The flow regime during the crystallization state and convection state on a vibrating granular bed

Advanced Powder Technology 20 (2009) 335–349

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Advanced Powder Technology journal homepage: www.elsevier.com/locate/apt

Original Paper

The flow regime during the crystallization state and convection state on a vibrating granular bed Shih-Chang Tai, Shu-San Hsiau * Department of Mechanical Engineering, National Central University, Chung-Li 32001, Taiwan, ROC

a r t i c l e

i n f o

Article history: Received 24 September 2008 Accepted 20 January 2009

Keywords: Granular motion Occupied percentage Concentration distribution Crystallization Convection Particle segregation

a b s t r a c t It is possible to utilize the large particle distribution on the free surface and inside a vibrating granular bed to understand the segregation phenomenon and the granular motion states in the vibrating bed. In this study we strive to analyze different flow regimes in a binary mixture in a granular vibrating bed. The granular temperature of large particles on the surface of the vibrating bed can be used to define the motion states, the crystallization state or the convection state. When the dimensionless vibration amplitude increases from 0.25 to 1.0, the granular motion transforms from the slowly stabilizing crystallization state to the strong convection state. When the amplitude of the dimensionless vibration increases from 1.25 to 2.5, the granular motion transforms from the fast stabilizing crystallization state to the unstable crystallization state. The percentage of large particles occupying the free surface and the concentration of these large particles are analyzed to understand the motion states. Ó 2009 The Society of Powder Technology Japan. Published by Elsevier BV and The Society of Powder Technology Japan. All rights reserved.

1. Introduction The rheology of granular systems has been widely investigated over the last decade as evidenced by the many published papers [1,2]. The rheological properties of granular suspensions are normally studied via molecular dynamics simulations [3] from which the ‘‘flow diagram” for the volume fraction or stress plane is derived. Recently the mechanisms for pattern formation in granular systems have been experimentally and theoretically studied, especially, those concerned with the relation between particle mobility and structure, in the quasistatic regime [4,5]. Since the pioneering work of Kosterlitz and Thouless [6] the properties of the melting transition of two-dimensional solids have been extensively studied. Continuous crystallization-to-liquid melting transition has been demonstrated by the measurement of two-dimensional melting in magnetic bubble arrays [7]. Several interesting phase transitions in granular system have been reported [8–15]. Olafsen and Urbach [14] investigated the transition from a hexagonally ordered solid phase to a disordered liquid in a monolayer vibrating spheres. Their experimental results showed strong similarity to simulations of the melting of hard disks in equilibrium, despite the fact that the granular monolayer was far from equilibrium, due to the effects of vibration force and interparticle collision and dissipation [16].

* Corresponding author. E-mail address: [email protected] (S.-S. Hsiau).

Clerc et al. [17] carried out a combined experimental, numerical and theoretical study of liquid–solid-like phase transitions that occurred in a vertically vibrating fluidized dense granular system. They characterized the dynamic behavior of the phase transition, while avoiding 2D effects such as curvature between the phase or crystal orientation interaction dependence. The crystallization phenomenon has also been studied by a number of researchers [18,19]. Reis et al. [18] investigated the crystallization of a uniformly heated quasi-2D granular fluid as a function of the filling fraction. They utilized the Lindemann ratio to define whether the granular bed was in the condition of crystallization or not. When the filling fraction U was 0.719, the Lindemann ratio of 0.15 indicated that the granular bed entered the crystallization condition. Related references showed that when the Lindemann ratio was between 0.1 and 0.15, the granular bed would be packed as crystallization [20,21]. The Lindemann ratio L was defined as the root-mean-square displacement of particles in a crystalline solid about their equilibrium lattice positions [22], divided by their nearest neighbor distance b. N 1 1 X L¼ ðDRi Þ2 b M i¼1

!1=2 ;

ð1Þ

where M is the number of particles; and the portion in brackets [. . .] denotes the average over the dynamic trajectories of the particles. The Lindemann criterion states that the crystal melts when L overcomes some ‘‘critical” (yet not specified a priori) value Lc [20]. Obviously, one would hope this latter quantity to be approximately

0921-8831/$ - see front matter Ó 2009 The Society of Powder Technology Japan. Published by Elsevier BV and The Society of Powder Technology Japan. All rights reserved. doi:10.1016/j.apt.2009.01.003

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Nomenclature a/d A AL AT CL,i CL CL,av Cs C D f g* L N Nu Nl

dimensionless vibration amplitude vibration acceleration, m/s2 area occupied by large particles, mm2 whole area of the free surface of the granular bed, mm2 concentration distribution of large particles in the ith layer of the granular bed concentration distribution of large particles average of concentration distribution of large particles segregation coefficient circumference of the cell, mm diameter of small particles, mm vibration frequency, Hz gravitational acceleration, m/s2 dimensionless Lindemann ratio number of particles amount of particles in the higher layer of the granular bed amount of particles in the lower layer of the granular bed

the same value for different pair potentials and thermodynamic conditions. In fact, the Lindemann ratio is not universal at all, with values spanning the range of 0.12–0.19. More specifically, Lc is reported to be 0.15–0.16 in a face-centered-cubic (FCC) solid and 0.18–0.19 in a body-centered-cubic (BCC) solid [22]. Moucka and Nezbeda [23] utilized the concept of the shape factor to define the difference between the fluid and crystallization. The shape factor f is the variable degree of the structure of the fluid and crystallization, defined as

f¼

C2 ; 4pS

ð2Þ

where C is the circumference of the cell and S is its surface area. Thus, when the shape factor was 1.103, the structure of the granular bed would form the regular hexagonally ordered solid phase. However, it has been found that the combined simultaneous use of the circumference and surface area to find a shape factor is sensitive to changes in the 2D structure and clearly marks domains made up of different figures. Particularly, it gives clear physical pictures of competition between more or less ordered domains and of the gradual building of a regular hexagonal arrangement in the region of the phase transition. The significant phenomena affecting the phase exhibited by the granular bed and the motion mechanism during vibration have been reported [10,11,14]. The motion mechanisms include crystallization [17–20] and convection [24–27] of the particles. Under some conditions, the flow of granular materials takes the form of convection cells [28]. The convection phenomena have drawn the interest of many researchers over the last decade. Gallas et al. [28], for example, used molecular dynamics method to study convection cells in a two-dimensional system. They found different types of convection cells that occurred because of the existence of walls. Taguchi [29] studied such convection cells numerically and proposed that they were induced by the elastic interactions between particles. Luding et al. [30] employed molecular dynamics methods to investigate convection cells in vibrating grains. Hsiau and Chen [31] experimentally studied the phenomena of the occurrence of convection cells in a two-dimensional box with 3-mm glass beads under different vibration intensities. They used image processing technology and particle tracking methods to measure and analyze the velocity fields and strength of the convection under different vibration conditions.

