Langmuir 1999, 15, 4351-4356
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Breakup of Fractal Flocs in a Turbulent Flow Motoyoshi Kobayashi,*,† Yasuhisa Adachi,‡ and Setsuo Ooi§ Doctoral Program in Agricultural Sciences and Institute of Agricultural and Forest Engineering, University of Tsukuba, Tennoudai 1-1-1, Tsukuba, Ibaraki 305-8572, Japan, and National Research Institute of Agricultural Engineering, Kannondai 2-1-2, Tsukuba, Ibaraki 305-8609, Japan Received June 25, 1998. In Final Form: March 16, 1999 A simple model of the breakup of a floc in a turbulent flow is proposed by analogy with that of a droplet. In the model, the strength of a floc is expressed as a product of the cohesive force between primary particles and the number of contacts between clusters in the floc. To confirm the validity of this model, three types of polystyrene latex flocs with different fractal dimensions were broken up in the turbulent flow generated with a Rushton type agitator in a mixing vessel. The relation of the maximum diameter of the floc vs the turbulent shear stress was confirmed to obey the scaling law predicted by the proposed model. The power of the relation was also consistent with theoretical prediction.
Introduction The physical properties of a floc suspended in an unstable dispersion are of fundamental importance in colloid science and its applications. So far, theoretical attempts made to reveal the process of floc growth are based either on von Smoluchowski’s rate equation1 for the growth kinetics or on the fractal concept2 for the structure of a floc. However, the theoretical treatment on the floc breakup suffers because the understanding of the floc strength against the exerted force is lacking. The strength of a floc against breakup ultimately determines the size of a floc that serves as a species for collective transportation of colloidal particles in the suspension. Thus, understanding the floc strength and the size in a real flow is crucial for the analysis of rheological properties of flocculated suspensions and for the efficient design of solid-liquid separation of colloidal materials. In the present study, we analyze the size of a floc in turbulent flow to clarify the expression on the strength of a floc against breakup. It can be considered that the breakup of a floc follows an analogy with that of a droplet.3 As for a floc, the main difference, namely, the geometrical structure of a floc, must be also taken into account. In the first attempt to relate the strength of a floc with its structure, reported by Tambo et al.,4,5 an expression was derived for the maximum diameter of a floc as a function of mixing intensity in a turbulent flow. It was assumed that the strength of a floc is a function of the porosity chosen uniformly inside the floc and changes with the floc diameter due to the increase of porosity. A later experiment, designed by Sonntag and Russel,6 studied the breakup of flocs in a simple shear flow * Corresponding author. Tel: 81-298-53-4862. Fax: 81-298-552203. E-mail:
[email protected]. † Doctoral Program in Agricultural Sciences, University of Tsukuba. ‡ Institute of Agricultural and Forest Engineering, University of Tsukuba. § National Research Institute of Agricultural Engineering. (1) von Smoluchowski, M. Z. Phys. Chem. 1917, 92, 129. (2) Lin, M. Y.; Lindsay, H. M.; Weitz, D. A.; Ball, R. C.; Klein, R.; Meakin, P. Nature 1989, 339, 360. (3) Shinnar, R. J. Fluid Mech. 1961, 10, 259. (4) Tambo, N.; Yamada, K.; Hozumi, H. Suidou Kyoukai Zasshi 1970, 427, 4, in Japanese. (5) Tambo, N.; Hozumi, H. Water Res. 1979, 13, 421.
