- Email: [email protected]
Aggregation and Breakup of Particles in a Shear Flow
JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.
187, 466–473 (1997)
CS964710
Aggregation and Breakup of Particles in a Shear Flow TERESA SERRA, JORDI COLOMER,
AND
XAVIER CASAMITJANA
Geophysical Fluid Dynamics Laboratory, University of Girona, Plac¸a Hospital, 6, 17071 Girona, Spain Received August 6, 1996; accepted November 19, 1996
An experimental study was carried out to investigate aggregation and breakup of particles in a Couette flow system with the inner cylinder rotating at constant speed. Experiments were conducted with monodisperse and polydisperse suspensions at different particle volume concentrations. Depending on both the shear stress and the particle concentration, three different regimes were found. In the first regime, when the concentration is less than a critical value, the final diameter of the aggregate is independent of concentration and depends only on the shear. When the concentration is higher, two new regimes are identified, with the final diameter of the aggregate depending on both shear stress and particle concentration. Also, the transition between these two regimes corresponds to the transition from laminar to turbulent flow. In addition, the final diameter of the aggregates in the turbulent flow regime is independent of the size of the primary particles and results to be is controlled by the Kolmogorov length scale. On the contrary, in the laminar flow regime the final diameter of the aggregates shows a light tendency to decrease as the diameter of the primary particles increases. q 1997 Academic Press Key Words: aggregation; breakup; concentration; shear; Couette; polystyrene latex.
INTRODUCTION
It is well known that, as a result of collisions, particles form aggregates that have a higher effective size than the primary particles. Aggregates have been observed from both inorganic sediments (1–3) and organic or planktonic particles (4). This increase in size is found to be especially relevant in lakes or seas where aggregates can account for the removal of particles in the epilimnion (5). Also, they play an important role in volcanic eruptions, where aggregate formation initiated in the atmosphere induces a premature fallout of fine-grained ash. As a result, when the aggregates are absorbed by the ocean, the settling velocity is higher than would be expected for dispersed ash particles (6). On the other hand, aggregation introduces, at the same time, modifications in the effective surfaces of the particles (Krishnappan et al., unpublished; 7, 8), which can be important carriers of pollutants that adhere to them. As the surface area of aggregates increases, more pollutants can be absorbed.
466
0021-9797/97 $25.00 Copyright q 1997 by Academic Press All rights of reproduction in any form reserved.
AID
JCIS 4710
/
6g1e$$$181
There are three different mechanisms that can account for the process of aggregation of a given set of particles. The first mechanism is Brownian coagulation and is relevant only for the smallest particles. A second mechanism that contributes to an increase in the collision frequency function is shear stress. Finally, sedimentation is relevant only if particles are of different size. In this case, the difference in the settling velocity due to the different sizes increases the chance of collision. Particles are in a continuous process of aggregation and breakup until eventually the steady-state size is reached. Although not completely understood, the rate of disaggregation is dependent on fluid shear, on collisions between particles, and on the probability of disaggregation after collision (2, 3). The effect of each collision frequency has been used to define ranges of sizes. Relevant collision frequencies can be assigned to different particle sizes; therefore, each frequency could be important at different stages of the aggregation/breakup process. Brownian motion is important for particles with diameters smaller than 1 mm, shear stress is important in the range 1–40 mm, and differential settling is important for particles larger than 40 mm (9). In the present study, for primary particles with a diameter between 2 and 5 mm, introduced in a Couette shear flow, with no differences between particle and fluid densities, the only relevant mechanism in the collision frequency is shear stress. Because of the sizes of the particles that have been used, the Brownian mechanism would be relevant only in the first seconds of the aggregation process and, therefore, it is disregarded. Also, the differential settling mechanism is irrelevant because the density of the solution is set to the same value as the density of the particles. A shear range between 25 and 195 s 01 was induced by means of a Couette flow system. The particle volume fraction was varied between 1.5 1 10 05 and 10.0 1 10 05 , corresponding to a mass concentration varying from 15 to 105 mg/liter. Ranges have been chosen as wide as possible and only limited because of technical reasons; however, that shear range covers a wide spectrum in laminar and turbulent flow. Other previous works (1, 10) have been concentrated in smaller parts of the spectrum and did not account for the coupling between the shear and particle concentration-dependent regimes of
02-17-97 21:14:03
coidas
AGGREGATION AND BREAKUP OF PARTICLES IN A SHEAR FLOW
FIG. 1. Schematic view of the experimental setup.
