1. Particle Simulations
Benjamin Glasser
Particle Simulations

2. Overview Physics of a collision Particle Simulations Continuum models
Experimental perspective
Instantaneous collisions
Sustained contacts
Particle Simulations
Hard particle models
Event-driven
Soft particle models
Time stepping
Continuum models
– There are 2 types of contacts: sustained and instantaneous
-Instantaneous contacts are called collisions.
-Continuum models similar to the Navier Stokes equations for momentum transfer are desired for particle modeling.
Particle Simulations

3. Why? To better understand, control, and optimize
Fluidization processes
Fluid bed reactors
Catalyst manufacture
Solids handling operations
Powder mixing
Hopper flow
Geophysical flows
Avalanches
Mudslides
Geophysical formations
Sand dunes
Martian topography
–Systems of particles of all sizes have similar physics, so models can be applied to systems of many length scales. That is why we can assume some physics of nanoparticles are the same as micronsized particles.
-It is desired to ultimately model granular systems in a similar manner as fluid systems.
-It is desired to both predict and understand granular phenomena such as avalanches and mudslides.
Particle Simulations

4. History Leonardo da Vinci (1452-1519) Charles de Coulomb (1736-1806)
first to device a simple and convincing experiment demonstrating dry friction.
Charles de Coulomb ( )
– Coulomb laws of dry friction between solids – would be extended to granular materials.
Michael Faraday ( )
examined how vibrations affect sand piles.
William Rankine ( )
examined friction in granular materials.
H. Jannsen (1880’s)
model of stresses in silos (granular material in a cylinder)
Lord Rayleigh ( )
further work on stresses in containers
Osborne Reynolds ( )
dilatency – expansion of material during shear
Ralph Bagnold (1950’s)
sand dunes, role of particle-particle interactions vs. fluid-particle interactions
Particle Simulations

5. Benefits Ability to see “inside” granular flows Relatively cheap
Permit theoretical investigations
Investigate transitions between fluid-like and solid-like behavior
Safer to run computer simulations
Validate granular experiments
Trace every particle
Part of a force chain?
Versatility to be used for similar systems
Quick answers
Industry pleasing
Manipulate parameters
Coefficient of restitution
Coefficient of friction
Since particles in granular systems are generally opaque, simulations allow you to “see inside” the granular flow to understand its behavior.
- Theoretical investigations such as the calculation of stresses and forces within granular flow systems are possible with simulations.
- Simulations provide the ability to change only one system parameter at a time and examine its affect on the flow system. This is generally not feasible for experimental systems. They allow you to test out theories via manipulation of properties.
Poschel and Scwager. Computational Granular Dynamics. Springer, New York, 2005.
Particle Simulations

6. A Simple Scenario
Two particles approach one another with known initial velocity in a frontal (normal) impact
Before: v1 m1 m2 v2
Conservation of momentum!
After: m1 u1 m2 u2
You begin by the simple scenario of the collision between two particles. It is desired to ultimately break down the entire granular flow into sums of 2-particle collisions.
- The conservation of momentum equation is written, and there are 2 unknowns (exit velocity of each particle). Another equation is required to determine these velocities.
It is very important to assume time of collision is very small and momentum is conserved. The another equation required is the Kinetic Energy Equantion.
2 unknowns (u1 and u2)
Require another equation
W. Goldsmith, Impact: The Theory of Physical Behavior of Colliding Solids, 1960
Particle Simulations

7. A Simple Scenario If kinetic energy were conserved (elastic spheres):
Then:
and
– A conservation of kinetic energy can be written to provide a second correlation in order to determine the exit particle velocities. This is the case for molecules.
-This equation is not valid, though, for granular particles due to inelastic particle collisions.
However, energy is not conserved
 inelastic collisions
Particle Simulations

8. Inelastic Collisions Initial velocity is v, rebound velocity is –ev
Unique to granular materials
Why?
Permanent deformation
Microcracks
Deformed surface
Acoustic Waves
Dissipated through heat
Inelastic collisions are defined when the rebound velocity is less than the initial velocity during a collision. In the example of a ball bouncing, the ball will not return to its original height after an inelastic collision.
-Inelastic collisions can occur due to permanent deformation of particles (such as brass beads getting “dimples” during collisions) or by energy loss through acoustic waves or other modes of energy loss.
Particle Simulations

