Floc breakup along centerline of contractile flow to orifice

Colloids and Surfaces, Elsevier

Science

56 (1991)

Publishers

13

13-23

B.V., Amsterdam

Floe breakup along centerline of contractile flow to orifice Ko Higashitani’, Nobufumi Inada and Toyohiko Ochi Department Kitakyushu (Received

of Applied Chemistry, 804 (Japan) 17 November

Kyushu Institute

1989; accepted

10 October

of Technology,

Sensuicho,

Tobata,

1990)

Abstract The breakup

process

of floes on the centerline

in converging

flow to an orifice is observed di-

rectly by stroboscopic photographs, using floes which are composed of a small number of visible particles. An axial symmetric test section is employed to observe the floe behavior before and after the orifice and a two-dimensional floes are broken in the velocity orifice, and that the maximum

one is used to observe floes within the orifice.

It is found that

acceleration region before the orifice, but not within and after the size i,,, of the floes which experienced the elongation rate (dVz/

dZ) is given by the following equation:

A model for the breakup of floes in an elongation flow is proposed, in which the floe breakup is assumed to be attributed to the difference of the drag force on the constituent particles at different locations. It is found that the variations in the size distribution are successfully predicted by this model.

of floes and the average floe size

INTRODUCTION

The coagulation or flocculation of colloidal particles is not desirable in many industrial processes in which materials such as electronic, magnetic and fine ceramic materials are manufactured from suspensions. The operation to break up floes is necessary in these processes. The deflocculation of floes should, however, be avoided in measuring the distribution of floe size by the Coulter counter and the cascade impactor in which floes pass through the orifice. In any case it is necessary to clarify the detailed mechanism of floe breakup in a flowing fluid to estimate the influence of deflocculation on these systems. The flow of fluids, such as shear flow, elongational flow and turbulent flow, can be used to deflocculate floes by exerting a hydrodynamic force on the constituent particles [l-4]. The contractile flow to an orifice was found to be effective in breaking up floes [ 5,6], but not many investigations on the floe ‘Author to whom correspondence

0166-6622/91/$03.50

should be addressed.

0 1991-

Elsevier

Science

Publishers

B.V.

14

breakup by contractile flow have been reported. Sonntag and Russel [ 61 found that floes of colloidal particles change their shape from spherical to elongated in a converging flow, and the average number of particles per broken floe is proportional to E -’ where E is an overall average strain rate. Yamamoto and Sugawara [7] reported that the average size of powders dispersed by a converging air flow is proportional to @V-o.’ where QVis the energy dissipation of the flow. Yuu and Oda [8] found that the change in particle size distribution of powders in a converging air flow can be evaluated by the population balance, assuming the breakup is due to the difference in the inertia and drag force on the constituent particles of different sizes. Kerekes [9] observed that pulp floes in a converging flow tend to deform and rupture in a tensile mode rather than in a shear mode. Results in these studies, except Kerekes’s, were derived from the macroscopic change in floe size distribution before and after the orifice. It is likely that direct observation offers further information on the detailed mechanism of floe breakup. In the contractile flow to an orifice, floes near the centerline and wall will be broken by the elongational and shear flows respectively. We presume that the floe breakup by the elongational flow is more significant because the flux of floes passing across the region where the elongational flow is dominant will be much larger than that across the region where the shear flow is dominant. In this study the breakup process of floes on the centerline in a converging flow was observed directly by stroboscopic photographs, using floes which are composed of a small number of visible particles. An axial symmetric test section was employed to observe the floe behavior before and after the orifice and a two-dimensional one was used to observe floes within the orifice. A model for the floe breakup in an elongational flow is proposed and the variations in the size of given floes and the average floe size are predicted by this model and compared with the experimental results. EXPERIMENTAL

Particles of a copolymer of styrene and divinylbenzene whose density is 1.05 g cmp3 are employed here. It is found by microscope that they are spherical in shape and of the average diameter 90 ,um with the standard deviation 10 pm. A 24 wt% glycerol solution whose density is the same as that of the particles is employed as the medium to avoid the effects of the inertia and sedimentation of particles. The agglomerates of these particles are prepared beforehand by stirring them in the glycerol solution with flocculants of 80 g rnp3 aluminum sulfate and 20 g rnp3 polyacrylamide of molecular weight 9.10’. Figure 1 shows a schematic drawing of the apparatus and the details of the test section. Two kinds of test sections are used. A three-dimensional (3D) cylindrical test section with an orifice 5 mm in diameter is employed to observe the behavior of the floes before and after the orifice in the axial symmetric

