Ratio of particle radii, r\/rx


Ratio of particle concentrations, C1/C2


Algal specific growth rate


assuming that the particles are impermeable spheres whose presence does not affect water motion, and that chemical attraction or repulsion has negligible effect:

where i and j are the particle indices, r is the radius of the ith particle, Vi is its fall velocity, Di its diffusivity, and y the average fluid shear.37

There are adjustments to these equations that account for fluid flow around the larger particle for the shear24 and differential sedimentation38 terms in what I will call the curvilinear approximation, as well as higher order terms that include greater hydro-dynamic detail as well as attractive forces.39,40 For example, considering the flow field around a larger particle when considering the rate of collision with a smaller for differential sedimentation leads to24

%Br = 4n(Di + Dj )(ri + rj ) ßijsh = 1.3 Y(ri + rj )3

where rj > ri. Similarly, the shear kernel becomes p2

The coagulation equations describe the rate of change of each size fraction in terms of the processes which change particle concentration.

= ^ ¿2 Pi,j-iCj-iCi - a^PijCjQ + sources - sinks (13.9)

where sinks can include loss from settling out of a mixed layer and sources can include algal growth and division.

The equations are modified when considering continuous distributions described with a particle size spectrum:

/ n(m, t)n(m', t)P(m, m')dm' + sources - sinks (13.10)

The integro-differential equations that result from using the number spectra require approximations to solve. Approaches include solving analytically after assuming that n = ar-b (the Jungian spectrum) and solving numerically after separating the spectrum into particle size regions in which the shape as a function of size is constant but the total mass in the region varies (the sectional approach of Gelbard et al.41).

One implication of fractal scaling is that aggregates are porous, a property which affects the flow through and around an aggregate. Li and Logan42,43 have documented the effect of this porosity on particle capture. Their results have been used to modify the coagulation kernels.44

The simple fractal relationship presupposes that a system is initially monodisperse (all particles the same). Jackson45 proposed that a consequence of fractal scaling is that rDf is conserved in a two-particle collision, in the same way that mass is. This was used to develop two-dimensional particle spectra that describe particle concentrations as functions of particle mass and rDf.

An important factor in determining whether two colliding particles combine is the stickiness a. Considering the probability of a contact causing two particles to combine, a is usually empirically determined or used as a fitting parameter (see below).

Observations on algal cultures have shown that it can vary with species and with nutritional status for any species with observed values ranging from 10-4 to 0.2 (see ref. 46).

Other issues which can affect net coagulation rates are the effect of non-spherical shape,24'47 and the breakup, or disaggregation, of larger aggregates from fluid forces that exceed the particle strength.48,49

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