the aggregation/breakup process. Both the shear and the concentration ranges used in the present study are characteristic of shallow lakes under storm conditions, so this study is also aimed at understanding the dynamics of particle aggregates in a natural environment. For example, in Lake Balaton, perturbations by episodic wind events produced shears and resulting bottom suspended sediment concentrations in the same size ranges (11) used in this work. These authors describe a storm event that lasted nearly 10 hours, therefore allowing the aggregation and breakup processes to occur. EXPERIMENTAL SETUP
Suspensions of latex particles were prepared in 1.29 M NaCl solution in ultrapure water (Milli-Q-Water, Millipore, Bedford, MA) at different volume fractions ranging from 1.5 1 10 05 to 10 1 10 05 . Diameters of the particles used were 2.00 { 0.06 and 5.0 { 0.3 mm. Latex particles (2 mm, Batch 5-376-46; 5 mm, Batch 488C) were purchased from Interfacial Dynamics Corporation (Portland, OR). The temperature was set at 23.0 { 0.57C. To avoid sedimentation effects, the density of the solution was set at the same value as that of the latex particles, r Å 1.055 g/cm3 . As a result of this salt dilution, the electric double layer was highly reduced to an approximate value of the Debye–Hu¨ckel length of 10 08 m. A schematic view of the Couette flow system (where suspensions were introduced) is presented in Fig. 1. This kind of apparatus was used because it generates a more isotropic turbulence than others such as blades in a jar (12). The outer cylinder was fixed, while the velocity of the inner cylinder could be adjusted. The inner cylinder, with a diameter of 160 mm, was made of stainless steel to avoid corrosion. The outer cylinder, with a diameter of 193 mm, was made
AID
JCIS 4710
/
6g1e$$$182
02-17-97 21:14:03
467
of Plexiglas to allow the fluid to be visible. The height of the outer cylinder was 360 mm. The width of the gap between cylinders was 16.5 mm, and the separation between them, at the bottom, was 15 mm. Rotation of the inner cylinder was maintained by a shaft. A digital controller ensured a constant velocity. The outer cylinder had two lateral openings for sampling. One of them was 5 cm from the bottom and the other was 5 cm from the top. Samples were taken from the bottom opening and passed through the optical cell of a particle size analyzer and were pumped back into the Couette system through the top opening. The flux throughout the tubes was laminar and always kept at a value much less than the shear into the gap so as not to disrupt the aggregates formed in the sheared main bulk fluid system. The residence time of the fluid in the tubes was very short (t* õ 0.09, where t* is the nondimensional time, as introduced in the next section. The volume of fluid in the tubes was always much less compared with the volume in the Couette system. Tests with a variation of the flow rates in the tubes showed no significant effect on the measured size distribution. The particle diameter distribution was measured with a laser Cis-1 Galai (Industrial Zone 10,500 Migdal Haemek, Israel). To measure the samples, a rotating laser is used to scan the sample; the interaction of the laser and the particle gives a pulse, and the width of the pulse is directly proportional to the particle size. This scattering technique affects only the pulse amplitude, not its width; therefore, it is not necessary to know the refractive index or the particle’s light absorption characteristics. This is an advantage compared with other techniques that use diffraction analysis. Two lens were used. Lens A covered the range from 0.5 to 150 mm and lens B covered from 5 to 600 mm. Aggregates formed with a particle fraction higher than 10.0 1 10 05 were impossible to measure because the solution was opaque to the laser beam that measured the particle diameters. The range of the shear was also limited by the Couette flow system used in the present study. To visualize characteristic sizes of the aggregates, samples from the sheared flow volume were deposited on a Millipore filter and then scanned using an electron microscope. Figure 2a shows a steady-state aggregate formed when the shear was 25 s 01 and the volume fraction was 2.5 1 10 05 . In Fig. 2b the concentration was the same but the shear was 70 s 01 , and in Fig. 2c, the concentration was 10.0 1 10 05 and the shear was 25 s 01 . The relationship between viscous forces and thermal forces is given by the Pe´clet number, Pe Å
6phGr 30 , KT
[1]
where h is the dynamic viscosity, K the Boltzmann constant, T the temperature, r0 the particle radius, and G the shear
coidas
468
AID
SERRA, COLOMER, AND CASAMITJANA
JCIS 4710
/
6g1e$$4710
02-17-97 21:14:03
coidas
469
AGGREGATION AND BREAKUP OF PARTICLES IN A SHEAR FLOW
FIG. 2. Scanning electron microscope photographs of aggregates formed at G Å 25 s 01 and f0 Å 2.5 1 10 05 (a), G Å 70 s 01 and f0 Å 2.5 1 10 05 (b), and G Å 25 s 01 and f0 Å 10 1 10 05 (c). The diameter of primary particles in all photographs was 2 mm. See scale and zoom magnification on the right.
stress. For the experimental study, the Pe´clet number was found to be 120 for particles with a diameter of 2 mm and 1800 for particles with a diameter of 5 mm. This ensures that aggregation due to Brownian movement can be neglected compared with the shear-induced aggregation (10). The flow between cylinders becomes unstable when angular velocity, v, exceeds the critical angular velocity (13), Vc Å n
S
3390(R 22 0 R 21 ) 4R 21 (R2 0 R1 ) 4
D
0.5
,
[2]
where R2 and R1 are the radii of the outer and inner cylinder, respectively, and n Å 0.9325 1 10 06 m2 /s is the kinematic viscosity of the fluid. The critical angular velocity obtained from [2] is Vc Å 0.087 s 01 (0.83 rpm). Instabilities generated when the velocity is higher could be clearly seen if a large quantity of larger particles were added to the fluid. Therefore, in our shear range, instabilities are always present. The angular velocity at which transition to turbulence takes place, vc , evaluated from van Duuren’s equation (14),
AID
JCIS 4710
/
6g1e$$$182
02-17-97 21:14:03
vc Å
3.16 1 10 5 (R2 0 R1 ) 0.7n R 2.7 2
[3]
gives a value of vc Å 9.8 s 01 (98 rpm). Although expression [3] was deduced for the case when the outer cylinder was rotating and the inner at rest, the result of Eq. [3] was taken as an estimate of the transition to turbulence. The angular velocities used in this study ranged from 40 to 211 rpm, therefore covering the transition from laminar to turbulent flow. Shear stress was determined in two ways depending on whether the flow was laminar or turbulent. For laminar flow, as an estimate of the mean shear, the average shear was GÅ
1 R2 0 R1
*
R1
g(r)dr,
[4]
R2
where g(r) is g(r) Å
coidas
2vR 21 R 22 (R 22 0 R 21 )r 2
[5]
470
SERRA, COLOMER, AND CASAMITJANA
s 01 . In the near shores of the Great Lakes, for example, the shear range induced from waves was found to vary between 100 and 400 s 01 (2, 3). Therefore the results obtained from our experiments can be extrapolated to environmental situations.
