Discrete particle-wall interactions in powder flow

0009-2509/87 $3.00 + 0.00

Chemical Engineering Science, Vol. 42, No. 4, pp. ?1>723, 1987.

Printedin Great Britain.

Pergamon JournalsLtd. DISCRETE

PARTICLE-WALL

INTERACTIOWS

3.J. Department

of

POWDER

FLOW

BRISCOE

Chemical

Imperial London

Abstract

IN

Engineering,

College, SW7

2BY.

This paper reviews the use of established contact mechanics as a means of modelling particle wall interactions particularly wall friction. The contact mechanical solutions adopted are first order and when these are combined with simple models of the particle-wall morphology they provide a means of predicting the wall traction and its variation with the average normal stress. These models are applied to the results obtained from three particle-wall studies and are shown to provide a reasonable account of the data. Introduction More stresses nuum

ance,

The

an

for

amenable

effective

stress

tions

and

interact

method

of

field

is

comparatively

upon

which

ton's des

recent not

this

realistic

Woodcock's similar

reasons and

for

complex.

of

most

Not

these

and

bulk

characteristics 42;4-I

the

of

the

variathe

the

presently

also

die.

costs

faced

with

that

this

there

situation

obviously,

the

contacts are

there also

processes

are

number

properties. also

have

at least

ratio

are

the

incredibly

of

by

surface,

Some

many

for

tic.

The

of

stochastic

a

Hill

this

nent

of $.

over

inter-

powder,

these

scalar

proper713

is

p is

stress

in

normal

diameter

to and

ratio

plane

of

the

the

ratio

the

of

in

flow

that

the

the

appa-

expo-

friction of

the

plane.

H its

plastic good

interface

a Jansen-Walkerll for

the

roughly

the

as

a constant

a restricted

predicts

some

the

of

as

intrinsic

characteris-

for

scale

is

the

make

as

remarkably

solution

defined

those

and

a

order

which

deformation

enhancement

will

some first

such

solution"

the

stress

sample

bulk

in

response and

and are

poly-

parallel

confined

experiments

condition

rent

the

sample

solid

between

apparent

coefficient

the

experimental

friction

approximations

friction

model

or

both

a polymeric

the

is

There

strains

stress-strain

the

the

wall

the

for

coeffcfent

characteris-

governed

of

analyses

stress:

occurs.

the

property.

predicting

interaction

potentially

but

large

are

cases

of

yielding

why

only

powder

the

as

1 exemplifies

different;

whilst

solids

avoided

and

a disc

rather

coherent

polymer

as

systems

propagate

Figure

necessity,

tem-

parti-

particle

transmitting

compressed

both

both

compressive

are

external and

interactions.

in

wtth

when

the

Of

for

conveniently

a solid

cohesive

wall

that

bodies'.

substantial

Dr.

be

for

In

aspect

of

is

case

of

a function

intrinsic

are

is

true

compared

the

point

plates

provi-

computing

often

homogenous

many

particle-wall

cannot

to

to

environment

generally

distances

is

mer

of

potential

current

and

powderas'.

Thorn-

the

is

configurations

have

which

This

effects

hence

this

are

walls

Dr.

sensitive time,

factor

behaviour

line

particle

purposes.

are

to me

involved

depressingly face

and

of

the

simply

solid-solid

processes tics

long

they

field

of

routine

occurs

key

First, laws

this

techniques

these local

exampleI.

with

simulations for

of of

slip

then

from

example also

wall

restrictions3.

It two

even

remote

but

the

approaches

modelling

in

mechani-

walls;

the

notable

familiar

quite

unacceptable

CES

an

approach

limitations: for

are

work'

only

most

the

I am

system.

of

and

second

containing

be as

cle-particle The

these

can such

perature4-6.

compli-

interior

interaction

rare

concentrated

the

the

the

particle

arrays,

the

in which

which

then

continuum

characteristics

theory

Discrete

within

ties

variables

is

is

Some

of

conti-

or

system

predictions

way

with

in

particles

notions.

the

propagation

stress

the

using

provide

variations

assembly

of

yield and

analysis

do

the

described

assembly

rheological

techniques

not,

are

example, to

or

than

powders

terms.

ascribed

cal

often in

the

forces

interface D is

the

height.

For

analysis

predicts

the

transmitted

the

nor

a

B. J. BRISCOE

714

ma1

stress

to

the

surcharge

with

reflected

in

the

parameters

have

before

now

but

friction tens.

on

the

and

tion

could

ous;

the

and

the

be

is

parallel The

tions. gates

in

and

cle-particle

is the

which

which

is

hence

obvicharac-

are

normal

stress

the

direc-

stress

clearly

the

condi-

by

applied

in

D/H a geomet-

reason

walls

powder

be

traction

The

the

manner

the

importance

of

propa-

critical

interest

in

e

parti-

-7

interactions.

