040603 Four topics for further development of dem to deal with industrial fluidization issues, ICMF plenary2004

Four Topics for Further

Development of DEM to

Deal with Industrial

Fluidization Issues

Masayuki Horio and Wenbin Zhang

Department of Chemical Engineering,

Tokyo University of Agriculture and

Technology,

Koganei Tokyo, 184-8588 Japan,

[email protected]

Come & Visit Tokyo Univ. A&T

at Koganei (25min from Shinjuku)

Chemical Engineers in ICMF

渦

year

Cap

acit

y i

n w

orl

d t

ota

l [%

]

year

Cap

acit

y i

n w

orl

d t

ota

l [%

]

From Burton to Fluid Cat. Cracking

Chemical Engineers’ Unforgettable

Memory

The FCC Development (1940－50)

air

air

steam

steam

steam

kerocene

& steam

kerocene

& steam kerocene

& steam

air

air air

product product product

product

product

FCC Plant development

in Catalytic Cracking of

Kerocene(1940－50)

Competition and Evolution

of Fluid Catalytic Plants in

1940-50

kerocene

& steam

The presence of

column wall makes

research much

easier

cloud

hail

artificial

plant

Natural Science and

Engineering Science

AICｈE Fluor Daniel Lectureship Award Lecture (2001)

Post

mdern

Era:

volcanic plateau

My background

-1974 Fixed/Moving Bed Reactors

and iron-making Processes

1974- Fluidization Engineering

75-99 Pressurized Fluidized Bed Combustion

Jets, Turbulent Transport in Freeboard

82-89 Scaling Law of Bubbling Fluidized Bed

89-92 Scaling Law of Clustering Suspensions

93- DEM Simulation

Waste Management, Material Processes

1997- Sustainability and Survival Issues

Biomass Utilization, Appropriate Technology

When Professor Tsuji et al. 1993 proposed an

excellent idea of applying the concept of

discrete/distinct element method of Cundall et al.

(1979) to fluidized beds borrowing the fluid phase

formulation from the two phase model,

I (Horio) almost immediately decided to join in the

simulation business of fluidized beds from

chemical engineers' view points.

This was because with his approach the real

industrial issues, such as agglomeration, gas

solid reactions and/or heat transfer, can be

directly incorporated into the model without the

tedious derivation of stochastic mechanics,

which is not only indirect but also sometimes

impossible from analytical reasons.

DEM, the last 10 years

DEM: Discrete Element Method

Fluid phase: local averaging

Particles: semi-rigorous treatment User friendly compared to Two Fluid Model & Direct

Navier-Stokes Simulation

•A new pressure/tool to reconstruct particle

reaction engineering based on individual

particle behavior

•Potential for more realistic problem definition/

solution

Our code development: SAFIRE

Simulation of Agglomerating Fluidization for Industrial

Reaction Engineering

Normal and tangential component of F collision

and F wall

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 is an extended Tsuji-Tanaka model

developed by TUAT Horio group

SAFIRE (Horio et al.,1998~)

(Non-linear spring)

t

t n t x

x F F m = n t F F m >

dt dx

x k F n n n n n h - D =

dt dx

x k F t t t t t h - D = n t F F m

km g = h 2 ( )

( ) 2 2

2

ln ln

p + = g

e e

w/wo Tangential Lubrication

w/wo Normal Lubrication

Soft Sphere Model with Cohesive Interactions

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

AGGLOMERATION COMBUSTION

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

SAFIRE

Achievements

AGGLOMERATION

■ Agglomerating Fluidization

by Liquid Bridging

through surface diffusion through viscous sintering by solidified liquid bridge

by van der Waals Interaction

by Solid Bridging

Coulomb Interaction

■ Size Enlargement

by Spray Granulation (Spraying, Bridging, Drying)

by Binderless Granulation (PSG)

■ Sinter/Clinker Formation in Combustors / Incinerators

in Polyolefine Reactors

in Fluidized Bed of Particles (Sintering of Fe, Si, etc.)

in Fluidized Bed CVD (Fines deposition and Sintering)

(Ash melting)

(Plastic melting)