PL PL,av r S TL TL,av TD TR uK

vK hui hvi

C

x f

U

percentage occupied by large particles average percentage occupied by large particles vibration amplitude, r = a/x2, m surface area of the cell, mm2 granular temperature of large particles, cm2/s2 average granular temperature of large particles, cm2/s2 time for large particles to rise to the free surface, s time at which large particles first occupy the maximum area of the free surface, s velocities of the kth large particle in the x-axis direction, cm/s velocities of the kth large particle in the y-axis direction, cm/s ensemble averaged velocity in the x-axis direction, cm/s ensemble averaged velocity in the y-axis direction, cm/s dimensionless vibration acceleration vibration radian frequency, 1/s shape factor dimensionless filling fraction

In this study we described several motion mechanisms, including self-organization, convection, geometrical void-filling and phase transitions that may occur due to collisions among the particles in a vibrating granular bed system. These mechanisms generated crystallization and convection states in the bed. The motion mechanisms also influence the segregation process in the granular bed. We investigated the relations between the formation of crystallization and motion mechanisms, such as the occurring time of crystallization structure and the formation condition of crystallization structure at different motion mechanisms. Besides, we employed the method of sampling granular bed ‘‘jelled” at a steady state allowing us to effectively examine the particle distributions inside the granular bed without involving expensive instruments.

2. Experimental Although the crystallization packing phenomenon has been widely and deeply studied in the past, the relation between the convection state and crystallization state has received less attention. Fig. 1a shows a schematic drawing of the experimental apparatus. A Techron VTS-100 (Vibration Test System; Techron Instruments Inc.) electromagnetic vibration system served as the vertical shaker. The shaker was vertically driven by sinusoidal signals produced by a function generator (Meter DDS FG-503) through a power amplifier (Techron 5530). The vibration frequency f and acceleration a were measured by a Dytran 3136A accelerometer fixed at the shaker and connected to an oscilloscope (TDS 210). The radian frequency x and amplitude r of the vibration were calculated using the relations x = 2pf and a = A/x2, respectively. The dimensionless vibration acceleration C was defined as C = A/g, where g is the gravitational acceleration. The vibration frequency used in the experiments was controlled at 20 Hz. The range of the dimensionless vibrated amplitudes a/d was between 0.25 and 2.5, where d is the diameter of the small nylon beads (d = 3.14 mm). In this study we performed some preliminary experiments at different vibration conditions (different vibration frequencies and vibration amplitudes). At the close vibration frequencies (10–60 Hz) there were similar motion regimes and state transitions. This indicated that the occurrence of the motion regimes and phase transitions at the close vibration frequencies depended on the intensity of the vibration amplitude [32–34]. The current experimental results could be applied to the different vibration conditions with vibration frequency relatively close to 20 Hz (10–

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337

shown in Fig 1b; and the total mass of the small particles in the tank was 236.14 g. The purpose of adding large particles was not for particle tracking purpose but for analyzing the variation of the segregation intensity of the particle bed. We have verified that the behaviors of self-organization, convection and geometrical void filling still occurred if the monosized granular materials were used. Adding large particles could enhance the motion state among particles but it did not seriously change the motion mechanism of the whole particle bed. Different motion mechanisms could generate different segregation intensity at different vibration conditions. The state of the segregation could be characterized by segregation coefficient. The image sensor of the digital video camera was set above the top of the granular bed to record the granular motion on the free surface of the granular bed as shown in Fig. 1a. The whole free surface of the granular bed (80 mm  80 mm) was visible through the viewing window. The average velocities in the directions of the Xaxis and Y-axis in each cell, hui and hvi, were averaged from at least 50 particle velocities as follows:

PN

K¼1 uK

hui ¼

hv i ¼

N PN

K¼1

N

vK

;

ð3Þ

;

ð4Þ

N is the total number of velocities used for averaging the mean values; and uK and vK are the velocities of the Kth particle measured from the two consecutive images containing the Kth particle. The fluctuation velocities in the two directions were calculated by 2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN 2 K¼1 ðuK  huiÞ ¼ ; N1

0 2 1=2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN 2 K¼1 ðv K  hv iÞ ¼ : N1

hu0 i1=2

hv i

ð5Þ

ð6Þ

The granular temperature TL was used to quantify the kinetic energy of the flow and could be calculated from

TL ¼ Fig. 1. Schematic drawings (a) of the experimental apparatus and (b) of particle packing in the granular bed.

