Studies of Diffusional Mixing in Rotating Drums via Computer Simulations

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Studies of Diffusional Mixing in Rotating Drums via Computer Simulations

G. Kohring

To cite this version:

G. Kohring. Studies of Diffusional Mixing in Rotating Drums via Computer Simulations. Journal de

Physique I, EDP Sciences, 1995, 5 (12), pp.1551-1561. �10.1051/jp1:1995216�. �jpa-00247158�

Classification

Physics

Abstracts

02.70c 46.10+z 62.20x

Studies of Diffusional Mixing in Rotating Drums via Computer

Simulations

G-A-

Kohring(~)

Central Institute for

Applied Mathematics,

Research Center Jülich

(KFA),

D-52425 Jülich,

Germany

(Received

24

August 1995, accepted

in final form 29

August 1995)

Abstract. Particle diffusion

m

rotating

drums is studied via computer simulations using a

fuit 3-D model which does net involve any

arbitrary

mput parameters. Trie diffusion coefficient for

surgie-component

systems agrees

qualitatively

with

previous experimental

results. On trie

other bond, trie diffusion coefficient for two-component systems is shown to be _a

highly

nonbnear function of trie rotation

velocity.

It is

suggested

that this results from _a

competition

bet~veen trie diffusive motion

parallel

to, ^and

flowing

motion perpendicular to trie _axis of rotation.

1. Introduction

Rotating

arums

perform

critical functions in many industrial _processes.

They

_are utilized for tasks such _as

humidification, dehumidification,

convective heat

transfer,

facilitation of

gas-sohd catalytic

reactions

and,

not the

least, ordinary mixing

_or

blending. Understaiiding

trie mech-

anisms behind

partide mobility

in

rotating

arums is _an

important step

towards

refining

the

efficiency

of such processes. In the absence of mechaiiical

agitators,

momentum _is

imparted

to the

partides only

ⁱⁿ trie

plane perpendicular

to the _rotation axis. Motion

parallel

to the axis of

rotation _occurs

through

momentum

changes

induced

by

trie collisions betweeii

partides.

Since these collisions _{occur more or} less

randomly

in

time,

this motion _is diffusive _in nature. Until

now most computer simulations bave

neglected

trie essential three-dimensional character of these

systems and,

via two-dimensional

simulations,

concentrated _on trie motion

perpendicular

to trie rotation _axis

iii.

In trie present work the diffusive motion _in the

longitudinal

direction

is examined

using

_a fuit three-dimensional mortel.

Previous

experimental

studies of diffusive motion _in

rotating cyhnders

have concentrated

on

single-component

systems, 1-e-, a

cylinder partially

filled with _a

single partiale

type [2,

3].

One-half of the

partides

_were

dyed

_a different color _in order to make them

distinguishable.

These diffusion

experiments

were

simplified through

trie _use of _an ideal

starting arrangement:

initially,

the colored

partides

_are

axially segregated

_so that mixing occurs

only through

the diffusive motion

parallel

to the rotation _axis. Under these ideal circumstances, some

aspects

^of

(*) g.kohring©kfa-juehch.de

@

Les Editions de

Physique

1995

the diffusive

mixing

_can be studied

analytically

and

Hogg

et ai. [2] obtained

good agreement

between the

theory

and

expenment.

The present

study

focuses _on diffusion in

two-component

systems. Diffusion in two-component

systems

^is

expected

to exhibit characteristics not found in

single-component systems,

because the former have been shown to exhibit spontaneous materai

segregation

_[4]. ^In this form of

segregation

several bands

parallel

to the rotation axis _are formed. These bauds altemate be-

tween the two material

types

and there is

normally

_an odd number of such bands. The nature

of this

phenomena

_is not

completely understood, however,

it is dear that diffusion must

play

an important ^rote. Indeed the results to be

presented

here demonstrate _a distinct difference between the diffusion in

1-component

and

2-component systems.

In the

following section,

a three dimensional mortel for

colliding,

visco-elastic

spheres

is

presented

and its numencal

implementation

_is discussed. The _next section describes the ex-

penmental arrangement. Following that,

trie results _are

presented

and discussed. The paper condudes with _some

speculative

comments on the mechanisms behind materai

segregation

in rotation arums.

2. A

Computational

Model for

Visco-Elastic, Mesoscopic, Spherical

Particles

The contact-force mortel

presented

here _is based _{upon a} visco-elastic mortel for the collision of two mesoscopic,

spherical partides developed

_over the last 100 years

by

Hertz

[5],

^{Johnson et} ai.