Industrial Issues & DEM

CHEMICAL REACTORS

Heat and Mass Transfer gas-particle particle-particle

Heterogeneous Reactions

Homogeneous Reactions

Polymerization

Catalytic Cracking

Partial Combustion

(with a big gas volume increase)

COMBUSTION / INCINERATION

Boiler Tube Immersion Effect

Particle-to-Particle Heat Transfer

Char Combustion

Volatile Combustion (Gas Phase mixing / Reaction)

Combustor Simulation

(high velocity jet)

Industrial Issues & DEM

ne

ck d

iam

ete

r, 2

(b) 1123K (a) 923K

2x

nec

k

ne

ck d

iam

ete

r

10 m m

SEM images of necks

after 3600s contact

Solid Bridging Particles (Mikami et al , 1996)

Sintering of

steel particles

in Fluidized

Bed Reduction

2x

ne

ck

700 800 900 1000 1100 1200 1300 0

5

10

15

20

25

30

Nec

k d

iam

ete

r 2

x

Calculated from

surface diffusion model

d p =200 m m d p =20 m m

Temperature [K]

Neck diameter determined from SEM images

after heat treatment in H2 atmosphere

Steel shot :dp=200m m, H2, 3600sSteel shot :dp=200m m, H2, 3600s

Model for Solid Bridging Particles

1. Spring constant: Hooke type (k=800N/m)

Duration of collision: Hertz type

2. Neck growth: Kuczynski’s surface diffusion model

D = D exp (-E /RT)

D =5.2x10 m/s, E = 2.21x10 J/mol (T>1180K)

3. Neck breakage

s = neck neck nc A F

t = neck neck tc A F

7 1

3 4 gd 56

/

= t r D T k

x g S

B

neck

0,s

0,s

s s -2 5

Kuwagi-Horio 1999

Kuwagi-Horio

6 m m

r g = 10 m m

Steel shot

200 m m

neck

Cross section

Surface Roughness and Multi-point Contact Kuwagi-Horio 1999

Kuwagi-Horio

Kuwagi-Horio

t= 0.438s 0.750s 1.06s 1.38s 1.69s

2.00s 2.31s 2.63s 2.94s 3.25s

1273K, u = 0.26 m/s, Dt=0.313s

Snapshots of Solid Bridging Particles

without Surface Roughness

0

Kuwagi-Horio 1999

d =200mm, T=1273K, u =0.26m/s

(a) Smooth surface (b) 3 micro-contact points (c) 9 micro-contact points

Fig.7 Agglomerates (or "dead zones") grown on the wall (t = 1.21 s).

p 0

(Case 1) (Case 2) (Case 3)

Agglomerates (or “dead zones”) grown on the wall (t = 1.21 s).

Kuwagi-Horio 2000

Kuwag

i-Horio

Intermediate condition

Agglomerates Sampled at t = 1.21s

(a) Smooth surface (b) 3 micro-contact

points

(c) 9 micro-contact

points

dp=200mm, T=1273K, u =0.26m/s

Weakest sinteringcondition

Strongest sinteringcondition

0

Kuwagi-Horio 1999

Kuwagi-Horio

xu

heat transfer coefficient (different for each particle)

external gas film

fluid cell

gQ

yu

gTε

pnh

pnT

particle

yv

xv

Energy balance

gpp k/dh=Nu ggg,p k/c=Pr μgpgp /dvu=Re μρ-

( ) ( )g

g,pgi

gigQ

c

1=

x

Tu+

t

T

ρ∂

ε∂

∂

ε∂

( ) ( )gpp

p

g TThd

16=Q -

ε-

rcp

p Pw)RT

E(expk=R

Gas phase :

( ) ( )STThHR=dt

dTcV gpprp

p

pp,pp --Δ-ρ

Particle :

2

1

p3

1

RePr6.0+0.2=Nu (Ranz-Marshall equation)

Poly-Olefine Reactor Simulation,

Kaneko et al. (1999)

t=6.0 sec t=9.1 sec t=8.2 sec

Hot spot

Particle circulation (artificially generated by feeding gas nonuniformly from distributor nozzles)