60 Hz). And in the experiments, the difference of motion mechanisms could affect the reproducibility of the experiments. In our study, the error bars at slowly stabilizing crystallization state, convection state and fast stabilizing crystallization state in Figs. 8 and 9 were very small. This meant that the reproducibility was high. At unstable crystallization state, the motion state was belonging to unstable granular motion, but the error bars of the data at unstable crystallization state in Figs. 8 and 9 were small. Therefore, the reproducibility at unstable crystallization state was also satisfied. The height, width and depth inside the tank were 120 mm, 80 mm and 80 mm. We stuffed the tank with two sizes of nylon beads (with mean diameters of 3.14 mm and 7.95 mm). The densities of the two kinds of particles were 1142 kg/m3. Before each experiment, a paper cylinder (with a height of 31.4 mm and a radius of 45 mm) was placed in the center of the container. A mass of 31.32 g of the large beads was fed into this paper cylinder (Fig. 1b). The filled mass was the same as the mass of the large particles to form one horizontal layer of beads in the empty container. The space outside the paper cylinder was then filled with small particles to the same height as that of the paper cylinder (31.4 mm). Finally the paper cylinder was removed. The tank was continually filled with small particles to the height of 60 mm, as

hu0 2 þ v 0 2 i : 2

ð7Þ

Since in the current study we follow the auto-correlation technique developed by Hsiau and Shieh [35], the correlation values of gray level derivatives are considered. The experimental errors of the velocities were reduced to within 1.5%. The granular motions were not significant during some of the experimental runs because of the relatively weak vibration conditions. It sometimes took a long time to capture images with noticeable granular motion on the free surface in succession. During the experiments, the large nylon beads that had been placed at the bottom of the tank would begin to move up to the surface of the granular bed at different dimensionless vibration amplitudes. The granular motion of the large particles was different during different motion states. The area occupied by large particle on the free surface, AL, can be found from the images. The whole area of the free surface of granular bed, AT, was known. The percentage of area on the surface of the granular bed occupied by large particles, PL, could be defined as follows:

PL ¼

AL  100%: AT

ð8Þ

We further probed into the relation between the granular concentration distribution inside the bed and the movement mechanisms of the particles. There were three movement mechanisms characterized. Every particle had influences on the flow development in its neighborhood and, simultaneously, its motion was also influenced by the surrounding flow. This feedback resulted in an

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organized motion of the flow. It was called self-organization. The granular materials may circulate with an outward velocity from the center of the free surface and a downward motion along the side walls. It was denoted as the ‘‘convection regime”. Geometrical void-filling mechanism was a local geometric effect leading to size segregation with small particles falling through voids and large particles rising up to the top. During the expansion phase of each shaking cycle all particles moved upwards, but during the compaction phase the small particles had higher probability falling through finding voids. This led that the probability of the small particles filling a void was greater than a large particle. If each sphere was packed into a hexagonal array locked in place by its six neighbor particles, this would be denoted as the ‘‘crystallization regime,” as shown in Fig. 2a and b. Konjac jelly powders were used to ‘‘freeze” the whole granular bed. The particle distribution could then be analyzed in three dimensions. The distribution of large particles inside the granular bed was measured along three different planes (Z = z/h, h = 20 mm and Z = 1, 2, and 3). The results help us understand the internal distribution of large particles. The concentration distribution of large particle in the ith layer CL,i is defined as the ratio of the number of large particle NL,i in the ith layer to the total number P of large particles 3i¼1 N L;i in the granular bed:

NL;i C L;i ¼ P3  100%: i¼1 N L;i

ð9Þ

The number of particles in the upper half layer of the granular bed Nu and the number of particles in the lower half layer of the granular bed Nl were calculated at different dimensionless vibration amplitudes. The segregation coefficient Cs of granular bed could be defined as

Cs ¼

Nu  Nl  100%: Nu þ Nl

ð10Þ

CS = 0 indicated the fully mixed state and CS = ±1 (±100%) the completely segregated states, with all the large particles located in the bottom-half (CS = 100%) or in the top-half (CS = +100%). However, many studies investigated the related topic about the continual increase of the vibration acceleration [36–39]. In this study, the movement behaviors of experiment conducted under the fixed vibration parameter and continuous increase of the vibration acceleration were similar. Most industry processes were operated under the fixed vibration condition. Thus, the current experiment utilized the experimental approach of the fixed vibration acceleration to perform all experiments. The experimental approach of this study could be closer to the processing of industry community than increasing vibration parameter continuously. Most motion states in three-dimensional granular beds are usually studied via numerical simulations or utilizing high-cost experimental devices, like nuclear magnetic resonance imaging (MRI) or X-rays. Our method of sampling the granular bed ‘‘jelled” at a steady state allows us to effectively examine the particle distributions inside the granular bed without involving expensive instruments. 3. Results and discussion 3.1. Four flow regimes The six plots shown in Figs. 3–6, respectively, indicate: (a) the variation of granular temperature TL with time at different dimensionless vibration amplitudes; (b) the percentage of large particles PL on the free surface of the granular bed as it varies with time at different dimensionless amplitudes; (c) the concentration distribution of large particles CL,i in the ith layer of the granular bed as it changes with time at different dimensionless amplitudes; (d) the variation of the segregation coefficient CS with time; (e) images of granular motion on the surface; and (f) the change with time of the concentration distribution of large particles at different dimensionless vibrated amplitudes. The four motion states observed in the experiments could be defined by the granular temperature TL, depending on the dimensionless vibration amplitudes (as shown in plots (a) in Figs. 3–6) as follows: slowly stabilizing crystallization state; convection state; fast stabilizing crystallization state and unstable crystallization state. The variation of the granular temperature on the surface of the vibrating bed could also be utilized to define the flow regimes as either the crystallization regime or the convection regime. The former was characterized as the state of granular motion where crystallization lattices formed and particles were packed, as shown in Fig. 2a and b. There were three types of crystallization regime, including the slowly stabilizing crystallization state, the fast stabilizing crystallization state and the unstable crystallization state. The details of the measured parameters indicating the four motion states (Figs. 3–6) were given below.

Fig. 2. Crystallization packing structures on the (a) X–Z plane and (b) X–Y plane.

