5.2.1 Fractal Aggregate Properties

The assumptions of aggregates as impervious, spherical objects have facilitated the development of particle interaction models and provided an obvious simplification of their geometric properties, defined by a single variable, the diameter, d. Perimeter, P, is then proportional to d, projected area, A, is proportional to d2, and volume, V, is proportional to d3. Under the coalesced sphere assumption, volume conservation can be easily computed in terms of changes in diameter, since when two particles collide and stick, the resulting volume is just the sum of the two original volumes and the diameter is found by assuming the resulting volume is again spherical. Other features of aggregation, including particle interaction terms and hydrodynamic interactions, also have been explored on the basis of spherical particles. In this approach aggregate density, p, is essentially constant and equal to the density of the primary particles from which the aggregate was formed, since p is defined as the total mass of the aggregate divided by its volume. In addition, porosity, $ is zero in this case.

In reality, aggregates are highly irregular, with complex geometry and relatively high porosity. Shape cannot be defined in terms of spherical, Euclidean geometry, and fractal geometry must be used instead. The primary geometric parameters of interest are the one-, two-, and three-dimensional fractal dimensions, which may be defined, respectively, by16,18

where l is a characteristic length for an aggregate, and D1, D2, and D3 are the one-, two-, and three-dimensional fractal dimensions, respectively. In general, l has been defined differently in different studies, but the most common definition, which will be used here, is to take l as the longest side of an aggregate. Note also that l takes the place of d in the Euclidean definitions of P, A, and V. Here, D1, D2, and D3 do not in general take integer values, as in Euclidean geometry. These fractal dimensions are obtained from the slope of a log-log plot between the respective aggregate property and l. In essence, fractal geometry expresses the mass distribution in the body of an aggregate, which is often nonhomogeneous and difficult to assess. Aggregates with lower fractal dimension exhibit a more porous and branched structure and, as shown below, have higher aggregation rates.

By taking into account the shape of primary particles and their packing characteristics in an aggregate, Logan15-18 derived various aggregate properties in terms of the fractal dimensions defined in Equation (5.1). The number of primary particles in an aggregate was shown to be

where N is the number of primary particles, f is a constant defined by f = ZH/Ho, Z is the packing factor, Ho and H are shape factors for the primary particles and the aggregate, respectively, and l0 is the characteristic length for the primary particles. The density of the primary particles is p0 and the volume of one primary particle is V0 = H0l3. The total solid mass in an aggregate, ms, is then (Np0V0), or ms

Using similar parameters, the aggregate solid density, ps, is calculated as the ratio of mass and encased volume of the fractal aggregate, defined as the combined volume of particles and pores within the aggregate, Ve = H l3. The aggregate solid density is then

Solid volume, Vs, is the volume associated with the primary particles, which is just Vs = N(H013). In addition, the porosity of the aggregate is <p = 1 - Vs/Ve, or

Finally, the total, or effective density of the aggregate is the total mass (solid plus fluid) divided by the encased volume, which can be shown to be

where pw is the water density. The net gravitational force for settling depends on the difference between p and pw, which can be written as

This last result demonstrates the intuitive idea that aggregate porosity should be important in controlling settling rate.

In the present experiments it was found that ♦ is related to size (discussed in the later part of this section), and indirectly to D2 and D3. Drag on an aggregate moving through the water column depends on the flow of water around and possibly through the aggregate, which in turn depends on overall shape, porosity, and distribution of primary particles within the aggregate structure. For example, flow through an aggregate with a uniform distribution of primary particles would be different from flow through an aggregate in which primary particles are more clustered, with relatively large and interconnected pore spaces, and these differences would lead to differences in overall drag (however, it should be noted that most researchers (e.g., ref. [17,18]) believe there is little or no flow through an aggregate). Even without flow through an aggregate, the distribution of primary particles would affect the manner in which an aggregate would move through the fluid.

Aggregate settling rate can be evaluated from a standard force balance between gravity, buoyancy, and drag,

where CD is a drag coefficient, ws is the settling velocity, and A is the projected area in the direction of movement (i.e., vertical). Since Ve is proportional to l3, A is proportional to l°2, and p depends on l°3-3, Equation (5.8) shows that w2 is proportional to (l°3 -°2 )/CD. Note that if Euclidean values are used for D2 and D3 (D2 = 2 and D3 = 3), then w;? is proportional to l/CD. If it is further assumed that laminar conditions apply, and the relationship for CD for drag on a sphere is used, CD a Re-1, where Re = wsl/v is a particle Reynolds number and v is kinematic viscosity of the fluid in which the settling occurs, then ws a i(d3 —d2+i) (5.9)

which shows that for a given l, larger D3 (more compact aggregate) facilitates faster settling, while larger D2 appears to inhibit settling. This is a somewhat contradictory result, since both D2 and D3 increase with greater compaction, and this contradiction may be a factor in explaining differing results reported for settling in the literature. However, in the case of the most compact (impermeable) aggregates (with D2 = 2 and D3 = 3), the relationship represented in Equation (5.9) converges to a typical Stokes' settling expression (for spheres) where the settling rate is a function of diameter squared. As shown below, results from the present study support an exponent in the settling relationship that is <2, suggesting that a fractal description of settling is needed.