TABLE 1 Shear Stress at Different Values of the Angular Velocity of the Inner Cylinder a
a
v (rpm)
G (s01)
40 56 75 105 125 165 211
25 32 50 70 90 135 195
RESULTS AND DISCUSSION
Calculated from [6]–[8].
and r is the radial distance. Integrating expression [4] we obtain GÅ
2vR1 R2 , R 22 0 R 21
[6]
where v is the angular velocity of the inner cylinder. In addition, turbulent shear has been estimated from
r GÅ
e , n
[7]
where e is the rate of energy dissipation per unit mass that can be scaled, using a characteristic velocity fluctuation u * and a characteristic length l, as e Ç u * 3 /l. It was assumed that l scales with the width of the gap, l Ç R2 0 R1 , and u * scales with the mean velocity uV (15), u * Ç uV , which is uV Å
2vR 21 (R 31 / 2R 32 0 3R1 R 22 ). 3(R 22 0 R 21 ) 2
[8]
It is important to point out that expressions [7] and [8] give an approximate estimation of the shear values. The exact values of the energy dissipated into the system, e, are somewhat difficult to determine. If continuity between laminar and turbulent shear is assumed, a proportionality constant, a, can be calculated. The value obtained was 0.03, which is much smaller than other values found in natural systems with isotropic turbulence. For example, Lazier and Mann (16) found a Å 0.37 and Bowen et al. (17) a Å 0.50, both in the open ocean. Turbulent shear was then found to depend on v 3 / 2 (see Eqs. [7] and [8]), whereas for the laminar case the dependence was linear (see Eq. [5]). The calculated shear stress that corresponds to the critical velocity found from [3] is 58 s 01 . In Table 1 the shear stresses calculated from Eqs. [6] and [7] are presented. It is clearly seen that in our study the calculated values ranged from 25 to 195
AID
JCIS 4710
/
6g1e$$$182
Figure 3 shows the evolution of the nondimensional diameter d/d 0 with nondimensional time t* for different shear rates, when the initial volume fraction is f0 Å 2.5 1 10 05 . d 0 is the diameter of the initial distribution of particles, which is 2 mm for all the cases represented in Fig. 3. As the characteristic diameter, d, we use the median of the size distribution (with a confidence of 95%) with respect to aggregate volume. t* depends on the shear, G, and on the initial fraction volume, f0 , as t* Å tGf0 . This nondimensional time represents a normalized number of collisions taking place in the system (10). At the beginning of the experiments shown in Fig. 3, at small t*, d/d 0 scaled well with t*. Here, the main factor that explains the aggregation process is the shear within the cylinder gap, according to Smoluchoski’s approach (18). For all the cases studied in Fig. 3 the steady-state size is reached for t* à 7. But, taking into account that t* depends on the product Gt, this means that the steady-state size is reached sooner when G increases. At larger t*, the breakup is more pronounced until it balances the aggregation, and the steady state is then reached. In addition, it is found that the larger the shear is, the faster aggregation occurs. As a result, the system reaches steady state more rapidly and the final size of the nondimensional diameter of the aggregates is smaller. This behavior is supported by other works such as those of Oles (10), who worked with latex particles, Lick and Lick (2), Burban et
02-17-97 21:14:03
FIG. 3. Evolution of d/d 0 with the nondimensional time t* at different shears and at f0 Å 2.5 1 10 05 .
coidas
AGGREGATION AND BREAKUP OF PARTICLES IN A SHEAR FLOW
FIG. 4. Representation of the final diameter of the aggregate, normalized to the diameter of the primary particles, d/d 0 , for different volume fractions, f0 , at different shear stresses, G. Here d 0 Å 2 mm.
al. (3), and Tsai et al. (1), who worked with sediment particles. Figure 4 depicts the relationship between d/d 0 and f0 at different shears. Depending on both shear stress and particle concentration, three different regimes were found. In the first regime, when the volume fraction is less than Ç6 1 10 05 , the final nondimensional diameter of the aggregate is independent of the volume fraction and depends only on shear. This regime was also found by Oles (10) with latex particles in a Couette flow system with a cylinder rotating between two static cylinders; however, when the concentration is larger, this result no longer holds. When the concentration is larger than Ç6 1 10 05 , two new regimes are identified depending on both the volume fraction and the shear. When the shear stress is less than 50 s 01 , if the volume fraction increases, the aggregation of particles is enhanced and the final diameter of the aggregate increases. On the other hand, if the shear is larger than 50 s 01 , an increase in the volume fraction enhances the breakup processes and the particle diameter shows a slight tendency to decrease. This is in accordance with the work of Burban et al. (3) and Lick and Lick (2), using sediment particles; however, these authors did not study the aggregation/ breakup process at the range of low shear rates as in our case. In addition, in the work described by Oles (10), the volume fraction was too small to show the behavior described in Fig. 4. As mentioned under Experimental Setup, it is important to point out that the transition between the two regimes coincides with the transition from laminar to turbulent flow, which, according to Eq. [3], was established at a shear of 58 s 01 . Therefore, a clear distinction in the
AID
JCIS 4710
/
6g1e$$$183
02-17-97 21:14:03
471
balance between aggregation and breakup is established, depending on whether the flow is laminar or turbulent. Experiments carried out with 5-mm particles (Fig. 5) show the same behavior as experiments carried out with 2mm particles (Fig. 4); however, in natural systems there is no suspension that contains only particles of one size. Therefore, to account for more realistic situations, experiments with polydisperse suspensions were carried out. The results of these experiments are presented in Fig. 6. The initial distribution of particles in the experiments, the results of which are presented in Fig. 6a, was a 75% 2-mm and 25% 5-mm particles; that of Fig. 6b was 50% 2-mm and 50% 5mm particles; and that of Fig. 6c was 25% 2-mm and 75% 5-mm particles. The mean initial diameters for these distributions were estimated to be 2.75, 3.50, and 4.25 mm, respectively. Results presented in Fig. 6 show the same three regimes found for the experiments carried out with monodisperse particles (Figs. 4 and 5); however, it is important to point out that the light tendency of d/d 0 to decrease in the turbulent regime found in Figs. 4 and 5 is not always observed. In addition, for a given G and f0 , we found similar values for the final diameter d of the aggregates in both monodisperse and polydisperse suspensions, in the turbulent regime. For example at G Å 70 s 01 and f0 Å 10.0 1 10 05 the value of the diameter d is around 20 mm for d 0 Å 5 mm (Fig. 5). For the same values of G and f0 , d is Ç21 mm for d 0 Å 3.5 mm (Fig. 6b). Taking into account that d/d 0 is related to the number of particles in the aggregate, if d is the same and d 0 is changing, this means that the final number of particles in the aggregate is different. This can be observed in the turbulent regimes in Figs. 4, 5, and 6. Therefore, the
FIG. 5. Representation of the final diameter of the aggregate, normalized to the diameter of the primary particles, d/d 0 , for different volume fractions, f0 , at different values of the shear stress, G. Here d 0 Å 5 mm.