generated in the gap between two concentric cylinders. They monitored the size and fractal dimension of flocs as a function of the shear stress of flow by small-angle light scattering. Their results were interpreted by the theory of Adler and Mills7 introducing the effect of permeability into the previous theory of Bagster and Tomi8 for the rupture of a spherical floc. However, the concept of permeability is not always consistent with continuous mechanics. Moreover, the assumption of a spherical shape of a floc is not realistic. On the other side, the experimental relation between the mean mass of flocs and the shear stress obtained for different ionic strengths by Sonntag and Russel6 was successfully normalized by the force predicted by the Derjaguin-Landau-Verwey-Overbeek theory. This result clearly demonstrated the significance of a cohesive force between primary particles. Another result that emphasizes the importance of cohesive force was obtained very recently by Yeung and Pelton.9 They conducted direct measurement of the strength of a floc using the micromechanical device. In this device, a single floc was pulled apart by two pipets under the view of a microscope and the force of floc rupture was directly obtained by the distortion of a cantilever. From the result of this measurement, they concluded that the strength of a floc does not change with its diameter but increases with its fractal dimension. It should be noted that this conclusion contradicts to the results of Tambo et al.,4,5 who always used mean-field approximation in their analysis. It is known that the fractal dimension of a floc increases with the number of contacts between clusters because of their rearrangements10-12 when the floc is formed (Figure 1). Also, it is quite reasonable to expect that the increase of the number of contacts results in higher strength. Therefore, the result of direct measurement can be regarded as evidence that the number of contacts between the clusters is important to the strength of a floc. That is, (6) Sonntag, R. C.; Russel, W. B. J. Colloid Interface Sci. 1986, 113, 399. (7) Adler, P. M.; Mills, P. M. J. Rheol. 1979, 23, 25. (8) Bagster, D. F.; Tomi, D. Chem. Eng. Sci. 1974, 29, 1773. (9) Yeung, A. K. C.; Pelton, R. J. Colloid Interface Sci. 1996, 184, 579. (10) Adachi, Y.; Ooi, S. Proc. World Congr. Chem. Eng. 1986, 3, 156. (11) Adachi, Y.; Ooi, S. J. Colloid Interface Sci. 1990, 135, 374. (12) Meakin, P.; Jullien, R. J. Chem. Phys. 1988, 89, 246.
10.1021/la980763o CCC: $18.00 © 1999 American Chemical Society Published on Web 05/19/1999
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Figure 1. Schematic illustration of the aggregation and breakup of a floc. Particles forming a cluster-cluster contact are painted darker.
the discrete number of cluster-cluster bond(s) and the magnitude of cohesive force dominate the strength of a floc. However, such a picture has not been successfully introduced into previous analyses. On the basis of the considerations mentioned above, we proposed a simple rule that the strength of a floc, Ffloc, is the product of the number of contacts between clusters, Nc, and the magnitude of the cohesive force between primary particles, f:
Ffloc ) fNc
Model The breakup of a floc in turbulent flow occurs when the hydrodynamic force acting on the floc, Fhyd, exceeds Ffloc. By analogy with the theoretical treatment on the breakup of a droplet,3 the criterion on the breakup is expressed as the equation
Fhyd/Ffloc g C1
(2)
where C1 denotes a constant. From the concept of the local isotropy of turbulence, Fhyd can be written in two ways depending on the condition of whether the diameter of a floc, df, is larger than the Kolmogorov microscale
η ) (ν3/)1/4
(3)
or not. They are
Fhyd ) C3F2/3df8/3
On the other hand, according to the rule formulated in the preceding section, Ffloc can be written as in eq 1. From eqs 1, 2, 4, and 5, the following relations are obtained for the maximum value of df,
(1)
To examine the validity of this expression, we studied the breakup of three types of flocs with different fractal dimensions in turbulent flow. The experimental result was found to be consistent with the proposed rule.
Fhyd ) C2µGdf2
Figure 2. Geometry of mixing vessel to generate turbulent flow.
for df e η for df . η
(4) (5)
where , µ, ν, and F denote the rate of energy dissipation per unit mass, viscosity, kinematic viscosity, and density of fluid, respectively. C2 and C3 are constants. G ()x/ν) is called the apparent shear rate in a turbulent flow with the dimensions of a velocity gradient. The range expressed by eq 4 is called the viscous subrange, in which the viscous shear stress is dominant, whereas the one expressed by eq 5 is called the inertial subrange, in which the dynamic pressure is dominant.
df,max ) C4 df,max ) C5
( ) µG fNc
-1/2
F2/3 fNc
-3/8
( )
for df e η
(6)
for df . η
(7)
where C4 and C5 denote some constants. Both of these equations result in one and the same power-law relation between df,max and .