[6],

^{Kuwabara and Kono} I?i and Mindlin [8]. This mortel is free of

arbitrary

_parameters in

the _sense that ail

input

variables _are materai

properties

^amenable to

expenmental

verification.

The mortel involves four

key

elements:

1) particle elasticity, 2)

_energy ^loss

through

internai

friction, 3)

attraction _on the contact surface and

4)

_energy loss

through

the action of frictional forces.

2.1. PARTICLE ELASTICITY. As demonstrated

by

H. Hertz

[Si,

^the deformation of elastic

partides

of finite extent leads to _a nonhnear

dependence

between the compressive force and the compression

length.

~~Î~~~'~ E~

il v(~Îj il v)) fi~~Î~ ~~'

~~~

where,

hj

₌ R~ +

Rj X~ Xj

_[,

(2)

and,

~°

ÎÎ ÎÎ

~~~

Here,

_R~

symbolizes

the radius of the1-th

particle.

E~

represents

the elastic modulus and _v~

the Poisson ratio.

X~

_is the

position

vector of the1-th

partide

aud

n(

indicates _a unit vector

poiuting

from grain

j

to

grain1 perpendicular

to the contact surface.

(Because

ail the forces descnbed here _are contact

forces, they

_{are nonzero}

only

if

h~j

> o-1

2.2. INTERNAL FRICTION.

Energy dissipation

due to the _viscous nature of the solid _par- tiffes _~vas first studied

by

Kuwabara and Kono

I?i

more than _a

century

^{after Hertz's work.}

They

obtained the

following

_expression for the non-conservative _viscous force

acting dunng

a

collision:

~~~~~~~

~B~ il a/Î~ÎÎj il a)) ~Ù~~Î~ ~~'

_~~~

where,

~~ 1111~

~~~

and,

~~

~

°~

j31~

+

ÎÎj'

^~~~

(~ and q~ are the coefficients of

viscosity

associated with volume deformation and shear.

v(

is the relative

velocity

of the

colliding partides

normal to the contact surface.

2.3. SURFACE ATTRACTION. When two surfaces _are

brought

into contact an attractive force due to the attractive part of the inter-molecular interaction arises. The _case of

sphencal partides composed

of molecules

interacting

via _a Lennard-Jones

potential

_was first studied

by

Hamaker [9] for non-elastic

grains.

It _was extended to the _case of elastic

grains by

Dahneke

[loi.

The form used here _was introduced

by

Johnson et ai. [6] for _a

general

molecular interaction characterized

by

the surface _energy,

W~j,

of the

contacting

materials.

~meso

~

~ ~°~

~'~13

~Î ~3 ~Î~J

^~~~

/~3/4

~l j~j

~3 3 E~

il v))

+

Ej il v))

R~ +

Rj

^{~3 ~3}

Note the small power,

1/4,

to which the reduced radius is raised. Since the

weight

of _a

spherical grain

is

equal

to

4/37rpgR~,

there will be _a

grain

size for which the

weight

^of a

particle

is

equal

in

magnitude

to this attractive force. Partides smaller than this critical size expenence this interaction _{as an} adhesive.

Typically,

this cntical _{grain size is on} the order of1 _mm

[11].

^For

grain

sizes much

larger

than about 1 _cm, this force _can be

neglected.

2.4. EXTERNAL FRICTION FORCES. The frictional forces which

develop

under conditions

of

slip parallel

to, or rotation about the normal _to the contact surface _were first studied

by

Mindlin

[8].

^Mindlin's

original expression

is

computationally expensive,

therefore _a

simplified

expression is used which amounts to

assuming

that _no

partial slipping

of the contact surfaces

occurs.

~~~~~~ ^°~~~

Î

G~

(2 ÎÎ ÎÎÎ

(2

_v~)

iÙ~Î~'~ ~~

~j

~~~

iJ

G~ is the shear modulus of the material and _{p is} the static friction coefficient. ôs is the

integrated slip

in the

sheanng

direction since the

partides

first _came into contact. ès _is allowed to increase until the

sheanng

force exceeds the hmit

imposed by

the static friction. At that

point

the contact

slips

and ôs _{is reset to}

zero.

Walton and Braun [8] were the first who

attempted

to

incorporate

Mindlin's friction

theory

into their simulations using what

they

called the

"incrementally slipping

friction mortel". Their mortel is

computationally

more

expensive

than

equation (8), however,

the

qualitative

results

appear ^to be the _same.

