The document discusses evaluating lubrication force on colliding particles for DEM simulation of fluidized beds. It summarizes previous work done by the author's group at Tokyo University of Agriculture and Technology on developing DEM models to simulate fluidized beds. It also discusses areas for improving the modeling of collision processes, including investigating realistic spring constants, restitution coefficients, and inter-particle forces like van der Waals, liquid/solid bridge, and lubrication forces. The author proposes revisiting classical lubrication theory originally developed for liquid-solid systems to inform how to model lubrication forces between colliding particles in DEM simulations.

1. Evaluation of lubrication
force on colliding particles
for DEM simulation of
fluidized beds
Wenbin Zhang , R. Noda and Masayuki Horio
Tokyo University of Agri. and Tech.
Koganei, Tokyo 184-8588, Japan
Sino-German Workshop, Beijing
Oct. 25, 2004

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11. Evaluation of lubrication force on colliding
particles for DEM simulation of fluidized beds
DEM in the 1990s
SAFIRE (Horio et al.,1998~)
Rupture joint h c
Attractive force Fc Surface/bridge force
(Non-linear spring)
kn Normal dumping h n w/wo Normal Lubrication
Normal elasticity
No tension joint Tangential dumping h t
Tangential elasticity k t
SAFIRE is an extended Friction slider m
Tsuji-Tanaka model w/wo Tangential Lubrication
developed by TUAT
Horio group.

12. COMBUSTION Spray Agglomerating AGGLOMERATION
Granulation/Coating Fluidization
FB
w/ Immersed Ash
Tubes : Melting
FB of Particles w/
Pressure Effect I-H
Solid Bridging van der Waals
Rong-Horio 1998 Tangential
2000 FB w/ Interaction
Kuwagi-Horio Lubrication
Immersed Iwadate-Horio Effect
1999
Coal/Waste Tubes 1998
Kuwagi-Horio
Combustion Parmanently
Rong-Horio 2000
in FBC Wet FB
1999
Mikami,Kamiya,
Fluidized Bed DEM Horio
Started from 1998
Particle-Particle Dry-Noncohesive Bed
Single Char Heat Transfer
Tsuji et al. 1993
Combustion Rong-Horio
in FBC Natural Phenomena
1999
Rong-Horio OTHER
1999 Lubrication
Force Effect
SAFIRE Olefine Scaling Law
Achievements Polymerization Noda-Horio for DEM Scaling Law
for DEM
PP, PE Structure of
2002 Computation
Computation
Kaneko et al. Emulsion Phase Kajikawa-Horio
2000～ Kuwagi-Horio
1999 2002～
Kajikawa-Horio
Catalytic Reactions
2001
CHEMICAL REACTIONS FUNDAMENTAL LARGE SCALE SIMULATION

13. SAFIRE 21
Contact force models
Normal force：Hertz model
Tangential force: Mindlin, Deresiewicz(1953)’s
‘no-slip’ solution
Particle size: PSD
Drag force: Extended Ergun CD

14. Model and Computing Conditions with PSD
Run1 Run2 Run3
Particle size [mm] 3.00 4.50/3.00/2.25 4.50/2.25
Particle number [#] 30000 2963/10000/23703 4444/35556
Volume fraction [-] 1 0.333/0.333/0.333 0.500/0.500
Sauter mean dsv=Σ(Ndp3)/Σ(Ndp2) = 3.00 mm
Particle vol. = 4.24×10-4m3,
Total particle surface area = 8.48×10-1m2
Young’s modulus: 80GPa, Poisson
ratio: 0.3, friction coefficient: 0.3
(Glass beads)

15. Computed Examples
Run 1 Run 2 Run 3
3.00mm 4.50 / 3.00 / 2.25 4.50 / 2.25 mm
mm
u0 = 1.438→2.938m/s (t<1sec), u0 = 2.938m/s (t≧1sec)

16. Now,
Looking for
a Realistic Collision Process Description
Looking for a solid basis for heat transfer & agglomeration modeling
Spring Restitution
constant ? coefficient ?
Inter-particle forces
Contact forces:
Field force: Near Contact •Van der Waals
Electrostatic force: force
force Lubrication force •Liquid and
solid bridge
force
•Impact force

17. Let’s revisit the Classical lubrication theory
(e.g. R. Davis)
basically for Liquid-solid systems
v1
analytical solution with r
H(r,t) h(0,t)
paraboloid p(r,t)
approximation
v2
 3
FL, (t )   2rp(r , t )dr  m R 2v(t ) / h
0 2
Lubrication force, FL
When h 0, FL infinity
Stokes Paradox: Two solid surfaces can never make contact
in a finite time in any viscous fluid due to the infinite
lubrication force when surface distance approaches zero
How shall we get along with the Stokes paradox,
practically or essentially?

18. What about
the Lubrication Force
in Gas-solid systems?
Gas
Lubrication force negligible ?
Can we overcome
“Stokes Paradox” ?

