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

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SAFIRE is an extended

Tsuji-Tanaka model

developed by TUAT

Horio group.

Surface/bridge force

Rupture joint h c

Attractive force F c

No tension joint

Normal elasticity k n

Normal dumping h n

Tangential dumping h t

k t Tangential elasticity

Friction slider m

SAFIRE (Horio et al.,1998~)

(Non-linear spring)

w/wo Tangential Lubrication

w/wo Normal Lubrication

DEM in the 1990s

Evaluation of lubrication force on colliding particles for DEM simulation of fluidized beds

SAFIRE

Achievements

I-H

1998

Ash

Melting

Olefine

Polymerization

PP, PE

Kaneko et al.

1999

Scaling Law

for DEM

Computation

Kajikawa-Horio

2000～

Natural Phenomena

Catalytic Reactions

CHEMICAL REACTIONS

Structure of

Emulsion Phase

Kajikawa-Horio

2001

FUNDAMENTAL LARGE SCALE SIMULATION

OTHER

AGGLOMERATIONCOMBUSTION

Coal/Waste

Combustion

in FBC

Spray

Granulation/Coating

Agglomerating

Fluidization

FB of

Solid Bridging

Kuwagi-Horio

1999

Tangential

Lubrication

Effect

Kuwagi-Horio

2000

Particles w/

van der Waals

Interaction

Iwadate-Horio

1998

Single Char

Combustion

in FBC

Rong-Horio

1999

Parmanently

Wet FB

Mikami,Kamiya,

Horio

1998

FB w/

Immersed

Tubes

Rong-Horio

1999

FB

w/ Immersed

Tubes :

Pressure Effect

Rong-Horio

2000

Particle-Particle

Heat Transfer

Rong-Horio

1999

Fluidized Bed DEM

Started from

Dry-Noncohesive Bed

Tsuji et al. 1993

Scaling Law

for DEM

Computation

Kuwagi-Horio

2002～

Lubrication

Force Effect

Noda-Horio

2002

I-H

1998

Ash

Melting

Olefine

Polymerization

PP, PE

Kaneko et al.

1999

Scaling Law

for DEM

Computation

Kajikawa-Horio

2000～

Natural Phenomena

Catalytic Reactions

CHEMICAL REACTIONS

Structure of

Emulsion Phase

Kajikawa-Horio

2001

FUNDAMENTAL LARGE SCALE SIMULATION

OTHER

AGGLOMERATIONCOMBUSTION

Coal/Waste

Combustion

in FBC

Spray

Granulation/Coating

Agglomerating

Fluidization

FB of

Solid Bridging

Kuwagi-Horio

1999

Tangential

Lubrication

Effect

Kuwagi-Horio

2000

Particles w/

van der Waals

Interaction

Iwadate-Horio

1998

Particles w/

van der Waals

Interaction

Iwadate-Horio

1998

Single Char

Combustion

in FBC

Rong-Horio

1999

Parmanently

Wet FB

Mikami,Kamiya,

Horio

1998

FB w/

Immersed

Tubes

Rong-Horio

1999

FB

w/ Immersed

Tubes :

Pressure Effect

Rong-Horio

2000

Particle-Particle

Heat Transfer

Rong-Horio

1999

Fluidized Bed DEM

Started from

Dry-Noncohesive Bed

Tsuji et al. 1993

Scaling Law

for DEM

Computation

Kuwagi-Horio

2002～

Lubrication

Force Effect

Noda-Horio

2002

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

Model and Computing Conditions with PSD

23

/ pp dd

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(Ndp

2) = 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)

Computed Examples

Run 1

3.00mm

Run 2

4.50 / 3.00 / 2.25

mm

Run 3

4.50 / 2.25 mm

u0 = 1.438→2.938m/s (t<1sec), u0 = 2.938m/s (t≧1sec)

Now, Looking for

a Realistic Collision Process Description

Spring constant ?

Restitution coefficient ?

