Radial Segregation in a Two-Dimensional Rotating Drum

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Radial Segregation in a Two-Dimensional Rotating Drum

Christian Dury, Gerald Ristow

To cite this version:

Christian Dury, Gerald Ristow. Radial Segregation in a Two-Dimensional Rotating Drum. Journal

de Physique I, EDP Sciences, 1997, 7 (5), pp.737-745. �10.1051/jp1:1997188�. �jpa-00247357�

Radial Segregation in

_a

Two-Dimensional Rotating Drum

Christian M.

Dury

and

Gerald

H. ltistow

(*)

Fachbereich

Physik, Philipps-Universitht Marburg,

Renthof 6, 35032

Marburg, Germany

(Received

19

August

1996, received in final form 23

December1996,

accepted 16

January 1997)

PACS.64.75.+g Solubility, segregation,

and

mixing; phase

separation PACS.83.70.Fn Granular solids

PACS.02.70.Ns Molecular

dynamics

and particle methods

Abstract. We simulate the size

segregation

of _a

binary

mixture of

granular

material in

a half filled two-dimensional

rotating

drum

using

the discrete elements method with linear contact forces. The

dynamics

of the segregation process is studied in detail _as function of the

angular velocity

of the drum. We propose an order parameter to characterize the final amount of

segregation

and the

segregation

speed which allows to compare systems of different sizes and with different material properties. We demonstrate its usefulness

by investigating

its time dependence _as function of the

angular velocity

of the drum and discuss the

interplay

between the thickness of the

flowing layer

and the _mean

particle velocity.

1. Introduction

The behavior of

granular

materials is of great

technological

interest

[lj,

and its

investigation

has _a

history

of _more than two hundred _years. Nevertheless the basic

physical understanding

of

granular

media is far from

being complete.

One of their most

intriguing properties

is their

tendency

to

segregate

^which is observed in _many industrial

particle handling situations,

such

as

transporting grains

_or

mixing pharmaceutical pills

_[2]. The

rotating-cylinder geometry

is

an

archetype

of _numerous devices used in industrial material

processing

where radial _segre-

gation [2-4]

can occur on short time scales and axial

segregation

is observed _on

larger

time scales [5]. ^{The mechanism of the}

segregation

_process is based _on the surface flow where small

particles get

^stuck

along

the inclined

plane

_more

likely

than the

larger

_ones, hence

accumulating

near the center of the

rotating

drum. It _was

argued

_e.g.

by

Hill and Kakalios [5] ^{that the}

angle

of _repose for different materials _or mixtures is the

triggering

mechanism for axial

segregation

but this could not be verified

(within

the _error

bars) by experiments

in _a two-dimensional

rotating

drum

using

disks of various sizes [6].

Many _parameters

_are involved in the _process of radial

segregation

^and

mixing,

^such as

size, shape,

mass [7], ^frictional

forces, angular velocity, filling

of the drum

[8, 9],

etc. In this paper we focus _on the

dependence

of the

angular velocity.

We _are

studying

the

segregation velocity

of _a half filled bidimensional drum

rotating

^with an

angular velocity

_w with radius R. The drum is filled with _a

binary

mixture of

particles

where the radii of the small

particles

_{are rs} and the radii of the

larger

_{ones are rL.} The first

part

of _{our paper} describes the numerical methods and

parameters

we use. In the second part

(*)

Author for correspondence

(e-mail: ristow©stat.physik.uni-marburg.de)

Q

Les

#ditions

_de

_Physique

₁₉₉₇

738 JOURNAL DE

PHYSIQUE

I N°5

a

quantitative analysis

of the

segregation

^{for different rotational velocities will be}

given

and the

interplay

of the thickness of the fluidized

layer

and the _mean

particle velocity

within it will be discussed.

2. Model

In order to

study

_our

system

with

computer

simulations _{we use} molecular

dynamics

for

granular

materials also known _as discrete element method

(DEM) [10].

In

experiments

with mustard seeds in _a

rotating

drum _one observes that most of the

particles

in the fluidized

layer

_are

sliding

and not

rotating [11].

^{In order to}

approximate

these conditions _we use

spherical particles

with

no rotation instead of

elliptical particles

with rotation. To avoid

close-packed

structures we use

particles

with different sizes in _our simulation. Whenever two

particles

I and

j

_are closer than the _sum of their radii

particle j

exerts a force _on

particle

I and vice _versa.