df,max ∝ -1/4
(8)
Experimental Section Floc Formation. Monodisperse polystyrene latex (PSL) spheres with a diameter of d0 ) 1356 nm and a density of 1.05 g/cm3 were used as the material of flocs in the breakup experiment. The PSL spheres were synthesized by means of aqueous polymerization without surfactant.13 Flocs were formed by coagulation of the PSL spheres with a simple electrolyte. That is, 1 mL of a PSL suspension was mixed with the same volume of a KCl solution. Three types of flocs were formed by changing the volume fraction of PSL particles in the suspension before mixing, φ0: 0.0182 (floc A), 0.00182 (floc B), and 0.0891 (floc C). Soon after initial mixing, a certain volume of a mixed suspension was sucked up into a sampler (a glass tube with a syringe) and left to age over 3 days to be fully coagulated. The concentration of KCl in the mixture was adjusted to 1.118 M, thus minimizing the effects of inertial, gravitational, and electrical repulsive forces exerted on and between particles. All distilled water and KCl solution were filtered using a Millipore filter with a mesh size of 0.22 µm before use. Mixing Apparatus. The turbulent flow for the breakup of flocs was generated by means of a Rushton type stirrer. The (13) Kotera, A.; Furusawa, K.; Takeda, Y. Kolloids Z. Z. Polym. 1970, 239, 677. (14) Kobayashi, M.; Adachi, Y. Trans. JSIDRE (Nougyoudoboku Gakkai Ronbunshu) 1997, 191, 111, in Japanese with English Abstract. (15) Adachi, Y.; Kamiko, M. Powder Technol. 1994, 78, 129. (16) Sonntag, R. C.; Russel, W. B. J. Colloid Interface Sci. 1987, 115, 390.
Breakup of Fractal Flocs in a Turbulent Flow
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Figure 3. Rate of energy dissipation per unit mass vs rotational speed of impeller Nr. is evaluated from the rate of 4kBT d03 128π dN(t) × coagulation using ) - RB + RT dt 3µ 8 15ν 0.18 A N(t)2, RT ) with Hamaker constant A ) 3 4 d0 36πµ 15πν 8 -21 1.0 × 10 J and Brownian coagulation kernel RB ) 0.5, where N(t), kB, and T denote the number concentration of flocs after time t elapsed from the beginning of coagulation, the Boltzmann constant, and the absolute temperature, respectively.
(
x
)
(
x
)
stirrer was installed into a cylindrical vessel with four equally placed baffles and with a lid to prevent the injection of air (Figure 2). The rate of energy dissipation per unit mass, , in the apparatus was evaluated by measuring the rate of coagulation of PSL particles in the vessel14 as a function of the rotational speed of the stirrer, Nr (Figure 3). Procedure. The breakup of flocs was carried out in the mixing vessel. When the piston of the syringe attached to the sampler was carefully pushed, the suspension containing coagulated flocs was gradually poured into the mixing vessel. The vessel was filled with a KCl solution of concentration equal to that in the sampler. This adjustment eliminates the effect of the density gradient, which may cause the breakup of flocs during the injection. Because of the high volume fraction of flocs C, their suspension was diluted by two steps. In the first step, the
suspension was diluted five times in a small bottle and then the diluted suspension was poured into the mixing vessel. The concentration of PSL particles in the vessel was set equal to a designed volume fraction, φb, by dilution. After being exposed to agitation, a small amount of suspension containing broken flocs was carefully sampled with a microslide (rectangular glass cell) for characterization with a microscope. All experiments were conducted in an air-conditioned room with a temperature at 20.0 ( 0.5 °C. Characterization of Broken Flocs. To analyze the strength of flocs in relation to their structure, it is necessary to obtain the size of flocs as well as the floc structure as a function of shear stress. For this purpose, broken flocs subjected to a certain mixing intensity were characterized in terms of the size distribution, the number concentration, and the morphology (the fractal dimension and the extent of elongation) under the view of the microscope. For the size distribution, flocs in the sampled suspension were photographed randomly. The maximum distance in the projected floc, Dm, was measured as an index of the size of the broken floc. The number of broken flocs in a specific volume was also counted through the microscope to obtain the number concentration of flocs. The fractal dimension, D, of flocs was determined by the method proposed previously15 using their size distribution and the number concentration of flocs. That is, if we suppose that all monitored flocs have the same value of D, the value of D can be obtained by solving the following equation of mass conservation: Nf
N(0) )
∑
n)1
( ) Dm,n d0
D
(9)
where N(0) and Nf denote the number of primary particles and flocs in the specific volume. As an index of the shape of the projected floc, we introduce the elongation factor, b, defined as
b ) Dm/Ds
(10)
where Ds is the shorter length of the rectangle circumscribed about the projected floc (Figure 1).