Ethylene polymerization Number of particles=14000

u0=3 umf

Gas inlet temp.=293 K

3umf 3umf 3umf 2.5umf 2.5umf

9.3umf 15.7umf

2umf 2umf

Idemitsu Petrochemical Co.,Ltd. Tokyo University of Agriculture & Technology

T [K] 293

343

393

(20℃)

(120℃)

Kaneko et al. 1999

Stationary solid revolution helps

the formation of hot spots.

particle temp. particle velocity vector

t=9.1 sec t=8.2 sec

particle temp. particle velocity vector

Uniform gas feeding Nonuniform gas feeding

3umf 3umf 3umf

15.7umf

2umf 2umf : Upward motion

: Downward motion Stationary circulation

Idemitsu Petrochemical Co.,Ltd. Tokyo University of Agriculture & Technology

A Rough Evaluation of

Heat Transfer Between Particles

Rong-Horio 1999

A B

0.4 nm

A B

radiation

particle-thinned film-particle

heat transfer

contact point heat transfer

when l AB < 2r + d : particle-particle heat conduction

convection

Four Topics for Further

Development of DEM

1. PSD

2. Large Scale Computation via

Similar Particle Assemblage Model

3. Surface Characterization and

Reactor Simulation

4. Lubrication Force and Effective

Restitution Coefficient

PSD Issue

Derivation of CD

corresponding to Ergun

Correlation and A Case Study

Master Thesis

by Nobuyuki Tagami

What We need for moving

from Uniform Particle

Systems to Non-uniform Ones

○ 3D Computation

○ Fluid-Particle Interactions

1) not from Ergun (1952) Correlation

2) not indifferent to particle arrangement

○ Contact Model with Particle Size Effect

Fookean to Herzean Spring

1. PSD

Today’s topic

Apparent Drag Coefficient

that corresponds to Ergun

Correlation

( ) ( ) ( )vuvu1.75ρd

με1150

d

ε-1

gρΔP/ΔLL/ΔP

fp

f

p

f*

-

-+

-=

-=D

velocityParticle:v

velocityFluid:u

densityFluid:ρ

fractionVoid:ε

diameterParticle:d

f

p

2

f2

p

pf

D

vuρd

F8C

-

p

0gερFnΔL

ΔPε fpf =+-- ( ) ( )/6dπ/ε1n

3p-=

(1) Bed Pressure Drop Correlation (Ergun(1952))

(2) Equation of motion for fluid (1D)

(3) Drag Coeff.

( )2.33

vuερd

με1200C

fp

fErgunD, +

-

-=

→ Apparent Drag Coeff.

1. PSD

Extension of C

D,Ergun

( )2.33

vuερd

με1200C

fp

fErgunD, +

-

-=

( )2.33

vuερd

με1200C

fp

fErgunD, +

-

-=

1. PSD

The Sum of Drag Force Consistent

with Ergun Correlation ?

dp1/dp2 [mm/mm]

Number of

particles

1.00 30000

1.50 /

0.750

4444 /

35556

( )( )0.5st1.122m/s

0.5st1.122m/s

0.811u

sPaμ18μ

1.204kg/mρ

2650kg/mρ

0

f

3f

3p

=

=

=

=

=1.00

0.75

1.25

Erguni,

Ci,

F

FErgunD,

Binary System

Error was within the Accuracy of

Ergun Correlation ±25%.

1. PSD

PSD Effect: A Case Study

23

/ pp dd

Run1 Run2 Run3

Diameter [mm] 3.00 4.50/3.00/2.25 4.50/2.25

Number [#] 30000 2963/10000/23703 4444/35556

Vol. Fraction 1 0.333/0.333/0.333 0.500/0.500

Surface to Volume Mean Diameter:

dsv=Σ(Ndp3)/Σ(Ndp

2) = 3.00 mm

Total solid volume = 4.24×10-4m3,

Total solid surface area = 8.48×10-1m2

Contact Force Model Normal：Hertz’ Model

Tangential: ‘no-slip’ Solution of Mindlin,

and Deresiewicz (1953)

Young’s modulus: 80GPa, Poisson ratio: 0.3, friction coefficient: 0.3

(Glass beads)

1. PSD

Comparison of the three cases

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)

