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where the Reynolds number Re is given by

ml where is the viscosity of the fluid and Rc,a is the radius of collision of the aggregate.

The use of Ap,a is rather difficult in practice because it is often hard to measure the actual projected area of aggregates due to (i) the geometry of typical settling experiments and (ii) limited image resolution typical in such experiments. Additionally, all of the uncertainties associated with aggregate projected area that were discussed earlier in the context of the volume obscuration method also apply here. The usual practice is to take Aa as the projected area of the sphere (or sometimes the ellipsoid) that encloses the aggregate being observed. This will overestimate the projected area because 'holes' in the projection of the aggregate will be included as part of the projected area. What this means in practice is that the correction factor n is as much a correction for projected area as for the drag coefficient. Adopting this practice, the projected area Ap a of the aggregate will be

Making use of an equation equivalent to Equation (3.2) for the mass of the aggregate:

we can substitute into Equation (3.30) to derive an expression that will tell us about the dependence of settling velocity on size and fractal dimension:

Rewriting with vt as the subject:

9ncML V Ro which implies that a log-log plot of settling velocity versus aggregate collision radius for a set of aggregates having a fractal dimension will yield a straight line with slope Dm - 1.

Now, let us critically examine the assumptions which underlie this analysis. If we have managed to accurately separate all of the size dependence of the system into the factor (Rc a/R0)Dm-1, then Equation (3.37) will be correct. The biggest problem with this is the parameter The first problem, of using settling velocity to determine an accurate value for the fractal dimension, is that of accurately knowing the Rc,a dependence of The second problem, of modelling settling velocity accurately requires a quantitative model for Clearly, the first of these two requirements is the less restrictive.

As explained above, the parameter is known as a permeability correction, but rightly accounts for two different effects: the extent to which the projected area of an aggregate is different to the projected area of a sphere having the same radius of collision, and the extent to which the drag coefficient of an aggregate is different to the drag coefficient of a sphere having the same projected area as the aggregate. Taking these two effects together, represents the extent to which the settling velocity of an aggregate is different to that of an impermeable sphere having the same drag force and radius of collision.

Looking at the meaning of in another way, if the hydrodynamic radius of an aggregate is the radius of a sphere that has the same settling velocity and drag force as the aggregate in question, then can be shown to relate the radius of collision to the hydrodynamic radius:

Is in fact constant with aggregate size as implied by Equation (3.38)? If it is, then this implies that the hydrodynamic radius is linearly related to the radius of collision, a conclusion that has considerable support in the literature [10, 66]. This would also mean that the dimensionless permeability of the aggregates is also constant with respect to aggregate size, a proposition that has had support recently [67, 68].

What are the issues relating to the shape of the aggregate, choice of size measurement and to preferential orientations in settling? Suppose that instead of the radius of collision of the aggregate we had preferred to use as a size characterization the radius of a sphere having the same projected area as the aggregate, RA,a. The drag force of the system would then be expressed as

2 PLVt2RA,a

The factor nR,a, which represents the projected area of the aggregate, is now strictly correct and the drag correction now represents a combination of drag coefficient correction and Reynolds number correction because we have used a different radius in the Reynolds number calculation. Since the different measures of aggregate radius are all linearly related to each other if the aggregates are fractal (it follows from self-similarity), the same conclusions regarding the scaling of settling velocity with aggregate size and the constancy of the factor are reached. Indeed, exactly the same conclusions are reached no matter which kind of radius or diameter we choose to use as a basis for the analysis.

The equations presented so far rely on the proposition that the projected area of the aggregate is proportional to the square of the radius we are choosing to characterize the aggregate size. As discussed in the previous section on the volume obscuration method, this is only the case for very large aggregates, and is never the case for aggregates with fractal dimension less than two [23]. Johnson et al. [23] point out that Equation (3.37) will have a different power-law dependence between settling velocity and aggregate size depending on whether the fractal dimension is less than or greater than two and additionally make a correction to the drag coefficient for somewhat higher Reynolds numbers | 0.1 < Re < 10 | . The characteristic length for the Reynolds number is taken as being the diameter of collision of the aggregate, in the absence of any better information.