When there exist a relative rotation about the axis

perpendicular

to the contact

surface,

then the frictional forces will induce _a moment to counteract this rotation. In Mindlin's

theory

this induced moment is descnbed

by

the

following equation

when

partial slipping

is

ignored:

~icouPie

= ~~~

8 Gi GJ

_~°

RIRJ

^~~~

/~3/2

~3 3 G~

(2

_vj +

Gj (2

v~) R~ +

Rj

^{~3 '}

ÎÎiÙ~Î~~ ~~ ~~~~~"~~~

~~~

uJ~

is the

velocity

about the axis

perpendicular

to the contact surface. ôa is the

integrated angular slip.

It

plays

the _same rote _as ôs does for

slip along

the shear direction.

2.5. THE COMPLETE MODEL. In addition _to the

couple

describe

bj, equation (9)

there wiÎÎ be torques induced

by

the action of the shear

forces, given

in

equation (8).

F[(~~~"~,

F]j~~°~

and

F(~~°

^{do not}

produce

any torques ^because

they

act

radially.

In total, the mortel described here

depends

_upon

eight

material

parameters. (In

addition

to the _seven identified

above,

the

density

of the material

making

_up the

partides

is needed in

order _to calculate the _masses used for

integrating

Newton's

equations

of

motion.) Although

ail of these parameters are iii

priuciple measurable,

their determinatiou _may, in _{some cases,} be

problematic. hleasuring

the internai

viscosities,

for

example,

is a difficult task.

Fortunately,

it is not necessary to know these parameters ^to

high

_accuracy in order to obtain

good

results from the simulations.

2.6. A NOTE _{ON THE} ALGORITHM. The

computational procedure

used here is _a

generaliza-

tion of the molecular

dynamics

method and consists of

calculating

the

positions

and velocities of each

partide

at every time

step [13].

^As a

prerequisite

the forces

acting

between ail

pairs

of

partiales

must be calculated

using

^the

procedure

described in the

proceeding

section. The

resulting

forces _are then

employed

to

integrate

the Hamiltoniaii form of Newton's

equations

of motion.

Modem

workstations,

and modem

parallel

_{computers, are moving} to cache based

systems

ⁱⁿ order to _overcome the _ever

increasing

_gap between processor

speed

and memory

speed. Using

such

systems

in _an eilicient _manner poses a different set of

problems compared

to those faced when using ^vector machines. In

particular

data

locality

and cache _{reuse are} the

major

_concems

on cache based systems.

The

algonthm

used in the

present

^studies

attempts

to addresses these issues. A

complete description

of the program ~vill be

given elsewhere,

here it suffices _to mention the

program's efficiency

_m terms of its execution

speed

_{on various}

platforms.

By

_way of _{companson, one} of the fastest 2-D programs mentioned in the hterature _runs

at 28.2

Kups (thousands

of

particle updates

_per

second)

_{on a} Sun

Sparc-10 [14].

That 2-D

program was

ported

to _a

Sparc-10

after

having

been

optimized

for _a vector machine.

By

companson, the

present

3-D program was

optimized

for data

locality

and cache _reuse and the

speed

is given in Table 1.

Table I. This table

gii~es

the _program

speed (m

thousands

of part~cie updates

_pet

second)

_on

i~arious processors. For these

test,

a dense

system consisting of

9261

particies

in _a 3-D cube with

periodic boundary

conditions _on aii sides _mas used.

processor

Speed (Kups)

Sparc-10

24.1

Sparc-20

37.5

SG R4400 41.3

IBM-Power2 45.2

3.

Description

of the

Experiment

The diflusional _processes inside _a

rotating

arum _can be studied

using

an

arrangement

^first

suggested by Hogg

et ai. [2]: a arum oriented

parallel

to the

ground

is

partially

filled iv.ith t~vo

types

of

distinguishable particles.

One

type

in the left half and _one

type

in the

right

half _as shown _in

Figure

1.

2R

g

C2

Fig.

1. Schematic

illustrating

the

experimental

set-up.

Initially

the two material components, ^Cl and C2

are well

separated.

As the arum rotates about its axis at _a constant

speed,

momentum is

imparted

to the

particles

in the

plane perpendicular

to the axis of rotation. Motion

along

the

longitudinal

direction _occurs

only through

random momentum

changes

induced

by

collisions between _par- tiffes.

Hence,

motion

parallel

to the axis of rotation _is diflusive.

The diffusion

equation

for the concentration of each comportent can be written as:

ôfif

^~ ôz2

where the natural definition of time for this system has been

used, namely:

the number of

revolutions, tif. Ci (z,tif)

_is the concentration of

component along

the rotation _axis.

Equation (10)

can be solved

subject

to

partide

conservation and to the initial condition discussed

above, yielding

_[2]:

1 ~ ^°° i

j~ ij2 2j~fif j~ ij