9. Coefficient of Normal Restitution
ε is the coefficient of normal restitution
Ratio of pre-collisional to post-collisional velocities
Change in Kinetic Energy
Function of approaching velocity
Common values:
Source: Jacques Duran, Sands, Powders, and Grains, Springer, 1999.
The coefficient of normal restitution is equal to 1 for an elastic collision. In the equation shown, u1-u2 is the difference in pre-collision velocities and v1-v2 represents post-collision velocities.
Glass spheres
100 cm/s
Steel spheres
Brass spheres
0.9
Particle Simulations

10. Source: Jacques Duran, Sands, Powders, and Grains, Springer, 1999.
Matrix Equations
Reference: System center of gravity
Particle - Particle
Particle – Wall (infinite mass, rigid body)
Source: Jacques Duran, Sands, Powders, and Grains, Springer, 1999.
Matrix equations can be written in order to solve for the unknown exit velocity of each particle after collision.
-Describing the collision between a particle and a wall is like a particle colliding with another particle of infinite mass.
u0: velocity of wall
u1: particle velocity
Particle Simulations

11. Normal Collision - No Friction
Only translational motion
x and y are components of linear momentum
ux = -evx
uy = vy
w1 = w0 w1 x y
– The simplest particle-wall collision example is when there is no rotation (spin, occurs when w1=0) in the particle velocity, and the wall has a frictionless surface. There is only translational motion.
-The normal vector to the wall is important.
-The change in momentum, X, is important since it can be used to calculate the force of collision. Force can be calculated by dividing momentum, X, by contact time.
Particle Simulations

12. Spin and Friction Spherical, Spinning Ball - Vertical Wall
Precollisional velocities vx vy w0
Compute: ux uy w1 w0 x y w1 x y
The next step to a more rigorous model is to incorporate the effects of spin and friction.
Particle Simulations

13. Normal Collision - Friction
m: coefficient of friction between the ball and the wall
Can distinguish between two cases based on m w1 x y
Case I:
Gliding velocity remains positive and non-zero
Source: Jacques Duran, Sands, Powders, and Grains, Springer, 1999.
Coulomb
–The first case of a normal collision is that in which the particle slips while in contact with the wall, and exits during this slip.
-In this case, friction does not affect the velocity normal to the wall by definition. Also coefficient of restitution has nothing to do with the slipping of the sphere.
-A relationship can be written using Coulomb’s law to relate the tangential and normal forces (tangential force= mu* normal force)
-In this situation, both rotation and slipping against the wall are accounted for and rotation and vertical velocity are equivalent from the walls point of view.
-Rolling occurs if velocity in y-direction is equal to a times mu.
Rotation
True when:
a: radius
4 Equations, 4 Unknowns (X, ux, uy, w1)
Particle Simulations

14. Normal Collision - Friction
Case II:
Gliding velocity drops to 0 during the collision w1 x y
Rotation
- The second case is where gliding velocity drops to zero during the collision and the particle experiences only rotation. Stop all tangential motion and go to rolling.
Pure rotation
True when:
Particle Simulations

15. Non-Frontal Collision with Friction1 2 v2 v1 n w1 w2
Two particles
Diameter d1 and d2
Mass m1 and m2
Unit vector normal to contact
Relative velocity at point of contact
A non-frontal collision is when the collision is not aligned with the previously defined x-y coordinate system. The particle approaches the wall from a different angle. Two vectors exist: Normal and Position.
Contribution from spin
Relative linear velocity
The magnitude of the relative velocity |vc| increases when the individual velocities point in opposite directions and when the rotations are in the same direction
Particle Simulations

16. Tangential Velocity - Rotational Motion1 2 v2 v1 vc g n
Normal component of vc
Tangential component of vc
Tangential unit vector
Angle of impact
From normal vector n to relative velocity vc
Normal collision: g = p
vc,, n
An angle of contact between the particles is defined.
-Each velocity contribution is broken into a normal and tangential component. The tangential component is due to friction.
Glancing collision: g = p/2 vc n
Particle Simulations