coidas
472
SERRA, COLOMER, AND CASAMITJANA
TABLE 2 Calculated Kolmogorov Length Scale (Eq. [9]) and the Median of the Distribution Measured at Different Values of the Shear Stress G (s01)
l (mm)
d (mm)
70 90 135 195
21 18 15 12
8-32 8-28 8-20 8-12
final size of the aggregate, d, seems to be dependent on flow and independent of d 0 . On the contrary, in the laminar flow regime, the final diameter of the aggregates shows a slight tendency to decrease as d 0 increases. The latter observation can be explained by the high compaction of the aggregates formed by the smallest particles. The higher the compaction of the aggregates, the more resistant they are at the same value of the shear rate, as proposed by Burban et al. (3). To assess whether the final size of the aggregate depends on flow or on diameter of the primary particles, the size of the smallest turbulent eddies within the fluid was estimated from the Kolmogorov length scale, l, lÅ
S D n3 e
1/4
,
[9]
where e Ç u * 3 /l, u * being calculated from Eq. [8]. The maximum calculated l was 21 mm. Therefore, d was found to be of the same order of magnitude as l; the experimental value of d obtained from Figs. 4, 5, and 6 varied between 8 and 32 mm (Table 2). Therefore, it has been proved that the main process that determines the final diameter of an aggregate is turbulent shear stress, as also found by Jankowski et al. (19). In Fig. 7 d/d 0 is plotted as a function of the shear G for different concentrations [(a) 1.5 1 10 05 , (b) 2.5 1 10 05 , (c) 5.0 1 10 05 , (d) 7.5 1 10 05 , (e) 10.0 1 10 05 ) of monodisperse particles. It is important to point out that the measurements did not cover the range of G between 0 and 25 s 01 . Here, d/d 0 should increase from an initial value, at G Å 0, where Brownian motion cannot be neglected. Thereafter, at larger values of G, there is a power decrease of d/d 0 as plotted in Fig. 7, as also proposed by van Leusen (20). In Oles (10), a linear dependence was found between d/d 0 and G; however, a power relationship seems to be more realistic bearing in mind that the limit value of d/d 0 at very
FIG. 6. Representation of the final diameter of the aggregate, normalized to the diameter of the primary particles, d/d 0 , for different volume
AID
JCIS 4710
/
6g1e$$$183
02-17-97 21:14:03
fractions, f0 , at different values of the shear stress, G, for polydisperse particle suspensions (a) d 0 Å 2.75 mm, (b) d 0 Å 3.5 mm, and (c) d 0 Å 4.25 mm.
coidas
AGGREGATION AND BREAKUP OF PARTICLES IN A SHEAR FLOW
473
FIG. 7. Representation of the final diameter of the aggregate, normalized to the diameter of the primary particles, d/d 0 , for different values of the shear G, at different volume fractions: (a) f0 Å 1.5 1 10 05 , (b) f0 Å 2.5 1 10 05 , (c) f0 Å 5 1 10 05 , (d) f0 Å 7.5 1 10 05 , and (e) f0 Å 10 1 10 05 .
high G must be, obviously, 1. In addition, Parker et al. (21) found that, due to the surface erosion of the aggregate, d depends on G 01 . We corroborated this result in the range 50–195 s 01 , corresponding to turbulent flow. REFERENCES 1. Tsai, C., Iacobellis, S., and Lick, W., J. Great Lakes Res. 13, 135 (1987). 2. Lick, W., and Lick, J., J. Great Lakes Res. 14, 514 (1988). 3. Burban, P., Lick, W., and Lick, J., J. Geophys. Res. 94, 8323 (1989). 4. Shimeta, J., Jumars, P. A., and Lessard, E. J., Limnol. Oceanogr. 40, 845 (1995). 5. Casamitjana, X., and Schladow, G., J. Environ. Eng. 119, 443 (1993). 6. Weisner, M. G., Wang, Y., and Zheng, L., Geology 23, 885 (1995). 7. Droppo, I. G., and Ongley, E. D., Water Res. 26, 65 (1992). 8. Axford, S. D. T., J. Chem. Soc. Faraday Trans. 92, 1007 (1996). 9. Logan, B. E., and Wilkinson, D. B., Environ. Sci. Technol. 25, 2031 (1991).
AID
JCIS 4710
/
6g1e$$$183
02-17-97 21:14:03
10. Oles, V., J. Colloid Interface Sci. 154, 351 (1991). 11. Luettich, R. A., Jr., Harleman, D. R. F., and Somlyo´dy, L., Limnol. Oceanogr. 35, 1050 (1990). 12. Pandya, J. D., and Spielman, L. A., Chem. Eng. Sci. 38, 1983 (1983). 13. Chandrasekhar, S., ‘‘Hydrodynamic and Hydromagnetic Stability.’’ Clarendon, Oxford, 1961. 14. Van Duuren, F. A., J. Sanit. Eng. Div. ASCE 94, 671 (1968). 15. Imberger, J., in ‘‘Limnology Now: A Paradigm of Planetary Problems, (R. Margalef, Ed.), pp. 99–193. Elsevier, Amsterdam, 1994. 16. Lazier, J. R. N., and Mann, K. H., Deep-Sea Res. 36, 1721 (1989). 17. Bowen, J. D., Stolzenbach, K. D., and Chisholm, S. W., Limnol. Oceanogr. 38, 36 (1993). 18. Friedlander, S. K., ‘‘Smoke, Dust and Haze.’’ Wiley–Interscience, New York, 1977. 19. Jankowski, J. A., Malcherek, A., and Zielke, W., J. Geophys. Res. 101, 3545 (1996). 20. Van Leusen, W., ‘‘Aggregation of Particles, Settling Velocity of Mud Flocs: A Review.’’ Springer-Verlag, Berlin/Heidelberg, 1988. 21. Parker, D. S., Kaufman, W. J., and Jenkins, D., ‘‘Characteristics of Biological Flocs in Turbulent Regimes.’’ Thesis, Sanitary Engineering Research Laboratory, Report No. 70-5, College of Engineering and School of Public Health, Berkeley, 1970.
