E L S E V 1 E R Powder Technology 96 [ 1998 ) 219-226

POWDER TECHNOLOGY

Motion state transitions in a vibrated granular bed

S.S. Hs iau * S.J. Pan Department o[Mechanical l,)~,ineerin,'4, ,,Vatio,al Central Unive#wio; Chung-Li 32054, 77ml an

Received 4 March 1906; revised 17 Seplcinber 1997

Abstract

According to the level of vibrational acceleration, the granular layers in a vertical shaker may have several types of motion: heaping, coherent, expansion, wave and arching. The periodical collisions between particles and the base plate of the shaker result in unstable dynamics and the five motion states, which are dependent on the amplitude of the vibrational acceleration. This study experimentally investigates the five states of motions and fire state transitions. © 1998 Elsevier Science S,A. All rights reserved.

Keywo*zls. Vibrated beds: Granular beds; Shakers

1. Introduction

Shakers are important industrial devices which mix [ I I, separate [ 2 ] and dry granular materials [ 3-5 ]. The motion of particles in the shaker is very complicated [61 and has interested many researchers. Jaeger and Nagel [ 7] gave a good review of related studies.

At the lower amplitudes of vibrational acceleration, some particles in the bed move: slowly upwards. After reaching a certain level, they avalanche along the bed surface and fornl a heap, as shown in Fig. 1 (a) (at a vibrational acceleration of 1.5 g, where g is the gravitational acceleration ). l,aroche et al. [ 8 ], Fauve et al. [ 9 I, Evesque and Rajchenbach [ 10 ], Cldment et al. [ 11 ], Pak et al. 112] and Hsiau and Pan [ 13] have studied this convect5 ve heaping phenomenon.

As the acceleration amplitude increases, the granular bed gets more energy and begins to jump. This is called a coherent phenomenon since all the: particles move like a solid body. as shown in Fig. 1 (b) ( at a vibrational amplitude of 2.9 ,~, ).

When the acceleration reaches a certain critical value, the coherent phenomenon disappears. Instead, an expansion phe- nomenon occurs, as shown in Fig. l ( c ) (at a vibrational acceleration of 4.5 g) . The particles occupy a large volume while vibrating, and those in the upper layers are very loosely packed when the bed expands I 1,2, 14].

When the acceleration increases to another critical value. the wave phenomenon appears, as shown in Fig. 1 (d) ( at a vibrational acceleration of 5.2 g) . The particles spread around

* Corresponding author. Tel.: ~- 886 3 426 7341: fax: + 886 3 425 4501: e-mail: sshsiau (a'cc.nc u.edu.lw

0032-5910/98/$19.00 (t-~ 1998 Elsevier Science S.A. All rights reserved. PII S0032-59 10( 97 )03377-9

the wave peaks. This phenomenon is due to the instability of the granular bed and has been studied recently by Miles and Henderson I 15], Pak and Behringer [ 16], Melo et al. [ 17] and Wassgren et al. [6].

The arching phenomenon, as shown in Fig. l ( e ) (at a vibrational acceleration of 6.7 g) , appears at a higher critical vibrational acceleration. Douady et al. [18], Rosato and Lan 119] and Hsiau and Wu [20] have investigated this phenomenon.

Melo cl al. [ 17] experimentally studied the parametric wave patterns of granular layers in a vertically oscillated cylindrical container. They flmnd that at a specific critical acceleration there was a well-defined transition from a flat surface to standing wave patterns. Melo [21 I described the pattern formation phenomena by the interaction of period doublings and standing waves. He also used a simple one- dimensional model to analyze the pattern transition which was quantitatively in agreement with his observations.

The live kinds of phenomena occurring in a vertical shaker were studied experimentally in this work. The transitions between the phenomena were also observed and analyzed.

2. Experimental setup

A vertical shaker driven by an eccentric drive system including a d.c. motor was used as the experimental facility. The experiments were performed in a rectangular container with adjustable width, 1.9 cm in depth and 29 cm in height. A vibrational amplitude (z .) of 5 mm was used. The vibra- tional frequency was adjusted by changing the rotational

5'.5. tt~ian. S.,I. Port / Pouder l>chmdo,Kv 96 (1998~ 219 220

(a)

220

ring

(b) ,herent

(c)

(d) . ve

;ion

(el g Fig. I. Photographs of the five motion phenomena: (a) heaping; (b) coherent; (c) expam, ion: ( d ) wave; (e) arching.

speed of the d.c. motor and could be measured by a HT-4000

OND SKKI tachometer. The amplitude of vibrational accel-

eration, ao, was found from ao=zoo) e (~o is the angular frequency) and could be non-dimensionalized by a gravita-

tional acceleration, F = a~,/g. The range of the dimensionless vibrational acceleration in this experiment was () < F < 7.