17. Momentum Change Linear change of momentum from a collision
Tangential contribution to angular momentum (torque)
Normal component of DP has no contribution
for i=1,2
Ii: moment of inertia of particle i
The change in angular momentum is the same for both particles
Particle Simulations

18. Outcome of the Collision
Making use of the equations, one can compute the outcome of the collision
Linear Velocity
Where: n t P D + =
The translation velocities before and after the collision are still related by this
–Solving the equations defined in the previous slides, a well-defined solution can be determined. This solution, though more complex, is as solvable as the simplest example of a particle hitting a wall.
Angular Velocity
Particle Simulations

19. Rolling
The previous equations describe all collisions on the basis of Coulomb’s Law (m) and e and nothing else
Ignored an important physical mechanism: rolling
Heuristic model to agree with experiments
Fire two spheres together with initial spin and examine the outcomes
Coefficient of tangential restitution, b
Equal to the smallest of two values:
b 0 – Rolling; b 1 – Sliding/Gliding w1 w2
There exists a friction associated with particle rolling. This is in addition to Coulomb’s Law.
-The coefficient of tangential restitution, , is defined. It serves as a fudge factor to quantify rolling, sliding, and gliding. Although we only have one parameter it can be separated into two o and 1.
Particle Simulations

20. Small values of g correspond to dry friction (previous result)
Critical Angle – g0
Applies for this form of the momentum equation
Applies for this form of the momentum equation
For g > g0:
Rolling regime
The critical angle, , is used to split the tangential restitution into rolling (o) and sliding (1) components.
- For an obtuse critical angle, you would expect sliding of the particles.
For g < g0:
Sliding/Gliding regime
Small values of g correspond to dry friction (previous result)
S.F. Forrester et al., Phys. Fluids, 6, p.1108 (1994)
Particle Simulations

21. Sustained Contacts Consider 2 identical spheres Mass, M Radius, R
Source: Jacques Duran, Sands, Powders, and Grains, Springer, 1999.
What about particles that are not completely rigid?
Spheres can deform
Can have interpenetration of the spheres, resulting in long contact times R R
Consider 2 identical spheres
Mass, M
Radius, R
relative velocity v x
Hertz’s elastic energy
- For particles that are not completely rigid, sustained contact modeling is used.
-It is assumed that there is a small amount of deformation, . It is up to us to decide when they deform. Metals may have a smaller scale of deformation.
-Elastic energy is stored during this deformation, and is given by Hertz’s elastic energy.
Stored by each sphere during contact
Fan and Zhu, Principles of Gas-Solid Flow, 1998
where
E – Young’s modulus
s – Poison’s ratio
Ratio of transverse strain to longitudinal strain
Particle Simulations

22. Sustained Contact Quantities
Upon impact the kinetic energy is converted into a reduced kinetic energy and stored elastic energy
Energy balance:
Velocity drops to zero when the two spheres have overlapped by a distance x0
The entire duration of the collision (up to x0 and recoil)
During collision Kinetic Energy (KE) gets converted to reduced KE and stored KE. By writing an energy balance, the duration of collision can be determined in terms of known and measurable quantities.
Important: t only depends weakly on v – exponent is 1/5. Thus the duration of impact is only a weak function of the initial relative velocity
Particle Simulations

23. Example Calculation Answer:
Consider aluminum beads (r = 2.70 g/cc) 1.5 mm in diameter moving towards one another at 5 cm/s.
E = 6 x 1011 dynes/cm2
s = 0.3
What is the duration of contact?
D=1.5 mm
Answer:
k = 7 x 1010 dynes/cm3/2
m = 4.77 x 10-3 g
t = 1.15 x 10-5 s
This example calculation of a collision between aluminum beads gives a reasonable approximation of contact time seen experimentally, at least from an engineering point of view. Therefore, this method is good enough to develop models.
Particle Simulations

24. Issues Exceed elasticity limit Energy dissipation Spin complications
Plastic, not elastic deformations
Model can be adjusted to handle this
Energy dissipation
Sound waves
Heat
Spin complications
Can handle similar to rigid spheres
Other issues are that cohesion and electrostatics have been neglected in this model.
Particle Simulations