The is

to

the

small

not

wall

the pla-

apply

for

constrained

different

as

the

also

would

avoided.

is

the

describes than

powder

the

or

this

significances

could

there

powder

and

response;

rather

where

teristically

it

same

we

to

stress

-{$}

p parameter

walls

hence

condition

to

the the

solution

ratios

normal

exp

compressive

In principle

Hill

ric

applied

a form

question

useful

to

now

arises

attempt

as

to whether

a treatment

of

par/

ticle-wall

interactions

intimate

interactions

realistic

is we

problem

of

stresses

because of

mine

value

the

ing of

of

these

gross

inherent

selected

data

of

ptinciples. discrete

ped

to

tions.

The

ling

on

the

in

order

wall

paper

for in

see

the

spite

the exa-

predict-

the

light

that

into

in

by will

models

they

do

behaviour

of

their Fig.

with

to

model

is

Is

this,

then

applied

Finally, as

notion

to

the

a means

of

develolimita-

of

and

2.1

Nominal General A

great

Point

of

model-

strains

between

of

the

mean

comments deal

is

with

contact

in

general

des-

the

one

to

be

and

we

of made

are will

stress,

for these

be the

P,

and

r;

Figure

an

elastic

the

real

parameters

parameters

about

friction

regions loaded

parameter

stress,

quantities

stresses

contact

derived

contact

A knowledge

adhesion

only

two

the

they

paper

shear

these

the

when

this

and

normal

interface

predictions known,

In

bodies In

about

least,

produced

solid

area

tact.

Problems

at

together4B5.

mean

traction,

Contact

terms

preoccupied

three

value

criptive

defines 2.

strain for HDPE and solid (upper)' and respectively.

of

is and

e Stress against PP as coherent powder (lower)'

1

of

a number

the

contact

assumptions

assessed

particle

mechanisms

illustrate

Following

its

a presentation

contact

particle-wall

systems.

approach

the surface

controlled

will

begins

introduce

particulate

with core

assumptions.

microcontacts

the

We

,

that

difficult.

This

flow

paper

the

faced

insights

during

The

be

be

certain

comments.

apowder

to

interactions

additional

particle

particle-particle

are

bulk.

their

admitted

certain

they

the

wall-particle

offer of

also

identifying

response

have

of

likely

will

of

internal

we

modelling

interactions Naturally

view

with

Recall,

interactions. the

in

the 2 conallows

magnitude

generated

in

the

of

71.5

Particle-wall interactions

influence shows

r-” Px

the

crossed

x

n

force

W c,

is

rougher

adhesive

are

several

approaches

the

2

Mean The

Fig.

then ing

be

confirmed

bodies.

from

by

difficult scaling

large

contacts

ments

in

contact

whilst

the the

friction

against

the

stored

14. to

contacts

the the

the

is

comparable

the

of

dimensions the

loped

size in

between

the

of

There topic4,15,16 to

the

Small is

but

S is

we

of

contacts

rigid

are

the of

envisaged.

the

deve-

adhesion

higher, group

described-

literature

will

confine

which

on

this

oursel-

concerns

the

an

the

distribution

value

is

n

of and

R we

has

is

rough

but

it

is

mainly

heights of

which

with

that with

are

contacts

apart.

the

pull

which

are

The

third

the

a variance

approximately

the

opposing

heights

produced

an the

is

asperities

others,

strain

a rough

adhesion

becomes

surfaces

varies

radii.

against

whilst

normally,

small

contacts

body

that

int-

behaviour

successfully

contacting

assume

the

of

with

This

been

types

elastic

contact

to

The

is

two

illustrates

impression

rough

interactions

If we

by

3b an

down

large

asperity

the

no

defined

scaling

surface

of

push

stored

the

I shows 3 B is 2

increases

larger

elastic

Some

distributed the

3c

together

surface-

Bodies

the

Three

have

is

radius

Table

dimensions. of

feature

surfaces

sphere

3c

contacts of

distribution

often

is

problem

for

independent

R

the

the

contact.

R assuming

counterface"-**.

important.

asperities rather

of

and

effective

a much

similar

Figure

as

on

term,

the

Figure

Figure

radii

additional

a

djmensions

the

reduced

is a constant

depending

of

of

the

(mainly

energy

geometry

of

adhesion

treated

the

while

of

in

2n

a plane.

asperities.

large

The

work

roughness

a contact

behaviour

roduces and

the

are

forces

adhesion

The

If

single

that

asperities

and

often

as

a

balance

equivalent

values

results. at

B is

curvature

the

and There

investi-

adhesion

free

of

to

bodies may

II:and

against

fibres

adhesive

this

100mJm-2.

idea

large

as

is

infer

have

the

particle-plane

where

area

calculated

of

section

we

the and

shown

point

extensive

narrow

contact

practicablel*-

This

is

latter

for

this

geometry

body.