Three kinds of glass beads, each having a density of 249(/

kg/m ~. were used in the experiment. The diameter (do) of

type 1 was 3.00 mm with a standard deviation of 0.16 mm

( 5.33% ), and an angle of repose of 25.30 ~. Type 2 particles

were very spherical and smooth, with a diameter of 2.96 mm

and standard deviation of 0.04 mm (1.35%), and an angle of repose of 19.63 °. Type 3 particles were smaller, with a diam-

eter of 1.28 mm and standard deviation of 0.03 mm ( 2.34% ),

and an angle of repose of 23.811 °. Type 1 particles were used in most of the current experiments. The other types were only

used for comparison of the state transitions.

The motion of the bed was recorded digitally by an image

processing system. The system included a Dalsa CCD image

sensor (up to 220 frames per second) with a 55 mm Micro-

3".S.H.viau. 3".J. P¢m / Powder 7echmdovv 96 (1998) 219 226 221

Nikkor 1'/2.8 lens, an image grabber board (Dipix P360F Power Grabber) , and a 150 watt tungsten halogen ligh! source.

There were about 8% o f b lack part ic les in the granular bed

to serve as tracers whiclL had exactly the same size and prop- erties as the background particles. The shift of each tracer could be measured by identifying its position in consecutive images. Hencc, a tracer velocity could be calculated by divid- ing its movement by the time delay between two consecutive images. In the current experiment, the coordinates were lixed in the bed, therelore the measured velocities were relative to the base plate of the granular bed. Using the above experi- mental technology to measure the velocities of particles in a granular flow with a known velocity, the error was within 2~ .

3. Experimental results

As described in Section I, the granular bed has diffi,~rcnt motion states according to the vibrational acceleration. Cor- responding to the appemance of the five motion phenomena, there are five critical values of dimensionless acceleration: ~ , 1~,. 1]~, l k , and 1~. The following sections describe the observations of the five: motion states and the transitions between the states.

3. I. Heapin A, phenomenrm

Evesque and Rajchenbach [10] found that the granular bed had enough energy to move when the vibrational accel- eration is increased up to 1.2 ;, (1;~-= 1.2). Some particles move upwards and some avalanche along the surface, result- ing in the bulk convection of the bed. Generally, a peak appears al one side of the container [ 9 I. This heaping phe- nomenon was recently studied by Hsiau and Pan [ 13]. We do not intend to present the details in this study.

3.2. Cohere,t I~henomemm

When the vibrational acceleration is increased up to l ' c ,

the particles receive more energy and can leave the base phtte, resulting in the formation of a gap [ I 1, However, the granular bed is still in a dense state. The whole bed is like a solid body andjumps up and down in the container. The linear densities ( number of particles per length) when the bed expands most can be measured from the images. Fig. 2 shows that the linear density varied with the vertical height in the bed with a width L of 19 cm and an initia,, bed height H of 3 cm. The linear densities are very unilorm at difi=erent bed heights and the maximum value is about 3.33 particles/cm. Thc particles in the highest level are looser due to the free llighls of particles: however, the linear density is similar to the lower levels since the flee flights are not very significant.

As described in Section 2, the particle velocities can be calculated from two consecutive images by dividing the dis-

4 ' " v - ~ ~ ' ' ' ' l . . . . I ' ' ' ' 1 ' ' ' ' i ' ' ' '

E 3

o t -

¢- (D ED

£2 L~ 2

o Expansion Phenomenon o Coherent Phenomenon

o c o o

u a o o

u

, , J , , 1 , , ~ , t , , I , , ~ , I , , , ,

1 2 3 4 5

Bed Layer (cm)

Fig. 2. Linear density plotted againsl the bed height in the coherent ( I'= 2.9 ) and expansion ( I ' = 4.2 ) states.

tance moved by each particle by the time. Fig. 3 ( F - 2 . 9 ) shows the measured veh)city distributions in the bed at three different times. In the figure, t denotes the time and T is one complete motion period which is equal to a vibrational period for the coherent state. The time when the bed is located at the lowest position is set as t = 0. The velocity distributions are very unitbrm when the bed jumps up and down, and the horizontal movements are weak.

