Energy transport in Granular systems

1. Energy transport in Granular systems Kiwing ToInstitute of Physics, Academia Sinica, Taipei OUTLINE General properties of granular systems Diffusion in granular gas Temperature profile in granular shocks Mixing dynamics in rotating drum Summary & Discussions Complex Dynamics in Granular Systems, Beijing, 31 May 2013

2. dilute rapid flow external forcing dissipation via friction and inelastic collisions immobile state Granular systems : collections of small, discrete constituents that interacting inelastically gas fluid solid few collision frequent contact dense, slow flow Energy flow quasi-static meta-stable glassy region No relative motion dynamic steady state maintained by external energy supply and overall dissipation energy transport within the system may give rise to interesting physical behaviors

3. Granular shocks in two-dimensional channel Mixing dynamics of slurry in rotating drum Guoqi Hu, Yinchang Li, MeiyingHou, K. ToPRE 81, 011305, 2010 C. C. Liao, S. S. Hsiau, K. ToPRE 82, 010302(R), 2010 Diffusion in two-dimensional granular gas WennanChen, Kiwing To PRE 80, 061305, 2009 granular systems with energy transport within different parts of the system

4. liquid medium : density rl viscosity h T FB FS density rd particle : radius a mass m v equation of motion Frand W weight, buoyance, viscous drag, mean square displacement terminal velocity diffusion coefficient Motivation : effect of fluid viscosity on dynamics e.g. : Brownian particle in a fluid dynamics in a fluid are slower in fluids of higher viscosities

5. Cataracting Introduction : Rotating drum J. Mellmann, Powder Technol. 118, 251 (2001).

6. Rotating Drum Experiment quasi-two-dimensional drum: 10 cm radius, 2.5 cm thick, 3 mm diameter glass beads 50% filling fraction drum on a rotor with horizontal axis and 2.3 rpm speed w=2.5 cm glass beads: 50% black 50% white glass beads distribution in the drum recorded by video camera at 30 fps

7. Rotating Drum Experiment quasi-two-dimensional drum: 10 cm radius, 2.5 cm thick, 3 mm diameter glass beads 50% filling fraction drum on a rotor with horizontal axis and 2.3 rpm speed w=2.5 cm glass beads: 50% black 50% white glass beads distribution in the drum recorded by video camera at 30 fps

8. Experiment q h flow speeds : 1000 fps fast camera particle track software v velocity profile mixing region surface velocity beads rotate with drum no relative motion thickness of mixing region

9. Experiment q h flow speeds : 1000 fps fast camera particle track software v mixing region beads rotate with drum no relative motion thickness of mixing region

10. Experiment average velocity in mixing region mean velocity and mixing region size mean velocity decreases with fluid viscosity mixing region size increases with viscosity

11. Experiment a: Ci=0 in cell of all black beads Ci=1 in cell of all white beads Mixing rate measurement 20x20 pixels ith cell number of pixel with value greater than a number of cells with beads, N mixing index fully mixed M=0 fully segregated M=0.5

12. Experiment Mixing rate measurement mixing rate increases with fluid viscosity

13. Diffusion prob. to change lane prob. to move down

14. Mixing average velocity in mixing region time spend in mixing region distance diffused G time between two entries to the mixing region R time to diffuse distance comparable to the drum fraction of the beads in the mixing region fraction of the beads in the mixing region time for all the beads to diffuse distance R is mixing rate

15. Microscopic dynamics q S. Courrech du Pont, et al, Phys. Rev. Lett. 90, 044301 2003. T, avalanche time

16. Microscopic dynamics equation of motion : St=a S. Courrech du Pont, et al, Phys. Rev. Lett. 90, 044301 2003. density ratio = = viscous-inertial transition ?

17. Granular shocks in two-dimensional channel Mixing dynamics of slurry in rotating drum Guoqi Hu, Yinchang Li, MeiyingHou, K. ToPRE 81, 011305, 2010 C. C. Liao, S. S. Hsiau, K. ToPRE 82, 010302(R), 2010 Diffusion in two-dimensional granular gas WennanChen, Kiwing To PRE 80, 061305, 2009 granular systems with energy transport within different parts of the system

18. single layer of 1 mm diameter steel ball confined exit dilute flow dense flow Q decreasing d Rapid (Dilute) Flow Slow (Dense) Flow

19. On reducing exit size Qout B Q0 Qin jam dense flow dilute flow Qd C D E d 0 d2 d1 Phys. Rev. Lett. 86, 71, 2001 Phys. Rev. E 71, 060301R, 2005

20. steel balls d=2 mm fps reservoir gate flow rate g =8cm particle trajectories g=1367 #/s dilute granular gas high mean flow speed granular fluid high density motionless granular solid running average (7 mm window)