25. Source: Jacques Duran, Sands, Powders, and Grains, Springer, 1999.
Particle Simulations
Follow trajectories of individual particles
Incorporate statics and dynamics
Methods
Particle dynamics
Hard Particles
Soft Particles
Cellular automata
Motion evolves according to simple rules based on lattice sites
Monte-Carlo
Analogous to molecules but change probabilities to match particles
Assumption of molecular chaos
Source: Jacques Duran, Sands, Powders, and Grains, Springer, 1999.
Source: Thorsten Poschel, Thomas Schwager, Computational Granular Dynamics, Springer, 2005.
Baxter and Behringer (1990) Cellular Automata of Granular Flows. Phys. Review A., 42,
Following the trajectory of individual particles is called discrete modeling. Cellular automata has been used for a hopper since a particle can move depending on the previous state of the particle.
Rosato et al. (1987) Monte Carlo simulation of particulate matter segregation. Powder Technology, 49, 59-69
Particle Simulations

26. Boundary Conditions Wall constructed from individual particles
Containers do not follow Newton’s equation of motion
Predetermined path as a function of time
Vibrated bed
Moving plate
Rotating vessel
Periodic boundaries
Can mimic infinitely-wide regions
Particle Simulations

27. Initial Conditions
Depending on the algorithm (predictor-corrector), you may need to define higher order derivatives
Most long-term behavior is independent of initial conditions
Often, random (non-overlapping) positions and velocities are assigned
Particle Simulations

28. Hard Particles Event Driven (ED) Strictly binary collisions
No integration of the equations of motion
More efficient
With gravity
Without gravity
Collision at (xc, yc + 1/2gt2)
Collision at (xc, yc)
v10
v20
For collisions with and without gravity, the time to collision is equal. Gravity does not have an impact on the calculation since it is implemented on all particles.
Without gravity – straight line paths
With gravity – parabolas
Time to collision is identical
Particle Simulations

29. Particle Motion Consider a particle Position vector xiy
xi0 = (x, y)
vi0 = (vx, vy)
Consider a particle
Position vector xi
Velocity vector vi
Initial position (t=0)
xi = xi0
vi = vi0
Undeterred position at time t di
Particle Simulations

30. Collision Prediction Two particles collide if:j
dj/2
di/2
Insert the equation of motion:
(same force on each particle, so the accelerations are equal as well and cancel)
-You can determine if and when two particles will collide in a granular flow system. Combine the equation of motion (force balance) with the distance required for collision as shown in the 2nd equation.
-If there exists a positive, real root- the particles will collide.
-When a collision occurs the norm of the position differences equal to the sum of the particles radii.
-If you know velocity and position, you will determine if they collide however, the probability of collision changes at every particle collision since collisions change trajectories.
Expand : Real root > 0 collision
Particle Simulations

31. Scheduling Can schedule the events for N particles in a box
4 walls of a box  4N events
N particles  N(N-1) events
An example stack from t=0
March to t = 0.1, execute the collision, then recalculate the stack
The whole stack, or
Just events with particles 5 or 7 (much faster)
time
wall event
particle event
type
0.1 no
yes
particle 5 and 7
0.25
particle 3 and wall 1
0.33
particle 2 and 5 …
-An Event is considered a Collision and every Collision changes the probability another particle will collide with another.
Particle Simulations

32. Gridding Predicting collisions between all particles wastes time
Black arrows will rarely collide
Divide region into cells
Search within cells for collisions
Include cell crossings as events
Track the cell location of each particle
-In order to reduce computational expense. A simulation divides the entire space into cells. At each collision, the probability of one particle hitting another is calculated. In order to reduce the calculation of a particle hitting another far away particle, the space is gridded to only allow close particles to collide, that is only particles that exist in the nearest neighboring cells.
Particle Simulations

33. Inelastic collapse
Left and right particle collide with the middle particles alternately until the motion of all three particles is zero
Poschel and Scwager. Computational Granular Dynamics. Springer, New York, 2005.
Only for a constant coefficient of restitution
Not valid for real materials
Experiments suggest e is a function of v
Particle Simulations

