs: Out-of-plane component of the mean impact velocity of the particles
%V=zeros(n,3); V=[vny vyA vyB]
[n,m]=size(V); % V is the velocity matrix; 1st column: relative normal x-
velocity, 2nd and 3rd columns: element A and element B y-velocity.
d=0.0063; % diameter
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for i=1:n a=V(i,1); if abs(a)<0.001 V(i,2)=0; %V(i,3)=0; end end
k=0; for i=1:n if V(i,2)~=0 %|| V(i,3)~=0 k=k+1; VyA(k,1)=V(i,2); %VyB(k,1)=V(i,3); end end
vyA2=mean(abs(VyA)); %vyB2=mean(abs(VyB)); vyA2s=std(abs(VyA)); %vyB2s=std(abs(VyB));
Appendix C.3. The calculation of coefficient of friction between spherical steel balls based on the linear tribometer measurements
Inputs: The horizontal force applied to a spherical steel ball by a similar ball and its vertical
displacements
Outputs: The tangential force applied to the spherical steel ball
% %T=input('tangential force vector:'); % %y=input('LVDT depth measurement:'); % %t=input('time:'); % n=length(T); % y0=min(y)*ones(n,1); % y1=(y-y0)/1000; % relative depth in mm % R=3.15; % radius in millimeters % for i=1:n % theta(i,1)=acos(1-y1(i,1)/R); % end % W=1; % for i=1:n % f(i,1)=T(i,1)*cos(theta(i,1))+W*sin(theta(i,1));
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% N(i,1)=-T(i,1)*sin(theta(i,1))+W*cos(theta(i,1)); % mu(i,1)=f(i,1)/N(i,1); % end % % t(1,1)=1; % % for i=1:n-1 % % t(i+1,1)=t(i,1)+1; % % end % plot(t,mu); % hold % plot(t,T,'r');
% predicting the tangential force measured by tribometer for ball-ball COF % measurements assuming a constant COF; theoretical approach mu=0.6; % constant COF W=1; % weight (N) s=0.9; d=6.3; % stroke and diameter in mm % theta0=asin(s/d); v=0.167; % velocity mm/s y1=0.026; y2=0.033; R=d/2; % depth magnitudes at both ends of the stroke theta1=acos(1-y1/R); theta2=acos(1-y2/R); n=100; for i=1:2*n+1 % theta(i,1)=-theta0+(i-1)*theta0/n; theta(i,1)=-theta1+(i-1)*(theta1+theta2)/(2*n); T(i,1)=(mu*cos(theta(i,1))-
sin(theta(i,1)))/(cos(theta(i,1))+mu*sin(theta(i,1)))*W; % tangential force
(N) t(i,1)=3.15/v*sin(theta(i,1)+theta1);
%t(i,1)=0.3636*(theta(i,1)+theta1)*180/pi; % time (s) end % t=theta*v; % conversion to time plot(theta*180/pi,T); figure plot(t,T);
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Appendix D: MATLAB Codes of Chapter 4
Appendix D.1. The calculation of local stress tensor, pressure and the coefficient of internal friction based on the DEM data to calibrate the equivalent properties needed in the continuum model
Inputs: Components of the normal and tangential forces applied to the steel particles in a
measurement location in a DEM simulation and the instantaneous velocities of particles
Outputs: The local stress tensor, local pressure and coefficient of internal friction in the
measurement location
% Multi layer (3D); No angle can be defined (unit vector instead)
[n,m]=size(V); % V is the velocity matrix (6 columns); 1st and 2nd and 3rd
columns: relative normal x-velocity, y-velocity and z-velocity, 4th, 5th and
6th columns: element A x-velocity, y-velocity and z-velocity.
d=0.0063; % diameter
% F is the average collision force matrix (6 columns) (over a single
collision duration); first and second, third columns: x, y and z components
of the normal force, forth, fifth and sixth columns: x, y and z components of
the tangential force.