~~~~'~~

2 ^~ _7r 2n

-1~~~

^~

L2~

^~~~

~

L

~Î

(11)

z lies _m the range

-L/2

< z <

L/2.

A similar result is obtained for the second

component.

In the

computer

simulations the average location of ail the

partides belonging

to _a

particular component

can be measured with _a

higher

accuracy than the

particle

concentration _{as a} function of

position.

From

equation (11),

the average

position

_is:

ce +1

_12~

i)~~~~~j

^~~~~

8L

~

^{~i ~} _eXp

< z >1~

/ (2n 1)~

L2

j13j

n=1

~2

_~

ifj

ç~

$

exp ~2

Equation (13)

is _a

good approximation

to

equation (12).

It _can

easily

be checked that ail the

higher

order terms

together

contribute

only

about

3%

_to the average ^at

tif

= 0. For

larger tif,

the

higher

order terms become

completely negligible. Hence, by measuring

< z > as a function of

tif

_one

can via

equation (13)

obtain the diffusion

constant,

D.

The

computer experiments

make _use of soft

spheres,

1-e-,

particles

with _an elastic

modulus,

E

m~

10~N/m~.

It is

possible

to

actually

manufacture and _use

partiales

with such low elastic moduli in

laboratory experiments

[6]. ^Indeed

they

_are valuable for

studying inter-partide

interactions. The purpose of

using

them _in the

present

simulations _{is to save}

computer time,

since the

integration

time step must be smaller than the collision time which varies hke t~ _m~

fi@

_[5] for

particles

of _mass

m.

Usiiig

_a normal value of E _m 10~~ would have increased the cpu time 300-fold.

Table II list the material parameters and Table III list the

tribological parameters

used in the present simulations. As _can be _seen the softness of the

partides

is reflected _not

only

_in the

elastic

modulus,

but also in the internai viscosities.

Table II. Materiai

parameters

used

for

the

soit spheres

in the

present

simulations.

(See

the tezt

for

_an

ezpianation of

the

symbois. )

Parameter Material-1 Material-2 Material-3

1-o _x 1-o x I.o _x

G 0.3 _x

106 N/m2

0.3 _x

106 N/m2

0.3 _x

106 l~/m2

v 0.25 0.25 0.25

(

5000 Poises 5000 Poises 5000 Poises

q 5000 Poises 5000 Poises 5000 Poises

p 1000

kg/m~

1000

kg/m~

1000

kg/m~

R 0.0036 _m 0.0024 _m 0.0030 _m

Table III.

Triboiogicai parameters

used

for

the

soit spheres

_m the

present

simulations. _i

j

mdicates the value the

parameter

takes when _a

particie of

materiai

type1

is in contact with _a

particie of

materiai

type j. (See

the tezt

for

_an

ezpianation of

the

symbois.)

Parameter 1 1 2 2-2

1-3, 2~3,

3-3

p o-1 0.2 o-à 0.2

W 0.2

J/m2

0.2

J/m2

0.2

J/m2

0.2

J/m2

The watts of the _{mixer are} also

composed

of

particles

in order to reduce the

complexity

of the force calculation. In the

present

simulations material-3 _was used

exdusively

for the arum watts and materais & 2 where used for the

particles

inside the arum.

Note,

the difference _{in p} for the three materials. Previous

expenmental

studies of

rotating

drums indicate that materials with

significantly

diflerent

angles

of _{repose, may}

segregate

^rather than mix

during

the _course of the

experiment

_[4]. In the

simulations,

diflerences _in the

angle

of _{repose are} achieved

by

_varying either _p, the coefficient of _static friction _or

by

_varying

lf,',

the surface _energy. Since _common

expenments

use mixtures of sand and

glass beads, varying

p, while

keeping

W _constant should

yield

results ~vhich _{are more}

readily comparable

with expenments.