34. 2D - Couette Flow U -U Particle Simulations
This is a shear flow example, where both walls are moving in opposite directions. The figure illustrates a “rope like” structure, where particles connect with each other. They group together in clusters due to lose of energy in the tangential direction.
Particle Simulations

35. 3D - Couette Flow Particle Simulations
Examples of high energy systems are fluidized beds, vibration and shear flows. These are cases were no particle sits on another particle.
Particle Simulations

36. Soft Particles Force-based Time stepping Small overlap allowed
Useful for
Statics
Dense quasi-static flows
To follow particles:
F is the normal force on particles
 proportional to amount of overlap x
Soft particle simulations are typically for cases where particles are sitting on particles. An example of this is a hopper flow. Force-based models allow for overlap.
Particle Simulations

37. Advanced Algorithms Acceleration from force Verlet Algorithm?
program termination
data output
initialization
predictor
force computation
corrector
Verlet Algorithm
Position from acceleration
Back out velocity
An advanced algorithm may determine the trajectory by solving Eulers’ equations or via a predictor or corrector.
Predictor – Corrector (left)
Predict future acceleration using previous position time derivatives
Acceleration from force
Adjust the predicted value
Poschel (2005) Computational Granular Dynamics. Springer, New York.
Particle Simulations

38. Contact Models Mutual compression of particles i and j
Spring and Dashpot
e may be related to kn and gn
Poschel (2005) Computational Granular Dynamics. Springer, New York.
Mutual compression of particles i and j
Schafer, Dippel & Wolf article
Particle Simulations

39. Force Calculation In addition to FnFw Fg F2 F3
In addition to Fn
Gravity
Wall forces
Interstitial fluid
Spin
Cohesion
Total of all is the resultant force on the particle, F
The overall force is determined by summing all the force variables between the three particles and wall.
Particle Simulations

40. More Contact Models Hertz Kuwabara and Kono Walton and Braun (right)
Poschel (2005) Computational Granular Dynamics. Springer, New York.
Walton and Braun (right)
More complicated models accounting for angled forces.
Can also include friction in the mechanism
Particle Simulations

41. Collections of spheres
Discrete Models
Shape Issues:
Spheres
Collections of spheres
Arbitrary surfaces
Simplest
Most Complex
Needles
Flakes
Modeling particles is simplest assuming they are spheres and gets much more complicated when modeling shapes which are not spherical. One possible way to model flakes and needles is by modeling them as a agglomeration of spheres.
Particle Simulations

42. Results: Constant inter-particle force Model
Optimized Condition: 20,000 particles, 2mm diameter, in an axially smaller rotating drum of 9 cm radius and 1 cm length are considered. The sidewalls are made frictionless to avoid end wall effects.
Inter-particle or particle-wall cohesive force is varied such that K=45-75
Avalanches start appearing at K = 30 and become bigger at K =45.
Distinct angles of repose are visible at top and bottom of “cascade” layer.
Fast-Flo Lactose
Size: 100 micron
RPM = 7
This is a rolling horizontal drum. The simulation is a Discrete Element Model. Experimentally the particles are not spheres but computationally they are. The angle of repose seems to be in good agreement accounting for the avalanching.
K = 45 ; RPM = 20
sp = 0.8 ; dp = 0.1 ;
sw = 0.5 dw = 0.5 ;
Credit: F. Muzzio
Particle Simulations

43. Results: Constant inter-particle force Model
Dynamic friction within the particles and the cohesion are increased to simulate the flow of more cohesive material. Wall friction is increased.
K = 60 ; RPM = 20
sp = 0.8 ; dp = 0.6 ;
sw = 0.8 ; dw = 0.8
K = 75 ; RPM = 20
sp = 0.8 ; dp = 0.6 ;
sw = 0.8 ; dw = 0.8
Avicel-101; Size : 50 mm; RPM = 7
Reg. Lactose; Size : 60 mm; RPM = 7
Applying cohesion to the experimental and computational simulation is facilitated. Experimental particle size is decreased computationally, and parameters are changed such as the friction between particles and particle-wall. Both show that the angle of repose increase as cohesion does.
Similarities:
Increase in size of avalanches.
Mixture of splashing and timid bulldozing.
Similarities:
Mixture of chugging and bulldozing.
Periodic Avalanches
Bigger Avalanches
Credit: F. Muzzio
Particle Simulations