vn=zeros(n,1); % absolute normal collision velocity (element A) ??? vt=zeros(n,1); % absolute tangential collision velocity (element A) ??? f=zeros(n,1); fn=zeros(n,1);
for i=1:n a=((V(i,1))^2+(V(i,2))^2+(V(i,3))^2)^0.5; if a>0.001 n1(i,1)=V(i,1)/a; n2(i,1)=V(i,2)/a; n3(i,1)=V(i,3)/a; vn(i,1)=V(i,4)*n1(i,1)+V(i,5)*n2(i,1)+V(i,6)*n3(i,1);
f(i,1)=sqrt((F(i,1))^2+(F(i,2))^2+(F(i,3))^2+(F(i,4))^2+(F(i,5))^2+(F(i,6))^2
); fn(i,1)=sqrt((F(i,1))^2+(F(i,2))^2+(F(i,3))^2); else V(i,4)=0; V(i,5)=0; V(i,6)=0; F(i,1)=0; F(i,2)=0; F(i,3)=0; F(i,4)=0; F(i,5)=0; F(i,6)=0; end
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end
% vn1=unique(vn); % %vt1=unique(vt); % % vn2=mean(abs(vn1)); % %vt2=mean(abs(vt1)); % % vn2s=std(abs(vn1)); % %vt2s=std(abs(vt1)); % % f1=unique(f); % f2=mean(f1); % fn1=unique(fn); % fn2=mean(fn1);
% added to the previous file
% for i=1:n % a=((V(i,1))^2+(V(i,2))^2+(V(i,3))^2)^0.5; % should the above filtered
velocities be used instead of these velocities ????????????? % n1(i,1)=V(i,1)/a; n2(i,1)=V(i,2)/a; n3(i,1)=V(i,3)/a; % end
k=0; Vol=0.02^2*0.009; % 0.009 m Volume of the measurement bin Tc=zeros(3,3); % collisional stress tensor TS=953; %2*47*10; % number of time steps (snap shots)
for i=1:n if V(i,4)~=0 || V(i,5)~=0 || V(i,6)~=0 k=k+1; Vx(k,1)=V(i,4); Vy(k,1)=V(i,5); Vz(k,1)=V(i,6); end end
vx2=mean(abs(Vx)); vy2=mean(abs(Vy)); vz2=mean(abs(Vz));
vx2s=std(abs(Vx)); vy2s=std(abs(Vy)); vz2s=std(abs(Vz));
% j=0; % for i=1:n % if F(i,1)~=0 || F(i,2)~=0 || F(i,3)~=0 || F(i,4)~=0 || F(i,5)~=0 ||
F(i,6)~=0 % j=j+1; % Fnx(j,1)=F(i,1); % Fny(j,1)=F(i,2); % Fnz(j,1)=F(i,3); % Ftx(j,1)=F(i,4);
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% Fty(j,1)=F(i,5); % Ftz(j,1)=F(i,6); % end % end
% considering all of the contact forces for i=1:n Fx(i,1)=F(i,1)+F(i,4); Fy(i,1)=F(i,2)+F(i,5); Fz(i,1)=F(i,3)+F(i,6); dX(i,1)=d/2*n1(i,1); dY(i,1)=d/2*n2(i,1); dZ(i,1)=d/2*n3(i,1);
Tc(1,1)=Tc(1,1)+Fx(i,1)*dX(i,1)/(Vol*TS); Tc(1,2)=Tc(1,2)+Fx(i,1)*dY(i,1)/(Vol*TS); Tc(1,3)=Tc(1,3)+Fx(i,1)*dZ(i,1)/(Vol*TS);
Tc(2,1)=Tc(2,1)+Fy(i,1)*dX(i,1)/(Vol*TS); Tc(2,2)=Tc(2,2)+Fy(i,1)*dY(i,1)/(Vol*TS); Tc(2,3)=Tc(2,3)+Fy(i,1)*dZ(i,1)/(Vol*TS);
Tc(3,1)=Tc(3,1)+Fz(i,1)*dX(i,1)/(Vol*TS); Tc(3,2)=Tc(3,2)+Fz(i,1)*dY(i,1)/(Vol*TS); Tc(3,3)=Tc(3,3)+Fz(i,1)*dZ(i,1)/(Vol*TS); end
Pc=-trace(Tc)/3; Tcs=(Tc+Tc')/2; % symmetric part of the stress tensor OK E=eig(Tcs); mu1=abs((max(E)-min(E))/(max(E)+min(E))); % effective coefficient of friction Tc0s=Tcs+Pc*eye(3); %T0=sqrt(((Tc0(1,1))^2+(Tc0(1,2))^2+(Tc0(1,3))^2+(Tc0(2,1))^2+(Tc0(2,2))^2+(T
c0(2,3))^2+(Tc0(3,1))^2+(Tc0(3,2))^2+(Tc0(3,3))^2)); T0s=norm(Tc0s,'fro'); % OK mu2=T0s/(Pc*sqrt(2));
Appendix D.2. The calculation of local shear rate based on the DEM data to calibrate the equivalent properties needed in the continuum model
Inputs: Components of the instantaneous velocities of particles in a measurement location and its
neighbor bins as described in Chapter 4 in a DEM simulation
Outputs: The local shear rate in the measurement location
% velocity gradient tensor and the inertial number calculation for multilayer
(3d) simulations % L components are calculated from the change of velocity from a bin to the % adjacent bins divided by the distance between the bins centers.