,?.3. 1-2rpansion phemmwmm

When the vibrational acceleration is increased to F~., the expansion phenomenon occurs. The particles in the upper levels have greater velocities, therefore they can fly to higher positions, resulting in the linear density being smaller than in the coherent state. The linear densities at different bed height levels are also shown in Fig. 2. It is found that the linear density decreases significantly with bed layer height in the container, which is not so in the coherent state.

Fig. 4 shows the velocity distributions at three different times when F = 4 . 2 . The figure is similar to Fig. 3 (coherent state) but the velocities are larger. The particles can reach higher positions than in the coherent state since the bed receives a greater amount of energy from the base plate. In this state, one motion period is equal to two vibrational per- iods and the particles leave the base plate twice per motion cycle.

3.4. }}"(ll'~? I)]lellomclir)l l

The wave phenomenon occurs after the vibrational accel- eration is increased to /~,. There are sine-like waves in the

222 S.S. Hsia u, S.,I, Pan / Powder Te{'hnology 96 (1998) 219-226

H (cm) +o

5 ~

t=(1/g)T 4 O

3O

2 O

6 O

5 O

t=(2/3)T 4 0

3 0

2 0 0

5 10

10

15

unit vec (m/sec)

L (cm)

t=(3/3)T

6 0

5 0

4 0

3 0

2 0 0

i

10 15

Fig. 3. Velocity distribulion for the bed in the coherent state ( 1"= 2.9

H (cm)

t= (1 /3 )T

7 0

6 0

50

4 3

3 0 5 10 15

unit vec (m/sec)

L (cm)

8 O

7 0

t = (2 /g )T +° 5 0

4 O

8 0

t=(3/3)T

' T

' T

T T t

I

1

I

7

/ ,'

f I

T !

5 10 15

7 0

6 0

5 0

4 0

0

i

5 !0 15

Fig. 4. Velocity distribution for the bed in the cxpan,'.;lon state ( 1 += 4.2 ).

S.S. Hsiau, ,%1. Pan/Powder Technolo,w 96 (1998) 219-226 223

free surface of the bed. The peaks become valleys after half a motion cycle. One motion cycle is equal to four vibrational cycles for this state. Fig. 5 shows the velocity distributions. Note that T denotes only half a motion period in the figure. In this state, particles leave the base plate once per half motion cycle ( two vibrational c~,cles) and the flight time is greater than one vibrational period. As the bed flies upwards, the particles in the upper levels are pushed by the lower particles, resulting in their faster movement. On the other hand, the particles in the lower lewfls are hindered by the upper parti- cles, so their velocities are smaller. The velocities in the bed are more uniform when the bed moves downwards.

Melo et al. [ 17 ] reported that for a thick granular bed ( H~ do > 7) the wavelength was independent of the bed height, but was inversely proporlional to the square of the vibration frequency. In the current experiment the container width was changed to investigate differences in the waves. It was found that for a container width of less than 7 cm ( H/d.~ < 23 ), there is only one wave. For a w d t h between 7 and 16 cm t 23 < H~ do < 53), there are two waves. Three waves could be observed for widths greater than 16 cm (H/do > 53 ). This result was not the same as that of Melo et al. [ 171 due to the differences in the dimensions of the particles, the containers and the bed heights used in the two experiments. Note that the container used in the experiment of Meto et al. was cylindrical and over 40 waves appeared in the: bed. The wave patterns observed in the cylindrical contaie~er were also different from those observed in this experiment.

The wave state is very unstable and can be disturbed easily. If the vibration is not steady or uni form, or the bed is disturbed by an external force, the bed can easily transform from the wave state to the arching state.