Results Cumulative frequency curves for Dm after exposure to the most vigorous agitation (Nr ) 10 rps) for 90 min are shown in Figure 4. Representative sizes Dm,99, Dm,95, and Dm,mean, which correspond to 99%, 95%, and numberaveraged Dm, respectively, are plotted against φb in Figure
Figure 4. Cumulative size distribution of broken flocs for various values of φb. Flocs A were broken in a mixing vessel with Nr ) 10 rps for 90 min.
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Figure 5. Representative size of flocs as a function of φb (square for Dm,mean, circle for Dm,99, and diamond for Dm,95). The data are extracted from Figure 4.
5. In these figures, it is indicated that size distribution curves do not change in the region where φb is below 10-6. Therefore, regrowth due to the collision between broken flocs can be negligible for these values of φb. Figure 6 shows the representative sizes as a function of time. From these results, it can be concluded that 90 min of agitation is sufficient to monitor the size distribution corresponding to the steady state. On the basis of these results, we carry out the analysis of the strength of flocs using the data taken for φb ) 5.3 × 10-7 (flocs A and B) and φb ) 5.1 × 10-7 (floc C) after agitation for 90 min. Typical size distribution curves taken for Nr ) 8.33 rps are shown in Figure 7. Similar curves are obtained at various values of Nr. The relation of µG vs Dm,99 as the maximum Dm is shown in Figure 8. From comparison of Dm,99 with the Kolmogorov microscale, it is demonstrated that the breakup of flocs always occurs in the viscous subrange. It is also found that flocs formed from higher φ0 exhibit higher strength. In Figure 9, the values of D obtained from the analysis of the size distribution (Figure 10) are plotted against mean floc mass, 〈i〉. As indicated in the figure, D decreases considerably with 〈i〉. It is also found that flocs formed from higher φ0 have higher values of D. In Figure 11, 〈i〉 is plotted against µG. As indicated in the figure, 〈i〉 decreases with an increase of µG. However, a considerable difference of 〈i〉 with respect to the condition of floc formation is observed.
Figure 6. Temporal evolution of floc size for floc B and φb ) 5.3 × 10-7. Symbols: square for Dm,mean, circle for Dm,99, and diamond for Dm,95. (a) Nr ) 10 rps. (b) Nr ) 0.5 rps.
Discussion In this section, we discuss the validity of the proposed model for the strength of flocs by normalizing the relation between the size of the flocs and the shear stress. In this analysis, we need values of Nc for each type of floc. They can be obtained from the analysis of D, which is a function of Nc. As demonstrated in Figure 9, the values of D decrease remarkably with 〈i〉 where 〈i〉 e 70 and approach unity in the smaller limit of 〈i〉. D ) 1 means that the object has a linear shape, and flocs broken by contractile flow at the orifice are reported to have elongated shape.16 Nevertheless, one cannot conclude that the decrease of D to unity at small 〈i〉 is evidence that the small flocs broken in the mixing vessel have a linear shape. The reason for the
Figure 7. Size distribution of flocs for Nr ) 8.33 rps.
decrease of D can also be found in the lower limit to apply the fractal to the projection of flocs. That is, a decrease of D in the scaling relation between Dm and 〈i〉 was also confirmed by table-tennis-ball simulation as well as Brownian coagulation of PSL spheres.17 This explanation is supported by the data for the elongation factor, b, in
Breakup of Fractal Flocs in a Turbulent Flow
Figure 8. 99% Dm, Dm,99, vs turbulent shear stress µG. Symbols: diamond for floc A, circle for floc B, and square for floc C. The solid line denotes the Kolmogorov microscale, η, given by eq 3.
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Figure 11. Mean floc mass 〈i〉 vs turbulent shear stress µG. Symbols source is as in Figure 8.
Figure 12. Mean elongation factor 〈b〉 vs mean floc mass 〈i〉. Symbols source is as in Figure 8. Figure 9. Fractal dimension D vs mean floc mass 〈i〉. Symbols source is as in Figure 8.
Figure 10. Calculated values of N(0) based on right-hand side of eq 9 for data from Nr ) 0.5 rps as a function of D. The values of D are determined by comparing these values to real N(0), N(0) ) 4.06 × 1011 (floc A, B) and 3.91 × 1011 (floc C).
Figure 12. As demonstrated in this figure, b is almost constant against 〈i〉 irrespective of the types of flocs.
