Experimental investigation into the influence of the particle size distribution upon the local mixing mechanisms in a flowing bulk material

39

Chemical Engineering and Processing, 33 (1994) 39-44

Experimental investigation into the influence of the particle size’ distribution upon the local mixing mechanisms in a flowing bulk material M. Peciar*, H. Buggischt Institut ftir Mechanische

and M. Renner

Verfahrenstechnik

und Mechanik,

Universittit

Karlsruhe

(TH),

76128 Karlsruhe

(Germany)

(Received September 20, 1993; in final form October 13, 1993)

Abstract The small-scale mixing of densely packed bulk material was modelled by Buggisch and Liiffelmann in 1989, starting from a diffusion approach which had been developed previously. An equation could be deduced from which the diffusion coefficient appears to be proportional to the shear rate. This model conception was confirmed through shearing experiments using a Couette apparatus. The model bulk consisted of vertically packed metal rods of various diameters, parts of which were marked by colours. The present paper investigates the influence of the particle size distribution upon the mixing process. It is to be expected that the distribution of the particle size has an influence upon the package structure of the bulk and hence, amongst other things, upon ‘the shear rate, and, eventually, upon the intensity of the diffusion. It has been possible to confirm this prediction through shearing experiments with the model bulk in the Couette apparatus.

Introduction

and modelling

In a previous paper [l], Buggisch and LMehnann modelled the small-scale mixing of densely packed bulk material consisting of non-cohesive, mechanically homogeneous particles during large-scale bulk deformation by means of a diffusion equation [2-71. For the particular case of a shear deformation, an equation could be deduced from which the diffusion coefficient appears to be proportional to the shear rate and the vacancy frequency 1. The vacancy frequency is defined as the ratio of the mean particle diameter to the mean distance between two vacancies which permits the transition of particles between two neighbouring particle layers. The vacancy frequency is, for the present, an unknown parameter that characterizes the packing structure ,of the granular material within the shearing zone. Since the value of this parameter is not predicted by the theory, it must be determined empirically. For this purpose, 1 may be varied in the numerical calculations until a satisfactory concordance between the calculated and the measured number density distribution of particles distinguished by a mechanically irrelevant attribute is found. In other words, 1 acts as a return current coefficient between the model and the experiment. *Permanent address: Slovak Technical University, Bratislava, Slovak Republic. tTo whom all correspondence should be addressed.

02552701/94/$7.00 SSDI 0255-2701(93)00500-4

In order to confirm the model, shearing experiments were carried out with a Couette apparatus, in which a model bulk with a particle size distribution was employed [l]. It is to be expected, of course, that the employed particle size distribution influences the local deformation rate and the vacancy frequency, and hence also the mixing process. At a narrow size distribution, a fairly organized, ‘crystalline’, structure of the bulk package is formed, whereas the bulk package with a wide size distribution is characterized by a more ‘amorphous’ structure. In comparison to an amorphous structure, a crystalline package has a much lower shearing capability. In this case, the shear rate prevailing in the proximity of the moving wall should have a high gradient before it quickly drops towards zero inside the bulk. This is equivalent to a high shear rate in wall proximity and an altogether small extension of the shearing zone. The shear rate in wall proximity should decrease and the shearing zone expand as the particle size distribution is widened. With regard to the influence of the particle size distribution upon the vacancy frequency, it is expected that the number of such vacancies is larger in a wide distribution than in a narrow one. The particle size distribution thus influences the diffusion coefficient in wall proximity via two opposing effects. On one hand, the shear rate has an increasing effect on the diffusion coefficient as the distribution becomes narrower, while a decrease in the vacancy frequency weakness this

0 1994 -

Elsevier Sequoia. All rights reserved

40

effect. At a sufficiently large distance from the moving wall, superposition of the two effects should therefore result in a diffusion which increases as the particle size distribution becomes broader. In the following, we report on experiments which confirm the preceding considerations regarding the influence of the particle size distribution upon the local deformation rate and the intensity of the diffusion.

