Collision Frequency Functions for Fractal Aggregates (from ref. [15,18])


Brownian motion Fluid shear

Collision Frequency Function

Differential sedimentation (1/3)-(1/D3)(2+(bTI-D2)/(2-bTI)) I (1/D3)((D3+bTI-D2)/(2-bTI)) *v0 \vi

Note: The parameters in these functions are: ¡i = collision frequency function (cm3 sec 1)

= Boltzmann's constant (1.38

10-16 gcm2sec-2K-1); T = absolute temperature (293 K); G = velocity gradient (sec-1); fiw = dynamic viscosity of water (0.01002 gcm-1 sec1); pw = density of water (0.99821 g cm-3); P0 = density of primary particle; g = gravitational constant (981 cm sec-2); £2 = aggregate area shape factor; a and bD = fractal functions depending on Reynolds number (a = 24 and bD = 1 for Re < 0.10); v = kinematic viscosity (0.01004 cm2 sec-1), and vj, vj = solid volume of i and j size class particles, and vq = primary particle volume.

place during aggregation. Microscale models are more helpful in this regard, and also helpful for present purposes is a more general conceptual description of aggregation, in terms of geometric properties (fractal dimensions) of the interacting aggregates. Several studies have already shown, for example, the effect of fractal dimension on collision frequencies,15,16,18 and similar results were found in the present study, as described later in this section.

The present conceptual model is based on general ideas presented in the literature describing aggregation processes, and is applied to a specific experiment in which an initially monodisperse suspension of primary particles, either spherical or at least with known fractal dimension, is mixed with or without coagulant addition. The model focuses on the initial stages of aggregation, before particles grow large enough that further growth may be limited by breakup. It is assumed that mixing speed and chemical conditions are constant during any given experiment.

Referring to Figure 5.1, the initial state of the suspension is characterized by initial values for average size and fractal dimension of the primary particles. Here, fractal dimension refers to either D2 or D3, and size refers to the longest dimension for an aggregate. As particles collide and stick, average size increases and fractal dimension decreases, according to processes discussed earlier in this section. For example, following a successful collision (i.e., one that results in the two particles sticking), the resulting volume is larger than the sum of the volumes of the two colliding particles, as additional pore space is incorporated in the aggregate.

As the process continues, both growth and breakup occur, but growth is faster. Eventually, a state, represented by point A in Figure 5.1, is reached in which there is a temporary balance between growth and breakup. During this period there may be some restructuring of the aggregates, as particles and clusters penetrate into the pore spaces of larger aggregates, not necessarily increasing size appreciably, but increasing

FIGURE 5.1 Conceptual model of temporal changes in fractal dimensions and average size (characteristic length) during initial stages of an aggregation process.

density, with a corresponding reduction in fractal dimension. With additional time, aggregates become more compact and average size may even decrease slightly, as particles and clusters that are only loosely joined break off and rejoin other aggregates in a more stable manner. The overall effect is a slight reduction in average size and a slight increase in fractal dimension. These changes also are illustrated with the floc sketches at the bottom of Figure 5.1.

The length of time in the initial phase (before point A) will vary, depending on chemical and mixing conditions, as well as the initial state of the suspension. In the experiments reported below, this phase lasts approximately 40 to 60 min. In a typical treatment process, the length of time allowed for mixing is on the order of several tens of minutes, so the later processes of restructuring and compaction are probably not significant. In natural systems particular conditions may last longer, and there is a greater chance particles will be in a near-equilibrium state.

5.3 EXPERIMENTAL SETUP 5.3.1 Image Analysis

Three sets of experiments were conducted in this study (Table 5.2), each one using an image-based analysis of aggregates. A nonintrusive imaging technique was used to capture images of aggregates and to analyze changes in aggregate properties with time. Using this technique, aggregates could be maintained in suspension and images were captured without sample extraction or any other interruption of the experiment. In one set of tests (Experiment Set 1, Section, the images were taken of the mixing jar containing suspensions immediately at the end of the flocculation step, assuming the particle shape and size did not change during the settling period. In another set of

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