Run3

Large particles become more mobile

receiving forces from smaller ones

1. PSD

Kuwagi K.a, Takeda H.b and Horio M.c,*

aDept. of Mech. Eng., Okayama University of Science,

Okayama 700-0005, Japan

bRflow Co., Ltd., Soka, Saitama 340-0015, Japan

cDept. of Chem. Eng., Tokyo University of Agri. and Technol.,

Koganei, Tokyo 184-8588, Japan

The Similar Particle Assembly (SPA)

Model,

An Approach to Large-Scale Discrete

Element (DEM) Simulation

Fluidization XI, May 9-14, 2004,

Ischia (Naples), Italy

2. SPA

1,940 1,960 1,980 2,000 2,0201.0E+1

1.0E+4

1.0E+7

1.0E+10

1.0E+13

1.0E+16

Year

Perf

orm

an

ce [M

FL

OP

S] Nishikawa et al. (1995)

Seki (2000)Oyanagi(2002)

Single processor for PC

Moore's Law

Fastest computer models

15 to 20 years

Development of Computer Pormance

How to deal with billions of particles?

TFM (Two-fluid model)

DSMC (Direct Simulation Monte Carlo)

Difficult to deal with realistic particle-particle and

particle-fluid interactions including cohesiveness

DEM (Discrete Element Method)

One million or less particles with PC in a practical

computation time

Hybrid model of DEM and TFM (Takeda & Horio, 2001)

Similarity condition for particle motion (Kazari et al., 1995)

Imaginary sphere model (Sakano et al., 2000)

2. SPA

Assumptions

(0. Particles are spherical)

1. A bed consists of particles of different species

having different properties, i.e. particle size,

density and chemical composition, and it has

some local structure of their assembly.

2. Of each group (species) N particles are supposed

to be represented by one particle at the center of

them. This center particle is called a

representative particle for the group.

3. The representative particles for different groups

can conserve the local particle assembly similar.

Similar Particle Assembly (SPA) Model

2. SPA

+

+

i’

Particle Coordination Scaling

i

original system m times larger system

Similar structure

x x+mDx

+

(a) (b)

(d)

Represented volume for N particles

+

m times larger system

of the same particles

as the smaller bed

(c)

A particle

x x+Dx

+

+ +

+

Preparation

(1) All particles are numbered: i=1~NT.

(2) Subspace:

(3) Group number of particles:

(4) Equation of motion for particle i:

( )ppk dG ,

( )( )kpipii Gdkk ,

gdFFdt

dvd pipi

ij

pijfii

pipi

p++=

p

33

66

Ffi: particle-fluid interaction force

Fpij: particle-particle interaction force

2. SPA

Governing Equations

gdFFdt

dvd

pipi

ijjpifi

i

pipi

p++=

p

3

''

''

*

''

*

'

'3

''66

pipi mdd ='

+=+

ijpijfi

ijjpifi

FFmFF 3

''

*

''

*

'

Equation of motion for original particle:

Equation of motion for m-times larger volume:

where

If ii vv =',

gdFFdt

dvd pipi

ij

pijfii

pipi

p++=

p

33

66

gdmFFdt

dvdm

pipi

ijjpifi

i

pipi

p++=

p

3

''

3

''

*

''

*

'

'3

''

3

66

( )cell

pi

f

pi

fPi N

ddF

--

-+-m

-=

vu)vu()1(75.1

)vu(1150

2

2

vu)vu(8

22

pillfDpi dCF --p

=

2. SPA

Computation Conditions for Case 1

Particles Geldart Group: D

Particle diameter: dp [mm ] (a) 1.0 (b) 3.0 (c) 6.0

Particle density: p [ kg/m3 ] 2650

Number of Particles (a) 270,000 (b) 30,000 (c) 7,500

Restitution coefficient 0.9

Friction coefficient 0.3

Spring constant: k [ N/m ] 800 (Dt=2.58x10-5s)

Bed

Column size 0.5×1.5m

Distributor Porous medium

Gas Air

Viscosity: mf [Pa s ] 1.75x10-5

Density: f [kg/m3 ] 1.15

.