3.1.1. The slowly stabilizing crystallization state The slowly stabilizing crystallization state (at a/d = 0.25 and 0.5) was mainly the resulted of the self-organization mechanism. The granular motion at a/d = 0.25 was very weak, making the short

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term observation of granular motion difficult. Thus the larger timescale plots of the granular temperature, occupied percentage and segregation coefficient at a/d = 0.25 are inserted in Fig. 3a, b and d, respectively. Due to the size ratio, it can be seen in Fig. 3b that some of the large particles, which were at the bottom initially, rose up to the free surface during the first 85 s (a/d = 0.5) or the first 5000 s (a/d = 0.25), resulting in a sharp increase of the occupied percentage of large particles. When more large particles occupied the free surface, the fluctuation velocities of large particles at the free surface were larger than those inside the bed where the particle motions were limited by the weight of the particles above. Thus the fluctuations and granular temperature of large particles increased sharply in the first 85 s at a/d = 0.5 and the first 5000 s at a/d = 0.25 as shown in Fig. 3a. However, some large particles sank into the bed from the top between 85 s and 200 s at a/d = 0.5 and between 5000 s and 7000 s at a/d = 0.25 so that the occupied percentage of the free surface decreased, as shown in Fig. 3b, resulting in the decrease of velocity fluctuations and granular temperature of large particles in Fig. 3a. The initial phenomenon of granular motion mentioned above was similar, regardless of how large the dimensionless vibrated amplitude was. Therefore the trends shown by the granular temperature TL and the occupied percentage PL in the initial stage were very similar to the other cases, as shown in plots (a) and (b) in Figs. 3–6. As the time continued to increase from 200 s to 550 s at a/ d = 0.5 and from 7000 s to 9200 s at a/d = 0.25, the large particle

a

distributed over the vibrating bed much uniformly than the distribution of large particles at other time and other vibration conditions, as shown in plot (ii) in Fig. 3f. It meant that large and small particles formed a good mixture at the vibration conditions at that time. The granular motion was stable, resulting in the granular temperature retaining its stable status, as shown in Fig. 3a. The concentration distribution in the third layer increased when a/ d = 0.25 and 0.5, as shown in Fig. 3c. The segregation coefficient consequently increased (from 100% to 14.7% at a/d = 0.5 and from 100% to 25% at a/d = 0.25), as shown in Fig. 3d. The arrangement of the small particles in a regular order (crystallization packing) is shown in plot (iii) in Fig. 3e. The crystallization packing grew from the bottom to the free surface between 550 s and 900 s at a/d = 0.5 and between 9200 s and 18,500 s at a/d = 0.25. Granular motion in the bed was limited by the crystallization packing, resulting in the decrease of granular temperature (Fig. 3a). This significant decrease in granular temperature was one specific phenomenon which could be observed in the crystallization regime. Because the structure of the crystallization packing was so stable it could not be easily destroyed by other motion mechanisms, it allowed large particles to rise to the free surface and not sink into the bed, as shown in plots (ii) and (iii) in Fig. 3e and plots (iii) and (iv) in Fig. 3f. The percentage occupied by large particles increased from 550 s to 900 s at a/d = 0.5 and from 9200 s to 18,500 s at a/d = 0.25 (Fig. 3b). The concentration in the third layer continued to increase (Fig. 3c) until the crystalli-

1.6 8E-05

a/d = 0.25

1.4

7E-05 6E-05

4E-05 3E-05

c 100

2E-05

0.8

1E-05

0.6

0

5000

10000

15000

a/d = 0.5

0.2 0

500

1000

1500

2000

2500

3000

0

3500

65

a/d= 0.5

80

50

60

45

40

0

500 1000 1500 2000 2500 3000 3500 4000

Time(Second)

Time(Second)

a/d = 0.5

100 80

40 35 30 P L (%)

25 20 15 10 5

65 60 55 50 45 40 35 30 25 20 15 10 5 0 0

a/d= 0.25

500

1000

1500

2000

Time (Second)

20

60 40

0 -20

20 0 -20

-40

-40 -60

-60 10000

15000

2500

3000

a/d = 0.25

-80 -100

-80 5000

0

20000

Time (Second)

0

Third Layer Second Layer First Layer

d 100

55

0

100 90 80 70 60 50 40 30 20 10 0

5000 10000 15000 20000

Time (Second) 60

P L (%)

ThirdLayer SecondLayer FirstLayer

90 80 70 60 50 40 30 20 10 0

Cs (%)

b

0

20000

Time (Second)

0.4

CL (%)

0

Cs (%)

TL(cm2/s2)

1

5E-05

CL (%)

T L(cm2/s 2)

1.2

5000

3500

10000

15000

20000

Time(Second)

-100 0

500

1000

1500

2000

2500

3000

3500

Time (Second)

Fig. 3. slowly stabilizing crystallization state: variations of (a) TL, (b) PL, (c) CL,i, (d) CS, with time for a/d = 0.25 and 0.5. (e) Images of granular motions, and (f) three-dimensional concentration distributions at a/d = 0.5.

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a/d = 0.5 e

f a/d = 0.5

2

1.5

1.5

Z

1

Y

4 1 2

X

x

3 4

4 1 2

y

X

1

x

4

(i) 0 s

2

1.5

4 1 2

X

3

x

2

3 4

1

X

1

3

x

10 9 8 7 6 5 4 3 2 1 0

2

1.5

4

4 1 2

X

3

x

2

3 4

(iv) 900 s

1

(V) 1800 s

y

2

y

1

3

2.5

2

1.5

Z

1

Y

y

3

2

(iii) 600 s

2.5

Z

1

Y

2

3

z

10 9 8 7 6 5 4 3 2 1 0

2.5

Z

4 1

y

(ii) 300 s

3

z

10 9 8 7 6 5 4 3 2 1 0

3

1

Y

3

2

2

1.5

Z

1

Y

3

2

2

2.5

z

Z

2.5

3

z

2.5

10 9 8 7 6 5 4 3 2 1 0

3

z

10 9 8 7 6 5 4 3 2 1 0

3

z

10 9 8 7 6 5 4 3 2 1 0

1

Y

4 1 2

y

X

3

x

3 4

1

(Vi) 3600 s

Fig. 3 (continued)

