Introduction

Particle aggregation is a complex process affected by various physical, chemical, and hydrodynamic conditions. It is of interest for understanding, modeling, and design in natural and engineered water and wastewater treatment systems. In natural aquatic

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systems such as lakes, rivers, and estuaries, particle aggregation is important because it controls the fate of both the particles themselves, as well as potentially hazardous substances adsorbed to the particles.1-5 In water and wastewater treatment, floccu-lation is used to produce larger aggregates that can more effectively be removed from the treatment stream by sedimentation and filtration.6-8 The growth of aggregates depends on the relative size of the colliding particles or clusters of particles, their number density, surface charge and roughness, local shear forces, and the suspending electrolyte. Specific factors that affect aggregation include coagulant dose, mixing intensity, particle concentration, temperature, solution pH, and organisms in the suspension.3,7,9 These factors contribute not only to changes in particle size and shape, but also affect flow around and possibly through the aggregate, with corresponding effects on transport and settling rates.

Historically, efforts to understand individual processes of aggregation have been based on relatively simple systems, assuming impervious spherical particles, with various mechanisms of particle interaction explained using Euclidean geometry. More recently, it has been recognized that aggregates are porous and irregularly shaped, and that these characteristics suggest different behavior than for impervious spheres. Fractal concepts have been adapted from general theoretical considerations originally discussed by Mandelbrot10 and later by Meakin.11-14 For specific applications in environmental engineering, much of the fundamental fractal theory for particle aggregation has been developed by Logan and his coworkers.15-18 Fractal theories have been used mainly as a quantifying tool for describing the structure of the aggregate, but several studies have also looked at the application of fractal characteristics as a means of analyzing the kinetics of aggregation.11,18

In addition to the assumption of impervious spherical particles, earlier studies also assumed that volume is conserved when two particles join (known as the coalesced sphere assumption). However, these assumptions are exact only for liquid droplets. When two aggregates collide, the resulting (larger) aggregate often has higher permeability than the parent aggregates, and the volume of the new aggregate is generally larger than the sum of the two original volumes. The overall goal of this study is to conceptualize and develop an aggregation model using fractal concepts, based on measurements from coagulation-flocculation experiments under a variety of environmental/process conditions, and to determine the potential impact of aggregate geometry on particle dynamics in natural and process oriented environments. The study is motivated by the idea that improvements in particle and aggregation modeling may be achieved by incorporating more realistic aggregate geometry; and fractal concepts are used to characterize the impacts of aggregate shape in relation to traditional models that have assumed spherical aggregates. In particular, incorporating realistic aggregate geometry is expected to provide improvements in our ability to describe such features as aggregate growth rates under different hydrodynamic and chemical conditions. Relationships between aggregate size and geometry, as characterized by fractal dimension, are sought, which can provide additional information for understanding and modeling particle behavior. To avoid potential problems associated with sample collection and handling, a nonintrusive image-based technique is used for the measurements.

This technique uses digital image analysis to obtain information for aggregates in suspension that can be used in model development. It is expected that results of this study will lead to a better understanding of particle behavior in aqueous suspensions, and will advance our capability to model aggregate interaction and transport.

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