2.I. NORMAL FORCE. For the force in normal direction _{we use} linear interactions with

damping:

F~)

=

-kz~re~([xi xj[

_(r~ +

rj)) iz~me~(v~ vj)fi.

Here _{x~, v~} denotes the

position

and

velocity

of

particle

I and h is the unit vector in normal direction fi

ii

(x~ xj).

The radius of

particle

I is _r~ and _{re~, me~ are} the reduced radius and

mass of the two

particles.

The stiffness of the material is characterized

by

_kz~ and _in and is

directly

related to the coefficient of restitution _£res via:

here

tc _the

The

algorithm

is stable for _a time

step

^of dt < tc

/8;

_to be _on the _save side _we choose the time

step

to be dt _:=

tc/15.

In order to save

computer time,

_we set

kn

to 10~ Pa which is about

one order of

magnitude

softer than vulcanite but the maximal

overlap

of two

particles

_was at most

0.3%

of the _sum of their

radii,

which is still realistic. To suppress

particle bouncing

on

the

surface,

which could disturb the

segregation

we choose the

damping

coefficient _in to

give

£res =

28.7%.

The coefficient of restitution has also _an effect _on the

angle

of _repose, shown in

Figure

I. With

decreasing

_£res ^the

angle

of _repose in the

rotating

drum increases

strictly monotonically.

The

parabolic

fit to the data

points gives

an

extrapolated

value of 39° in the limit _{£res ~} 0. For these values of _our

parameters

^{the time} step is dt

= 0.85 x

10~~

s.

2.2. SHEAR FORCE. For the shear force _we put _upon contact a linear

spring

between the two

particles

which results in _a

restoring

force. If the

restoring

force would be

larger

than Coulomb's frictional force the

spring

is _cut and _we switch to

sliding

^friction with Coulomb's law

~(

"

~~~gll I(~l

_~J

lil~l)

~~Il

ks /

(Vi Vj)I(~)d~, /ll~$I)

Here I is the unit vector in shear direction I I fi. With this frictional law _we do not need to have _an artificial

roughening

of the

boundary

walls _to have static friction.

For the friction coefficient _{~ we} have ~ = 0.4 and for the

spring

constant

ks

^of the

tangential

spring

we use

ks

₌ 100 Pam.

40

30

j

I

A ₂₀

to

0

0 20 40 60 80 100

Coefficient of restitution l'll]

Fig.

I. The

dependence

of the

angle

of repose

(6)

on the coefficient of restitution _£res for _an

angular velocity

of _{w =} 2 Hz. The solid line is _a

parabolic

fit to the data points.

2.3. PARAMETERS. In _our simulation _{we use a}

binary

mixture of N

= 1600

particles

with radius _rs

= 0.038 _cm and _rL

" 0.064 _cm, the fraction of the small

particles

is

f

= 45%. Our

drum with radius R

= 3.5 _cm is half filled and the

packing

fraction _{is ~}

= 0.77. The

particles

and the inner wall of the drum _are made of the _same material described

by

the parameters above.

3.

Rotating

Drum

We _are

starting

with _a half filled drum in order to minimize the effect of

mixing [8,9]

shown in

Figure

2a and then let the drum _rotate with

angular velocity

w. We choose _w in _a

region

with continuous surface flow

just

^{above where} we would get ^{distinct avalanches}

[12].

^After

some revolutions _we

get

a

clearly segregated

_core

composed

out of small

particles

drawn _as black circles shown in

Figure

^2b. In order to characterize the

segregation quantitatively

_we

investigate

the

density

of small

particles

at a

given

distance _r from the _center of the drum which is similar to the method used in

[13].

3.I. DENSITY PROFILE. To obtain the

density profile

for small

particles

_we divide the disc

segment

^into concentric shells with width /h and inner radius R~

= ilh of shell I where I =

0,1,.

,

(Rllh)

1. To

quantify

the

quality

of the

segregation

_we normalize the measured

density by dividing by

_ps

"

~1/(~r])

which is the theoretical

density

if

only

small

particles

_were

present.

The

probability density

_pa

(Ri)

is thus defined _as:

pa

(Ri)

_:=

~~~na(R~).

~1

Here

na(R~)

is the number of small

particles

in shell I with width /h divided

by

the _area

A =

) _[(R~

₊

_{/h)~ R))}

~ of shell I.

p~(Ri)

_ranges from 0 to I and in the

unsegregated

_case

740 JOURNAL DE

PHYSIQUE

I N°5

ja) 16)

Fig.