The

cylindrical

arum used in the

present

simulations _was 12 _cm

long

aria 8 _cm in

diameter, giving

_an

aspect

^{ratio of1-à- At the}

beginning

of the

simulation, equal

_masses of

partides

from materials _or 2 _were

placed

in the arum _as described above. The arum _was then rotated at

a constant

angular speed.

A

typical

simulation involved about 1000

partides

inside the arum

and 1000

partides making

_up the arum watts.

4. Results and Discussion

As _a first step ^the self-diffusion for

particles composed

of material is studied.

(This

is doue

by giving

each

partide

_a

tag corresponding

to that half of the arum iii which it

started.)

The average

position

in the

longitudinal

direction

(denoted by: z)

is

plotted

in

Figure

2 _{as a} function of the number of arum revolutions for _two different rotation

speeds.

As to be

expected,

the

particles

diffused faster at

higher

rotation

speeds.

A further

understanding

of the diffusion _{process can} be obtained

by studying

aiiimated videos of the

rotating

arum. The videos show quite

dearly

that diffusion _is initiated at the free surface of the

granulate,

diffusion in the bulk

plays

_a subordinate rote for the _case of _a

single species. Basically, partides brought

_up to the free surface _via the action of the drum's rotation execute a random walk

parallel

to the axis of rotation _as

they

roll

along

the surface.

From

Figure

2 and similar

plots

the diffusion constants _can be extracted. A

plot

of the self-diffusion coefficient _{as a} function of rotation

speed

is shown in

Figure

3. This data _agrees

qualitatively

with the

experimental

results of Rao et ai. [3].

Quantitative agreement

is not

expect

^{because Rao} et al. _are

using partides

with different material

properties.

The _case of two

diffusing species

has not, to _our

knowledge,

been studied

experimentally.

Figure

4

depicts

the average

position

of the

partides composed

of materai 1 _as

they

diffuse

through particles composed

of materai 2. Note the

qualitative

differeiice betiv.een this

figure

and

Figure

2. For the

two-species situation,

the diffusion _rate is

iiearly

the _same for the _two rotation

speeds

and in fact it is

slightly

^smaller at the

higher

rotation rate. This countenntuitive

phenomenon

exists over a range of rotation

speeds

_as illustrated in

Figure

5.

o.1

2.0 Hz +

10.0 Hz +

fi

O.oi

il

v +

+Éj,3u,((

~~ +

Î,~

, ^{+- +}

++

0.001

0 5 10 15 20

N

Fig.

2. Self~diffusion: average position _{as a} function of the number of revolutions tif for two different rotation

speeds.

o.ooi

o. oooi

ce~

a

1

le-05

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

~ lrev/sl

Fig.

3. Diffusion coefficients for self-diffusion _{as a} function of trie number of revolutions.

o .1

2.0 Hz .

10.0 Hz +

à

o.oi il

0.001

0 5 10 15 20 25

N

Fig.

4.

2-Component

diffusion: _average

position

_{as a} function of trie number of revolutions tif _for

two different rotation

speeds.

Again,

animated videos shed _some

light

_on the _processes

taking place.

Since the

partides

of material 2 have _a

larger

coefficient of static friction than those of materai

1,

^their

angle

of

repose is

larger,

^which means that

they

will tend _to roll downhill _onto the

particles

of material 1. The

partides

of material 2 are also smaller than those of materai

1,

hence

they

_are able to

diffuse

through

material 1 trot

only

at the

surfaces,

but also in the bulk.

(As layers

of species ¹

move

past

^each other

they

_{open up} voids

through

which the smaller

particles

_cari

move.)

For the

two-species

_case, diffusion in the bulk

plays

_a

larger

rote than for the

single-species

_case.

Another indicator of the enhanced

mobility

of the smaller

particles

_{is given}

by

the ratio of the total kinetic _energy of the small

partides

to that of the

larger partides. Figure

6 shows

o.ooi

f

0.0001

~~

~ i 1

1

ie-05

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

~ lrev/sl

Fig.

5. Diffusion coefficients for

2-comportent

diffusion _{as a} function of trie number of revolutions.

0.8

0.7

°°~

j

d~ O.S

(

~'~

i

(

0.3

0.2

o.1

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Qlrev/s)

Fig.

6. Trie ratio of trie average kinetic energy for

partiales composed

of material 2 to that for

partiales composed

of material 1.

this ratio _{as a} function of the rotation

speed.