the

next

are

The

to

a plane

surface to

of

is

of

monofilaments

Adhesion

ves

of

it

work

data fine

adhesion

thermodynamic mutual

contacting

monofila-

on

the

surface

which

of

contact

particle

The

particle.

using

straightforward

on

have

a

behaviour

size

which

the

of

asperities

is

resort

orthogonal

behaviour

is not

we

a sphere

troublesome

Scaling predict

fine

as

particles

we

a surface

radius

is

mechanical

312

the

these

B R S where

between

contact

decreases.

the

the

Then

S is

could

of

to

to

of

contact

then

due

descreases

can

contact-

incorporating to

measurement

is

hence

produce

same

on

small

or

Two

variables

predictions

experiments

operations

monofilaments.

nominally

such

and

elastic work

two

titania

available

) and

surface

ranging

stress

of

3a

their

force

I

monofila-

titania

surface

the

(Figure

proportional

Manipulating

generally either

interface value

the

significance

elastic)17-ia.

F/A

contact.

the

that

separate

of

higher

measured

model,

?rn=

a function

the

assume

to

Table

terephthalate

the

perfectly

**m

as

contactl'.

point

IF

topography.

required

The

we

*W/A

surface

polyethylene

ments,

gate Pm

of

by with

are u, then the

B. J. BRISCOE

716

SOLID

BODIES

I

A

(>

dWn

64

/‘“.

(b)

>

>

n&v3

PARTICLE

I

nL2/3

I

.

d-cl

Fig.

=3/ *Rv

o; E is

adhesive mean

force

and

radius.

the governing VRV o/RS

stingly the

now,

modulus;

(high will

E)

None

The

the

For solids

of

two

the

greater adhesion. is

not

these

behaviour

E

is

case

roughness

for

which

may

sive

forces

and

Friction

adhesion

a dominant

If

will

occur

At

gross

sliding

upon system

the

the

do

how

contact

how-

adhearea

contacts.

force

is

applied

Figure

3b

and

as

critical

will

character

and

in first

some

to

occur

Contacts

depicted

sed14.

they

may They

as the

particle

Point

a tangential

contacts slip

guides

influence

do

which

interactions.

for

of

contacts

a few

may

stresses

of

introduce be

particle-wall introduce

Nor

systems6,24.

asperity

ever

its

smooth

body

in many few

Intere-

stronger the

treat in

. treatments

a

and

a function

the

adhe;T21ess23;

is

is

the

a given

with

other

is

was

R

energies,

each

parameter

adhesion

feature

The

B R S where

that

lower

this

decrease

inelastic

such

modulus.

oppose

adhesion

the

surfaces.

as

Schematic illustration of solid and particle array contacts

3

elastic

elastic,

E

magnitude

the scales

asperity

surface

ARRAYS

the value

force

is

increa-

of

the

and,

of

transmission

contacts,

the

the

micro-

commence force

to

c,

force,

depending

motion

will

be

717

Particle-wall interactions

apparently nuous. of

the

ward

continuous

sliding

since

the this

it

contacts

often

as

ribution

forces

Larger

damped

The in

in

is well

require tions

First

sorts

of

or

is

friction

the

real

area

is

and

hence

with

this

process

work-

this

is

A,

tive

pressure, the

z varies

P, current with

says

two

of

variables is

the

one

context. P as 29

As

is

equate

the

applied

the

area

contacts elastic

(equivalent

given the

normal inclu-

Poisson's

strictly W*

to

correct

(WC+ We

load14.

A is

by

total

constant

and

not

Then

force.

fibre

WA)

further

unaltered

by

the

as

F = zA

contact

zo +

=

is

aP

and

P = W/A

we

have

if

the

Hertzian

2/3

zo(DR) 2/3 - c w*

that

The "Adhe-

R and dies

WC

may

and

D,

other

z/3

W*

+

+

aW*

(3)

aW*

determined a can

c is

from be

adhesion

obtained

stu-

from

a constant.

friction Figure

yield

static

quantities The

two on

contact

of

fact

4 shows

friction

polymeric equation

of

as

with

W

in

the

load

and

The

they

particle First

tion.

fibres

we

applied

is

points

may

relevant

systems

in

may

nx2

defined

the

range by

by

for

good. be

in

W

based

briefly

the

treat-

a following

consider load

mean

load

a prediction

are

a limited index

experimental

agreement

additional

introduced ment

the

F against

3I*.