coidas
187, 466–473 (1997)
CS964710
Aggregation and Breakup of Particles in a Shear Flow TERESA SERRA, JORDI COLOMER,
AND
XAVIER CASAMITJANA
Geophysical Fluid Dynamics Laboratory, University of Girona, Plac¸a Hospital, 6, 17071 Girona, Spain Received August 6, 1996; accepted November 19, 1996
An experimental study was carried out to investigate aggregation and breakup of particles in a Couette flow system with the inner cylinder rotating at constant speed. Experiments were conducted with monodisperse and polydisperse suspensions at different particle volume concentrations. Depending on both the shear stress and the particle concentration, three different regimes were found. In the first regime, when the concentration is less than a critical value, the final diameter of the aggregate is independent of concentration and depends only on the shear. When the concentration is higher, two new regimes are identified, with the final diameter of the aggregate depending on both shear stress and particle concentration. Also, the transition between these two regimes corresponds to the transition from laminar to turbulent flow. In addition, the final diameter of the aggregates in the turbulent flow regime is independent of the size of the primary particles and results to be is controlled by the Kolmogorov length scale. On the contrary, in the laminar flow regime the final diameter of the aggregates shows a light tendency to decrease as the diameter of the primary particles increases. q 1997 Academic Press Key Words: aggregation; breakup; concentration; shear; Couette; polystyrene latex.
INTRODUCTION
It is well known that, as a result of collisions, particles form aggregates that have a higher effective size than the primary particles. Aggregates have been observed from both inorganic sediments (1–3) and organic or planktonic particles (4). This increase in size is found to be especially relevant in lakes or seas where aggregates can account for the removal of particles in the epilimnion (5). Also, they play an important role in volcanic eruptions, where aggregate formation initiated in the atmosphere induces a premature fallout of fine-grained ash. As a result, when the aggregates are absorbed by the ocean, the settling velocity is higher than would be expected for dispersed ash particles (6). On the other hand, aggregation introduces, at the same time, modifications in the effective surfaces of the particles (Krishnappan et al., unpublished; 7, 8), which can be important carriers of pollutants that adhere to them. As the surface area of aggregates increases, more pollutants can be absorbed.
466
0021-9797/97 $25.00 Copyright q 1997 by Academic Press All rights of reproduction in any form reserved.
AID
JCIS 4710
/
6g1e$$$181
There are three different mechanisms that can account for the process of aggregation of a given set of particles. The first mechanism is Brownian coagulation and is relevant only for the smallest particles. A second mechanism that contributes to an increase in the collision frequency function is shear stress. Finally, sedimentation is relevant only if particles are of different size. In this case, the difference in the settling velocity due to the different sizes increases the chance of collision. Particles are in a continuous process of aggregation and breakup until eventually the steady-state size is reached. Although not completely understood, the rate of disaggregation is dependent on fluid shear, on collisions between particles, and on the probability of disaggregation after collision (2, 3). The effect of each collision frequency has been used to define ranges of sizes. Relevant collision frequencies can be assigned to different particle sizes; therefore, each frequency could be important at different stages of the aggregation/breakup process. Brownian motion is important for particles with diameters smaller than 1 mm, shear stress is important in the range 1–40 mm, and differential settling is important for particles larger than 40 mm (9). In the present study, for primary particles with a diameter between 2 and 5 mm, introduced in a Couette shear flow, with no differences between particle and fluid densities, the only relevant mechanism in the collision frequency is shear stress. Because of the sizes of the particles that have been used, the Brownian mechanism would be relevant only in the first seconds of the aggregation process and, therefore, it is disregarded. Also, the differential settling mechanism is irrelevant because the density of the solution is set to the same value as the density of the particles. A shear range between 25 and 195 s 01 was induced by means of a Couette flow system. The particle volume fraction was varied between 1.5 1 10 05 and 10.0 1 10 05 , corresponding to a mass concentration varying from 15 to 105 mg/liter. Ranges have been chosen as wide as possible and only limited because of technical reasons; however, that shear range covers a wide spectrum in laminar and turbulent flow. Other previous works (1, 10) have been concentrated in smaller parts of the spectrum and did not account for the coupling between the shear and particle concentration-dependent regimes of
02-17-97 21:14:03
coidas
AGGREGATION AND BREAKUP OF PARTICLES IN A SHEAR FLOW
FIG. 1. Schematic view of the experimental setup.