Wassgren et al. [6] defined a Froude number as the ratio of the vibrational velocity amplitude and the square root of the product of the gravitational acceleration and the wave amplitude, Fr= : . (g r / )~ / : , where r / i s the wave amplitude. They found that the Froude number was well correlated with the dimensionless acceleration amplitude for a granular bed of [ixed height. The current experiment investigated the influ- ences of the acceleration amplitude and the initial bed height on the wave amplitude. For a container of width 19 cm, Fig. 6 shows that the wave amplitude varied with the dimensionless acceleration amplitude for beds with different initial bed heights, H. The wave amplitude increases with the accelera- tion amplitude due to the bed receiving higher energy from the base plate. The wave amplitude also decreases with increase in the initial bed height ( the case with more particles in the bed) under the same level of vibrational acceleration, because the energy received by each particle is smaller for the thicker bed. Keeping increasing the initial bed height, the wave phenomenon will disappear eventually. Fig. 7 shows the wave Froude number as a function of the vibrational acceleration amplitude. The Froude number is independent of the acceleration amplitude. For the beds with initial heights of H/do= 10. 20, and 30, the Froude numbers are very close to 1.35, 1.50, and 1.74, respectively.

H (cm)

t=(1/3)T

7 0

6 0

5 0

3 0 5 10 15

>

unit vecl

(m/sec)

L (cm)

t=(2/3)T

9 0

8 O :

7 0 ~

6 0

5 0

4 0

I t

) r'

, s

10 15

++I +0!

t=(3/3)T ;

7+ I 6C,

5 O

0 10 15

Fig. 5. Ve loc i ty d i s t r ibu t ion for the bed in the w a v e state ( F = 5. I ).

~,4 5,'.S. Hsiau. S..I. l'att ~Prowler Teclmolo~,,~ 96 (1998) 219 220

17.5

E E 15.0

E <~ 1 2 . 5 -

> c~

10.0

7.5

4,5

n H : 3 Cm

H = 6 c m

o H : 9 c m

D C

a

[3

D

A

Z~ z&

z&

O 0

O

0

C

C

o o

o c i L

L h , I i i i , I ,

5 . 0 5 . 5

Vibrational Acceleration (F) Fig. 6. Wa,,e amplitude plotted againsl the vibrational accclcrathm for dif ferenl initial bed heights.

2.0 I i [

1.5

..Q

E ;~ 1.0

" O

o L L

0.5

0.0

o O O O O O O O

D O D f ] ~ O D D

o o c

D H = 3 c m

H = 6 c rn

o H = 9 c m

I L , ~ , l J ~ J , I ,

4 . 5 5 . 0 5 . 5

Vibrational Acceleration ([') Fig. 7. Wave Froude number plotted againsl the ,,ibrational acceleration I'o differcnl initial bed heights.

H (cm) 7~:

6O

t = (1 /3 )T ~

4 3

3 9

7 :;

6 3

t=(2/3) -r ~o

4 0

3 0

/

/ / /

; / ,,i

E, 10

/ /

p , / . / /

5 10 15

5

unit vect

(m/sec)

L (cm)

t= (3 /3 )T

70

6 0

5 0

4 C

3 C

,/ ./

/~ .I / /

5 tC 15

F ig . ~. V e l o c i t y d i s t r i b u t i o n for t he b e d in the a r c h i n g s t a l e ( F = 6 . 4 ).

S.S. H~iau, 5'.,/. Pan / Powder Teclmoh*,<v 96 (1998) 219-226 225

3.5. Arching t)henomeno,2

Arches are formed in the bed when the dimensionless acceleration amplitude is greater than 11,~. As shown in Fig. I ( e ) , the area where particles move slowly is called a node and the area where particles have greater velocities is called an anti-node. Convective motion occurs on each side of a node. Fig. 8 shows the velocity distributions at three different times. The motion period T is four times the vibra- tional period. As opposed to the other states, therc are signif- icant horizontal movements in the arching state. The bed leaves the base plate twice during a motion period, and the longer llight time is about two vibrational periods.

Since the upper limit of the vibrational acceleration for the current apparatus is 7 g, there may be other new phenomena at higher accelerations. However, no such phenomena seem to have been reported in the literature.

3.6. Mot ion xtate tran.~ili,'mx

As the vibrational acceleration is slowly increased, the motion state changes from heaping, to coherent, to expansion. to wave, and tinally to arching. The critical transitional accel- erations between the states can be observed. After the bed reaches the arching state~ it does not follow the original route back through the previous; states when the vibrational accel- eration is decreased.