Three

sensi-

interesting

a matter

be

z. and

studies;

contact

mean is

7

F =

conditions.

which

it

it

frictional

both

the

Modulus

but

adopted.

a point

elastic

to

only

potentially

but

the

A is

Although

contact

mechanisms are

W * is

a a

contact

the

the

magnitude

is

a plate);

,I1 where

that

of

fibres

sufficient

tangential

using

asumptions

The

contact

For

on

W A is

losses

subtle

shear

the

of

1.

by

the

that

stress

where

one

orthogonal

Young's

zo', for

directly

assumptions

a combined

E the

area

contact

the

a sphere =/94 D is

real

from

for

ro,

constants

measured

Table

x(DW*R)

geometry

assume

the

interface

of

this in

of

is

be

several

do

where

at

termed

terms;

I will

assume

scaled

will

the

shear

to many

is

and

assumption.

many

other

deformation

and

all

sometimes

of

a function

interface

in

rs.

we

friction for

is

apparent

a reasonable

simply

and

the

calculated

be

system

The

can

ratio.

two

the

generally

it

fric-

contact

section

contains

are

energy

overall by

this

product

stress,

the

model,

stated the

The

and

(2)

characteristic

cannot

load,

expe-

dissipation

6 and

-

interface.

ding

adhesive

energy

accounts is

there

subsurface

interface

Model",

area,

is

by

fric-

approximaby

upon

are

contact

con-

Adhesive

contact.

the

In

deal

but

to

partially

area27,28.

sion

that

interface

of

scaled

interface

and

mechanism;

the

describes

and

large

a'

given

to maximum

in

exp(a'P)

0

described

pronounced

proved

friction27*28.

near

more

more

It does

been

T-T’

providing

motion.

assume

attributed

very

term

we

is

typi-

many

such

assumptions has

is

have

or

velocity

or

depending

which

naturally

practiced12-14.

value

deformation

observed

mean

temperature,

sliding

and

maximum

dist-

the

(constant

For

a probability

systems

numerous whose

rience.

tion

of

fibre

velo-

follow

show

their

modelling

the

The

which

not

to

always

are

(1)

variables

occur,

distribution

and

will

the

will

be

contacts

contacts

discontinuities

tacts

will

a gamma

Gaussian;

asperity

of

probably and

+aP-

0

apriori

friction25-26.

will

z-z

nature

possible

a system

events.

pattern

of

normally

discontinuities

force

a stochastic

tion

the

the

straightfor-

a knowledge

of

disconti-

of

quite

of

stick-slip

frictional

heavily

not

requires

dependence

calL4.

is

is

behaviour

fibre

not

modelling

motion

although

predict

city

or noticeably

Mathematical

sec-

variation

F

examining

F = RWn

.30

An

718

B. J. BRISCOE

not

be

explored

in

the

asperity

contacts2q~35.

Discrete

Particle-Wall

context

of

multiple

Interaction

in

Assemblies Figure

10

usefully rough

O-

0

W/UN

which of

tion

40

Fig.

4

Mean frictional force, F, against load for fine contacting polymer monofilaments (diameter ca. 2Op1n)~'.

examination be

of

a function

equation of

the 213

ratio

as

values

of

n

+ 1 whilst

Next

we

+

2/s*

likely

result

point

nor

ness

on

large

if

adhesion

of

in

realistic

case

In

3d

Figure

where the

for

is

u = 0 and same

u = 0 but

curve

surface

Figure

3c

increases pressure

case

number

with

load

unity32.

W/p0

po,

The

z ensures

that

contact n>

n'

are

on

mean

n

is

contact

3d

is

n is

and

n

so-called plastic

mean

asperi-

the

constant unity

near

as

the

A is

In general n

ranges

pressure but

this

from

feature

will

is

the

3c

is

approp-

Case

3d

(3g)

predicts

(3f)

result

will

produce

cases

contacts

then

both

will

be

that

cases

if

Particle

will

be

for

in

the

vertical

direction

In

the

friction

plane.

stress

at

while

the

the

wall

contacts

and

we

supports

the

same

will

be

not

because

of of

the

the

powder

neglect

the

naturally

the

particle

treat

the

cess-

We

equation

(3)

area

with

in which the now

wall.

to

a suitable normal

Three

discussed.

whole

We

must autonomy

contacted normal

shared

by

each

wall

of

the

the contact

load.

locally

near

include

the

the

are

to We

occurring

as

F

assumptions

or

discontinuities

write

3i

individual

resulting

assembly.

now

are or

have

they

either

dissipation can

the

purpose.

that

motion

nearly

A and

major

stresses

through

both

describe

then

the

This gene-

in

the

the

interface

shall

also

fluctuation wall

and

a quasi-static out

description load

pro-

extensions

N particles

examples

3c

pro-

plastically

to

fraction

the

is

3h

sum

The

is

case

also

particles

assume

the

F is

to

this

that

A

load.

Three

assume

case

that

deform

the

is

described

The

the

that

to

array.

adopted

been

particles

they

contributions

wall

of

that

predict

step

A a

produce

proportional next

that

has

the

case

a function

contacts.