Experimental

details

Apparatus The apparatus

employed is depicted schematically in Fig. 1. The cylindric model granulate (3) is placed in a concentric ring around the rotational axis (4). The bulk is delimited towards the axis by a rough rotating wall (5). The outer boundary consists of a fixed outer rubber wall (6) through which a radially directed pressure load can be applied to the model granulate by means of four clamps. These clamps compress the rubber wall via a pressure plate (2) which can only expand in a radial direction, i.e. towards the bulk. The complete apparatus, including the drive and the control unit regulating the rotational velocity, is illustrated in Fig. 2. Model granulate

We used a two-dimensional model granulate for the experimental investigation into the diffusion process.

Fig. 1. Sectional

drawing

Fig. 2. Complete

set-up.

of the Couette

apparatus.

The model granulate consisted of 20-mm length cylindrical steel rods of various diameters. These metal rods were packed vertically and filled into the shear cell. A proportion of them were marked by colours in order to determine the diffusion process. Particle size distributions

The particle size distributions employed (Fig. 3) were composed of rods with diameters of 2.0, 2.5, 3.0 and 4.0 mm, respectively. The distributions were characterized by the relative number X,,, of rods with a diameter of 2.5 mm, which represented the major part of each bulk. Seven distributions (Fig. 3(a) -3(g)) ranging from X,,, = 0.5 to X,,, = 1.O were investigated. For the distributions X,,, = 0.5 to X,., = 0.9, the total number of N,, the number of N4,,, of 4.0 mm particles and the surface porosity E were kept constant. With the distributions X,,, = 0.95 and X,,, = 1.0, this was no longer possible and hence only rods with diameters of 2.5 mm and 4.0 mm were used in this case. Here, the number of rods with a diameter of 4.0 mm was defined in such a manner that the surface porosity would correspond as closely as possible to the values of the distributions X2.5 = 0.5 to x,,, = 0.9. Experimental

procedure

and analysis

For the determination of the velocity profiles, the rotationally symmetrical shear flow occurring in the Couette apparatus during shearing was recorded by means of a video recorder. Next, the video films were analyzed on a monitor. For this purpose, the area of the shear cell containing the granulate was divided into circular rings by drawing concentrically equidistant rings. On each of these rings, the velocity of a sufficiently large number of particles was determined, and their mean values allocated to the radial coordinate of each ring. To allow experimental investigations into the mixing process, a part of the model granulate contained in a concentric ring (rO< r < r,) around the axis of symmetry of the apparatus was colour marked (Fig. 4). The quantity r, (Fig. 4) represents the inner radius while rl is the radius of the outer boundary of the two-dimensional granulate package. A photograph was taken of this initial situation. After the shear process began, the instantaneous mixing conditions at various times t = wt were photographed. To allow a quantitative analysis of the number density profiles, photographs of the Couette cell were again divided in concentric rings around the rotational axis. Then, the number of marked particles and the total number of particles in each interval were counted. The ratio of both values yielded the number density A. The results obtained for A represent mean values for each ring and were allocated to the mean radius of each ring.

41 1 ”

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1.0 -

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I

I u

f 5 0.8 -

i 5

2

o 0.6 5

0” 0.6 5

ii! 0.4 .k

f 0.4 ?+

.E 0.2 ij B L 0.0

I 2

1

particle

(a)

0

5 5

< $

I

3 diameter

I

I 4 D, [mm]

I

I

I

I

0.8

x

A

I

1.0

5 particle

(b)

1.0

0.8

t 5 0.8 %

0.6

6 0.6 5

9 P 0.4 k

I

I

_ I

I

1.0

0

diameter

!