Snapshots of Dry Particles

(c) SPA bed (representative particle, dp’=6.0mm)

0.262s 1.05s 1.31s 0.528s 1.58s 1.84s 2.10s 2.36s 2.62s 0.790s

(a) Original bed (dp=1.0mm)

(b) SPA bed (representative particle, dp’=3.0mm)

Average height of dry particles

initially located in the half lower region

p=2650kg/m3, Column : 0.5×1.5m, u0=1.2m/s

0 1 2 3 4 50

0.1

0.2

0.3

0.4

Time [s]

Avera

ge h

eig

ht

of

low

er

half

set

part

icle

s [

m]

dp=1.0mm (Original bed)

dp'=3.0mm (SPA bed)

dp'=6.0mm (SPA bed)

Dry

decreasing U0

dp=1.0mm (Original bed)(fluid cell: 134x333)

(fluid cell: 22x56)

+ u0: decreasing u0: increasing

Snapshots of Wet Particles (V=1.0x10-2)

(c) SPA bed (representative particles, dp’=6.0mm)

0.262s 1.05s 1.31s 0.528s 1.58s 1.84s 2.10s 2.36s 2.62s 0.790s

(a) Original bed (dp=1.0mm)

(b) SPA bed (representative particle, dp’=3.0mm)

Average height of wet particles

initially located in the half lower region

p=2650kg/m3, Column : 0.5×1.5m, u0=1.2m/s

0 1 2 3 4 50

0.1

0.2

0.3

0.4

Time [s]

Avera

ge h

eig

ht

of

low

er

half

set

part

icle

s [

m]

dp=1.0mm (Original bed)

dp'=3.0mm (SPA bed)

dp'=6.0mm (SPA bed)

Wet

decreasing U0

dp=1.0mm (Original bed)(fluid cell: 134x333)

(fluid cell: 22x56)

+ u0: decreasing u0: increasing

Comparisons of umf

(a) Dry particles (b) Wet particles

Umf from Wen-Yu correlation = 0.57m/s

0 0.2 0.4 0.6 0.8 1 1.2 1.40

2,000

4,000

6,000

8,000

10,000

U0 [m/s]

DP

[P

a]

dp'=3.0mm

dp'=6.0mm

dp=1.0mm

Umf = 0.72m/s dry

0 0.2 0.4 0.6 0.8 1 1.2 1.40

2,000

4,000

6,000

8,000

10,000

U0 [m/s]

DP

[P

a]

dp'=3.0mm

dp'=6.0mm

dp=1.0mm

Umf = 0.70m/swet (V=1.0x10 )-2

2. SPA

CPU time for real 1s on Pentium 4 2.66GHz

Dry [s] Wet [s]

Original bed

(dp=1mm)

27,300

(7hrs 34min)

27,600

(7hrs 39min)

SPA bed

(dp’=3mm)

1,760

(29min)

1,870

(31min)

SPA bed

(dp’=6mm)

426

(7min)

508

(8min) 1/55 1/64

1/15 1/15

2. SPA

Computation Conditions for Case 2

Column 0.156x0.390m

Nozzle width 4mm

Particle (original)

dp 1.0mm

p 2000, 3000 kg/m3

Gas Air

f 1.15kg/m3

mf 1.75x10-5Pa s

p=3000kg/m3

p=2000kg/m3

Fig: Initial state

.

Single bubble fluidization of two-density mixed particles

15m/s (0.482s)

0.7m/s 0.7m/s

Single Bubble Behavior of Two-Density Particles

t=0.056s t=0.111s t=0.167s t=0.223s t=0.278s

(a) dp=1.0mm (original bed)

(b) dp’=2.0mm (SPA bed)

p=3000kg/m3 p=2000kg/m3

Single Bubble Behavior of Two-Density Particles

t=0.278s t=0.557s t=0.835s t=1.114s t=1.392s

p=3000kg/m3 p=2000kg/m3

(a) dp=1.0mm (original bed)

(b) dp’=2.0mm (SPA bed)

Vertical velocity distributions of particle

and gas phases along the center line

0 0.5 1 1.5 2 2.5 30

0.02

0.04

0.06

0.08

0.1

0.12

0.14

z [

m]

Particle velocity averagedin each fluid cell [m/s]

Originalbed

SPA model

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Original bed

SPAmodel

Gas velocity [m/s]

0.12

0.10

0.08

0.06

0.04

0.02

0

Z

[m]

0.12

0.10

0.08

0.06

0.04

0.02

0

Z

[m]

(b) t=0.111s

0 0.5 1 1.5 2 2.50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

z [

m]

Gas velocity [m/s] Particle velocity averagedin each fluid cell [m/s]

Original bed

SPA model

0 0.05 0.1 0.15 0.2 0.25 0.30

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Original bed

SPA model

Bubble region(No particles exist.)