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% L12, L21, L23, L32 are close to zero (=zero assumption) and the rest 5
components are non zero. Average velocity in 6 bins or 4 bins (second
version) % and 3 directions should be calculated
% The average values should be calculated in each time step for all the % directions (3D)
% L11=(Vxright-Vxleft)/dx; L13=(Vxtop-Vxbottom)/dz; % L33=(Vztop-Vzbottom)/dz; L22=(Vybackward-Vyforward)/dy; % L31=(Vzright-Vzleft)/dx; L12=(Vxbackward-Vxforward)/dy; % L32=(Vzbackward-Vzforward)/dy; L21=(Vyright-Vyleft)/dx; % L23=(Vytop-Vybottom)/dz; dx=dz=8*d=50 dy=18 in mm % last components should be much smaller than others
% think to include the bin H Vx, Vy, Vz to compare the results of the
velocity % gradient for smaller distances i.e. 25 an 9 mm, respectively. average the % velocities per time step? % OR (second version) % L11=(Vxright-Vx)/dx; L13=(Vxtop-Vx)/dz; % L33=(Vztop-Vz)/dz; L22=(Vybackward-Vy)/dy; % L31=(Vzright-Vz)/dx; L12=(Vxbackward-Vx)/dy; % L32=(Vzbackward-Vz)/dy; L21=(Vyright-Vy)/dx; % L23=(Vytop-Vy)/dz; dx=dz=4*d=25 dy=9 in mm % Vx, Vy and Vz are velocity % components in H (previous)
% for each point e.g. H [n,m]=size(V); % has 18 columns; 1-3 left, 4-6 right, 7-9 bottom, 10-12 top,
13-15 forward, 16-18 backward (x,y,z) velocities, n: number of time steps 18
DAQ % OR % 1-3 H, 4-6 right, 7-9 H, 10-12 top, 13-15 H, 16-18 backward (x,y,z) % velocities 12 DAQ % if the first and the second versions give the same results, the second is % preferable as it is less time consuming %dx=.05; dz=0.05; dy=0.018; % in meters dx=.02; dz=0.02; dy=0.009; % in meters L11=zeros(n,1);L12=zeros(n,1);L13=zeros(n,1);L21=zeros(n,1);L22=zeros(n,1);L2
3=zeros(n,1);L31=zeros(n,1);L32=zeros(n,1);L33=zeros(n,1); % velocity
gradient tensor components Lm=zeros(3,3); % average stress gradient tensor % L=zeros(3*n,n); D=zeros(3*n,n); Ddev=zeros(3*n,n); gamma=zeros(n,1); for i=1:n L11(i,1)=(V(i,4)-V(i,1))/dx; % velocity gradient tensor components at
each time step L12(i,1)=(V(i,16)-V(i,13))/dy; L13(i,1)=(V(i,10)-V(i,7))/dz; L21(i,1)=(V(i,5)-V(i,2))/dx; L22(i,1)=(V(i,17)-V(i,14))/dy; L23(i,1)=(V(i,11)-V(i,8))/dz; L31(i,1)=(V(i,6)-V(i,3))/dx; L32(i,1)=(V(i,18)-V(i,15))/dy; L33(i,1)=(V(i,12)-V(i,9))/dz; Lm(1,1)=Lm(1,1)+L11(i,1)/n; Lm(1,2)=Lm(1,2)+L12(i,1)/n;
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Lm(1,3)=Lm(1,3)+L13(i,1)/n; Lm(2,1)=Lm(2,1)+L21(i,1)/n; Lm(2,2)=Lm(2,2)+L22(i,1)/n; Lm(2,3)=Lm(2,3)+L23(i,1)/n; Lm(3,1)=Lm(3,1)+L31(i,1)/n; Lm(3,2)=Lm(3,2)+L32(i,1)/n; Lm(3,3)=Lm(3,3)+L33(i,1)/n; L=[L11(i,1), L12(i,1), L13(i,1); L21(i,1), L22(i,1), L23(i,1); L31(i,1),
L32(i,1), L33(i,1)]; % velocity gradient tensor at each time step D=(L+L')/2; % symmetric velocity gradient tensor components Ddev=D-trace(D)/3*eye(3); % Deviatoric rate of strain tensor at each time
step
%gamma=sqrt(((Ddev(1,1))^2+2*(Ddev(1,2))^2+(Ddev(2,2))^2+2*(Ddev(1,3))^2+2*(D
dev(2,3))^2+(Ddev(3,3))^2)); gamma(i,1)=sqrt(2)*norm(Ddev,'fro'); % shear rate at each time step %
or *sqrt(2) tr(D)=0 Kamrin's thesis %D0=det(dev(L)) end gammaave=mean(gamma); % average shear rate gamma_std=std(gamma);
Dm=(Lm+Lm')/2; % symmetric velocity gradient tensor components Ddevm=Dm-trace(Dm)/3*eye(3); % Deviatoric rate of strain tensor at each time
ste gammam=sqrt(2)*norm(Ddevm,'fro'); % shear rate at each time step % or
*sqrt(2) tr(D)=0 Kamrin's thesis DmT=trace(Dm); % is it equal to zero? Dilatancy=-DmT/(gammam/sqrt(2)); % Tk should be studied (calculated to br compared to Tc), the previous code
can still be used for Vtot and % GTp
