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anism in size range

Note: The base case is a numerical simulation using the sectional approach with all coagulation mechanisms and particle settling possible for all particles. The "no settling" cases result when there is no loss of particles by settling out of a layer.

Source: From Burd and Jackson, Environ. Sci. Technol., 36, 323, 2002.

anism in size range

Note: The base case is a numerical simulation using the sectional approach with all coagulation mechanisms and particle settling possible for all particles. The "no settling" cases result when there is no loss of particles by settling out of a layer.

Source: From Burd and Jackson, Environ. Sci. Technol., 36, 323, 2002.

could not be considered separately. They were able to reproduce the scaling results only when they omitted particle settling and imposed only one coagulation mechanism in a given size range. Thus, the simple analysis is not necessarily correct.

13.3.2 Rectilinear and Heterodisperse

Many of the simple relationships derived from coagulation theory implicitly assume that the systems are initially monodisperse. It is made when assuming that particle number is proportional to volume for all particles or, more basically, in the linearizations that are made to derive the simplified equations. The effect of the monodisperse assumption can be tested by assuming that there are initially two particle sizes and making similar simplifications.

13.3.2.1 Critical Concentration

The simplicity of the formulation and its lack of dependence on particle radius suggest that it could be used to predict a critical concentration for mixed assemblages of phytoplankton, where no one particle type dominates. An expanded version of Equation (13.13) for two particles is dCa 2

—— = IbCb - a^bbCj? - aftabCaCb dt b where the subscripts "a" and "b" are used to distinguish the two particles.

At steady state, dCa/dt = 0 and the first part of Equation (13.18) reduces to

= Ca,Cr 8f3 Cb = Ca,Cr 8Va Cb where X = ra/rb, Va = 4nr3a, and Vb = 4nr^. Expressing the concentrations in terms of volumes and performing similar manipulations for Cb yields:

(1 + k-1)3 CVb = CV,cr----CVa where CVa = VaCa and CVb = VbCb represent the volumetric concentrations of the two particles, and CV,cr is the critical concentration for homogeneous distributions if Ma = Mb (Equation 13.15). The problem is that if ra = rb, CVa and CVb cannot both be at steady state and have positive values: a larger particle is more likely to collide with a smaller particle than vice versa. Thus, prediction for the simple monodisperse system is not appropriate for the polydisperse system. It does, however, provide a simple estimate.

13.3.2.2 Estimating Stickiness

The problem of using relationships derived for monodisperse systems to describe the fate of heterodisperse systems extends to the method used to estimate stickiness.

A modified version of Equation (13.16) to describe the effect of collisions between particles with two different sizes is then:

= _2 a ( 3 Y 8 fa Ca Ca + 3 Y 8fa3Ca CA _ a 3 Y (fa + fb)3Ca Cb

= _ —(Cv,a Ca + Cv,bCb) _ — (4 n f3) (1 + k)3 Ca Cb n n \3 /

4ay ( (1 + k)3 n . Cv,aCa + Cv,bCb +----CaCv,b | (13.20)

where X = ra/rb, CV,a = 4/3nr^Ca, and CV,b = 4/3nrbCb. We would like to put this into the form of Equation (13.16) in order to compare the heterodisperse and monodisperse cases. Noting that the total volumetric concentration is related to the volumetric concentrations of the components by

Substituting the results from Equation (13.21) into Equation (13.20) yields dCT 4ay ( (1 + X)3

Cv,tCt — Cv,bCa — Cv ,a Cb +----Ca Cv,b dt n V ' ' 4

4ya C C / - rbCbCa + ^3CaCb - ((1 + X)3/4)Car3Cb ' n V,T T\ r3C2 + rbCbCa + r|CaCb + ^C2

n A (ra/rb )3(Ca/Cb) + 1 + (ra/rb)3 + (Cb/Ca ) J— Cv,tCt (13.22)

where a' = a(1 — 4(1 — X — X2 + X3)/(X3n + X3 + 1 + n—1)), n = Ca/Cb, and X = ra/rb. a' is the stickiness coefficient that would be estimated by using the procedure for the monodisperse system. The value of a' depends on the relative sizes and concentrations of the two particles (Figure 13.3).

13.3.3 Curvilinear

Models of coagulation in planktonic systems have expanded in their use of both ecological and coagulation descriptions. Recent models use more sophisticated coagulation kernels, calculation schemes, fractal scaling on mass and ecological dynamics (Table 13.4). The kernels in Equations (13.7) and (13.8) are used in the following calculations.

13.3.3.1 Simple Algal Growth

The effect of changing the coagulation kernel on model results can be seen by comparing the results from a simple model of exponential algal growth run with

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