If

they or

is

also

to W.

such

n'

the

to W and

rough

are

the

migra-

as

case

to of

in

have

particle

a solid

F e Wn'where Z/3.30 this a/r o(DWR) ' as autonomous discrete

way

2/3

dependence

the

we

completely

riate;3f. 2/3 but

wall

A is

I

If

W

in

where

restriction

interface,

on

transmission

The

particles

restricted

rally

the

2/3.

the

case

n is

the

In

constant

that

everywhere34. n' to W where

unity.

case

intermediate

remains

then

at

contacts

Figure

the

such

pressure

the

is

this In

occurs

stress,

proportional to

'ig;

precursor.

contact

simply

as

if

and

3e

the

asperity

For

Figure

are

sphere32*33.

remains

constant

in

shows

3c.

particle

smooth

asperities

friction

case

asperities

and

as

predicted

flow

the

asperity

shown

3e

of

Figure

likely

the

be

of

is

The

a fairly

asperieties

a larger

the

deformation ty

of

remains

Archard

all

is

neighbour

to 3d;3g.

can

the

for

distribu-

solids3';

Figure

rough-

earlier

a normal

a less the

per

predicted number

rough

all

height;

where

of

a

of

no

of

and

For

have

proportional

the

planes

planes.

which

deformation

access

portional

values

is neither

which

shown

small

influence

discussed

height

will

large

consider

The

terms

asperity

for

contact

was

contacts

tion

the

elastic3'.

A,

For

should

n will

= a;

however.

scale

n

that

a/c

always

a,

W

3 shows

analogy the

against

equivalent

I

I __

-ve

(3a)

the

between

against

terms

for

is

solids

arrays

5 k

3 shows

drawn

to per

unit

of

is

transmitted

of

this

the

approach

719

Particle-wall interactions

Wall

Traction

in

Figure an

5 shows

experiment

tion

of

force

or

at or

to

The

Figure

x10

with

the

stress

is

computed this

total

is

the

motion of

of

the

particles

kiln

is

fixed

zero of

hence

powders

silica

data

glass

sphere

The

load

these

are

are

and

radii.

the

aa

is

of

data

accurately fall are for The

on

pass

Measure S; Fig.

normal 5.37

wall

stress

P against

that to

the of

mass

0.2

the

adhesive

trivial

linear

and

The whilst

a smooth

experiments

6

through

the

is

in

Fig.

diameter

cohesive.

listed

rough

of

length

its

load

the

restraint

that

not

for

assuming

only

I 8

the

disconti-

data

a function

normal

data

indices

data

smaller

the

of

many

shown

indicating

component the

sets

(radius

rotation34.

are

kiln;

(5OOmm)

profile

character

the

as

the

Both

common

stress

friction, of

kiln, in

(47.71mm). the

wall

types

rough

transducer

7 are

origin

two

spheres.

during

there

Figure

migra-

with

rather

a diaphragm

tor-

wall

cohesive

glass

hydrostatic

In

the

experiment

interface

normal

detectable

nuities.

from The

described;

diameter)

a smooth

a horizontal

studies

smooth

of fric-

frictional

of

for

a typical

particle

Although

the

wall

over

estimated

minimum

are

of

repose.

induce

very

6 is

obtained

is

results

and

move

form

section

the

The

of

the

particles

silicas

they

the

wall

with

tion. of

in

angle

is designed sliding

a schematic

kiln33*34.

the

the

Kilns

to measure

when

surface

cylinder

que

used

powders

curved

Rotating

the

Table

2 for

using

tubes

particles

2 2;

curve.

of

produce

U 5

W/N

Fig.

5

Section of a rotating ing particles.

kiln

contain-

F against particle load, W, for apparatus shown in Figure 537. sand particles are rough whereas glass spheres (b,s) are quite smooth.

Fig.

7

load

indices

particles

near

yield

unity

indices

whilst near

the 0.85.

smooth

the The the

720

B. J. BRISCOE

These

data

lows.

As

the

reases

so

does

A*

scales

particle

the

the

interpreted mass

we

in

as

the

fol-

kiln

inc-

apparent

number

case,

vails

be

contact area, A*; 7/g with W and hence

approximately

so will valent

may

of

(d),

case

anticipate

If the

contacts. (g)

that

in

Figure

F will

be

equi-

3 pregiven

by34

neously wall

7/9

+aW

stresses

rithmic

coordinates

mustard

seeds,

an

extension

requires

the

described of

of

range

shown is

smooth

plate

include

particles multiple

3g

3 and

consistent silica

or

the

condition ensure

also

allow

F as

IY.

to

an

the may

have

of

case n

be

convenient

in

of

W and

demonstrated rough to use

as

particle the

model

0.80

silica

i or

unity

c,

the

P is

close

to

(4)

has

stress

this

mustard

the used

for

index

dilating previously

n is

are

index

ranges and

case

expected

value

which

of

ca.

has

been

on

single

for

smooth

measurements

predicted

rough

and

polyethylene

predicted

particle

the

normal

seeds

The

range

the

c singularly hence

glass

load

model

case

the

system,

as

during

The

direct

contacts

glass

from

0.92

particle

to

0.95

radius.