The following experiment uses the container with a lixed width of 19 cm and an initial bed height of 3 cm. Fig. 9 show~ the state transitions for type I glass beads. In the ligure, there are five areas corresponding to the live motion states. The heaping, coherent, expansion, wave and arching phenomena are denoted by the symbols I to V, respectively. The state transitions are clearly shown by the solid line in the figure.

I:Heaping

II:Coherent

I l l :Expansion

IV:Wave

V:Arching

4 O

t ~ -g r n

' ' I i

V 5 ~ .... T - -

i / / / '

A Iv_

/ //

3 [ ~ ' . /

l t . . " , , , I , , I , , , , I i , I t ~ , , I , , ,

1 2 3 4 5 6

Vibrational Acceleration (F)

Fig. 9. State transitions for parlicles o f l y p c I.

The bed begins to heap when F > / ] , = 1.2. The coherent and expansion states occur when I ' > I~,= 2.2 and F > l]~ =4 . I. The critical accelerations fl)r the wave and arching phenom- ena arc / \ v = 5 . ( ) and ~ \ - 6 . 2 . The motion state follows another state-acceleration relation to return when the accel- eration is decreased. The wave phenomenon does not appear in the return mute. The state changes directly' from arching to expansion when the acceleration is smaller than the critical value of F - 4 . 4 . This is a hysteresis phenomenon.

Using the same particles ( type I ) but changing the initial bed height (H/d, = 10, 20, and 33.33). the critical accelera- lions of the five states are shown in Fig. 10. It is found that the critical values are smaller for the case with more particles in lhe container ( larger H/do). though the differences are not significant.

Comparing the three different types of glass beads and lixing the initial bed height at 3 cm, the cowesponding critical accelerations are shown in Fig. 11. The wave and arching phenomena are not observed for type 2 particles. Type 2 beads have a smaller angle of repose and a smaller standard devia- lion of the diameter. This kind of particle is highly spherical and smooth. The collisional and frictional forces among par- ticles and between particles and the side walls seem to play important roles in the appearance of the wave and arching states. However, it is necessary to perform more investiga- tions and measurements to dra,a, the above conclusion.

The convective motion of Ihe air may play an important role in the states. However. the force from the air to the particles is much lower than the body lorce, so the force of the air should not be important for the existence of the states. The collisional and frictional forces among particles and

50

4 5

4 0

-o ~ 35 2-

c -

+-' 30 O -

a

-g 25 cfl

20

15

10

10

' ' ' ' I ' ' ' ' I ' ' ' ' I ' ' ' ' I ' ' ' '

Heaping

Coherent

, Expansion

' Wave

' Arching

o

=

. . , I , . . , I , . . , I , , . . 1 , ~ , , I . . , ,

2 . 0 3 .0 4 .0 5 .0 6 .0 7.0

Vibrational Acceleration (F) Fig. 10. Critical accelerations lor Ihe tixc states for different initial bed hcigllts.

226 S.S. ttsiat¢, S,.I. Prm / Powder TectmOlOLt O' 96 (I 908) 219 226

5 . 0 . . . . I . . . . I . . . . I ' ' . . . . . . I ' ' ' ' Acknowledgements

4 . 0

ID

13_

3 . 0

r-

eD c-

2 . 0

1 .0

j Heaping

k Coherent

,,,' E):pansion { Wave

i , Aiching

" i n n l I n i l u i n n n n | , n ~ , | n n n a l n n ,

1 . 0 2 . 0 3 . 0 4 . 0 5 . 0 6 . 0

Vibrational Acceleration (F) 7.0

Fig. I I. Critical accelerations for the f ive s lates l , r difl'erenl types or

particles.

between particles a3d the side walls are believed to be the key factors.

4. Conclusions

Five motion states are observed in the vibrated granular bed. The heaping, coherent, expansion, wave, and arching phenomena appear when the vibrational acceleration is increased through the corresponding critical values. A hys- teresis phenomenon exists. The wave and arching states are not observed for the very spherical, smooth particles. Colli- sional and frictional li)rces play important roles in generating the wave and arching phenomena. The velocity distributions for the states have been measured. To understand the internal physics of these phenomena, it is necessary to obtain more data concerning the velocity distributions and the conveclive motion.

The authors would like to acknowledge the support from the National Science Council of the R.O.C. for this work through Grants NSC83-0410-008-018 and NSC84-2211-E- 008-036.

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