2. 2D-drum: small particles _are drawn _as filled circles and

large particles

_{as open} circles.

a) Snapshot

of the drum

right

before the first avalanche.

b) Snapshot

of the drum after

rotating

t ₌ 60 _s with

angular velocity

_w

= 1.0 Hz, _i.e. after 9 rotations.

0.6

(a)

0.6

(b)

p

0.4 0.4 q

0.2

0 0

0 ₁ 2 3 0 1 2 3

R~

jcmj

R~

jcmj

Fig.

3.

Density profiles

for

Figure

2:

a)

_at t ₌ 0 _s;

b)

after

segregation, giving

a value of _q

= 0.39.

The dotted line denotes the

density

for the

unsegregated

case po.

the

probability density

is in all shells

(2 IN /~) (rs /R)~

_which

_gives

for _our values po = 0.228. In

Figure

3a we show the

density profile

of

Figure

2a and the dotted line denotes po. The

density profile

fluctuates around po as

expected.

In

Figure

3b the

density profile

of the

configuration

in

Figure

2b is

given,

here _{one can}

clearly

_see the

segregated

_core of the small black

particles

where the value of _{p is}

approximately

0.5. On

top

^of the core, R~ < 0.25 _{cm, we can see} the fluidized

layer composed

of

large

and small

particles.

The

probability density

for _our

segregated configuration

cannot reach the maximum

possible

value of one, because the

segregated

core

of small

particles

is not

really

_a half disk and _on

top

of it there is

always

_a fluidized _zone which is

composed

^also out of

large particles.

Therefore the maximum realistic

possible

value for _{p is} about 0.7. Further _away from the center of the

drum,

R~ > 2 _cm, the

density

^of the small

particles

is below the value of the

unsegregated

case and _even

drops

to less than po

/2

for R = 3 _cm. In this

region

the

larger particles

concentrate as

already

visible in

Figure

2b. It is

0.6

(al

0.6

(b)

Pend pend

0.4 0.4

0.2

~

0.2

0 0

0 20 40 60 0 20 40, 60

t isi ^t isi

Fig.

4. Evolution of the

density

of small particles at the outer most shell:

a)

_uJ

= 0.75 Hz and

b)

w = 2.5 Hz. The dotted line denotes the

density

for the

unsegregated

_{case po.}

interesting

to note that

right

at the wall

boundary

the

density

of small

particles

increases

again

which _we found in all _our simulations for different model

parameter ks,

_{~, in} and

regardless

of the

angular velocity

of the drum. Its

height

_{pond °" PA}

(R /h)

_seems to be related to the

quality

of the

segregation

since values of _pond close to _or above _po

always

_gave not well

segregated

_cores. This could be due to _some finite size effects and

depends

_on the thickness of the fluidized _zone. Even

though

the

angle

of _repose is

nearly

constant in _our range of _w

[12];

8 _m

33°,

the thickness of the fluidized _{zone grows} with _w

[14,15]

and _so the small

particles

cannot

segregate

fast

enough through

the fluidized

layer

to the solid block and will end _up at the wall. In

Figure 4,

_we show the time evolution of _pond for two different

angular

velocities

w = 0.75 Hz

(a)

and _w

= 2.5 Hz

(b), respectively.

For lower values of _w, the

depletion

of the small

particles

in the outermost shell is faster and the final value around which pond fluctuates is lower. This indicates that the maximal achievable amount of

segregation

is

higher

for smaller _w which will be verified and

quantified

in the next section

using

_an

appropriate

order

parameter.

3.2. ORDER PARAMETER. To compare the

quality

and

speed

of the

segregation

^{of different}

runs with different

parameters

^{it is desirable} to have

only

_one order

parameter

to

investigate.

We choose _as such _an order

parameter

_q the _{sum over} all

positive

deviations of _pa

(Ri)

above the _{mean po} normalized with

respect

to the

ideally segregated

_case, where all inner shells I ₌

0,

, c are

composed only

out of small

particles

and in the _outer shells I ₌ c+

I,

,

(Rllh)

I there are

only large particles.