(a) t=0.056s

Similar Particle Assembly (SPA) model

for large-scale DEM simulation

Validations (comparisons with the original) Non-cohesive particles

>Slug flow occurred at the beginning of fluidization: similar

>Bubble diameter: almost the same

>Bubble shape: not clear with large representing volume

>Umf: fair agreement

Cohesive particles: the same tendency as the above

Binary (density) System:

>Bubble: similar

>Particle mixing: similar

SPA concept: promising. 2. SPA

Measurement

of

Stress-Deforemation Characteristics

for a Polypropylene Particle

of Fluidized Bed Polymerization

for DEM Simulation

M. Horio, N. Furukawa*, H. Kamiya and Y. Kaneko

*) Idemitsu Petrochemicals Co.

3. More Realistic Surface Characterization

Computation conditions

Particles

Number of particles nt 14000

Particle diameter dp 1.0×10-3

m

Restitution coefficient e 0.9

Friction coefficient μ 0.3

Spring constant k 800 N/m

Bed

Bed size 0.153×0.383 m

Types of distributor perforated plate

Gas velocity 0.156 m/s (=3Umf)

Initial temperature 343 K

Pressure 3.0 MPa

Numerical parameters

Number of fluid cells 41×105

Time step 1.30×10-5

s

Snapshots of temperature distribution in PP bed

(without van der Waals force)

0 7 150 7 150 7 15 ΔT [K]

Snapshots of temperature distribution in PP bed

(with van der Waals force)

Ha = 5×10-19 J

Ha = 5×10-20 J

0 7 150 7 150 7 15 ΔT [K]

Experimental

determination of

repulsion force

3. Surface Characterization

Polymerization in a Micro Reactor

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 10 20 30 40 50 60

Diameter[mm]

Time [min]

The micro reactor PP growth with time

Optical microscope images

5 min 10 min 15 min 20 min 30 min 60 min 1 min 0 min 2 min

Catalyst TiCl3

Pressure 0.98 MPa

Temperature 343 K

Reactor stage φ14 mm


Force-displacement meter

1

1

2

34

7

5

8

6

10

9

1

1

2

34

7

5

8

6

10

9

1: material testing machine’s

stage

2: electric balance

3: table

4: polypropylene particle

5: aluminum rod

6: capacitance change

7: micro meter

8: nano-stage

9: x-y stage

10: cross-head of material

testing machine


Repeated force-displacement characteristics

of a polypropylene particle

10-6

10-5

10-4

10-3

10-8 10-7 10-6 10-5

Force [N]

Displacement [m]

dp = 597μm

1st10-6

10-5

10-4

10-3

10-8 10-7 10-6 10-5

Force [N]

Displacement [m]

dp = 597μm

1st

2nd

2nd

10-6

10-5

10-4

10-3

10-8 10-7 10-6 10-5

Force [N]

Displacement [m]

dp = 597μm

1st

2nd2nd

3rd

3rd

FE-SEM images: whole grain and its surface

k ~100 N/m Fdp0.5x1.5 (Hertzean spring)

x

dp=597mm

Repeated force-displacement

characteristics of a polypropylene particle (maximum load from first cycle)

10-6

10-5

10-4

10-3

10-8 10-7 10-6 10-5

Force [N]

Displacement [m]

dp = 487μm

1st

1st

10-6

10-5

10-4

10-3

10-8 10-7 10-6 10-5

Force [N]

Displacement [m]

dp = 487μm

1st

2nd

1st

2nd

10-6

10-5

10-4

10-3

10-8 10-7 10-6 10-5

Force [N]