19. Examining classical lubrication theory
from gas-solid systems’ view point
dh
v1
H (r, t )  h(0, t )  r 2 / R
 v(t )  (v1  v2 )
dt H(r,t) h(0,t) p(r,t)
r rotated paraboloid
dv approximation
m  F (t )   FL v2
dt  3
FL , (t )   2rp(r , t )dr  m R 2v(t ) / h
0 2
Assumptions in classical lubrication theory are
invalid in gas solid systems
 Initial gap size h0 is assumed to be much smaller
than particle radius
 Upper limit of integration of pressure for
lubrication force is extended to infinity
 Paraboloid approximation of un-deformed surface
 Fluid is treated as a continuum

20. Breakdown of classical theory
in gas-solid systems
Check h0 effect in Classical check the integration limit
Theory for dp=50 mm, FCC
Ratio of L,0to other forces

R
FL, R   2rp(r , t )dr
Ratio of lubrication force FL,R/FL,¡Þ
10 1.0
0
8 0.9 
6 FL,0/Fd FL,   2rp(r , t )dr
0.8 0
4
0.7
FL,0/G Classical theory
2
F
0.6
0
0.5
0.01 0.1 1
h0/
R 0.4
0.0 0.2 0.4 0.6 0.8 1.0
• h0 as lubrication effect area Relative initial distance h0/R
• Adoption of h0=R
Classical theory is valid
• lubrication effect area is much
larger in gas-solid systems than
only for very small h0.
in liquid-solid systems

21. Numerical solutions
for pressure distribution on a colliding
sphere compared with classical theory
Pressure
h0=0.01R h0=0. 1R h0=R
Relative radial distance r/R
numerical for a sphere analytical with paraboloid approximation
H (r , t )  h(0, t )  2R  2 R 2  r 2 H (r, t )  h(0, t )  r 2 / R
• Pressure decays to zero much faster with
paraboloid approximation
• Contribution of pressure in the outer region to the
lubrication force may play an important role

22. Case studies on minimum approaching distance
• Assuming that minimum surface distance equals to surface roughness
• Whether the fluid remains as a continuum is determined by the relative
magnitude of surface distance to mean free path of fluid molecules
Case 1: hmin>l0 Case 2: hmin<l0 Case 3: hmin ~ Z0
Surface
1/10 R 1/1000 R smooth
roughness
Fluid
Continuum Non-continuum Non-continuum
treatment
Adhesive
Neglected Neglected Important
Force
Typical
FCC GB Ideal smooth GB
particles

23. Collision process with lubrication effect
Force:
FL/FL0 Velocity: v/v0
• Physical contact happens when hmin equals to surface roughness
• Energy dissipation in both approaching and separating stages
• Lubrication force increases quickly when surfaces approach closer

24. Even with Lubrication force we can avoid
“Stokes Paradox”
Case 1: hmin>l0
l0=mean free
path
 Increase of lubrication force stops when roughness
makes contact
 To realistic particles with surface roughness, stokes
paradox is practically avoided

25. Avoidance of “Stokes Paradox”
Case 2: hmin<l0
l0=mean free
path
 Increase of lubrication force is slowed down in close
approaching distance
 Treatment of fluid as a non-continuum helps us avoid the
infinite lubrication force

26. Avoidance of “Stokes Paradox”
Case 3: hmin ~ Z0 0.0
Z0=repulsi ve
molecular distance Van der Waals force: important!!
collapse distance hcollaplse
 Magnitude of van der Waals force increases more rapidly
when h -> 0
 A critical collapse distance hcollaplse is defined to indicate the
adhesive force dominant region (~10-9m)
 Consideration of adhesive force in the last approaching stage
saves us again from Stokes Paradox essentially

27. Collapse distance contours
ˆ
ˆ
ESt 2 
1 l0 ˆ 2
hcollapse ln
6 1ˆ h
( St  h0 ln 0 ) ˆ h
h  , St 
mv0 ˆ
, E
E

AR
4 Z0 ˆ
hcollapse 4 16.3 l0 6mR 2 2
1 2 6Z 0 mv0
mv0
2
 The collapse distance decreases with increase of
Stokes number
 It increases with the increase of Hamaker constant A

28. Effective Restitution Coefficient
• Lubrication effect is actually a kind of damping effect, causing
kinetic energy dissipation during both approaching and
separating stage
• Restitution coefficient can be regarded as a criterion for
evaluating the lubrication effect on collision process
*
1  ec * ec: restitution coefficient due Ste
e  ec  Ste to deformation. Suppose ec=1 e  1
2St St
mv0
Stokes Number: St  Ratio of particle inertia
6mR 2 to viscous force
*
Critical Stokes
*
mvc mve
St 
*
St 
*
6mR 2 6mR 2
c e
Number
• vc* is called “critical contact velocity”. If initial velocity v0 <vc*, particles
cannot make contact in the approaching stage
• ve* is called “critical escape velocity” . If v0 <ve*, particles cannot escape
from the lubrication effect area and will stop during the separation stage

29. Calculated examples and discussions Case 1: FCC
surface
roughness:
1/10 R
 Under the same approaching velocity v0, the effect of
lubrication force on larger particles is less significant than on
smaller particles

30. Calculated examples and Case 1: FCC
discussions w/ different roughness
Ste*
 Collisions with Stokes numbers less than Ste* result in a
restitution coefficient to be zero, consequently causing cluster and
agglomeration to occur.
The independent effects of particle size and approaching velocity on the
coefficient of restitution can be included in the consideration of Stokes numbers

31. Calculation examples and discussions Case 2: GB
GB, surface roughness:
1/1000 R
 Consideration of non-continuum fluid weakens the
lubrication effect and thus increases the values of
the restitution coefficient

32. Calculation examples and Case 3: smooth
discussions GB
Lubrication effect is most significant in case 3 since
particles can approach much more closely and the
effect of non-continuum fluid is more significant.

33. Concluding Remarks
 By numerically extending classical lubrication
theory into gas-solid systems, semi-empirical
expressions for lubrication force are proposed.
 Evaluation of lubrication effect on collision
process are made in terms of restitution coefficient.
 Stokes Paradox is avoided by considering surface
roughness, non-continuum fluid and van der Waals
force.
 Further research should be aiming at
incorporating lubrication force and an effective
restitution coefficient into DEM simulation in the
near contact area.

34. From Burton to Fluid Cat. Cracking
Chemical Engineers’ Unforgettable
Memory
The FCC Development (1940－50)
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