We

will value

-

1

been

previous-

contact

it

is

0

2

asperity

flow

the

there

2!

+a)

a mean

the

for

0.7

indices

alone

absolute

viable

although

is

unity.

this

0.72

i but

T;

where

glass for

that case

equation the

h in

dynamic

1 and

assembly;

The

The

1,

a smooth by

sensing

The

wall

to

particles-

is

result.

for

of

contacts.

load

The

is near

confirmed

produce

expression:

F=W(F

unity.

the

of

of

be

that

case

embodied

a prediction

to

capable

mean

to

the

polyethylene metallic

averaged

(i.e.

applied.

i as

pellets

likely

close

correspond

the

cases

c at

is

be

logo-

The

flow).

and h or

computed

autonomy,

hence

are

by

may

for

will

index

the

and

to near

decreases

induced

value

The

observed

and

model

the

value

suggests

and

a function

For

The

for

a knowledge

values

reasonable.

hence

that

The

from

wall.

that

decreases

stress

the

appropriate

contacts,

not,

at

evidence

index

equitable

calculated

rough

particles

will

shown

is

with

h,

n

experimental

asperity

Figure

of

which

are

an

is

measurements.

respectively

and

a smooth

particle

static

on

particles,

spheres

which

many

Figure

walls3'.

three

by

tangential

plotted

for

glass

of

is

assumptions

particles

227,34.

the

in

the

estimated

the

spheres

depicted

of

be

a and

to

the

value

in Table

close

case

by

may

c' and

of

This

(3).

including

load

effective

load

equation

adoption

earlier

sharing The

of

silo

detected

transducer action

and

data4's4'

the

are for

as

pellets,

normal

in model

typical

(flow)

(4)

the

8 shows

data F = C'W

measure

contact of

the

stress

softer

of

the I

two

contacting

particles

bodies.

and

wall,

For

F

hard

s a W and

-2

solids, a may

be

I

L

obtained

by

studying

single

particle

by

Particles

wall

-1

0

1nP

2

contacts31,37,38. Wall Smooth

Friction Silo Dr.

loped

Generated

Tuzun

clever

at

Fig.

Walls and

his

transducers

associates which

will

have

deve-

simulta-

8

Xn z against transducers polyethylene; b.s. smooth

.in P sensed by in silo walls. m.s. mustard glass spheres41.

p.e. seed,

Particle-wall

may

account

for

predictions riment

by

porosity

the

based invoking near

wall

case

f the

analogue

model

adopts

such

that

2R<

S 7

for

the

glass

polyethylene average

lute

3R

of

dilatancy point

is

that

ced

a contact

that

produced

distribution cases

model

of

included.

The

the

dilatancy

asperity

predicted

to

the

is

abso-

P /

a

interesting produ-

similar

to

coherent

body

heights.

In

increase

with

a

both

0

towards

Fig.

9

Traction The

Figure

During

manifest where

in p is

normal

the an

transmittal

now

be

ratio

imagined between solid;

and

g.

case

practice planes

to

simulated

ing

by

and

is

an

data

plotted

on

Figure

9 for

pacted

in

ranges

from

this

these

the

equations and

(1)

hence

is

case

pins

the

data

and of

for

16%

limit system

(3)

is

are

n

of the

very

expected

systems

w/w

to

a term small; value

com-

water.

8/9

in

n

of

IJ 8/9

and

the

height

of

in

the

tion

3 is

from

compact

n'for

compact

be

approach

generation

similar

being

to

the the

and

when

the

onset

of case

a good

change

the (1).

account

friction

when

in

n

fric-

temperature. then as

be

used

to

a function

using

account of

the

a modified

along

the

lines

out-

Section11?43. to

that

incorporating

friction

accurately

due

context

Introductory

adopted

is

the

analysis

a relationship

in

may

friction

the

Janson-Walker

be

predicts

data

increases

temperature

of

0)

under

may

this

counterfa-e

wall

n'

with

(a =

area

why

gross

(e).

case

provides

value

also

with

lined

in

case

contact

to

also

absolute

These

may

or

the

a situation

increases

this

cooking

model

unity.

ca-

reason

c situation;

the

way;

incorporated

same

creates

for

The

elastic

However

contacts

the

flow

smooth

d.

for

increasing is

for

defined

load.

plastic

for The

given

of

819

tion

is

a well

n'is

slid-

co-ordinates42. of

I

contact

that

of 9

the

the

predicted

The

Figure

compact

in

case

over

Z/3 case

Archard2'

of

o ~1 0;

e.