Field force: Electrostatic

force

Contact forces: •Van der Waals

force •Liquid and solid bridge

force •Impact force

Near Contact force:

Lubrication force

Looking for a solid basis for heat transfer & agglomeration modeling

Inter-particle forces

Let’s revisit the Classical lubrication theory (e.g. R. Davis)

basically for Liquid-solid systems

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?

htvRdrtrrptFL /)(2

3),(2)( 2

0, m

Lubrication force, FL

When h 0, FL infinity

v1

v2

H(r,t) h(0,t) r

p(r,t)

analytical solution with paraboloid approximation

What about the Lubrication Force in Gas-solid systems?

Lubrication force negligible ? Can we overcome “Stokes Paradox” ?

Gas

Examining classical lubrication theory from gas-solid systems’ view point

v1

v2

H(r,t) h(0,t) r

p(r,t)

)()( 21 vvtvdt

dh

LFtFdt

dvm )(

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

RrthtrH /),0(),( 2

rotated paraboloid approximation

htvRdrtrrptFL /)(2

3),(2)( 2

0, m

Breakdown of classical theory in gas-solid systems

• h0 as lubrication effect area

• Adoption of h0=R

• lubrication effect area is much larger in gas-solid systems than in liquid-solid systems

R

RL drtrrpF0

, ),(2

0

, ),(2 drtrrpFL

Classical theory is valid only for very small h0.

check the integration limit

FL,0/G

FL,0/Fd

0.01 0.1 1

0

2

4

6

8

10

Ra

tio

of

F L,0

to

oth

er

forc

es

h 0 / R

0.0 0.2 0.4 0.6 0.8 1.0

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Ratio of lubrication force FL

,R

/FL

,¡Þ

Relative initial distance

h0/R

Classical theory

Check h0 effect in Classical Theory for dp=50 mm, FCC

Numerical solutions for pressure distribution on a colliding sphere compared with classical theory

Pre

ssu

re

Relative radial distance r/R

h0=0.01R h0=0. 1R h0=R

numerical for a sphere analytical with paraboloid approximation

• 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

2222),0(),( rRRthtrH RrthtrH /),0(),( 2

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

Surface

roughness 1/10 R 1/1000 R smooth

Fluid

treatment Continuum Non-continuum Non-continuum

Adhesive

Force Neglected Neglected Important

Typical

particles FCC GB Ideal smooth GB

Case 2: hmin<l0 Case 3: hmin ~ Z0 Case 1: hmin>l0

Collision process with lubrication effect

• 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

Force:

FL/FL0 Velocity: v/v0

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

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

Avoidance of “Stokes Paradox”

Avoidance of “Stokes Paradox”

Case 3: hmin ~ Z0

Z0=repulsi ve

molecular distance

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

Van der Waals force: important!!

collapse distance hcollaplse

0.0

Collapse distance contours

The collapse distance decreases with increase of Stokes number It increases with the increase of Hamaker constant A

)3.16

ˆlnˆ

4

1(

ˆ

6lnˆ

4

1ˆ 00

2

0

02 hhSt

hh

Z

lStE

collapse

collapse

0

ˆl

hh ,

2

0

6 R

mvSt

m ,

2

002

0

6

2

1ˆ

mvZ

AR

mv

EE

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

St

Ste e

*

1

2

0

6 R

mvSt

mStokes Number:

Ratio of particle inertia

to viscous force

Critical Stokes Number

2

**

6 R

mvSt c

cm

2

**

6 R

mvSt e

em

• 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

*

2

1e

cc St

St

eee

ec: restitution coefficient due

to deformation. Suppose ec=1

Calculated examples and discussions Case 1: FCC

Under the same approaching velocity v0, the effect of lubrication force on larger particles is less significant than on smaller particles

surface roughness: 1/10 R

Case 1: FCC w/ different roughness

Collisions with Stokes numbers less than Ste* result in a

restitution coefficient to be zero, consequently causing cluster and agglomeration to occur.

Calculated examples and discussions

Ste*

The independent effects of particle size and approaching velocity on the coefficient of restitution can be included in the consideration of Stokes numbers

Calculation examples and discussions Case 2: GB

Consideration of non-continuum fluid weakens the lubrication effect and thus increases the values of the restitution coefficient

GB, surface roughness:

1/1000 R

Case 3: smooth 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.

Calculation examples and discussions

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.

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