44. Uniform Binary System Comparison of model and experiment
Optimized Condition: 10,000 red and 10,000 green particles of same size (radius: 1mm) are loaded side by side along the axis of the drum. The drum of radius 9 cm and length of 1 cm is considered. The sidewalls are made frictionless to avoid wall effects.
Inter-particle or particle-wall cohesive force is varied such that K=0 – 120
Glass beads (40 mm)
RPM=10
K=0
RPM = 20
Colored Avicel (50 mm)
RPM=10
K=60
RPM = 20
The computational-time, though, of the simulation is not the same as the experimental real time. The patterns that form are perhaps illustrating that although particles such as Avicel are not spherical they do not need to be modeled as non-spheres.
No avalanches.
Mixes well in 3-4 revolutions
Avalanches appearing
Slower mixing
Credit: F. Muzzio
Particle Simulations

45. Non-uniform Binary System
Experiment: Blue (30 mm) and Red glass beads (50 mm) of equal mass are axially loaded side by side in a drum of radius = 7.5 cm and length 30 cm.
Simulation: 8000 blue particles (1mm) and 2370 red particles(1.5mm) of same density are loaded side by side along the axis of the drum. Red and blue particles of are of the same total mass.
RPM =12
RPM =20
Inter-particle Force Model
FRR = KRR WB (red-red pair)
FRB = KRB WB (red-blue pair)
FBB = KBB WB (blue-blue pair)
WB is the weight of a blue particle.
When particle size changes, particles of different sizes segregate. This is due to size, as voids are created and particles smaller than this void enter this space where larger particles cannot.
Non-cohesive binary mixture : Axial Size Segregation is evident in both the simulation and experiments.
(K.M.Hill et.al Phys. Rev. E.,49,1994).
Credit: F. Muzzio
Particle Simulations

46. Example –A Particle/Wall Contact
2D Disk - Flat, Vertical Wall
R = 0.5 mm
kn = 10 N mm-3/2
Dt = 0.25 s
m = kg
vp0 = (vx0, vy0) = (1 mm s-1, 0 mm s-1)
xp0 = (xx0, xy0) = (1.2 mm, 1 mm)
xw = 0 (line from origin to +∞)
Here is an example of a particle contacting a wall and bouncing back from the wall. x y x y
Particle Simulations

47. Results Wall Collision!! Particle Simulations
The simple physics are illustrated. This intuitively makes sense as the particle is moving its velocity does not change until it hits the wall, when the collision occurs an overlap occurs and the direction of movement changes in the opposite direction.
Particle Simulations

48. Hard vs. Soft When to use hard (event) instead of soft (force)
Average collision duration << Time between collisions
Granular gases  cosmic dust clouds
Unknown interaction force
Non-linear materials
Complicated particle shapes
Can experimentally determine pre- and post-collisional velocities
Poschel (2005) Computational Granular Dynamics. Springer, New York.
Particle Simulations

49. Drawbacks Computationally expensive Dynamic issues
Dt << t to calculate forces
20,000 particles
Real time of seconds to minutes
Dynamic issues
Strain hardening
Contact erosion over time
Drawbacks of these simulations is their computational expense. Two parameters must be sacrificed, the real-time of the simulation and the number of particles inside of the simulation.
Particle Simulations

50. Continuum
Discrete particles replaced (averaged out) with continuous medium
Quantities such as velocity and density are assumed to be smooth functions of position and time
Volume element (dv) contains multiple particles
Time (dt) should be large compared to time required for a particle to cross dv
For Continuum models a large enough length scale averages out particles looking at velocity of groups rather than individual particles. This is only viable if there exists a large enough group of particles.
Truesdall, C. and Muncaster, R.G. (1980) – Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatonic Gas. Academic Press, pp. xvi.
Particle Simulations