the aggregation/breakup process. Both the shear and the concentration ranges used in the present study are characteristic of shallow lakes under storm conditions, so this study is also aimed at understanding the dynamics of particle aggregates in a natural environment. For example, in Lake Balaton, perturbations by episodic wind events produced shears and resulting bottom suspended sediment concentrations in the same size ranges (11) used in this work. These authors describe a storm event that lasted nearly 10 hours, therefore allowing the aggregation and breakup processes to occur. EXPERIMENTAL SETUP
Suspensions of latex particles were prepared in 1.29 M NaCl solution in ultrapure water (Milli-Q-Water, Millipore, Bedford, MA) at different volume fractions ranging from 1.5 1 10 05 to 10 1 10 05 . Diameters of the particles used were 2.00 { 0.06 and 5.0 { 0.3 mm. Latex particles (2 mm, Batch 5-376-46; 5 mm, Batch 488C) were purchased from Interfacial Dynamics Corporation (Portland, OR). The temperature was set at 23.0 { 0.57C. To avoid sedimentation effects, the density of the solution was set at the same value as that of the latex particles, r Å 1.055 g/cm3 . As a result of this salt dilution, the electric double layer was highly reduced to an approximate value of the Debye–Hu¨ckel length of 10 08 m. A schematic view of the Couette flow system (where suspensions were introduced) is presented in Fig. 1. This kind of apparatus was used because it generates a more isotropic turbulence than others such as blades in a jar (12). The outer cylinder was fixed, while the velocity of the inner cylinder could be adjusted. The inner cylinder, with a diameter of 160 mm, was made of stainless steel to avoid corrosion. The outer cylinder, with a diameter of 193 mm, was made
AID
JCIS 4710
/
6g1e$$$182
02-17-97 21:14:03
467
of Plexiglas to allow the fluid to be visible. The height of the outer cylinder was 360 mm. The width of the gap between cylinders was 16.5 mm, and the separation between them, at the bottom, was 15 mm. Rotation of the inner cylinder was maintained by a shaft. A digital controller ensured a constant velocity. The outer cylinder had two lateral openings for sampling. One of them was 5 cm from the bottom and the other was 5 cm from the top. Samples were taken from the bottom opening and passed through the optical cell of a particle size analyzer and were pumped back into the Couette system through the top opening. The flux throughout the tubes was laminar and always kept at a value much less than the shear into the gap so as not to disrupt the aggregates formed in the sheared main bulk fluid system. The residence time of the fluid in the tubes was very short (t* õ 0.09, where t* is the nondimensional time, as introduced in the next section. The volume of fluid in the tubes was always much less compared with the volume in the Couette system. Tests with a variation of the flow rates in the tubes showed no significant effect on the measured size distribution. The particle diameter distribution was measured with a laser Cis-1 Galai (Industrial Zone 10,500 Migdal Haemek, Israel). To measure the samples, a rotating laser is used to scan the sample; the interaction of the laser and the particle gives a pulse, and the width of the pulse is directly proportional to the particle size. This scattering technique affects only the pulse amplitude, not its width; therefore, it is not necessary to know the refractive index or the particle’s light absorption characteristics. This is an advantage compared with other techniques that use diffraction analysis. Two lens were used. Lens A covered the range from 0.5 to 150 mm and lens B covered from 5 to 600 mm. Aggregates formed with a particle fraction higher than 10.0 1 10 05 were impossible to measure because the solution was opaque to the laser beam that measured the particle diameters. The range of the shear was also limited by the Couette flow system used in the present study. To visualize characteristic sizes of the aggregates, samples from the sheared flow volume were deposited on a Millipore filter and then scanned using an electron microscope. Figure 2a shows a steady-state aggregate formed when the shear was 25 s 01 and the volume fraction was 2.5 1 10 05 . In Fig. 2b the concentration was the same but the shear was 70 s 01 , and in Fig. 2c, the concentration was 10.0 1 10 05 and the shear was 25 s 01 . The relationship between viscous forces and thermal forces is given by the Pe´clet number, Pe Å
6phGr 30 , KT
[1]
where h is the dynamic viscosity, K the Boltzmann constant, T the temperature, r0 the particle radius, and G the shear
coidas
468
AID
SERRA, COLOMER, AND CASAMITJANA
JCIS 4710
/
6g1e$$4710
02-17-97 21:14:03
coidas
469
AGGREGATION AND BREAKUP OF PARTICLES IN A SHEAR FLOW
FIG. 2. Scanning electron microscope photographs of aggregates formed at G Å 25 s 01 and f0 Å 2.5 1 10 05 (a), G Å 70 s 01 and f0 Å 2.5 1 10 05 (b), and G Å 25 s 01 and f0 Å 10 1 10 05 (c). The diameter of primary particles in all photographs was 2 mm. See scale and zoom magnification on the right.
stress. For the experimental study, the Pe´clet number was found to be 120 for particles with a diameter of 2 mm and 1800 for particles with a diameter of 5 mm. This ensures that aggregation due to Brownian movement can be neglected compared with the shear-induced aggregation (10). The flow between cylinders becomes unstable when angular velocity, v, exceeds the critical angular velocity (13), Vc Å n
S
3390(R 22 0 R 21 ) 4R 21 (R2 0 R1 ) 4
D
0.5
,
[2]
where R2 and R1 are the radii of the outer and inner cylinder, respectively, and n Å 0.9325 1 10 06 m2 /s is the kinematic viscosity of the fluid. The critical angular velocity obtained from [2] is Vc Å 0.087 s 01 (0.83 rpm). Instabilities generated when the velocity is higher could be clearly seen if a large quantity of larger particles were added to the fluid. Therefore, in our shear range, instabilities are always present. The angular velocity at which transition to turbulence takes place, vc , evaluated from van Duuren’s equation (14),
AID
JCIS 4710
/
6g1e$$$182
02-17-97 21:14:03
vc Å
3.16 1 10 5 (R2 0 R1 ) 0.7n R 2.7 2
[3]
gives a value of vc Å 9.8 s 01 (98 rpm). Although expression [3] was deduced for the case when the outer cylinder was rotating and the inner at rest, the result of Eq. [3] was taken as an estimate of the transition to turbulence. The angular velocities used in this study ranged from 40 to 211 rpm, therefore covering the transition from laminar to turbulent flow. Shear stress was determined in two ways depending on whether the flow was laminar or turbulent. For laminar flow, as an estimate of the mean shear, the average shear was GÅ
1 R2 0 R1
*
R1
g(r)dr,
[4]
R2
where g(r) is g(r) Å
coidas
2vR 21 R 22 (R 22 0 R 21 )r 2
[5]
470
SERRA, COLOMER, AND CASAMITJANA
s 01 . In the near shores of the Great Lakes, for example, the shear range induced from waves was found to vary between 100 and 400 s 01 (2, 3). Therefore the results obtained from our experiments can be extrapolated to environmental situations.