21. shock speed y u=59 mm/s

22. anisotropic gas crystalline solid fluid y reference frame co-moving with shock random close pack density nc=0.267 mm−2 y=y’-ut ; vy=vy’+u density profile energy profile mean flow KE thermal KE energy transfer from mean flow to thermal

23. probability density symmetric uni-modal asymmetric bimodal velocity [m/s]

24. type-2 type-1 probability density velocity [m/s]

25. 1 2 density [mm-2] 2 2 position, y [mm] Tx [mJ] 1 0 0 type-1 particles 1 type-2 particles 2 Tx increases after collision

26. Tx [mJ] 1 0 0 type-2 particle velocity distribution is symmetrical can be treated as isotropic granular gas momentum balance random close pack density

27. Granular shocks in two-dimensional channel Mixing dynamics of slurry in rotating drum Guoqi Hu, Yinchang Li, MeiyingHou, K. ToPRE 81, 011305, 2010 C. C. Liao, S. S. Hsiau, K. ToPRE 82, 010302(R), 2010 Diffusion in two-dimensional granular gas WennanChen, Kiwing To PRE 80, 061305, 2009 granular systems with energy transport within different parts of the system

28. mean square displacement MSD = log ( MSD ) slope = 1 log ( t ) Unusual diffusion in quasi-two-dimensional granular gas MSD = 6Dt diffusion constant in 2 dimension MSD = 4Dt in 1 dimension MSD = 2Dt

29. diameter, s mean free time typical speed, v number/area, r mean free path area swept per unit time = 2sn collision rate = 2rsn total molecule number in fixed area A Unusual diffusion in quasi-two-dimensional granular gas 2s diffusion constant [m2/s] v D inversely proportional to N usual diffusion

30. Unusual diffusion in quasi-two-dimensional granular gas granular gas : a collection of ‘molecules’ interact with each other with inelastic collision quasi-two-dimensional: two dimensional projection from three dimensional motion plastic ball diameter: 6 mm mass: 0.12 gm vibration frequency: 20 Hz amplitude: 1.84 mm camera resolution: 1024x1024 frame rate: 1000 fps

31. vibrating stage 12.5 mm 300 mm Unusual diffusion in quasi-two-dimensional granular gas get trajectory each particles measure MSD

32. MSD = 4Dt diffusive motion MSD = <(Dx)2> = <(vt)2> = <v2> t2 ballistic motion N =500 slope=2

33. NTg 26 mJ 1116 mJ 500 1000 N =1000 D=100 mm/s2 N =500 D=75 mm/s2

34. 4D [mm2/s] diffusion increases with N for N < 1000 Tg [10-9 J] N

35. Langevin equation : ball dynamics motion in vertical direction motion in horizontal direction collision with top or bottom (type-1) collision with other balls (type-2) small velocity fluctuation in horizontal direction large velocity fluctuation in horizontal direction top and bottom act on the balls like a viscous fluid balls may gain kinetic energy in the horizontal direction due to inelastic collision

36. type-2 collision : type-1 collision : with other balls with top/bottom large perturbation to horizontal velocity small perturbation to horizontal velocity effective temperature T2 effective temperature T1

37. enter HSS with speed v2 decay to LSS in t area swept = 2s v2t high speed state, T2 decay time (life time), t type-2 collision type-1 collisions diffusion in LSS with rate D1 , duration t1 area swept = 4pD1t1 total area swept in 1 second, low speed state, T1 diffusion constant D1 type-2 collision rate = f f, fraction of beads in the HSS

38. particle trajectory and speed high speed state low speed state

39. low speed high speed high speed Two state model ball excited to HSS after colliding with another ball ball in HSS relaxes to LSS in t2 due to collision with top and bottom 1800 500 200 50

40. =130 mm/s, t =100ms, D1 = 5.5 mm2/s, D2 =100 mm2/s T1 = 54 nJ, T2 =4400 nJ Two effect temperature baths collision with other ball collision with top or bottom temperature bath-2, T2 temperature bath-1, T1 effective granular temperature

41. energy supply energy dissipation quasi-two-dimensional granular gas travelling granular shock 1800 500 200 50 velocity [m/s] Summary rotating drum fluid solid fast slow thermal mean Non-equilibrium granular system

43. Introduction Dry granular materials consist of small discrete solid constituents, e.g., sand, rocks, snow, salt, grains, pills, milk powder, styrofoam, …, etc. Important in agricultural, pharmaceutical, food industries:storage, transport and manipulation of grains, seeds, tablets, ore, chemical powder, … Mechanics of individual component is known but their collective behavior is not fully understood.