15

lb) fd

l--44--c+

(4

r-B

a

C

blc\d(elflg

I + 54 lo+

500 I80 75550+

(mm)

(a)

A C E F H J

Reservoir Head tank ElectromagnetkJ Test section Roller pump NOZZk

L Camera N Stroboscope Fig. 1. Schematic test section;

B Pump 0 Valve valve G Flowmeter I Floe tank K Microscope M Lamp house

a Ib[c(dle]f\glh 350 100 350104 A Nozzle

14

IO

80

2.5

(mm)

B Orifice

C Confined liquid drawing of apparatus

(b) two-dimensional

and dimensions

of test sections:

(a) three-dimensional

test section.

contractile flow. The test section is covered with a rectangular water reservoir to minimize the effect of optical error due to the curvature of the cylinder. A two-dimensional (2D ) test section with an orifice of 10~2.5 mm2 cross-section area and of 100 mm length is also used to observe the floe behavior within the orifice. The contraction ratios of the 3D and 2D test sections are l/100 and l/ 8 respectively. The glycerol solution flows into the test section F from the head tank C and the flow rate is measured by the flowmeter G at the exit. When a steady state is reached, the solution with floes in the tank I is injected from the nozzle J into the center of the test section with the same velocity as the axial velocity. The behavior of floes around the orifice is observed by taking stroboscopic photographs using the camera L with the microscope K. All the experiments are carried out in a temperature-controlled room at 22 + 1 ‘C. RESULTS

Figure 2 shows the size distribution of floes injected from the nozzle. The i on the abscissa indicates the apparent number of the constituent particles. The apparent number means the number of particles which are identified in the stroboscopic photographs. We consider this number is not far from the real number of particles, because the number of constituent particles is not large enough for them to be hidden completely behind the neighbouring particles.

16

I

o.”

2

3 4 i (-1

5

6

Fig. 2. Size distribution of floes injected upstream trations of i-fold floes and total floes respectively.

from the nozzle: ni and N are number concen-

aQ.4.0 7

-

-

6 -

(cm%.)

Theoretical 0

Experimental

;;504N

__

0-2.0

-1.5

-1.0 -0.5 Z (cm)

Fig. 3. Variation in the centerline

0.0 velocity of the converging

flow to 3D orifice.

The coincidence of the distributions in the figure indicates that particles injected upstream can be assumed to have the same size distribution independent of the flow rate Q. Figure 3 shows the variation in the axial velocity V, on the centerline of the entry flow into the 3D orifice. Here the abscissa 2 is directed towards the flowing direction and the origin is located at the orifice surface on the upstream side. It is clear that the velocity increases abruptly just before the orifice. This variation in flow velocity is measured by taking stroboscopic photographs of

17

the movement of a single particle. The velocity is compared with the numerical result computed by the finite element method elsewhere [lo]. The agreement between them confirms the adequacy of this velocity variation. Figure 4 shows the variation in the average number of constituent particles, ( i) , and also a typical reduced axial velocity V,/ V,, normalized by the upstream velocity V,,, for the sake of comparison. The value of (i) changes abruptly just before the orifice where V,/ V,, changes, but is unchanged after the orifice. However, these results do not. indicate whether floes are broken within the orifice or not, because their behavior within such a thin orifice cannot be observed. The behavior within the orifice is observed using the 2D orifice. Figure 5 shows the variation in (i) around the 2D orifice. It is found that all the floes are broken before the orifice as in the case of the 3D orifice, but floes are not broken within and after the orifice, irrespective of Q. These results imply

‘“I/-.jF; -20-10

o

20 30 40 Z (cm)

IO

50

60

Fig. 4. Variations in the number average of constituent particles in a floe for various flow rates for 3D orifice.

4_ -!-__L_-_ -15 -6

-5

0

5

IO 15 20

25

koriflce4 Z (cm)

Fig. 5. Variations in the number average of constituent particles in a floe for various flow rates for 2D orifice.

18

Q

I 0.5

a

Q /

I I

I

5 IO dv,/ dz

08 I

I

50

100

500

k’)

Fig. 6. Number of constituent particles in floes which experienced given elongation rates.