D: [mm]

I

I

ig 0.4 g

.g 0.2 z L 0.0

0.2

% L 0.0 2

1

3

particle

Cc)

I

diameter

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t K

Iz

2 particle

Cd)

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I

1

5

4 Di [mm]

I I

1.0

L

diameter

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’

3

_ I

4

I

5

Di [mm]

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1

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_-

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0.8

6 0.6 El $t 0.4 t .E 0.2 z L 0.0

i particle

u

f $

7T 6

diameter

-_ 1

5

_

I

I

2

particle

I

3 diameter

4 D, [mm]

,

5

D, [mm]

(f)

Di [mm]

Fig. 3. Particle size distributions: (a) X 25 =OS; (b) A’,, = 0.6; (c) X2, = 0.7; (d) X2,, = 0.8; (e) X2., = 0.9; (f) A’,., = 0.95; and (g) x2,, = 1.0.

1.0

0.8

0.6 5 s a? 0.4 t

.g 0.2 2 b 0.0

(d

particle

diameter

TABLE 1. Values of the return current ceofficients B and C for the description of the velocity profiles for the various particle size distributions X,,, Value of X, 5 ro = 35mm

0.5-0.8

r, = 9omm r, = 5omm

Coefficient B Coefficient C

=2OD

0.9

0.95

1.0

0.45

0.51

0.58

1.14

14.36

14.39

14.20

14.04

w = 0.6 s-1

The measured function: Fig. 4. Determination tions.

of the velocity

profiles

and mixing

condi-

Results Velocity and shear rate profiles

For distributions with X,,, = 0.5 to X,,, = 0.8, no significant influence of the particle size distribution upon the radial velocity profile u(r) was established. Hence, a mean velocity profile was determined (Fig. 5(a)) for these distributions. Figures 5(b)-(d) illustrate the velocity profiles for the distributions X,,, = 0.9 to Xz,5 = 1.0.

1.2

I

1

I

values fitted the following exponential

where (0 if the angular velocity of the rotating cylinder and D relates to the diameters of those rods occuring with the highest relative frequency. In our case, the value of D was 2.5 mm. The values for the return current coefficients B and C are summarized in Table 1. In Fig. 6 all the radial velocity profiles outlined in Fig. 5 have been combined, thereby allowing an effective comparison. It is quite clear that in close proximity to the wall the velocity drops faster, and quickly approaches zero inside the bulk as the distribution becomes narrower.

I

I

1.0 <

0.8 3 ,”

0.6

-

0.4

-

0.2

-

0

measured

-

u/rr,u=axp[-B(r/D-C)]

values

measured -

values

u/row=exp[-@r/D-C)]

>

O.O 14

16

18

20

22

r/D

(a) 1.2

I

1.2

I

I

I

r/D

(b) I

I

I

I

1.0 c

3 ,o

0.8

-

0.6

-

0.4

-

0.2

-

0

measured

-

u/r~=exp[-B(r/D-C)]

values

0

measured

-

u/r~=.xp[-E(r/D-C)]

values -

‘=

O.O 14

I 16

20

I 22

r/D

(4 Fig. 5. Radial

18

velocity

profiles for various

18

size distributions:

(a) X ,,=0.5-0.8;

22

r/D

(d)

particle

1 20

(b) X,.,=0.9;

X,,, =0.95;

and(d)

X,, = = 1.0.

43

3 L

A

0.6

>

0.4

-

0.2

-

-

disk.

0.5-0.8

--

disk.

0.9

.‘.. .distr.

0.95

--

1.0

disk.

_

-

0.4

0.0

-

i....... .._.._........._....__.... ..__.......

I

14

I

I

16

IIll

I

18

20

I

22

I

I

24

_

I, 26

28

r/D

Fig. 6. Combined size distributions.

radial

velocity

profiles

-

of the

disk.

various

particle

Fig. 8. Number density as a function of the radius after a shearing time T = 250 for various particle size distributions.

0.5-0.8

--

disk.

0.9

...’

disk.

0.95

disk.

1.0

0.6

-

disk.

--

distr.

....

distr.

A

0.4

1

1

0.2 0.0 I

14

I

I

16

I

1

18

I

I

20

I

I

22

I

I

24

I

26

28

r/o

Fig. 7. Profiles of the non-dimensional distributions investigated.

shear rate for the various

Fig. 9. Number density as a function of the radius after a shearing time r = 500 for various particle size distributions.