zation packing structure was distributed all over the bed. The segregation coefficient increased to 70% at a/d = 0.5 and to 50% at a/ d = 0.25 (Fig. 3d). After the crystallization packing structure filled the granular bed after 900 s (a/d = 0.5) and 18,500 s (a/d = 0.25) as shown in plots (iv) and (v) in Fig. 3e and plots (v) and (vi) in Fig. 3f, all parameters reached a steady state, including the granular temperature, occupied percentage, concentration distribution and segregation coefficient. The granular temperatures were about 0.14 cm2/s2 (a/d = 0.5) and 2  105 cm2/s2 (a/d = 0.25). The percentage of the surface of the granular bed occupied by large particle reached a steady state of about 35% at a/d = 0.25 and 0.5 and the fluctuation in the quantity of the occupied percentage was smaller than 20% at a/d = 0.5 and smaller than 10% at a/d = 0.25, as shown in Fig. 3b. The concentration distributions of large particles in the third layer at a steady state were between 60% and 80% at a/d = 0.5 and between 60% and 70% at a/d = 0.25, as shown in Fig. 3c. The size segregation reached a stable value with the

segregation coefficients between 70% and 80% at a/d = 0.5 and between 40% and 60% at a/d = 0.25 when the granular temperature reached a steady state. The speed of generating the crystallization packing structure (slowly stabilizing crystallization state) was the slowest. 3.1.2. The convection state When the dimensionless vibration amplitude increased to 0.75, 0.875 and 1.0, the motion was transformed into the convection state. The convection state was mainly the result of the convection mechanism. The initial phenomena of granular motion in the first 20 s at a/d = 1.0, the first 200 s at a/d = 0.875 and the first 400 s at a/ d = 0.75 (Fig. 4a and b) were similar to what occurred during the first 85 s at a/d = 0.5(Fig. 3a and b). The granular temperature and the occupied percentage increased sharply with time and then decreased. The plot of the granular temperature at a/d = 1.0 is inserted into Fig. 4a to make for convenient observation of the parameters.

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When the time increased from 20 s to 2000 s (a/d = 1.0), from 200 s to 1200 s (a/d = 0.875) and from 400 s to 3450 s (a/d = 0.75), more large particles became distributed throughout the convection cell which extended to the four corners of the bed, as shown in plots (ii) and (iii) in Fig. 4e and plots (ii) and (iii) in Fig. 4f. It was due to the small particles generating four convection rolls in four corners making large particles and exhibiting a circulating motion outward from the center of the free surface and downward along the side walls. Large particles were distributed over the inner part of convection rolls in the four corners. Because the mass of large particles were larger, it was not easy for large particles to generate the movement. Thus the large particles were distributed over the inner part of the convection roll. The convection state was illustrated by the three-dimensional concentration distributions of large particles (Fig. 4f).The granular temperature of the large particles increased quickly (Fig. 4a). The convective motion during the convection state caused the concentration distribution of large particles to reach a stable state after a short time, as shown in Fig. 4c. Most of the large particles were distributed over the convection cell which was in the upper layer of the bed. The size segregation increased to about 60% at a/d = 1.0, 50% at a/d = 0.875 and 40% at a/d = 0.75 (Fig. 4d). When the convective motion stabilized during the convection state (after 2000 s at a/d = 1.0, 1200 s at a/d = 0.875 and 3450 s at

a3

a/d = 0.75) all the parameters reached the stable stage, including the granular temperature, the occupied percentage, the concentration distribution and the segregation coefficient. The granular temperature was about 0.28 cm2/s2 for a/d = 0.75, 1 cm2/s2 for a/ d = 0.875 and 4.16 cm2/s2 for a/d = 1.0, larger than that at the crystallization state (TL ffi 0.14 cm2/s2 at a/d = 0.5; see Fig. 3a). In the convection state most of the large particles were inside the bed, as shown in Fig. 4e and f. The percentage of the surface occupied by large particles was at its lowest value (PL < 10% at a/d = 1.0), compared with the other states, with fluctuations in quantity of occupied percentage of less than 15%, as shown in Fig. 4b. Because the vibration intensity was not strong enough, some of large particles were still distributed over the bottom of the particle bed at the vibration amplitude of 0.75 and 0.875. When the vibration intensity was strong enough, the particles at the bottom of the particle bed at the vibration amplitude of 1.0 could participate in the movement of the convection cell and could be distributed over the upper positions of convection cell. Thus the concentrations of large particles at the bottom of particle bed at the vibration amplitude of 1.0 were less. Most of the large particles were distributed throughout the convection cell between the second layer and the third layer of the granular bed, so that the concentration distributions in the upper two layers were close (as shown in Fig. 4c). This could have caused the segregation coefficient to be higher (between 40% and

c 100

2

2

TL(cm /s )

2

1.5

100 90 80 70 60 50 40 30 20 10 0

Third Layer Second Layer First Layer

Third Layer Second Layer First Layer

CL (%)

CL (%)

2.5

90 80 70 60 50 40 30 20 10 0

0

500 1000 1500 2000 2500 3000 3500 4000

0

500 1000 1500 2000 2500 3000 3500 4000

Time(Second)

Time(Second)

100

1

Third Layer Second Layer First Layer

90 80

CL (%)

70

0.5

60 50 40 30

0

20

0

500

1000

1500 2000 Time (Second)

2500

3000

3500

10 0 0

b 65

1500

2000

2500

3000

3500

4000

d 100

55

80

50

60

45

40

40

20

35

Cs (%)

30

100 80

0

60 40

25

-20

20

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15

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20

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Fig. 4. convection state: variations of (a) TL, (b) PL, (c) CL,i, (d) CS, with time for a/d = 0.75, 0.875 and 1.0. (e) Images of granular motions, and (f) three-dimensional concentration distributions at a/d = 1.0.

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60%; see Fig. 4d). The large particles were more uniformly mixed in the upper two layers. The granular temperature of large particles increased with the increase of a/d (Fig. 4a). The occupied percentage decreased with the increase of a/d (Fig. 4b). This can be explained by most of the large particles being distributed throughout the convection cell in the four corners of the bed at the stronger convection mechanism (Fig. 4e and f); the distribution of large particles could diffuse to the lower layers of the bed during the stronger state of convection.