TABLE 1 Shear Stress at Different Values of the Angular Velocity of the Inner Cylinder a
a
v (rpm)
G (s01)
40 56 75 105 125 165 211
25 32 50 70 90 135 195
RESULTS AND DISCUSSION
Calculated from [6]–[8].
and r is the radial distance. Integrating expression [4] we obtain GÅ
2vR1 R2 , R 22 0 R 21
[6]
where v is the angular velocity of the inner cylinder. In addition, turbulent shear has been estimated from
r GÅ
e , n
[7]
where e is the rate of energy dissipation per unit mass that can be scaled, using a characteristic velocity fluctuation u * and a characteristic length l, as e Ç u * 3 /l. It was assumed that l scales with the width of the gap, l Ç R2 0 R1 , and u * scales with the mean velocity uV (15), u * Ç uV , which is uV Å
2vR 21 (R 31 / 2R 32 0 3R1 R 22 ). 3(R 22 0 R 21 ) 2
[8]
It is important to point out that expressions [7] and [8] give an approximate estimation of the shear values. The exact values of the energy dissipated into the system, e, are somewhat difficult to determine. If continuity between laminar and turbulent shear is assumed, a proportionality constant, a, can be calculated. The value obtained was 0.03, which is much smaller than other values found in natural systems with isotropic turbulence. For example, Lazier and Mann (16) found a Å 0.37 and Bowen et al. (17) a Å 0.50, both in the open ocean. Turbulent shear was then found to depend on v 3 / 2 (see Eqs. [7] and [8]), whereas for the laminar case the dependence was linear (see Eq. [5]). The calculated shear stress that corresponds to the critical velocity found from [3] is 58 s 01 . In Table 1 the shear stresses calculated from Eqs. [6] and [7] are presented. It is clearly seen that in our study the calculated values ranged from 25 to 195
AID
JCIS 4710
/
6g1e$$$182
Figure 3 shows the evolution of the nondimensional diameter d/d 0 with nondimensional time t* for different shear rates, when the initial volume fraction is f0 Å 2.5 1 10 05 . d 0 is the diameter of the initial distribution of particles, which is 2 mm for all the cases represented in Fig. 3. As the characteristic diameter, d, we use the median of the size distribution (with a confidence of 95%) with respect to aggregate volume. t* depends on the shear, G, and on the initial fraction volume, f0 , as t* Å tGf0 . This nondimensional time represents a normalized number of collisions taking place in the system (10). At the beginning of the experiments shown in Fig. 3, at small t*, d/d 0 scaled well with t*. Here, the main factor that explains the aggregation process is the shear within the cylinder gap, according to Smoluchoski’s approach (18). For all the cases studied in Fig. 3 the steady-state size is reached for t* à 7. But, taking into account that t* depends on the product Gt, this means that the steady-state size is reached sooner when G increases. At larger t*, the breakup is more pronounced until it balances the aggregation, and the steady state is then reached. In addition, it is found that the larger the shear is, the faster aggregation occurs. As a result, the system reaches steady state more rapidly and the final size of the nondimensional diameter of the aggregates is smaller. This behavior is supported by other works such as those of Oles (10), who worked with latex particles, Lick and Lick (2), Burban et
02-17-97 21:14:03
FIG. 3. Evolution of d/d 0 with the nondimensional time t* at different shears and at f0 Å 2.5 1 10 05 .
coidas
AGGREGATION AND BREAKUP OF PARTICLES IN A SHEAR FLOW
FIG. 4. Representation of the final diameter of the aggregate, normalized to the diameter of the primary particles, d/d 0 , for different volume fractions, f0 , at different shear stresses, G. Here d 0 Å 2 mm.
al. (3), and Tsai et al. (1), who worked with sediment particles. Figure 4 depicts the relationship between d/d 0 and f0 at different shears. Depending on both shear stress and particle concentration, three different regimes were found. In the first regime, when the volume fraction is less than Ç6 1 10 05 , the final nondimensional diameter of the aggregate is independent of the volume fraction and depends only on shear. This regime was also found by Oles (10) with latex particles in a Couette flow system with a cylinder rotating between two static cylinders; however, when the concentration is larger, this result no longer holds. When the concentration is larger than Ç6 1 10 05 , two new regimes are identified depending on both the volume fraction and the shear. When the shear stress is less than 50 s 01 , if the volume fraction increases, the aggregation of particles is enhanced and the final diameter of the aggregate increases. On the other hand, if the shear is larger than 50 s 01 , an increase in the volume fraction enhances the breakup processes and the particle diameter shows a slight tendency to decrease. This is in accordance with the work of Burban et al. (3) and Lick and Lick (2), using sediment particles; however, these authors did not study the aggregation/ breakup process at the range of low shear rates as in our case. In addition, in the work described by Oles (10), the volume fraction was too small to show the behavior described in Fig. 4. As mentioned under Experimental Setup, it is important to point out that the transition between the two regimes coincides with the transition from laminar to turbulent flow, which, according to Eq. [3], was established at a shear of 58 s 01 . Therefore, a clear distinction in the
AID
JCIS 4710
/
6g1e$$$183
02-17-97 21:14:03
471
balance between aggregation and breakup is established, depending on whether the flow is laminar or turbulent. Experiments carried out with 5-mm particles (Fig. 5) show the same behavior as experiments carried out with 2mm particles (Fig. 4); however, in natural systems there is no suspension that contains only particles of one size. Therefore, to account for more realistic situations, experiments with polydisperse suspensions were carried out. The results of these experiments are presented in Fig. 6. The initial distribution of particles in the experiments, the results of which are presented in Fig. 6a, was a 75% 2-mm and 25% 5-mm particles; that of Fig. 6b was 50% 2-mm and 50% 5mm particles; and that of Fig. 6c was 25% 2-mm and 75% 5-mm particles. The mean initial diameters for these distributions were estimated to be 2.75, 3.50, and 4.25 mm, respectively. Results presented in Fig. 6 show the same three regimes found for the experiments carried out with monodisperse particles (Figs. 4 and 5); however, it is important to point out that the light tendency of d/d 0 to decrease in the turbulent regime found in Figs. 4 and 5 is not always observed. In addition, for a given G and f0 , we found similar values for the final diameter d of the aggregates in both monodisperse and polydisperse suspensions, in the turbulent regime. For example at G Å 70 s 01 and f0 Å 10.0 1 10 05 the value of the diameter d is around 20 mm for d 0 Å 5 mm (Fig. 5). For the same values of G and f0 , d is Ç21 mm for d 0 Å 3.5 mm (Fig. 6b). Taking into account that d/d 0 is related to the number of particles in the aggregate, if d is the same and d 0 is changing, this means that the final number of particles in the aggregate is different. This can be observed in the turbulent regimes in Figs. 4, 5, and 6. Therefore, the
FIG. 5. Representation of the final diameter of the aggregate, normalized to the diameter of the primary particles, d/d 0 , for different volume fractions, f0 , at different values of the shear stress, G. Here d 0 Å 5 mm.