I

The non-dimensional

1.0

shear rate

____._

I

I

I

I

d u

0.8

dr 0 r

was determined (Fig. 7).

\

0.6

by means

of the

velocity

profiles

Dzj3usion process

The number densities at three different times are depicted in Figs. 8- 10 respectively. The shear rates and the number densities for the various distributions confirm our initial considerations. For r/D < 15, the shear rate is lower at a wide distribution than at a narrow distribution. When r/D > 15, however, the situation is reversed, i.e. the shear rate decreases as the distribution becomes narrower. As predicted by the model, the particle size distribution influences the shear rate and, hence, the intensity of diffusion. The marked particle zone at time t = 0 extended to r-,/D = 20. Hence, the border between the marked and unmarked particle zone occurs in the zone in which both the change in the shear rate and the vacancy frequency intensify the diffusion as the size

I

-5.

‘..,,

K_-r._ 0

I

._._._. ___

-

disk.

OS-O.8

--

disk.

0.9

....

disk.

0.95

_

-

A 0.4 0.2 0.0

\

,__ _7 _...__ ......._.. _;

t--,

III

14

16

II

I 18

III

II

20

22

I 24

9

III 26

28

r/D

Fig. 10. Number density as a function shearing time z = 750 for various particle

of the radius after size distributions.

a

distribution becomes broader. The highest diffusion therefore was obtained for the distributions X,,, = 0.50.8; the lowest for the distribution X,,, = 1.O. In order to quantify the influence of the vacancy frequency parameter upon the diffusion process, the results obtained from these experiments will be adapted in a future paper to the model developed by Buggisch and LGffelmann.

44

Acknowledgment

We would like to thank the Deutsche Forschungsgemeinschaft for their financial support within the scope of the Sonderforschungsbereich 2 19 ‘Silobauwerke und ihre spezifischen Beanspruchungen’, which enabled Dr. M. Peciar to stay as a guest scientist in Karlsruhe.

B. C D

Di D* N

NO r r. 6 rm U

relative particle 3.0, 4.0)

1

vacancy frequency surface porosity non-dimensional deformation rate non-dimensional time angular velocity of the driving hub

&

?c 5 =

cot

w

frequency

(i = 2.0, 2.5,

References

Nomenclature A

Xi = N,/N,

number density of marked particles return current coefficients of the velocity profiles diameter of particles occuring at highest relative frequency particle diameter (i = 2.0, 2.5, 3.0, 4.0) mean particle diameter number of particles with a diameter of i mm (i = 2.0, 2.5, 3.0, 4.0) total number of particles radial coordinate driving hub radius radius of bulk expansion initial radial spread of marked particles time average particle velocity

1 H. Buggisch and G. LBffelmann, Theoretical and experimental investigations into local granulate mixing mechanisms, Chem. Eng. Process., 26 (1989) 193. 2 E. Schliinder, Particle heat transfer, Proc. 7th Znt. Heat Transfer Conf., Munich, 1982, Hemisphere, Washington, DC, 1982, Vol. 1, p. 195. W. Miiller and H. Rumpf, Das Mischen von Pulvem in Mischem mit axialer Mischbewegung, Chem.-Zng.-Tech., 39 (1967) 365. K. Sommer, Mechanismen des Pulvermischens, Chem.-Zng.Tech., 49 (1977) 305. J. Bridgwater, W. S. Foo and D. J. Stephens, Particle mixing and segregation in failure zones - theory and experiment, Powder Technol., 41 (1985) 147. S. B. Savage and D. K. K. Lun, Particle size segregation in inclined chute flow of dry cohesionless granular solids, J. Fluid Mech., 189 (1988) 311. C. S. Campbell and D. G. Wang, Effective conductivity of shearing particle flows, Proc. Znt. Heat Transfer Conf., San Francisco, 1986, Hemisphere, New York, p. 2567.