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° Ems — mud, column a VIS, Ems '89 □ VIS, North Sea x VIS, Ems '90

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FIGURE 12.5 Relation between settling velocity and floc size (Equation (12.5)) for self-similar flocs.

in the ocean (not shown here). The overall trend of the data points seems slightly steeper than the fit with nf = 2, because no data are available in the lower right corner of the graph, that is, small ws and large Df. However, when the individual data sets are studied, the slopes agree better with nf = 2.

Figure 12.5 shows that for particles with a diameter up to a few 100 ^.m, the data can be properly represented with the fit ws a Df, that is, with an average fractal dimension, nf = 2. At floc diameters beyond Df = 1 mm, Equation (12.5) predicts a rapid deviation from a simple power law behavior because of the increasing role of the particle Reynolds number. We can expect that the validity of (12.5) becomes limited at floc sizes beyond a few millimetres.

Winterwerp9 has developed a Eulerian flocculation model, which is identical to a Lagrangian model (Winterwerp8) in case of no advection and constant sediment concentration. This flocculation model is used in Sections 12.2 and 12.3 of this chapter to analyze the effects of the flocculation time, and reads:

dt where c is the suspended sediment concentration by mass, G is a measure for the shear rate at the smallest turbulence length scale (G = ^Je/v), kA (m2/kg) is a dimensional aggregation parameter and kB(sec1/2/m2) is a floc break-up parameter. Here we have assumed that the fractal dimension nf attains its mean value of nf = 2.

For small values of Df, the first term on the right-hand side of (12.7), that is, the aggregation term dominates, whereas for large Df, the second term, that is, the break-up term dominates. From this flocculation equation, an equilibrium floc size Df,e is obtained for dDf /dt = 0. A mathematically trivial but physically unsound and unstable solution is Df,e = 0. In this case, small particles always grow. The other equilibrium solution for the Lagrangian model reads:

kA C

knVo

which is a stable equilibrium, as for Df < Df,e, the flocs grow and for Df > Df,e, the flocs break up. For the chosen fractal dimension, nf = 2, the differential equation (12.7) can be easily solved analytically, if the sediment concentration by mass c is constant. First, a time scale parameter T' is defined:

The solution of Equation (12.7) then has the implicit form:

in which Do is the floc size at t = 0. This solution describes the aggregation/floc-break-up process for flocs initially either smaller or larger than the flocs at equilibrium size.

From (12.10), a time constant for flocculation Tf can be defined in the case that the initial floc size D0 is much smaller or much larger than the equilibrium value, yielding the maximum time scales of the aggregation and floc break-up processes:

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