Displacement [m]

dp = 487μm

1st

2nd

3rd

3rd

1st

2nd

FE-SEM images: whole grain and its surface

Fdp0.5x1.5 (Hertzean spring)

x

dp=487mm

FE-SEM image of the top particle after

three times pressing


Particle surface morphology changes by

collisions

Plastic deformation in the case of PP

Hertz model stands OK

Experimental Determination of Cohesion

Force: Now on going


4. Lubrication Force

Lubrication Force and

effective Restitution

Coefficient

W. Zhang, R. Noda and M. Horio

Submitted to Powder Technology

Realistic collision process

Interparticle forces

Fluidization behavior

Spring constant Restitution

coefficient

? ?

Field force:

Electrostatic force

Contact force:

Van der Waals force

Liquid and solid bridge force

Impact force

‘Near Contact’ force:

Lubrication force

Heat transfer, agglomeration


Classical lubrication theory

For Liquid-Solid Systems; Tribology, filtration etc.

Why not in Gas-solid systems?

Lubrication force negligible ?

Introduction of “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

Can we avoid the paradox practically or essentially?


Davies’ development of lubrication theory to gas-solid systems

v1

v2

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

p(r,t)

)()( 21 vvtvdt

dh+-=-=

LFtFdt

dvm -=-= )(

• identical and elastic

• head-on collision

• rigid during approaching

Assumptions in classical lubrication theory

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 undeformed surface

Fluid is treated as a continuum

RrthtrH /),0(),( 2+= 22 )/(2

3),(

Rrh

Rvtrp

+=

mhvRdrtrrpFL /

2

3),(2 2

0, pmp ==

Examination of the assumptions in gas-solid systems

0.01 0.1 1

0

2

4

6

8

10

Ra

tio

of

FL

,0 t

o o

the

r fo

rce

s

h0/R

FL,0/G

FL,0/Fd

Order-of-magnitude estimation

• FCC particles: 50mm, v0=ut, at 20C

• Comparison of initial lubrication

force to other forces

• Particle radius as “near contact

area” or “lubrication effect area”

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

=R

RL drtrrpF0

, ),(2p

=0

, ),(2 drtrrpFL p

accurate

more reasonable with large

lubrication effect area

R: particle

radius

h0: initial

separation

Numerical solutions for pressure distribution P

ressu

re

Relative radial distance r/R

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

numerical

analytical with paraboloid

approximation

• Pressure decays to zero much more slowly than that with paraboloid

approximation

• Contribution of pressure in the outer region to the lubrication force

may play an important role

• Numerical calculations for lubrication force are needed

Avoidance of “Stokes Paradox”

• Assume 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

0.0 0.2 0.4 0.6 0.8 1.00

5

10

15

20

25

Ra

tio

of

lub

ric

ati

on

fo

rce

to

init

ial

va

lue

FL

,0 a

t h

0

Ratio of surface distance h/h0

approaching

detaching

hmin/h0

contact

• FCC particle: 50mm, v0=ut/5

• Fluid: Continuum

R

h

R

h

F

FhK

anaL

numL 2

,

,

1 lg035.0lg281.0041.1)( --==

+-== Rhh

vRrpdrhFR

anaL

11

2

32)( 2

0, pmp

Surface roughness of FCC is observed

to be one tenth of particle radius

Maximum lubrication force is reached

when roughness make contact

To realistic particles, stokes paradox is

avoided


Case 2: hmin<l0 • Particles in this case have relatively smaller roughness

• Non-continuum fluid effect should be

considered in the last stage of approaching

• Maxwell slip theory (Hocking 1973) was adopted

1E-8 1E-7 1E-6 1E-5 1E-41E-11

1E-10

1E-9

1E-8

1E-7

1E-6

Non-continuum fluid

Continuum fluid

Lu

bri

ca

tio

n f

orc

e F

L (

N)

Surface distance h (m)

v0=ut/2

v0=ut/5

• GB particle: 50mm, v0=ut/5

• Fluid: Non-continuum

R

h

R

h

F

FhK

slipanaL

slipnumL 2

,,

,,

2 lg009.0lg082.0309.1)( --==

( ) ( )