51. Continuum
Stay in this region where the average quantities are equal to the bulk
10-10
10-6
10-5
10-4
10-3
10-2
10-1
10-7
10-8
10-9
r = mass/volume
l (in meters)
Particle Simulations

52. Continuum Simulation for Granular Flows
Average over a small region in a granular flow.
Consider a mass balance over a stationary volume element of size ΔxΔyΔz.
Rate of mass accumulation = rate of mass in – rate of mass out Δx Δy Δz
Granular Flow
Dividing by ΔxΔyΔz and taking limits, we obtain or
An area contacting particles is made up of particles and of empty regions. Therefore the density of the volume is proportional to the volume fraction and the density of the particles. Getting an equation gives us how mass varied with time. If mass changed more particles came in than came out, or vice versa.
We can extend this approach to calculate momentum and energy balance.
Particle Dynamic Simulations – Limited by computational resources
Continuum Simulations with physically realistic closures
(Use a set of equations for design, control and optimization)
More efficient design, scaling & control through Continuum modeling
Particle Simulations

53. Pseudo-Thermal Energy Balance
Hydrodynamic Model
Mass Balance
u solids velocity
v solids fraction
T granular temperature
Momentum Balance
Pseudo-Thermal Energy Balance
Do momentum balance to get velocity. The equation can be expected to be non-Newtonian. Faster particles that bounce are easier to move than stationary particles. This can be solved using a pseudo-thermal energy balance (which describes how fast they are moving compared to their neighbors). The granular temperature is the flucuating particle velocity not the real temperature.
Boundary Conditions (Johnson and Jackson, 1987)
Force Balance
Energy Balance
Particle Simulations

54. Computational Approaches
Steady State Simulations
Linear Stability Analysis
Steady state solution is augmented by a small perturbation.
The small perturbation has a periodic form.
time
s = - i  determines the rate of growth or decay of the perturbation waves.
Transient Integration on Linear Instabilities
( Anderson et al. 1995)
Direct Integration
Bifurcation
Analysis
Bubble formation in a gas-particle system.
Particle Simulations

55. Couette and Channel Geometry
Channel Flow u0
Couette Flow
Particle Simulations

56. Steady State Solutions for Couette Flows
The structure of fully developed solutions (a) Particle volume fraction, (b) pseudo-thermal temperature, (c) axial velocity.
Broken lines: wall is a source of energy.
Solid lines: wall is a sink of energy.
(Nott et al. 1999)
Particle Simulations

57. Linear Stability for Couette Flows
Eigenfunctions Associated with Solids Fractions
Instability dominated by symmetric patterns
Instability dominated by anti-symmetric patterns
(Alam and Nott, 1998)
Particle Simulations

58. Transient Integration in Couette Flows
Skipped
(Wang and
Tong
1998)
Particle Simulations

59. Couette Flow with Binary Particles (Steady State)
A: large/heavy particles
B: small/light particles
(R=dA/dB M=mA/mB) u0 y x
Non-uniform Solids
Distribution
Species Segregation
Non-linear Velocity
This continuum model can consider particles of different size and tell us if segregation depends on particle size. This model assumes free flowing particles with no collisions.
-u0
(Equal Density: R=2, ep=0.9
Mean solids fraction=0.1)
Particle Simulations

60. Steady State Solutions for Channel Flows
u* dimensionless velocity
T* dimensionless granular temperature
v solids fraction
Solid line: wall is a sink of energy.
Broken line: wall is a source of energy.
(Wang et al. 1997)
Particle Simulations

61. Linear Stability for Channel Flows
Eigenfunctions Associated with Solids Fractions
Instability dominated by symmetric patterns
Instability dominated by anti-symmetric patterns
(Wang et al. 1997)
Particle Simulations

62. Transient Integration in Channel Flows
Solids Fraction Distribution in the Channel
Transient Integration for symmetric patterns
Transient Integration for anti-symmetric patterns
(Wang and Tong 2001)
Particle Simulations

63. Channel Flow with Binary Particles (Steady State)
A: large/heavy particles
B: small/light particles
(R=dA/dB M=mA/mB) y x
(Equal Density: R=2, ep=0.9
Mean solids fraction=0.1)
Particle Simulations