Conclusions And Recommendations

The focus of this study is on developing relationships between aggregate size and shape, and transport and growth behavior. Experiments were conducted to demonstrate the dependence of aggregate properties on fractal dimensions, consistent with the supposition that improvements could be made in describing aggregate characteristics and behavior in aqueous solutions using a fractal-based analysis, rather than relying on the common approach of assuming the aggregates are impervious spheres. An image-based analysis was used to develop basic geometrical properties of aggregates in suspension, from which properties such as fractal dimension could be derived. Basic properties such as density and porosity were shown to be functions of size, which also was correlated with fractal dimension. Along with relatively high porosity, these results show that the solid sphere assumption has obvious drawbacks, although it clearly has facilitated progress in aggregation theory.

To develop a full model of particle transport and behavior in an aqueous system, an extended version of Equation (5.10) would serve as a reasonable starting point, with additional terms added according to the needs of a particular application. Those terms might include breakup, settling, and internal reactions, or sources and sinks of material. The present experiments were designed to shed light on the particle aggregation and settling mechanisms, and although progress has been made, the results here are far from complete. For example, there is still uncertainty in the volume calculations, and methods are needed for nonintrusive measurement of aggregate volume. Much work remains to be done to complete a full model description, particularly with respect to settling.

From a modeling point of view, an approach needs to be developed that can incorporate changes in fractal dimensions directly into the aggregation modeling framework (Equation 5.10). Once fractal dimensions are known, they can be used to calculate collision frequency functions (Table 5.1) which are needed in the aggregation calculations. Experimental results were shown to be consistent with the conceptual model, and indicated that the basic Smoluchowski approach (Equation 5.10) could still be used, but that collision frequencies should be updated as functions of time, as fractal dimensions change. For the present tests fractal dimensions were supplied to the model using a simple empirical fit to the observed data, but ideally these values would be developed in a separate submodel. At this stage, the only approach available to accomplish this is with a DLA or RLA-type model, which would account for each individual aggregate (or at least some representative number of aggregates) and keep track of changes in fractal dimensions. While theoretically appealing, this approach is at best cumbersome for general modeling purposes, and is computationally unrealistic, even with today's computer capabilities. Nonetheless, as a research tool, this type of model may be useful and is worth a more detailed look.

Values for j are considerably different when fractal dimensions are taken into account, and an unexpected result of incorporating better resolved j in the calculations has been the realization of the need for further evaluation of a, in particular as a time-varying parameter that depends not only on the chemistry of the solution and the surface of the aggregate, but also on aggregate structure. In other words, noting that a is generally used as a fitting parameter, and that aggregation is controlled by the product of a and j, then changes in j suggest that corresponding changes in a will be required. In addition, since j changes with time, it may be expected that a should change with time. This makes intuitive sense, as the likelihood of two colliding aggregates sticking together should be greater when they have a rougher surface, compared with a smooth sphere. So far, very little research has been directed at a finer description of collision efficiency, and this is an important area for further improvement of aggregation models.

The conceptual model developed here has been useful in explaining the different stages of aggregation for the present experiments and, more importantly, the relationship between fractal dimensions and different aggregation processes. Although designed with the specific conditions of the present experiments in mind, the model provides a framework within which one may conceptualize the correspondence between fractal dimensions and physical changes occurring among the aggregates. The model is easily extended to consider situations where an initially monodisperse suspension is not required, as the basic ideas are also applicable when a change in mixing or chemistry is induced in a system where at least the initial size distribution is known.

A property of particular interest is the settling rate, and results shown in Figure 5.13 indicate that Stokes' law is inadequate for estimating settling for fractal aggregates. Particular problems with the application of Stokes' law include determination of appropriate aggregate size and density, as well as the assumption of laminar flow around the aggregate, and the associated formulation of the drag coefficient. This last point also applies for the fractal-based expression (Equation (5.9)), since the correct formulation for drag coefficient has not been demonstrated, only assumed. The results shown in Figure 5.13 suggest the dependence of ws on l is different than in Equation (5.9), and this difference may be due to an inadequate description of the drag coefficient. The drag coefficient assumed in Equation (5.9) comes from the expression for a sphere in laminar flow, but it would be more consistent to assume CD as a function of fractal dimensions, which characterize general aggregate shape and would introduce the impact of different shapes on settling characteristics. Along with collision frequency and efficiency, this is another specific area where fractal dimensions can improve our ability to model particle aggregation and behavior in suspensions.

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