Particle Simulations

1. Particle Simulations Benjamin Glasser Particle Simulations

2. Overview • Physics of a collision • Experimental perspective • Instantaneous collisions • Sustained contacts • Particle Simulations • Hard particle models • Event-driven • Soft particle models • Time stepping • Continuum models http://www.ux1.eiu.edu/~cfadd/1150/07Mom/Images/NewtPend.gif http://www.ux1.eiu.edu/~cfadd/1150/07Mom/Images/boing.gif 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 http://www.rrdc.com/images/ph_peru_rockslide_lrg.jpg http://photojournal.jpl.nasa.gov/jpegMod/PIA02405_modest.jpg Particle Simulations

4. History • Leonardo da Vinci (1452-1519) • first to device a simple and convincing experiment demonstrating dry friction. • Charles de Coulomb (1736-1806) – Coulomb laws of dry friction between solids – would be extended to granular materials. • Michael Faraday (1791-1872) • examined how vibrations affect sand piles. • William Rankine (1820-1872) • examined friction in granular materials. • H. Jannsen (1880’s) • model of stresses in silos (granular material in a cylinder) • Lord Rayleigh (1842-1919) • further work on stresses in containers • Osborne Reynolds (1842-1912) • 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 http://www.nature.com/nature/journal/v435/n7045/images/4351041a-f1.2.jpg Poschel and Scwager. Computational Granular Dynamics. Springer, New York, 2005. Particle Simulations

6. m1 m2 v2 m1 m2 v1 u2 u1 A Simple Scenario Before: Two particles approach one another with known initial velocity in a frontal (normal) impact Conservation of momentum! After: 2 unknowns (u1 and u2) Require another equation W. Goldsmith, Impact: The Theory of Physical Behavior of Colliding Solids, 1960 Particle Simulations

7. and A Simple Scenario If kinetic energy were conserved (elastic spheres): Then: 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 http://www.mathworks.com/access/helpdesk/help/techdoc/math/ballode.gif 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. Particle Simulations

10. Matrix Equations • Reference: System center of gravity • Particle - Particle • Particle – Wall (infinite mass, rigid body) Source: Jacques Duran, Sands, Powders, and Grains, Springer, 1999. u0: velocity of wall u1: particle velocity Particle Simulations

11. w1 y x Normal Collision - No Friction • Only translational motion • x and y are components of linear momentum • ux = -evx • uy = vy • w1 = w0 Particle Simulations

12. w0 w1 y y x x Spin and Friction • Spherical, Spinning Ball - Vertical Wall • Precollisional velocities • vx • vy • w0 • Compute: • ux • uy • w1 Particle Simulations

13. w1 • True when: y x Normal Collision - Friction m: coefficient of friction between the ball and the wall Can distinguish between two cases based on m Case I: • Gliding velocity remains positive and non-zero Source: Jacques Duran, Sands, Powders, and Grains, Springer, 1999. Coulomb Rotation a: radius 4 Equations, 4 Unknowns (X, ux, uy, w1) Particle Simulations

14. w1 True when: y x Normal Collision - Friction Case II: • Gliding velocity drops to 0 during the collision Rotation Pure rotation Particle Simulations

15. w2 w1 2 n 1 Relative linear velocity Contribution from spin v2 v1 Non-Frontal Collision with Friction • Two particles • Diameter d1and d2 • Mass m1and m2 • Unit vector normal to contact • Relative velocity at point of contact 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. 2 n 1 g vc vc,, n n vc v2 v1 Tangential Velocity - Rotational Motion • 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 Glancing collision: g = p/2 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. Angular Velocity D = D + D t n P P P Outcome of the Collision Making use of the equations, one can compute the outcome of the collision Linear Velocity Where: The translation velocities before and after the collision are still related by this Particle Simulations

19. w1 w2 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: • b0 – Rolling; b1 – Sliding/Gliding Particle Simulations

20. Critical Angle – g0 Applies for this form of the momentum equation Applies for this form of the momentum equation For g > g0: Rolling regime 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. x R where E – Young’s modulus s – Poison’s ratio R Sustained Contacts 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 Consider 2 identical spheres • Mass, M • Radius, R relative velocity v Hertz’s elastic energy Stored by each sphere during contact Fan and Zhu, Principles of Gas-Solid Flow, 1998 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) 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. Answer: k = 7 x 1010 dynes/cm3/2 m = 4.77 x 10-3 g t = 1.15 x 10-5 s Example Calculation 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 Particle Simulations

24. Issues • Exceed elasticity limit • Plastic, not elastic deformations • Model can be adjusted to handle this • Energy dissipation • Sound waves • Heat • Spin complications • Can handle similar to rigid spheres Particle Simulations

25. 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, 1017-1020 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. With gravity Without gravity v10 v20 Collision at (xc, yc) Collision at (xc, yc + 1/2gt2) Hard Particles • Event Driven (ED) • Strictly binary collisions • No integration of the equations of motion • More efficient • Without gravity – straight line paths • With gravity – parabolas • Time to collision is identical Particle Simulations

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

30. j dj/2 di/2 i Expand : Real root > 0 collision Collision Prediction Two particles collide if: Insert the equation of motion: (same force on each particle, so the accelerations are equal as well and cancel) 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) Particle Simulations

32. Divide region into cells • Search within cells for collisions • Include cell crossings as events • Track the cell location of each particle Gridding Predicting collisions between all particles wastes time • Black arrows will rarely collide 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. U -U 2D - Couette Flow Particle Simulations

35. 3D - Couette Flow Particle Simulations

36. x 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 Particle Simulations

37. initialization predictor force computation corrector data output ? program termination Advanced Algorithms • Acceleration from force • Verlet Algorithm • Position from acceleration • Back out velocity • 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 • 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 Particle Simulations

39. F2 Fw F3 Fg Force Calculation • In addition to Fn • Gravity • Wall forces • Interstitial fluid • Spin • Cohesion Total of all is the resultant force on the particle, F Particle Simulations

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

41. Simplest Spheres Collections of spheres Arbitrary surfaces Needles Most Complex Flakes Discrete Models Shape Issues: 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 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 Similarities: • Mixture of chugging and bulldozing. • Periodic Avalanches • Bigger Avalanches • Similarities: • Increase in size of avalanches. • Mixture of splashing and timid bulldozing. Credit: F. Muzzio Particle Simulations

44. Uniform Binary SystemComparison 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 Colored Avicel (50 mm) RPM=10 K=60 RPM = 20 K=0 RPM = 20 • Avalanches appearing • Slower mixing • No avalanches. • Mixes well in 3-4 revolutions 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 = KRRWB(red-redpair) FRB = KRBWB(red-bluepair) FBB = KBBWB(blue-bluepair) WBis the weight of a blue particle. 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. y y x x 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 = 0.148 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 +∞) Particle Simulations

47. Wall Collision!! Results 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 • Dt << t to calculate forces • 20,000 particles • Real time of seconds to minutes • Dynamic issues • Strain hardening • Contact erosion over time 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 Truesdall, C. and Muncaster, R.G. (1980) – Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatonic Gas. Academic Press, pp. xvi. Particle Simulations