150 s at a/d = 1.5 and the first 26 s at a/d = 1.25) was similar to that which occurred in the first 85 s at a/d = 0.5 (Fig. 3a and b). The granular temperature and the occupied percentage first increased sharply with time and then decreased, as shown in Fig. 5a and b. When the time increased from 150 s to 250 s at a/d = 1.5 and from 26 s to 45 s at a/d = 1.25, most of the large particles became distributed relatively uniformly over the third layer of the vibrating bed. The geometrical void-filling mechanism caused the large particles rise to the upper layers. The stabilizing of the granular motion resulted in the granular temperature becoming stable, as shown in Fig. 5a. The concentration distributions in the third layer increased at a/d = 1.5 and 1.25 (Fig. 5c). Therefore, the segregation coefficient increased (becoming close to 100%), as shown in Fig. 5d. The arrangement of small particles in a regular order (crystallization packing as shown in plot (i) in Fig. 5e and plot (ii) in Fig. 5f) caused the crystallization packing structure to grow from the bottom to the free surface between 250 s and 350 s at a/d = 1.5

3.1.3. The fast stabilizing crystallization state When the dimensionless vibrated amplitude increased to 1.25 and 1.5, the motion transformed from the convection state to the fast stabilizing crystallization state. The fast stabilizing crystallization state was mainly the result of the geometrical void-filling mechanism. The initial phenomenon of granular motion (in the first

e a/d = 1.0

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Fig. 5. fast stabilizing crystallization state: variations of (a) TL, (b) PL, (c) CL,i, (d) CS, with time for a/d = 1.25 and 1.5. (e) Images of granular motions, and (f) three-dimensional concentration distributions at a/d = 1.5.

and between 45 s and 270 s at a/d = 1.25. The granular motion in the bed was limited by this packing, resulting in a decrease of the granular temperature (Fig. 5a). Note the significant decrease of granular temperature at the time between 550 s and 900 s at a/d = 0.5 (Fig. 3a) observed in Fig. 5a. The speed of the generation of crystallization packing was faster during the fast stabilizing crystallization state than during the slowly stabilizing crystallization state. This was because the geometrical void-filling mechanism had a stronger driving force for the generation of crystallization packing than did the self-organization mechanism. The occupied percentage and concentration distribution of large particles also reached the stable state faster. The segregation coefficient was 100% (Fig. 5d). When the granular bed was filled with the crystallization packing structure after 350 s at a/d = 1.5 and 270 s at a/d = 1.25, as shown in plots (ii) and (iii) in Fig. 5e) and plots (iii) and (iv) in Fig. 5f, all parameters reached the steady state, including the granular temperature, the occupied percentage, the concentration distribution and the segregation coefficient. It can be seen in Fig. 5e and f shows that the motion quickly reaches the stable crystallization state, which consolidates the granular packing structure in the granular bed, so the fluctuation in the quantities of granular temperature (Fig. 5a) and occupied percentage (Fig. 5b) became smaller. The percentage of large particles occupying the surface of the granular bed was about 55% at a/d = 1.5 and about 60% at

a/d = 1.25. The fluctuation in percentage occupied was the smallest (<10%) in Fig. 5b, resulting from the bed being filled so fast with the crystallization packing structure. All large particles were distributed throughout the upper layers of the bed, so the concentration distribution of large particles in the third layer quickly increased to 100% (Fig. 5c). This could cause the segregation coefficient be close to 100%. In the crystallization state, the concentration distribution in the third layer (due to the geometrical void-filling mechanism) had a higher value than that under the self-organization mechanism and reached the steady state faster. The speed at which crystallization packing was generated was fastest during the fast stabilizing crystallization state and the structure of crystallization packing was solider. 3.1.4. The unstable crystallization state At dimensionless vibration amplitudes of 2.0 and 2.5, the structure was in the unstable crystallization state between the convection state (governed by the convection mechanism) and the crystallization state (governed by the geometrical void-filling mechanism). The crystallized packed structure could be destroyed by the convection mechanism several times during the unstable crystallization state, resulting in instability of the motion. In other words the crystallization state and convection state could occur in turns, as shown in plots (i), (ii) and (iii) in Fig. 6e and plots (ii), (iii) and (iv) in Fig. 6f.

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= 1.5

= 1.5 10 9 8 7 6 5 4 3 2 1 0

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Fig. 5 (continued)

The initial phenomenon of granular motion at a/d = 2.0 in the first 100 s was similar to that shown in the first 85 s at a/d = 0.5 (Fig. 3a and b). The granular temperature and the occupied percentage first increased with the increase of time and then decreased, as shown in Figs. 6a and b. As the time increased from 100 s to 240 s, the granular motion stabilized under the geometrical void-filling mechanism, and the granular temperature remained stable as shown in Fig. 6a. The small particles became arranged in regular order (crystallization packing structure) which grew from the bottom to the free surface between 240 s and 270 s. A significant decrease in the granular temperature (mentioned above) could be observed, as in Fig. 6(a). The crystallization packing structure filled the granular bed from 270 s to 420 s, with large particles distributed over the upper layer of the bed as shown in plot (i) in Fig. 6e and plot (ii) in Fig. 6f. The percentage occupied by large particles was about 55%. The concentration distribution in the third layer was about 93%, and the segregation coefficient was close to 100%.