number

the

with

particle

where

where

water

compacts

original

is

contacts; of

normal

readily

ended

data

the

f

contact

curvature

load

a

upon

in

lower

and

case

planes

to

values

particular

situand

compacts

presence

the

may

difficulties.

counterfaces;

mean

applied

maize

logarithmic

asperity

experimental

a surface

several

glass

betwixt

spherical

approximation

a plane

the

intermediate

slide

The

and is

powder

t””

W/N

#l/D

topograhy

compacts

spherically

over

0.0239

of

PT

P A the

wall

good

to

produces

for

of

and

alignment

sliding

corresponding

For

not

in

tangential

example

substrates.

remains

form

of

are

a term

flowing"

attempt

friction

suitable

this

is

in

which

wall.

some

"free

It

the

The be

as

the

stress

for

because

wall

of at

to a

depicted

tractions

ratio

surcharge.

coherent

is

PT/PA

normal

or

ation

The

experiment wall

average

stress

stress

Compaction

compaction

1 generates

0.9

Friction against load for spherically shaped maize compacts sliding against glass plates42. The load indices are shown.

unity. Wall

/

for

also

when

processes

which

a rough

of

is

n'

same

is

by

n and

the

force

section

an 0.98

frictional

term

S,

R is

prediction

the

or

appropriate.

to

The

0.2

state;

predicts

0.9.

the

721

expe-

parameter

0.85

radius.

a reasonable value

and

and

active

c is

from to

(4)

dilatancy

the

case

0.75

particle

yields

bed in

a dilatancy

of

between

equation

local

the

This

range

discrepancies

upon

interactions

studies

predicted. developed

given data the

If in

derived

PT/P,

Currently to model

equa-

ratio this

the

wall

B. J. BRISCOE

722

and

barrel

zones

in

these

traction single

examples

averaged

The

is

in

their

tractions

are

discrete

interactions

dered

a continuum.

as

Concluding

simple

nical

parameters

wall

serious,

but the

equality

in

each

adopted model

The

major

load

index,

of

wall

of

the rough

tacts not

only

also

in

of

interactions, relatively

the

has

been

value

and

indicate

established theory

as

to

to

the

been

gene-

particle

the

cohe-

assem-

Rc/pm 9 4-7 5.5 2.1

Load indices for particles sliding in rotatln kilns. Particle size 97 ca. 300 pm . Kiln

sand .. .. ballotini glass spheres

radius/mm

n expt

26.71 40.61 47.71

1.07 1.12 0.99

26.71 40.61 47.71

0.89 0.84

calnN 1 1 1

) :

0.837 0.835

very

mechanics

of The

demonstrate intrinsic mechanics

5.

3 for 6.

rough

con-

recent

years

friction

but

7. 8.

phenomena. to

2.

4.

significant of

in

1.

treat-

analogy

contact

References

3.

of

Figure

ideas

of

do

absolute

close

predict

subject

the

interesting

lubrication

contact a means

the

presented

method

the

bodies

unexplored.

paper

of

been

models

modelling

these the

of

Ti02/%

A

continuum

has

The

of

have

and

origins of

focuses understand-

Autoadhesion of monofilaments of various titania content; Rc is the calculate effective radius. Fibres have similar diameters ca. 20~.

Particle

weak-

measured

contacts;

order

2

is

and

tool

this

is

solid

adhesion

application

the

of

description

in

our

contribution

the

friction.

between

Table

each

traction

contact

approach

for

interactions

developments

from

indiscriminate

this

There

contact

assumptions

investigative

coherent

response

of

mecha-

critical,

bearing

microscopic or

methodology it

models

friction.

the

paupacy

this that

response..

feature

example.

not

the

least

0 0.05 1.0 1.5

the

between The

mean

of

, although

the

of

friction

particle

well

tribological

1

interac-

contact

the

are

load

validate n

wall

interface

model

bulk

account

with

Table

consi-

modelling

The

somewhat

the

to

ing

is

established

apparently

the

and

as

an

approach

for

values

capacity. invoking

contributions

using

discrete

here

rally

such the

adhesion

in

the

flow

not

blies.

intrinsi-

bulk

averaged

contact.

for the

the the

or

sums

accounted

from

of

sive

counterfaces.

mean

and

particle

at

smooth

adopt

neas

upon

ing

by

particle

upon

produced and

model;

minds

bulk

predictive

the

our

the

therefore

treated

certain

virtues

stress

of

but

of

focussed

tractions

stress

use

some

the

Clearly

interactions.

has

ca.

survey

has

powders

to

In

Remarks

This tions

necessary

are

wall

feed/compaction extrudets.

to describe

models

limited wall

the food

characteristics

compact. cally

it

parameters

transmission

The

in

screw

The

9.

particle/wall this

paper,

intention the

of

interpreting

this

potential

limitations and

is

11. 12.

of

friction of

10.

particle

13.