The sohd fine indicates the

expected

ratio if the _average

velocity

of

partides belonging

to both _{species were}

equal.

As _can be _seen, the measured ratios lie above this

hne,

thus the smaller

partides

_{are moving, on average,} with

higher speeds

than the

larger partides.

In other

words,

the diffusion in this

two-component system

is

being

driven

by

the smaller

partides.

At

larger

rotation

speeds partides

roll clown the surface faster

leaving

less time for motion

parallel

to the rotation _axis.

Hence,

the diffusion _{processes are} slowed because their _are fewer

partides

from

species

2

rolhng

onto those of _species 1.

Eventually,

_as the rotation

speed

increases, the entire free surface will fluidize

allowing

for increased

mobility

in the

longitudinal

direction and the diffusion coefficient increases.

If the rotation

speed

is increased still

further,

_one

eventually-

enters the

centrifugal

_regime

where the

partides

_are held _to the inner surface of the arum via

centrifugal

forces. In this

regime

ail diffusion must _come to a hait. This

implies

that there must be _a maximum diffusion coefficient _{at some} rotation

speed larger

than those which coula be studied here.

Likewise,

as the rotation

speed

decreases towards _zero the

mobility

of the small

partides

must also decrease until the diffusive motion

stops, consequently,

there must also be _a maximum

diffusion coefficient at _some rotation

speed

much smaller than those which coula be studied here.

5.

Summary

and Conclusions

A

computational

mortel for the collision of two visco-elastic

spheres

which is

independent

of

arbitrary

parameters ^{has been}

presented. Using

this mortel the diffusion of

partides

in

a three-dimensional

rotating

arum has been studied. The

major

^result is that the diffusion coefficient for _a

two-component system

has _a much richer structure than

anticipated by studying

a

single-component system.

^{This is}

primarily

due to the

larger mobility

of the smaller

partides compared

to that of the

larger partides.

One

phenomena

in

rotating

arums which has been

given

_a

good

deal of attention in the literature _is the

spontaneous segregation

of two material

types [4].

^In a

typical experiment,

one

component

^{is smaller and}

rougher land

thus has _a

larger angle

of

repose)

than the other.

After about 5 minutes of

rotation,

the two materials

spontaneously segregate

^into a series of bands

parallel

to the rotation _axis. The

composition

of the bands altemates between

high

concentrations of

component

and

high

concentrations of

component

2. In most _cases there _is

an odd number of

bauds,

with the

smaller, rougher partides

located at the erras of the arums.

The mechanisms which create this banded structure are

fairly

well understood. A statistical fluctuation in the concentration of the

smaller, rougher component yields

_a local increase in the

angle

_{or repose.} The

larger particles

will tend _to roll _away from such _a

locality,

thus

depleting

their numbers _in the

region

of the fluctuation which drives the

angle

of _repose to _even

higher

values.

Now,

it is trot

completely

understood

why

the bauds should be stable, _{smce as} shown

here, partide

diffusion will tend to

destroy

the banded pattem.

This

phenomena

_{occurs on a} time scale which is _too

large

to be simulated with

present

resources,

however,

the above results _can be

applied

to this

problem.

The

present

work has shown that diffusion in

two-component systems

is driven

by

the

mobility

of the smaller

partides.

Therefore _it is

possible

to uiiderstand the

stability

^of band

pattems

as follows. Let _a band of

large particles

exist between two bands of smaller

partides.

If the rate at which the small

particles

diffuse

through

the

larger particles

is

large enough,

then the small

partides

from _one baud _can diffuse

through

the band of

large partides

and

i~eplenish

the baud of small

partides

on the other side. Now if the current of small

partiales

is

conserved,

then diffusion of the

large particles

into the _region

occupied by

the small

partides

will be

depressed

due to

geometrical

factors.

This idea

implies

that

only systems

with _an odd number of bands _are

stable,

_as is indeed

reported

iii the literature.

(There

has been

only

_one

reported experiment yielding

_{an even} number of bands

[4],

however

they

dia not _{use a}

simple cylindrical arum,

rather

they

used

a more

complicated

arum

shape

which enhanced the

angle

of _repose of the

larger particles.)

Fuitliermore it

explains why early

expenments daimed that the

segregation

started with the smaller

partides

at the erras of the druni. Indeed any other

arrangement

^would

necessanly

be

unstable.

It may be

possible

to test these and other ideas _on spontaneous ^materai

segregation

m the

near future

through

the _use of

high performance computer systems.

References

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