Because the variation of D against 〈i〉 obtained in our previous research on the floc formation17 corresponds to that of the present research for the flocs breakup, it can be considered that the shear stress causes no rearrangement of clusters in the floc (which will make flocs more compact18). If we generalize this result, one can conclude that the increase of D during agitation in the smaller 〈i〉 region19 was caused by the regrowth, which enhances the face-to-face collisions20 (Figure 1). The values of D for flocs A-C in the region where 〈i〉 g 70 are 1.79, 1.68, and 1.91, respectively. By comparing these values of D with that of table-tennis-ball simulations,11,17 where D ) 1.69 (Nc ) 1) and 1.82 (Nc ) 2) in the same region, we determine Nc ) 1 for floc B and 2 for floc A, respectively. There are no computational data comparable to the value of D for floc C. However, it is natural to suppose Nc ) 3 for floc C because a floc with higher D should have a generally higher value of Nc. A difference of the data for Dm,99 vs µG is indicated in Figure 8. However, when µG is scaled by the estimated values of Nc, the data of Dm,99 vs µG collapse onto a single curve (Figure 13). Similarly, 〈i〉 vs µG can be scaled as (17) Adachi, Y.; Kobayashi, M.; Ooi, S. J. Colloid Interface Sci. 1998, 208, 353. (18) Lin, M. Y.; Klein, R.; Lindsay, H. M.; Weitz, D. A.; Ball, R. C.; Meakin, P. J. Colloid Interface Sci. 1990, 137, 263-280. (19) Oles, V. J. Colloid Interface Sci. 1992, 154, 351. (20) Doi, M.; Chen, D. J. Chem. Phys. 1989, 90, 5271.
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and Russel16 although the experimental conditions are different. All of these findings confirm the validity of the model proposed by us on the breakup and the strength of a floc. Conclusions The theoretical model on the breakup and strength of a floc in turbulent flow is proposed by analogy with the breakup of a droplet. In the proposed model, we make the hypothesis that the floc strength is a product of the cohesive force between primary particles and the number of contacts between clusters. To confirm the validity of this model, PSL flocs coagulated by 1:1 electrolyte were broken up in the mixing vessel. The experimental results show good agreement with the theoretical prediction.
Figure 13. 99% Dm, Dm,99, vs scaled turbulent shear stress µG/Nc. Symbols: diamond for floc A (Nc ) 2), circle for floc B (Nc ) 1), and square for floc C (Nc ) 3).
Figure 14. Mean floc mass 〈i〉 vs scaled turbulent shear stress µG/Nc. Symbols source is as in Figure 13.
indicated in Figure 14. The convergence of various results into a single curve means that the strength of a floc is controlled by the number of contacts between clusters. In addition, the result of a power relation, Dm,99 ∝ (µG/Nc)-0.535, obtained experimentally shows good agreement with the prediction of eq 6. It is also worth noting that the powerlaw relation, 〈i〉 ∝ (µG/Nc)-1, is identical to that of Sonntag
Acknowledgment. This work is financially supported by a Grant-in-Aid for Scientific Research (09460105) and the Research Project of the University of Tsukuba. The authors are also thankful to Prof. Atsushi Tada for his guidance throughout this research. Nomenclature b ) Dm/Ds, elongation factor C1, C2, C3, C4, C5 ) some constants d0 ) diameter of a primary particle df ) diameter of floc df,max ) maximum value of df D ) fractal dimension Dm ) maximum distance in the projected floc Dm,99, Dm,95, Dm,mean ) 99%, 95%, and mean values of Dm Ds ) shorter length of the rectangle circumscribed about the projected floc f ) magnitude of the cohesive force between primary particles Ffloc ) strength of a floc Fhyd ) hydrodynamic force acting on a floc G ) (/ν)1/2, apparent shear rate in a turbulent flow i ) (Dm/d0)D, number of primary particles in a floc (floc mass) N(0) ) number concentration of primary particles Nc ) number of contacts between clusters Nf ) number concentration of flocs Nr ) rotational speed of the stirrer ) rate of energy dissipation per unit mass φ0 ) volume fraction of particles before mixing with a KCl solution φb ) volume fraction of particles in the mixing vessel η ) Kolmogorov microscale µ ) viscosity of the fluid ν ) kinematic viscosity of the fluid F ) density of the fluid LA980763O