O-0’

I ; 3.0. \ 0”

-1.0

-0.5 Z (cm)

0.0 orifice

Fig. 7. Change in the dimension of Z direction of fivefold floes.

that the floe breakup is attributable to the drag forces exerted on the particles by the elongational flow in the region of the velocity acceleration. It is possible to evaluate the elongation rate of the axial flow, (dV,/dZ), from the data of velocity variation shown in Fig. 3. Figure 6 shows the sizes of the floes which exist at given elongation rates. It is clear that all the data exist under a single line independently of Q, and the maximum number of particles in a floe i,,, is given by the following equation:

imax =7.0(dV,/dZ)

--OJ

(1)

It is clear that the size of the floes which experienced a given elongation rate is not determined only by the elongation rate. This means that the breakup of

19

floes depends not only on the elongation rate but also on the interparticle attractive forces which will have a distribution in strength. It is observed that floes are deformed in the velocity acceleration region. In the case of twofold floes, the particles line up in the flow direction. Figure 7 shows typical deformations of fivefold floes, where Dz/DO is the reduced floe size in the 2 direction, normalized by the size of constituent particles DO. If five particles line up completely, Dz/Do should be five. Since Dz/D, becomes almost five as the floes approach the orifice, particles are presumed to line up in the 2 direction before the breakup if the number of constituent particles is small enough. It is also found that the position of floes where the deformation begins shifts upstream as Q increases. DISCUSSION

Here we consider a simple model for the breakup process of floes on the centerline in contractile flow. First, we consider a floe composed of two equal spherical particles in a medium whose density is identical with that of the particles, as shown in Fig. 8. The particles are assumed to line up to the flow direction as suggested by the results in Fig. 7. Since there is no difference in density between the particles and the medium, it is plausible to assume that effects of the inertia of particles relative to the medium, and sedimentation, are negligible, and the center of gravity of floes moves with the medium. Hence the breakup of floes is considered to be caused by the difference between the drag forces exerted on the front and rear particles by the elongational flow. The following equations hold for linked particles A and B: 3nlu;lD(U,-UG)-F=O

(2)

-3qdD(UG-&)+F=O

(3)

-3 .-.-.-.-. >k Z<

0

Fig. 8. Schematic drawing of proposed model.

20

where U,, Us and Uo are the velocities of particles A, B and the center of gravity of the floe respectively, D is the particle diameter, il (which is 0.645 for the particles of a doublet [ 111) is the correction factor for the Stokes’ drag force exerted on the particles owing to the existence of the neighbouring particle, fi is the viscosity of the medium and F is the interparticle hydrodynamic force. Since the flow on the centerline is an elongational flow and the particle size is small, ( UA - UG) and ( UG - U,) are given approximately as follows: U, - U, =Z,(dUo/dZ)

(4)

U, - U, =ZB (dUo/dZ)

(5)

ZA and Zn are the distances between the center of gravity and the centers of particles A and B respectively. Using Eqns (2 ) - (5 ) and ZA + Zg = D, F is given by the following expression: F= (3/2)7@12(dUo/dZ)

(6)

The particles in this study are flocculated by the van der Waals force and the bridging force of polymers. When F, given by Eqn (6), becomes greater than the sum of these interparticle attractive forces, the floe will break up. Hence the value of F at the floe breakup is equal to the maximum interparticle force. This means that the interparticle force can be evaluated by substituting the value of (dUo/dZ) at the point of floe breakup into Eqn (6). A similar argument can be carried out for n-fold floes (n> 3) which are composed of equal-sized particles and line up in the flowing direction as a chain. According to Gluckman et al. [ 11],1 for the particles in the chain depends on their location, but here we assume A is the same as that for twofold particles for the sake of simplicity. Then the interparticle force at the ith bond between the ith and (i+ 1)th particles from the front particle, Fi, can be derived as follows, regarding a linear floe as consisting of the front i particles and that of the rear (n-i) particles as the particles A and B in Eqns (2) and (3) respectively: Fi= (3/2)i(n--i)n~D2(dU,/dZ)

(7)

Now we consider the breakup of n-fold particles. Suppose the breakup probability P of a twofold particle is known to be P=f(F), then the probability for the breakup at the ith bond is given by i-l

Pi=k~lIl-f(F~)If(F,)