Trans.I.Chem.E., 60, R.M. Nedderman, 1982, 259. C. Thornton and D.J. Barnes, Acta Mechanica (in press), 1985. L. Woodcock and M.F. Edwards, Powder Techn. Pergamon Tech., 85; Particle Press 1985. K.L. Johnson, "Contact Mechanics", C.U.P., 1985. D. Tabor, J. Colloid and Interface 2. Sci., 58, 1977, B.J. Briscoe, Phil.Mag. E, 1981, 511. K. Ridgeway and K.J. Tarbuck, Chem.Eng. 1147. Sci. 23_, 1968, D.P. Isherwood, Plastics and Rubber Processing and Applications 2, 1982, 253. B.J. Briscoe and R.W. Nosker, Wear, 95, 1984, 241. R. Hill, "The Mathematical Theory of Plasticity", O.U.P., London., 1950. D.M. Walker, Chemical Engineering Science 21, 1966, 975. B.J. Briscoe and S.L. Kremnitzer, J.Phys.D.Appl.Phys. 12, 1979, 505. B.J. Briscoe, M-3. Adams and T.K. Wee in A.C.S. Symp. Ser. No. 287, 1985, 375.

Particle-wall

14.

15. 16.

17.

18. 19.

20. 21.

22. 23. 24. 25. 26. 27.

28.

29.

30.

31.

32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42.

43.

M.J. Adams and A.T. B.J. Briscoe, J.Phys.D.Appl.Phys. 18, 1985, Winkler, 2143. D. Maugis, J.Mat.Sci., 20, 1985, 3041. Aspects of D. Maugis, in 'Microscopic Adhesion and Lubrication.', ed. J.M. Elsevier, 1982. Georges. V.M. Muller and Yu. P. B.V. Derjaguin, J. Colloid Sci., 53, 1975, Toporov, 314. K. Kendall and A-D. K.L. Johnson, Proc.Roy.Soc. A324, 1971,301. Roberts, V.S. Yushchenko and B-V. V.M. Muller, .I. Colloid Sci., 77, 1980, Derjaguin, 91. Fuller and D. Tabor, Proc.Roy. K.N.G. 1975. 327. sot A345. of the K.L. Johnson in "The Mechanics Contact Between Deformable Bodies.', ed. A.D. de Pater and J.J. Kalker, Delft University Press, 1975. G.A.D. Briggs and B.J. Briscoe, J.Phys.D.Appl.Phys. 10, 1977, 2453. and D. Tabor, N. Gane, P.F. Ptealtzer e, 1974, 495. Proc.Roy.Soc. J.Phys.D.Appl.Phys. 8, K. Kendall, 1975, 1449. B.R. Dudley and H.W. Swift, Phil.Mag. 40, 1949, 849. D.M. Rowson, Wear 31, 1975, 213. B.J. Briscoe and D. Tabor in "FundamenSuch and tals of Tribology". ed. N.P. K. Saka, MIT Press; Cambridge, Mass. 1980. 733. in "Friction and TracB.J.- Briscoe, tion", ed. D. Dowson, Westbury House, Press Guildford 1981, 81. I.P.C. Briscoe and A.C. Smith, Reviews on B.J. Deformation Behaviour of Materials III 151. No. 3, 1980, M.J. Adams, J.A.K. Amuzu and B.J. J.Phys.D., J.Appl.Phys. (in Brlscoe, press). J.A.K. Amuzu, B.J. Briscoe and M. J.Phys.D.Appl.Phys. 2, 1976, Chaudri, 133. Proc.Roy.Soc.Lond. E, J.F. Archard, 1958. 190. Proc.Phys. H.G. -Howell and A.S. Lodge, 1954, 89. sot. 676, J. Greenwood in "Rough Surfaces", ed. T. Thomas, Longmans, London, 1982. N. Adams, J.Appl.Polymer Sci. 1. 1963, 2075. H. Radwan and M.J. B.J. Briscoe, Can.J.Chem.Eng. 61, 1983, 769. Streat, L. Pope and M.J. Adams, B.J. Briscoe, Powder Techn. 37, 1984, 169. B. Scruton and R.F. B.J. Briscoe, Proc.Roy.Soc. A.333, 1973, 99. Wilks, IJ. Tiiiziinand R.M. Nedderman, Chem.Eng. Sci., 1985, 40, 337. Powder U . Tiiziin and R.M. Nedderman, 3;, 1982, 27. Techn. and B.J. Briscoe, U. Tiizlin. M.J. Adams Chem.Eng.Sci. (in press). M.S.D. Fernando and A.C. B.J. Briscoe, J.Phys.D.Appl.Phys. l8_, 1985, Smith, 1085. "Englargement and Stanley-Wood, N.G. Compaction of Particulate Solids", Butterworth (1983).

interactions

723