To illustrate this

procedure,

we have shaded the _area that enters our calculations in

Figure

3b. In the

ideally segregated

case the inner half disk has radius of

Rc

= rs 2

IN /~

which

gives

for _our values

Rc

= 1.64 _cm. Cast into _a

formula,

we define _our order parameter _q as:

~ °"

) ^~~ij

^~

(PA (Ri)

_{PO) 8}

_(PA lRl)

_PO)

1=0

with

8(x)

_as the Heaviside

Step

function. For

calculating

the normalization constant _cz~

we choose for convenience the width of the shells /h to be _an

integer

^{fraction of the drum}

radius R and of

Rc.

So _we

get

^{for the}

probability density

in the

ideally segregated

_case

742 JOURNAL DE

PHYSIQUE

I N°5

o..5 o..5

o o

~~ o

~

04 '

/~ ^O o

O

~ q

q

o o '

~

8

o o

o_3

~ o

°

°~ ~'

o o

0.2 0.2

0.1

(a)

0.1

(b)

0 0

0 20 40 60 0 20 40 60

t jsj ^t psi

Fig.

5. Time evolution of _our order parameter

q(t). a)

For _w

= 0.75 Hz. The solid line is the fitted _curve ~vith _q«

= 0.395 and _T

= 15.4 _s.

b)

For _{w =} 2.5 Hz. The solid line I; the fitted _curve with _q«

= 0.275 and _T

= I-I _s and the dotted line is the time evolution of _{a more}

segregated

initial

configuration.

psa(Ri)

=

8(Rc Ri)

and the normalization constant cn is:

cn .=

~j(I _{po)8(Rc R~)}

=

(I po) )

"

l j

~~~) ) fi.

~~~ ^~1 ~

In

Figure

5 _{we compare} the time evolution of the order parameter

q(t)

for _runs ~v.ith different

angular

velocities and find that

q(t)

saturates in both _cases. The fluctuations of the order parameter ^{in the saturated}

phase

_are also seen in

experiments

_[3] and _are therefore most

probably

not an artifact of _our method. Both

graphs

can

easily

be fitted to an

exponential

law of the form:

q(t)

= qc~

(I _e~~/~)

where _qc~ denotes the

quality

of the

segregation (or

^final amount of

segregation)

and _T the characteristic

segregation

time. When

using

the data

given

in

Figure 5b,

_we

get

^for w = 0.75 Hz

(a)

^{values of} qc~ = 0.395 and _T

= 15.4 _s whereas for _w

= 2.5 Hz

(b)

values of _q~

= 0.275 and

T = 7.7 _s. This states that _even

though

the initial

segregation speed

is

higher

for

higher

values of _w, the final amount of

segregation

is less. In order to half the characteristic

segregation

time T, w has to be increased

by

_a factor of 3. Also shown in

Figure

5b _as dotted line is _a test run

where _we started with _{a more}

segregated configuration.

The value of _q decreases in time and verifies that the order

parameter

indeed does not reach _a value of I but rather

approaches

_qc~.

3.3. DYNAMICS _{OF THE} SEGREGATION PRocEss. But what is the exact

dependence

of the

quality

of the

segregation

and the characteristic

segregation

time _on the

angular velocity

of the

rotating

drum? We start with _a

rotating speed

_w

just

above the distinct avalanche

regime [12]

which

gives

_{w =} 0.5 Hz for _our simulation

parameters

^{and increase} w in steps of 0.25 Hz _up to 2.5 Hz. The

corresponding

values of _qc~ and _{T are}

given

_as function of _w in

Figures

6a and

b,

respectively.

With

increasing

_{w, qc~} decreases

only slightly

in the

beginning

_up to _{w =} I Hz but for

higher

values of _w the decrease becomes _more

pronounced.

The

segregation

mechanism is

0.5 25

k~ ₌ 100000 +- k~ = 100000

k~ = .50000 * k~ ₌ 50000

0A ^~h Q _4~

~

^{20 j}

qm *

~

j ^T [sj

* ~ ^'

0.3 4l 1.5

* 4l

0.2 10

0.I

(aj

_.5

(bj

0 0

0 1 2 3 0 1 2 3

~ [Hzj _w [Hzj

Fig.

6. Dependence _on the

angular

velocity of

q(t,

uJ):

a)

the coefficient _q« (uJ) ^{for the}

quality

of the

segregation, b)

the coefficient

T(w)

for the characteristic

segregation

time. The solid line

corresponds

to T(uJ) ₌ ^{13.7 W~~~~}

based _on the surface flow and with _no surface flow there is _no

segregation.