The probability

n-1

n

m=i+1

of no breakup

[l-f(Fm)l

(8)

of the floe is given by

n-1

P,,=~~lMFdl

(9)

21

Since the probability for a floe to break up into more than two pieces is observed to be very small, this probability is assumed to be negligible. These equations enable us to estimate the probable breakup process of the floes which have experienced an elongation flow. Figure 9 shows the breakup probability P of twofold particles obtained by observation vs. the value of F evaluated by Eqn (6). The result indicates that there exists a distribution in the strength of the interparticle adhesive force as suggested in Fig. 6. Since this interparticle adhesive force will be the same as that between constituent particles in larger floes, this correlation is applicable to estimate the breakup probability of large floes, if F is known.

0.5

I

2

5

IO

20x

0’

F (dyne) Fig. 9. Breakup probability

of twofold particles vs. interparticle

force exerted hydrodynamically.

1.0

0 123456 i (-1 Fig. 10. Comparison between the experiment and the theoretical prediction of the distributions of the floes of given size which experienced the converging flow. N, and ni are initial and deflocculated number concentrations of i-fold floes respectively.

22

2.0

I ____-1 i

1

‘s5-

‘!

\

ppEJ -+-I

1.0’-15L

I

-10

-5 2 (cm)

Fig. 11. Comparison of the variation the theoretical prediction.

-0

in number average of floe size between the experiment

and

Figure 10 shows the size distributions of floes which were initially fourfold, fivefold and sixfold particles upstream and experienced a deformation rate of 8.0 s-l. Solid lines in the figure indicate the prediction by Eqns (7)-(g) with P=f(F) shown in Fig. 9. The reduction in the number of floes of the initial size indicates that the larger the floes are, the more easily they are broken. It is also found that floes tend to break up at the middle rather than being eroded one by one. These trends are quantitatively predicted by the above theory. Figure 11 shows the axial variation of the average size of floes. The solid line indicates the theoretical prediction obtained by applying the above model to the floes whose upstream size distribution is given in Fig. 2. It is clear that the measurement is well predicted by the model. These agreements between the prediction and the experiment in Figs. 10 and 11 indicate that the proposed breakup model is adequate and that floes with a small number of constituent particles are deformed almost linearly and broken before the orifice, because of the difference in the drag force on the constituent particles. CONCLUSIONS

The following conclusions are drawn for the breakup of the floes consisting of small numbers of particles on the centerline near an orifice in contractile flow. (1) Floes are elongated and lined up almost linearly towards the flowing direction, and then broken before entering into the orifice where the flow velocity increases abruptly. However, they are not broken within and after the orifice. (2) The maximum number of constituent particles remaining in the floes which experienced a given elongation rate is given by the following equation:

23

(3 ) The proposed model for the floe breakup predicts the axial variation of floe size distributions quantitatively if P=f(F) is known. (4) The floe breakup is considered to be attributed to the difference in the hydrodynamic drag force exerted on the constituent particles located at the different axial positions. ACKNOWLEDGMENT

We thank Mr. T. Ishii for experimental assistance.

REFERENCES 1 2 3 4 5 6 7 8 9 10 11

R. Sonntag and W. Russel, J. Colloid Interface Sci., 113 (1986) 399. C.F. Lu and L.A. Spielman, J. Colloid Interface Sci., 103 (1985) 95. G.C. Ansell, E. Dickinson and M. Ludvigsen, J. Chem. Sot., Faraday Trans. 2, 81 (1985) 1269. J.P. Pandya and L.A. Spielman, J. Colloid Interface Sci., 90 (1982) 517. Y. Kousaka, K. Okuyama, A. Shimizu and T. Yoshida, J. Chem. Eng. Jpn., 12 (1979) 152. R. Sonntag and W. Russel, J. Colloid Interface Sci., 115 (1987) 390. H. Yamamoto and A. Sugawara, Kagaku Kogaku Ronbunshu, 9 (1983) 183. S. Yuu and T. Oda, AIChE J., 29 (1983) 191. R.J. Kereker, TAPPI Proc. of Papermaker Conf., 1983, p. 129. N. Inada, M.S. Thesis, Kyushu Institute of Technology, 1987. M.J. Gluckman, R. Pfeffer and S. Weinbaum, J. Fluid Mech., 50 (1971) 705.