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water column with initial height Z0 above the bottom of the water column can be obtained from r Tr Ç Tr

in which ws is the settling velocity and a" = a'DpgA/v(^ 3 sec-1 for Dp = 4 ^m). The residence time Tr can be solved directly from this equation through integration of the simplified flocculation model (12.7) to yield

Zo ws,e

Df,e Dfe

Do Df

in which ws,e is the equilibrium settling velocity for equilibrium floc size (e.g. Equation (12.5) and (12.8)). The maximum settling velocity, as a function of a limited residence time, follows from (12.12) and (12.13) (t = Tr):

ws,o

The maximum settling velocity predicted with Equation (12.14) is plotted in Figure 12.7 for the kA and kB values given before, a suspended sediment concentration of 1 g/l, and Zo = 1,2, and 4 m, respectively. It is shown that for small values of G, the flocs cannot attain equilibrium conditions, given by Equation (12.8) because of the limited residence time. If the concentration would decrease by a factor 1o,

Shear rate parameter, G [sec 1]

FIGURE12.7 Effects of limited residence time on the relation between floc size and shear rate (Equation 12.14, after Winterwerp, Continental Shelf Res., 22, 1339-1360, 2002) compared with data by Van Leussen (PhD thesis, University of Utrecht, The Netherlands, 1994) measured in settling column.

Shear rate parameter, G [sec 1]

FIGURE12.7 Effects of limited residence time on the relation between floc size and shear rate (Equation 12.14, after Winterwerp, Continental Shelf Res., 22, 1339-1360, 2002) compared with data by Van Leussen (PhD thesis, University of Utrecht, The Netherlands, 1994) measured in settling column.

the shear rate at which equilibrium would be possible, would increase by the same factor. We have also plotted some data presented by Van Leussen,5 which appear to be properly modeled by Equation (12.14). As the experiments by Van Leussen were carried out at various sediment concentrations, the settling velocity in Figure 12.7 is divided by c to allow mutual comparison.

Note thatFigure 12.7 is qualitatively similar to the diagram by Dyer in Figure 12.2. We conclude that the left part of this graph is affected by a limited residence time, or similarly by too long flocculation times.

We can substantiate these observations further, and investigate when the classical diagram by Van Leussen7 of flocculation processes in the water column, sketched in Figure 12.3 is correct. It shows larger flocs higher in the water column, where turbulent shear is relatively small, and smaller flocs near the bed, where turbulent shear is high.

This diagram can of course only hold when the flocculation time is small when compared to the mixing and settling time of sediment. As the settling velocity of mud flocs is generally of the order of a few 0.1 to 1 mm/sec, or smaller, the vertical mixing time h2/ez (where h is the water depth and ez is the vertical eddy diffusivity) is generally much smaller than the settling time h/ws. Moreover, the time scale for floc break-up is almost always smaller than the time scale for aggregation, as follows from Equation (12.11). Hence, to compare flocculation time with residence time, we may compare aggregation time with settling time only. The relevant aggregation time is the time necessary to form larger flocs with size Du higher in the water column through aggregation of smaller flocs with size Dl originating from the lower part of the water column. The latter are more or less in equilibrium with the hydrodynamic conditions near the bed as G (and often c) is large. Also, for the present analysis, we assume that the vertical gradient in suspended sediment concentration is small.

The flocculation time Tf for Du > Dl is obtained from Equation (12.11):

where Gu is assumed to be represented by the mean value of the shear rate in the upper 25% of the water column. We assume further that Dl is the floc size in equilibrium with the shear rate Gl in the lower 25% of the water column. Gu and Gl are found from averaging G, using Nezu and Nakagawa's approximation 21 of the dissipation rate e = u*(1 - z)/khZ, where Z = z/h and u* is the shear velocity:

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