5.2.2 Model Development

Models of suspended sediment transport are important for evaluating efficiency of removal in treatment plant operations and also in predicting the distribution of suspended load and associated (sorbed) contaminant fluxes in water quality models. These models require some description of aggregation processes and must simulate changes in particle size distribution. Aggregation models may generally be classified as either microscale or macroscale. An example of a microscale model is the classic diffusion-limited aggregation (DLA) model,19,20 or one of its various derivatives such as reaction-limited aggregation (RLA). Such models have an advantage in that they consider particle interactions directly, and allow examination of individual aggregates. However, they are generally not very convenient for incorporation into more general sediment transport and water quality models.

Macroscale models address general properties of the suspension, and not individual aggregates. The most well-known macroscale modeling framework was originally described by Smoluchowski,21 and it considers mass conservation for aggregates in different size classes. A basic form of the equation may be written in discrete form as where nk represents the number of aggregates in size class k, t is time, a is collision efficiency, j(i, j) is the rate at which particles of volumes V and Vj collide (collision frequency function), and i, j, and k represent different aggregate size classes. The first summation accounts for the formation of aggregates in the k class, from collisions of particles in the i and j classes. The second summation reflects the loss of k-sized aggregates as they combine with all other aggregate sizes to form larger aggregates. Additional terms such as breakup, settling, and internal source or decay may be added on the right-hand side of Equation (5.10), or terms may be dropped, depending on the processes of importance for a given application. For simplicity, these terms are neglected in the present discussion.

In the discrete form suggested by Equation (5.10), size distribution is determined simply by the number of particles, nk, in each of the k size classes considered for a given problem. Separate equations are written for each size class and the interaction terms determine how the size distribution changes over time. Various forms of this equation may be incorporated into more general advection diffusion type models written to evaluate the distribution of sediment and associated (sorbed) contaminant in water quality models (e.g., ref. [22]).

Major assumptions of the Smoluchowski approach (Equation (5.10)) are that only two particles take part in any single collision, particles follow rectilinear paths (i.e., the particles move in a straight line up to the collision point), and particle volume is conserved during the agglomeration process (again, the coalesced sphere assumption). The rectilinear assumption tends to over predict aggregation rates, while the coalesced sphere assumption under predicts them.3 In reality, as the coagulation of solid particles proceeds, fluid is incorporated into pores in the aggregates that are formed, resulting in a larger collision diameter than the coalesced sphere diameter.23 However, this process is not explicitly included in the traditional model.

The collision frequency function, j(i, j), reflects the physical factors that affect coagulation, such as temperature, viscosity, shear stress, and aggregate size and shape. The three major mechanisms that contribute to collisions are Brownian motion or perikinetic flocculation, fluid shear or orthokinetic flocculation, and differential settling. The total collision frequency is the sum of contributions from these three transport mechanisms, i+j=k i+j=k

where ¡Br, ¡Sh, and ¡DS are the contributions due to Brownian motion, fluid shear, and differential sedimentation, respectively. If the colliding particles are submicron in size, Brownian motion is appreciable. However, with larger particles, Brownian motion becomes less important.24 In traditional methods, i is calculated from constant parameters describing aggregation kinetics, assuming spherical particles. In other words, there is no dependence on the actual shape and size of the aggregates in calculating i. Formulas have been developed to calculate i based on fractal geometry for each of the three above-mentioned transport mechanisms (Table 5.1). These expressions are based on solid volume of the aggregate, Vs, defined previously. The degree to which the values determined from Table 5.1 differ from those determined using the traditional approach assuming spherical aggregates depends on how far the respective fractal dimensions are from their Euclidean counterparts. The functions in Table 5.1 reduce to corresponding traditional estimates when Euclidean values are used, but in general they produce larger values for i .18 The present results (Figure 5.12) also confirm this relationship.

In the experiments described below, collision frequencies and, as a secondary effect, collision efficiencies, are examined as they depend on geometric characteristics of the interacting particles. Density also is shown to be dependent on particle size, which in general is a function of time. Conceptual Fractal Model of Aggregation

Although useful for general modeling purposes, the Smoluchowski model does not provide a basis for developing insight into the details of the physical processes that take

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