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FIGURE 5.10 Aggregate properties as a function of size for lake water suspensions and measurements by Amos and Droppo32 for (a) density and (b) porosity. Bars represent the observed range of data.

FIGURE 5.11 Total collision frequency (fi) as a function of particle size (l) at different times of aggregation for latex suspensions mixed at G = 20 sec-1.

FIGURE 5.11 Total collision frequency (fi) as a function of particle size (l) at different times of aggregation for latex suspensions mixed at G = 20 sec-1.

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FIGURE 5.12 Comparison of total collision frequency function calculated according to fractal (fi(D)) and Euclidean (fi(3)) formulas at different times of aggregation.

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FIGURE 5.12 Comparison of total collision frequency function calculated according to fractal (fi(D)) and Euclidean (fi(3)) formulas at different times of aggregation.

increase is seen in the values of fi for the fractal dimension corresponding to 30 min, relative to the values for 10 min. This is due to the effect of larger collision radii for aggregates that sweep out a larger volume and provide greater opportunity for collisions with other particles.

It also should be noted that the fi values shown in Figure 5.11 are significantly greater than the values that would be calculated based on Euclidean dimensions. For example, setting D2 = 2 and D3 = 3 in the formulas in Table 5.1 gives fi values consistent with traditional models that assume spherical particles. A comparison of fi calculated using the Euclidean values (fi (3)) and fi from the fractal formulas (fi(D)) is shown in Figure 5.12, where it can be seen that order of magnitude and greater increases are obtained using the fractal calculations. Chakraborti25 provides a more detailed comparison of different calculation results for a range of particle sizes.

### 5.4.1.5 Settling Velocity

Using results from Experiment Set 1, the exponent in Equation (5.9) was found to be 1.96 for initial conditions, 1.73 at the charge neutralization dose, and 1.47 for sweep floc, for the lake water samples. Corresponding values for the montmorillonite suspensions were 1.82, 1.70, and 1.62, respectively. Although the exponent in these results becomes smaller for higher alum dose, l also increases with each successive coagulation stage, suggesting more rapid collision rates.35 Thus, larger aggregate size appears to offset the effect of lower fractal dimension and helps to explain the common observation from water treatment practice that sweep coagulation results in better (lower turbidity water) settling than charge neutralization.

However, in order to fully evaluate settling, the aggregate density also must be taken into account. With increasing size (l), results from Figure 5.10(a) show that density decreases, which should reduce ws (Equation (5.9)). Calculations based on the present samples show that ws increases with increasing l, but at a slower rate than would be expected using a traditional (Stokes-based) model (Figure 5.13). Data in this figure were obtained by taking images of samples from the Buffalo River, which were allowed to passively settle in the mixing jar. The images were double exposed so that each aggregate was pictured twice. The difference in locations of each of the aggregate pairs was then divided by the known time interval between the two images to obtain settling rate. These rates were then plotted as a function of aggregate size. The best fit line drawn through the data has ws a lL3°, an exponent value clearly less than 2 and, in fact, even less than the exponent suggested in Equation (5.9) (discussed earlier). Thus, an additional factor must be affecting the settling, which may be due to changes in density or changes in the drag coefficient, as previously suggested. The net effect of these various factors is unknown in general, and further data are needed to determine the correct relationship for settling.

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