The crystallization packing structure was destroyed by convective motion between 420 s and 700 s, as shown in plot (ii) in Fig. 6e and plot (iii) in Fig. 6f. The granular temperature of the large particles increased (Fig. 6a) because the granular motion (under the convection mechanism) was very random. This caused a larger fluctuation in the velocity and granular temperature. The large particles sank into the bed due to the convection mechanism. The percentage occupied by large particles decreased to 15% (Fig. 6b) and the concentration distribution in the third layer of the bed decreased to 55% (Fig. 6c), resulting in a decrease in the segregation coefficient to about 55% (Fig. 6d). When the time increased from 700 s to 800 s, there was another significant decrease in the granular temperature (as mentioned above) observed (Fig. 6a), resulting from the growth of the crystallization packing structure from the bottom to the free surface. This structure filled the granular bed after 800 s with large particles distributed over the upper layer of the bed, as shown in plots (iii) and (iv) in Fig. 6e and plots (iv) and (v) in Fig. 6f. The percentage

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Fig. 6. Unstable crystallization state: variations of (a) TL, (b) PL, (c) CL,i, (d) CS, with time for a/d = 2.0 and 2.5. (e) Images of granular motions, and (f) three-dimensional concentration distributions at a/d = 2.0.

occupied by large particles was about 55%. The concentration distribution in the third layer was about 100%. The segregation coefficient was close to 100%. The granular motion showed the same trend at a/d = 2.5 as that at a/d = 2.0, that is the crystallization state and convection state could occur in turns many times (the unstable crystallization state), resulting in complicated flow behaviors with serious variation of physical parameters. The percentage occupied by large particles fluctuated most seriously during the unstable crystallization state among all motion states because of the instability of the granular motion. This unstable motion was mainly the result of the continuous transformation between the convection state and the crystallization state. The percentage of large particles on the surface could exceed 30%, as shown in Fig. 6b. This also showed that the destruction of crystallization packing due to the convection mechanism and the formation of crystallization packing due to the geometrical void-filling mechanism could compete during the unstable crystallization state. The structure of crystallization packing was the most unstable among all the crystallization states, including slowly stabilizing crystallization state, fast stabilizing crystallization state and unstable crystallization state. 3.1.5. Relations between the flow regime and the motion mechanism As the dimensionless vibrating amplitude increased from 0.25 to 1.0, the motion state in the granular bed slowly transformed

from the slowly stabilizing crystallization state (self-organization mechanism, as shown in Fig. 3) to the strong convection state (convection mechanism, as shown in Fig. 4). When the dimensionless vibration amplitude increased from 1.25 to 2.5, the motion state in the granular bed was transformed from the fast stabilizing crystallization state (geometrical void-filling mechanism, as shown in Fig. 5) to the unstable crystallization state (phase transition mechanism, as shown in Fig. 6). The unstable crystallization state occurred because the intensity of the vibration caused the variation in the solid fraction of granular bed which then generated a transition between the convection state and crystallization state. It was found that fluctuation in the occupied percentage was smaller during the convection state than during the crystallization state. Although the granular temperature was larger and increased faster during the convection state, the motion mechanism was still more stable. 3.2. Time-average variations at the stable motion varied with the a/d We take the time-average of the granular temperature TL, occupied percentage PL and concentration distribution CL in the stable stage, then plot these averaged parameters TL,av, PL,av and CL,av against the dimensionless vibration amplitude; see Figs. 7–9. Fig. 10 also shows the variation of time when the first large particle rose to the free surface TD, and the time when large particles first

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a/d = 2.0

a/d = 2.0

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(vi) 3600 s

Fig. 6 (continued)

occupied the maximum area at the free surface TR, with the dimensionless vibration amplitude. The transitions between different states shown in the figures are discussed below. 3.2.1. Transition from theslowly stabilizing crystallization state to the convection state When the dimensionless vibration amplitude increased from 0.25 to 1.0, the granular temperature increased as a/d increased, as shown in Fig. 7. However, the granular temperature TL,av increased significantly from a/d = 0.75 to 1.0. As shown in Fig. 8, the values of PL,av are close to 40% when a/d = 0.25 and 0.5, but they drop to about 10% when a/d = 0.75, 0.875 and 1.0. This was the result of the flow regimes being transformed from the crystallization regime (slowly stabilizing crystallization state) into the convection regime (convection state). Because the crystallization packing could cause that large particles to rise to the free surface and not sink, the PL,av was much larger during the slowly stabilizing crystal-

lization state (PL,av ffi 40%) than during the convection state (PL,av ffi 10%). The higher CL,av in the third layer of the slowly stabilizing crystallization state (CL,av > 60%) turned into the lower CL,av in the third layer during the convection state (CL,av < 60%), as shown in Fig. 9. The values of TD and TR had a larger difference (see Fig. 10) because the increase in the dimensionless vibration amplitude caused a transformation of the motion state. The result was that the voids between particles became larger and the motions of the particles much more violent. The time for large particles to rise to the free surface was reduced. Both TD and TR decreased sharply from the slowly stabilizing crystallization state to the convection state. 3.2.2. Transition from the convection state to the fast stabilizing crystallization state When the dimensionless vibration amplitude increased from 0.75 to 1.5, the granular temperature TL,av dropped suddenly

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5

0

5

Γ

10

15

The Slowly Stabilizing Crystallization State The Convection State

4

The Fast Stabilizing Crystallization State

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2

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TL,av (cm /s )

The Unstable Crystallization State

2

1

0 0

0.5

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a/d Fig. 7. Variation of the time-averaged granular temperature TL,av with the dimensionless vibration amplitudes.

100

0

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Γ

10

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90

The Convection State The Fast Stabilizing Crystallization State

80

The Unstable Crystallization State

P L,av (%)

70 60 50 40 30 20 10 0

0

0.5

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1.5

2

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3

a/d Fig. 8. Time-averaged percentage PL,av of large particles occupying the free surface plotted against dimensionless vibration amplitudes.

