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Computer Simulation of Granular Matter:
A Study of An Industrial Grinding Mill
By John Drozd
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Outline• Introduction and motivation.• Event driven algorithm and formulae.• Crushing forces.• Discussion and analysis
– Verification– Parameters– Steady state– Mean square displacement– Circulation
• Conclusions and recommendations.
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Granular Matter• Granular matter definition
– Small discrete particles vs. continuum• Granular matter interest
– Biology, engineering, geology, material science, physics.– Mathematics and computer science.
• Granular motion– Energy input and dissipation.
• Granular experiments– Vibration
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Small AmplitudeSurface Waves
3-Node Arching
Large AmplitudeSurface Waves
C. Wassgren et al. 1996
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Other Phenomena in Granular Materials
• Shear flow• Vertical shaking• Horizontal shaking• Conical hopper• Rotating drum• Cylindrical pan
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Cylindrical Pan Oscillations
Oleh Baran et al. 2001
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HarrySwinneyet al. 1997
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Harry Swinney et al. 1997
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Raw Feed
Grinding Media, Rods
VibratorMotor
GroundProduct
Vibratory Drum Grinder
IsolationSpring
Reactor Springs
Vibratory Drum Grinder
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Goal
• Find optimum oscillation that results in a force between the rods which achieves the ultimate stress of a particular medium that is to be crushed between the rods.
• Minimize the total energy required to grind the medium.
• Mixing is also important.
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Typical Simulation
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Event-driven Simulation Without Gravity
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Event-driven Simulation With Gravity
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Lubachevsky Algorithm
• Time values – Sum collision times
• Heap data structure– Disk with smallest time value kept at top
• Sectoring– Time complexity O (log n) vs O (n)– Buffer zones
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Rod 1 collides with Rod 2 with a collision time = 58
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Rod 2 collides with Rod 3 with a collision time = 124
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Rod-Rod Collisions
• Treat as smooth disk collisions• Calculate Newtonian trajectories• Calculate contact times
• Adjust velocities
02 2222 ijijijijij rtbtv
||2/ ijijijji vrbvv
ijijij vrb
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A Smooth Disk Collision
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Coefficient of Restitution• e is calculated as a velocity-dependent restitution
coefficient to reduce overlap occurrences as justified by experiments and defined below
• Here vn is the component of relative velocity along the line joining the disk centers, B = (1)v0
, = 0.7, v0=g and varying between 0 and 1 is a tunable parameter for the simulation.
0
0
,,1
vvvvvB
ven
nnn
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Rod-Container Collisions 2
222
21 rRytgtvytvx yx
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Typical Simulation
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Circulation• The net circulation of the whole system was
calculated by first calculating the angular velocity of the vortex about the center of mass of the system and then using the formula
• where is the angular velocity of rotation or vorticity of the system and rmax is the distance from the farthest disk to the center of mass of the system.
2max22 rdAdA
2 V
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Circulation• The angular velocity of the vortex was calculated using the
formula i
ii
i
akvvaN
ˆˆ11
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Typical Time Averaged Velocity Field
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Net Circulation () vs Time (t)
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Measuring Disk Disk Forces as Collision Energies
• For a disk-disk collision, the collision energy can be calculated as
• where , and is the relative normal velocity between the disks before a collision.
2,2
1relneffc vmE
diskeff mm21
relnv ,
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Measuring Disk Container Forces as Collision Energies
• For a disk-container collision, the collision energy can be calculated as
• where and is the dot product of the velocity of the disk before the collision and a unit vector of the container surface normal , that is calculated as
•
2,2
1relneffc vmE
diskeff mm relnv ,
n̂
22,yx
tdydvyxv
vyx
reln
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Measuring Forces as Collision Energies• These collision energies can be compared to the modulus
of toughness of the material that is to be crushed between the disks.
• The modulus of toughness is defined as a strain-energy density, u, taken to the strain at rupture R using the formula
R
o xx du
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Parameters for Simulation
• The program was run using the parameters (g, , , y, Ay, e0, eW) and a simulation that produced a realistic motion was selected (y=126 rad/s= 20 Hz, Ay=1.5 cm, e0=0.4, eW=1.0).
• By a realistic motion, we mean that the disks would cluster together at the bottom of the container within a relatively short period of time with few overlaps.
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Total Kinetic Energy (KE/M) vs Time (t)
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Power Spectrum of Total Kinetic Energy ( P() )
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Mean Square Displacement Plots: Mixing Times
• Fixed time origin t0 formula:
• Moving time origin t‘ formula:
N
iiitt
tytrtytrN
r1
002 1
0
t
N
iii
ttt
tytrtytrNN
r1
'
2 ''11
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Mean Square Displacement (< r2 >) versus time (t)
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Phase Diagram: Amplitude (A) vs Frequency (y)
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Time Averaged Net Circulation () vs Frequency (y)
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Non-dimensional Parameter (/(Ay2 y)) vs Frequency (y)
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Circulation at Sloped Part of Curve
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Circulation at Leveling Off Part of Curve
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Velocity Field Snapshot for Typical Simulation
Axis of Symmetry
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0, 2
/ 2
3/2
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Number of Different Disks that Experience Mean Collision Energy (ni) i times vs Time (t)
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Conclusion
By quantifying the crushing forces in terms of collision energies and studying circulation and mixing, this thesis has outlined a thorough systematic approach to studying the grinding mill industrial crushing problem.
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Next Steps• Finer test matrices and calibrating results with
experiments• Compare the percentages of the medium that was
crushed and compare to the n1, n2, … and saturation times for various simulations using different amplitudes and frequencies of oscillation.
• Determine the frequency and amplitude for an optimum oscillation, that is, to minimize the energy required to get a well-mixed container with all the rods experiencing the mean (threshold) collision energy. This is the solution to the task outlined in this thesis.
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Recommendations for Future Work
• Incorporating horizontal container oscillations.• Incorporating rod rotations.• Using a mixture of different sized rods.• Using different quantities of rods and different
container sizes.• Study situations with different rod-wall boundary
conditions: multiple vortices.• Parallelization and sectoring using many numbers
of rods.
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Acknowledgements
• Brad Smith and Kirk Bevan• Dr. Oleh Baran• Dr. Ivan Saika-Voivod• Dr. Sreeram Valluri• Drs. Peter Poole and Robert Martinuzzi• NSERC