coidas
472
SERRA, COLOMER, AND CASAMITJANA
TABLE 2 Calculated Kolmogorov Length Scale (Eq. [9]) and the Median of the Distribution Measured at Different Values of the Shear Stress G (s01)
l (mm)
d (mm)
70 90 135 195
21 18 15 12
8-32 8-28 8-20 8-12
final size of the aggregate, d, seems to be dependent on flow and independent of d 0 . On the contrary, in the laminar flow regime, the final diameter of the aggregates shows a slight tendency to decrease as d 0 increases. The latter observation can be explained by the high compaction of the aggregates formed by the smallest particles. The higher the compaction of the aggregates, the more resistant they are at the same value of the shear rate, as proposed by Burban et al. (3). To assess whether the final size of the aggregate depends on flow or on diameter of the primary particles, the size of the smallest turbulent eddies within the fluid was estimated from the Kolmogorov length scale, l, lÅ
S D n3 e
1/4
,
[9]
where e Ç u * 3 /l, u * being calculated from Eq. [8]. The maximum calculated l was 21 mm. Therefore, d was found to be of the same order of magnitude as l; the experimental value of d obtained from Figs. 4, 5, and 6 varied between 8 and 32 mm (Table 2). Therefore, it has been proved that the main process that determines the final diameter of an aggregate is turbulent shear stress, as also found by Jankowski et al. (19). In Fig. 7 d/d 0 is plotted as a function of the shear G for different concentrations [(a) 1.5 1 10 05 , (b) 2.5 1 10 05 , (c) 5.0 1 10 05 , (d) 7.5 1 10 05 , (e) 10.0 1 10 05 ) of monodisperse particles. It is important to point out that the measurements did not cover the range of G between 0 and 25 s 01 . Here, d/d 0 should increase from an initial value, at G Å 0, where Brownian motion cannot be neglected. Thereafter, at larger values of G, there is a power decrease of d/d 0 as plotted in Fig. 7, as also proposed by van Leusen (20). In Oles (10), a linear dependence was found between d/d 0 and G; however, a power relationship seems to be more realistic bearing in mind that the limit value of d/d 0 at very
FIG. 6. Representation of the final diameter of the aggregate, normalized to the diameter of the primary particles, d/d 0 , for different volume
AID
JCIS 4710
/
6g1e$$$183
02-17-97 21:14:03
fractions, f0 , at different values of the shear stress, G, for polydisperse particle suspensions (a) d 0 Å 2.75 mm, (b) d 0 Å 3.5 mm, and (c) d 0 Å 4.25 mm.
coidas
AGGREGATION AND BREAKUP OF PARTICLES IN A SHEAR FLOW
473
FIG. 7. Representation of the final diameter of the aggregate, normalized to the diameter of the primary particles, d/d 0 , for different values of the shear G, at different volume fractions: (a) f0 Å 1.5 1 10 05 , (b) f0 Å 2.5 1 10 05 , (c) f0 Å 5 1 10 05 , (d) f0 Å 7.5 1 10 05 , and (e) f0 Å 10 1 10 05 .
high G must be, obviously, 1. In addition, Parker et al. (21) found that, due to the surface erosion of the aggregate, d depends on G 01 . We corroborated this result in the range 50–195 s 01 , corresponding to turbulent flow. REFERENCES 1. Tsai, C., Iacobellis, S., and Lick, W., J. Great Lakes Res. 13, 135 (1987). 2. Lick, W., and Lick, J., J. Great Lakes Res. 14, 514 (1988). 3. Burban, P., Lick, W., and Lick, J., J. Geophys. Res. 94, 8323 (1989). 4. Shimeta, J., Jumars, P. A., and Lessard, E. J., Limnol. Oceanogr. 40, 845 (1995). 5. Casamitjana, X., and Schladow, G., J. Environ. Eng. 119, 443 (1993). 6. Weisner, M. G., Wang, Y., and Zheng, L., Geology 23, 885 (1995). 7. Droppo, I. G., and Ongley, E. D., Water Res. 26, 65 (1992). 8. Axford, S. D. T., J. Chem. Soc. Faraday Trans. 92, 1007 (1996). 9. Logan, B. E., and Wilkinson, D. B., Environ. Sci. Technol. 25, 2031 (1991).
AID
JCIS 4710
/
6g1e$$$183
02-17-97 21:14:03
10. Oles, V., J. Colloid Interface Sci. 154, 351 (1991). 11. Luettich, R. A., Jr., Harleman, D. R. F., and Somlyo´dy, L., Limnol. Oceanogr. 35, 1050 (1990). 12. Pandya, J. D., and Spielman, L. A., Chem. Eng. Sci. 38, 1983 (1983). 13. Chandrasekhar, S., ‘‘Hydrodynamic and Hydromagnetic Stability.’’ Clarendon, Oxford, 1961. 14. Van Duuren, F. A., J. Sanit. Eng. Div. ASCE 94, 671 (1968). 15. Imberger, J., in ‘‘Limnology Now: A Paradigm of Planetary Problems, (R. Margalef, Ed.), pp. 99–193. Elsevier, Amsterdam, 1994. 16. Lazier, J. R. N., and Mann, K. H., Deep-Sea Res. 36, 1721 (1989). 17. Bowen, J. D., Stolzenbach, K. D., and Chisholm, S. W., Limnol. Oceanogr. 38, 36 (1993). 18. Friedlander, S. K., ‘‘Smoke, Dust and Haze.’’ Wiley–Interscience, New York, 1977. 19. Jankowski, J. A., Malcherek, A., and Zielke, W., J. Geophys. Res. 101, 3545 (1996). 20. Van Leusen, W., ‘‘Aggregation of Particles, Settling Velocity of Mud Flocs: A Review.’’ Springer-Verlag, Berlin/Heidelberg, 1988. 21. Parker, D. S., Kaufman, W. J., and Jenkins, D., ‘‘Characteristics of Biological Flocs in Turbulent Regimes.’’ Thesis, Sanitary Engineering Research Laboratory, Report No. 70-5, College of Engineering and School of Public Health, Berkeley, 1970.
coidas