+

++++-

++=

Rh

lRhlRh

h

lhlh

l

vRF slipanaL

00

002

0

2

,,

6ln6

6ln6

12

pm

=

h

l

l

vRF slipanaL

0

0

2

,,

6ln

2

pm

l0>>h

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


Case 3: hmin is comparable to Z0

• When the surface distance can be approached to the dominant range

of van der Waals force, -----

1E-10 1E-9 1E-8 1E-7 1E-6 1E-5 1E-4-1.4x10

-6

-1.2x10-6

-1.0x10-6

-8.0x10-7

-6.0x10-7

-4.0x10-7

-2.0x10-7

0.0

2.0x10-7

Forc

es F

(N

)

Surface distance h (m)

FL

hvw

Ftotal

Forc

es F

(N)

Fvw

• GB particle: 50mm, v0=ut/10

• Fluid: Non-continuum

)()( vwL FFtFdt

dvm --=-=

212h

ARFvw -= A: Hamaker constant

Magnitude of van der Waals force

increases more rapidly when h -> 0

A characteristic distance hvw 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

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

pm=where Ratio of particle inertia to viscous force

Critical Stokes Number 2

**

6 R

mvSt c

cpm

=*

2

** 2

6c

ee St

R

mvSt ==

pm

• vc* is called “critical contact velocity” under which particles cannot make

contact due to the repulsive lubrication force in the approaching stage

• ve* is called “critical escape velocity” under which particles cannot escape

from the lubrication effect area and will cease during the separation stage

)()( min0

* hfhfSte -=

f(h): characteristic function

+-

+-

+=

Rh

h

Rh

h

Rh

hhf 32

1 ln004.0ln079.0ln962.0)(

0

0

2

0

0

2

0

26

1ln6

1ln636

161ln6

36

1)(

l

R

h

R

Rh

l

l

Rh

h

l

l

hhf -

+-

++

++-

+

+=

Case 1

Case 2,3

Examples and discussion

20 30 40 50 60 70 80 90 100 110

0.0

0.2

0.4

0.6

0.8

1.0

Re

sti

tuti

on

co

eff

icie

nt

e

Diameter of FCC particles dp (mm)0.1 1 10 100 1000

0.0

0.2

0.4

0.6

0.8

1.0

Re

sti

tuti

on

co

eff

icie

nt

e

Stokes Number St

Case 1: FCC, hmin/h0=1/10

ut

ut/2

ut/5

ut/10 ut/20

ut/50

umf hmin/h0=1/20

hmin/h0=1/10

hmin/h0=1/5

Case 1: FCC, different roughness

Under same approaching velocity, effect of the lubrication force on larger

particles is less significant than on smaller particles

The independent effects of particle size and approaching velocity on the

coefficient of restitution can be included in the consideration of Stokes numbers

Collisions with Stokes numbers less than Ste* result in a restitution coefficient

to be zero, consequently causing cluster and agglomeration to occur

Examples and discussion

20 30 40 50 60 70 80 90 100 110

0.0

0.2

0.4

0.6

0.8

1.0

ut

ut /2

ut /10

ut /50R

es

titu

tio

n c

oe

ffic

ien

t e

Diameter of GB d (mm)

20 30 40 50 60 70 80 90 100 110

0.0

0.2

0.4

0.6

0.8

1.0

ut

ut /5

ut /20

Re

sti

tuti

on

co

eff

icie

nt

e

Diameter of smooth GB dp (mm)

Case 2: GB, solid line: with slip, dotted

line: without slip Case 3: GB, solid line: with slip and van der

Waals force, dotted line: without slip

Consideration of non-continuum fluid weakens the lubrication effect and thus

increases the values of the restitution coefficient

The lubrication effect is more significant in case 3 since particles can approach

much more closely so that the effect of non-continuum fluid may be more

significant

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 according to 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.


Industrial Development and

Fundamental Knowledge

Development need each other

Wishing much frequent

Exchange and Collaboration

between Physical/Mechanical

Scientists and Chemical

Engineers

In Japanese very

old folk song

Ryojin-Hisho:

Asobi-wo sen-to-ya

Umare-kem.

(Were’nt we born

for doing fun?)