(Fig. 7) when a/d was between 1 and 1.25, indicating the transition of the flow regime. In Fig. 8 we see that the values of PL,av were close to 10% at a/d = 0.75, 0.875 and 1.0. They increased to between 60% and 70% at a/d = 1.25 and 1.5. This resulted from the transformation of the flow regime from the convection regime (convection state) into the crystallization regime (fast stabilizing crystallization state). Crystallization packing caused the large particles to rise to the free surface and not sink into the bed. The PL,av was larger during the fast stabilizing crystallization state (60% < PL,av < 70%) than for the convection state (PL,av ffi 10%). The lower CL,av in the third layer during the convection state (CL,av < 60%) turned into the higher CL,av in the third layer during the fast stabilizing crystallization

state (CL,av ffi 100%), as shown in Fig. 9. The TD and TR increased with the increase of a/d in the transition from the convection state into the fast stabilizing crystallization state (Fig. 10), because the driving force of large particles rising to free surface was weaker under the crystallization regime than that under the convection regime. Thus, both TD and TR could decrease slowly with the increase of the a/d. 3.2.3. Transition from the fast stabilizing crystallization state to the unstable crystallization state When the dimensionless vibrated amplitude increased from 1.25 to 2.5, the granular temperature increased slowly with the

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110

0

Γ

5

10

15

100 90 80

CL,av (%)

70 60 50 40 30 20 10 0 -10

0

0.5

1

1.5

2

2.5

3

a/d Fig. 9. Time-averaged concentration distribution CL,av of large particles at different heights in the granular bed at different dimensionless vibration amplitudes at (a) the : second layer; 4: first layer); (c) the fast slowly stabilizing crystallization state (j: third layer; : second layer; h: first layer); (b) the convection state (N: third layer; stabilizing crystallization state (: third layer; : second layer; }: first layer); (d) the unstable crystallization state (d: third layer; : second layer; s: first layer).

7000

0

Γ

5

100

6000

0

10

5

Γ

15

10

15

90 80

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Time (Second)

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4000

3000

70 60 50 40 30 20

2000

10 0

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1.5

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2.5

3

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1000

0 0

0.5

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1.5

2

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3

a/d Fig. 10. Time for large particles to rise to the free surface TD at (a) the slowly stabilizing crystallization state (h); (b) the convection state (4); (c) the fast stabilizing crystallization state (}); (d) the unstable crystallization state (s), and the time at which large particles first occupied the maximum area of the free surface TR at (a) the slowly stabilizing crystallization state (j); (b) the convection state (N); (c) the fast stabilizing crystallization state (); (d) the unstable crystallization state (d) as it varied with the dimensionless vibration amplitude.

increase of a/d, as shown in Fig. 7. Because the flow regimes remained in the crystallization regime from a/d = 1.25 to 2.5, the curve of the granular temperature TL,av followed the same trend. Thus TL,av versus a/d seemed continuous from a/d = 1.25 to 2.5. However, from Fig. 8, the occupied percentage PL,av increased from a/d = 1.25 to 1.5 and decreased from a/d = 2.0 to 2.5. It was resulted from the transformation of the motion states from the fast stabilizing crystallization state into the unstable crystallization state. The

structure of crystallization packing during the unstable crystallization state was unstable, resulting in larger fluctuation of the PL,av during the unstable crystallization state than that at the fast stabilizing crystallization state. The PL,av was smaller during the unstable crystallization state (PL,av < 55%) than during the fast stabilizing crystallization state (PL,av > 55%). The lower fluctuation of the CL,av in the upper two layers during the fast stabilizing crystallization state (the error bar of CL,av was close to 0%) turned into the higher

S.-C. Tai, S.-S. Hsiau / Advanced Powder Technology 20 (2009) 335–349

fluctuation of CL,av in the upper two layers during the unstable crystallization state (the error bar of CL,av was between 10% to 30%), as shown in Fig. 9. This was resulted from that the continuous transformation between convection state and crystallization state during the unstable crystallization state. The TD and TR could slowly decrease with the increase of a/d from the fast stabilizing crystallization state to the unstable crystallization state (Fig. 10), because the unstable motion caused the voids between the particles to become larger and the motions of particles more violent. This reduced the time for large particles to rise to the free surface. 3.2.4. Variation of parameters with the motion state and the motion mechanism The variation of the physical properties, (e.g., the occupied percentage and granular temperature) helps us to understand the motion in the granular bed (including the trends of motion type and particle distribution). The resultant occupied percentage of the free surface of the granular bed (Fig. 8) and the values of concentration distribution of large particle in the third layer of the granular bed (Fig. 9) had a similar trend. 4. Conclusion The granular temperature and the motion state have specific relations. Observing the relationship between the percentages occupied on the surface and the concentration distribution of large particles in the granular bed can be further used to analyze the granular motion. Therefore, the level of density (solid fraction of particles) in the granular bed is what decides whether a crystallization state or convection state is formed or not. When there is a crystallization state formed in the granular bed, a significant decrease of the granular temperature can be observed in the plot (a). When the motion in the granular bed is in the crystallization state, the large particles rise to the upper bed. There will be better size segregation and a higher segregation coefficient. When the motion is in the convection state, the segregation coefficient is lower. As amplitude of the dimensionless vibration increases, the space between particles becomes larger and the motions of the particles become much more violent, resulting in a decrease of TD and TR. However TD and TR increase during the transition from the convection state to the fast stabilizing crystallization state. Acknowledgements Financial support from the National Science Council of the ROC for this work through projects NSC 96-2221-E-008-072 and NSC 95-2221-E-008-135-MY2 is gratefully acknowledged. References [1] G.D.R. Midi, On dense granular flows, Eur. Phys. J. E14 (2004) 367–371. [2] P.G. de Gennes, Reflections on the mechanics of granular matter, Physica A 261 (1998) 267–293. [3] D.S. Grebenkov, M. Pica Ciamarra, M. Nicodemi, A. Coniglio, Flow, ordering, and jamming of sheared granular suspensions, Phys. Rev. Lett. 100 (2008) 078001. [4] E. Kolb, J. Cviklinski, J. Lanuza, P. Claudin, E. Cle´ment, Reorganization of a dense granular assembly: the unjamming response function, Phys. Rev. E. 69 (2004) 031306. [5] E. Kolb, C. Goldenberg, S. Inagaki, E. Cle´ment, Reorganization of a twodimensional disordered granular medium due to a small local cyclic perturbation, J. Stat. Mech. 7 (2006) P07017.

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