Therefore it is clear that in the limit _{w ~} 0 _we

get

_T ~ cc, because for lower and lower _{w we are} in the discrete

avalanche

regime

and the

separation

of the avalanches grows more and _more. The best fit for

T(uJ) by

_{a power} law

gives:

T(uJ)

₌ 13.7

uJ~°'~~

The natural time scale for the

segregation

_process ^is the average

frequency

_a

particle

is

exposed

to the

surface,

I.e. the number of revolutions.

Defining

_{nR i=}

T(w)w/(2~),

we can

rewrite the above

equation

^for

T(w)

and obtain

nR = 2.18

w°

_~~,

which is shown in

Figure

7a _as solid line where the data

points

from

Figure

fib _are

replotted

in this dimensionless fashion. One _{can see} that for

higher

_{w we} need _more revolutions to achieve

segregation.

In order to discuss the influence of the thickness of the fluidized

layer

_on the characteristic

segregation

time

T(w),

_{we vary our} simulation

parameter ks

which is related to the surface

roughness

of the material

being

used. How this influences the

layer

thickness

df

is shown in

Figure

7b _as function of _w for the three different values ks ₌

50000(*),100000(o)

and 150

000(.),

The thickness increases with

decreasing ks,

In _our

regime

of _w, the thickness of the fluidized _zone is

increasing

with

increasing angular velocity [14,15].

This _means that for

higher angular

velocities the small

particles

need _more time to

segregate through

the fluidized

layer

onto the solid block than for lower _w, therefore the final amount of

segregation

cannot be

as

good

_as for low values of _w. Even

though

the

segregation speed

is better for

higher

_w which is due to the

greater throughput

of material in the fluidized _zone it will need _more revolutions to achieve the saturated state than for lower

angular

velocities.

The effect of _a

larger

^fluidized zone on the final amount of

segregation

_qc~ and the initial

segregation speed

_T is demonstrated in the

Figures 6a,

fib and 7

by adding

_some data

points

for

ks

₌ 50 000. The dotted line

corresponds

to the _power law fits

T'(w)

= 10.6

w~°.~

and

n[

= 1.69

w°.~, respectively,

which verifies that _a

larger

fluidized

layer corresponds

to a faster

744 JOURNAL DE

PHYSIQUE

I N°5

4 0.8

k~ =100000

~~ ~~~~~

i

3 0.6

nR(uJ)

dj[cm]

2 °.4

i

(~)

^~'~

lb)

0 0

0 1 2 3 0 1 2

~ iHzj

~iHzj

Fig.

7.

a)

Number of characteristic revolutions _nR to achieve

segregation

and

b) dependence

of the thickness of the fuidized _{zone on} ks for the values ks

= 50

000(*),100 000(o)

and 150

000(.),

initial

segregation speed (lower

value of

T).

But from

Figure 6a,

one also reads off that this

means a lower value for the final _amount of

segregation.

Thus

decreasing

the internal friction

of the

grains

_seems to have the _same effect _as

increasing

the rotation

speed

of the drum.

4. Conclusions

Radial

segregation

is well observed

experimentally

in

rotating

drums and _we have studied its

dynamics

and its

dependence

_on the external

angular velocity

of the drum and the thickness of the fluidized

layer.

We found that the

quality

of the

segregation

decreases with

increasing

angular velocity

and the characteristic

segregation

time decreases. The latter _{seems to}

diverge

in the limit _w ~ 0.

We

speculate

that in three

spatial dimensions,

radial

segregation

is _more

easily

to

achieve,

since the

empty

_spaces between the

particles

_are connected

by

_a network and small

particles

could traverse

through

^it which will lead to _a better

segregation.

Also in three dimensions small

particles colliding

with

larger

_{ones can} be deflected

parallel

to the direction of the rotational axis and therefore the

velocity

in direction of the downwards flow is reduced. Hence the

particles

have _more time _to

segregate

^until

they

hit the wall. These

hypotheses

could be tested

by performing

real three-dimensional numerical simulations and

investigating

the diffusion constants in different directions.

Acknowledgments

We would like to thank S. Grossmann for

enlightening

discussions and

gratefully acknowledge

financial

support by

the Deutsche

Forschungsgemeinschaft

and _a generous grant ^of computer time

by

the HRZ

Marburg.

References

ill Jaeger Y, Nagel

Y and

Behringer R-P-, Phys. Today

4

(1996)

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