As explained by Meakin et al. [11], the reality with respect to projected area dependence on aggregate size is rather more complex than two simple cases depending whether the fractal dimension is larger or smaller than two. Small aggregates tend to present more projected area than would be expected, because of the finite size of the primary particles. This would lead to lower settling velocities for small aggregates than might be expected from Equation (3.37), which without correction would lead to an overprediction of Dm. As with the volume obscuration analysis, the way to account for this may well be to have the projected area well characterized in terms of G, B, y and S from computer simulations for aggregates with different size and structure.

3.4.1.2 Experimental approaches

These kinds of experiments are without exception carried out in a column of fluid, usually of the same composition as that from which the aggregates were sampled. The aggregates are introduced into the top of the column and one or more microscopes are used to measure the settling velocity. Farrow and Warren describe a floc density analyser [69] which may be used to determine the fractal dimension. Nobbs et al. [70] review many of the practical aspects involved in performing this type of experimental investigation.

There are a number of difficulties with the settling approach in practice. For large flocs, the settling velocity and induced drag may be large enough to restructure the aggregates under investigation because of the extremely weak mechanical strength of the aggregates [24]. Conversely, the settling velocities usually have such a small absolute value that stray convection currents can render the measurement meaningless. For this reason, control of fluid movement in the column is absolutely critical. Settling columns are usually temperature controlled by recirculated air [70] or water [71]. Particular care needs to be taken with illumination - continuous, high-intensity light sources can easily add heat to the system and induce currents. Strobe-type illumination avoids this problem and additionally can provide very clear images, as the high but brief intensity allows a narrow aperture (good depth of field) and short exposure time (no motion blur). Introducing the flocs to the top of the settling column can also induce fluid motion in the column, both through mechanical disturbance and by introducing too much sample, which effectively creates a density gradient in the column. Using an inverted pipette to introduce aggregates one at a time without disturbing the liquid in the column is a difficult skill to master, but is the best way to avoid this kind of problem. An alternate approach to the problem of induced currents might be to introduce small 'tracer particles', density matched to the suspending fluid. This allows the background fluid motion to be visualized and corrected for.

3.4.2 Chord Length Measurement

Focused beam reflectance measurement (FBRM) is a relatively new on-line particle measurement technique that measures chord lengths using a scanning laser. The instrument is designed as a robust probe, tipped with a sapphire window through which a circularly scanning laser is focused. Particles that approach the window reflect the laser back through the instrument's optics and the duration of the reflected signal is converted to a chord length using the known scanning speed of the laser.

Although the technique has been used to monitor flocculation [72, 73], it appears that it has not yet been used to determine the fractal dimension of these systems. This should be a relatively straightforward matter: if one assumes that the instrument samples the aggregates in an unbiased manner, then the same kind of analysis as was applied for the volume obscuration techniques can be used. If the suspension flowing past the instrument can be assumed to have a constant solids loading and the chord length distribution can be assumed to be related to the aggregate size distribution in a direct way, then the frequency of chord acquisition and the length of the chords will effectively tell us the concentration of aggregates of different sizes. This gives us the apparent solid volume fraction, which, combined with the known true solids loading, tells us the aggregate density and hence the fractal dimension.

Kovalsky and Bushell speculate that it might even be possible to measure the fractal dimension directly from the texture of the reflected laser [74].

3.4.3 Particle Counting Techniques

A very direct way of measuring the fractal dimension is to have direct measurement of the size of individual aggregates and of the amount of solid that comprises them, a technique sometimes referred to as the particle concentration technique

[75]. Stoll et al. [76] and Jackson et al. [77] report this kind of analysis using aperture impedance particle sizing. Aperture impedance particle sizing works by passing a suspension of particles in electrolyte through a small orifice with a voltage applied across it. Solid particles passing through the orifice reduce the amount of electrolyte available for current flow and hence change the impedance of the orifice. The change of impedance is more or less directly related to the solid volume of the particle passing through the orifice with little dependence on particle morphology. Jackson et al. [77] used two aperture impedance methods to determine the solid volume distribution of aggregating algal particles and additionally used image analysis to measure the size distribution of the same particles. A mass fractal dimension was then assumed that led to the best match between the two methods of size analysis, returning a value of 2.3. Sterling et al. [78] used a similar approach to examine the structure of crude oil and clay aggregates, except that they used forward light scattering instead of image analysis to determine the size distribution.

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