Floe fractal dimension

Physical properties, such as floe densities, settling velocities or the diffusion of compounds inside aggregates and floes, depend on their fractal dimensions, which have to be calculated. The floe fractal dimensions can be determined directly from simulation, since each monomer and particle position is repeatedly computed during a simulation, allowing the mathematical definition of a fractal to be applied directly. 

Dyer, K.R., and Manning, A.J. (1999) Observation of the size, settling velocity and effective density of floes, and their fractal dimensions. Neth. J. Sea Res. 41, 87-95. 

Fig. 18. Fractal dimension of activated sludge floes 1 (a) and 2 (e) (initial gray-level images) Binary image (b) and EDM image (c) of floe 1 Calculation of the fractal dimension of floes 1 (d) and 2 (f)...

Finally, in a related dynamical, though nonrheological context, Adler (1987) determined the drag exerted on fractal two-dimensional floes possessing various fractal dimensions. He found a very weak effect (if indeed any effect) of this particular parameter on the drag for spatial dimensions ranging from 1.5 to 2.0. 

The strong-, weak- and intermediate regimes are all a product of the elastic constant of the basic mechanical unit (the floe, the links between the floes, or a combination of both) and the number of these units present in the direction of the externally applied force (Shih et al. 1990). Therefore, the fractal dimension defines to the size of the clusters. A large fractal dimension represents a large cluster that translates to less cluster-cluster interactions per unit volume and a decreased elastic modulus. At high volume fractions, cluster size decreases and the number of cluster-cluster interactions increases, and thus the elastic constant also increases. 

The dependence of the elastic modulus on protein concentration has been used to establish the framework of fractal geometry. Bremer et al. (1990) indicated that a cluster of protein molecules would possess a fractal nature if the power-law dependence exists between the amount of floes in the cluster and the radius of the cluster. In addition, the magnitude of this power, signified as the fractal dimension, D, would be below 3. The elastic constant of protein aggregates could be described as a function of the aggregate volume fraction ... 

Spicer et al. [21] analysed 0.87 p,m particles in a stirred tank system similar to that used by Kusters et. al. and found, using the volume obscuration method, mass fractal dimensions of 2.1 for small floes increasing to 2,5 for larger floes, consistent with the findings of Oles. The increase was attributed to shear-induced restructuring. 

Figure 3.10 The number of boxes versus box size from the floe shown in Figure 3.9. The slope of the curve as log L goes to zero returns the fractal dimension. Da = 1.79 for this case, which is somewhat lower than the mass fractal dimension of 1.88 due to the finite size effects analysed by Nelson et al. [41].

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 floe 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. 

One important conclusion can be made the spatial disposition of particles in floes results from the biopolymer/particle concentration ratio in addition to the biopolymer conformations. In particular, flocculation processes with rigid biopolymers resulted in the formation of a regular network characterized by fractal dimensions that were higher than those obtained on the basis of the classical DLCA or RLCA models (Figure 4.16). Despite the highly loose structure of the aggregate that was formed, the increase in fractal dimension reflected the high order of particles in such networks. 

The fractal nature of bacterial aggregates (floes) has implications for transport processes, including movement of dissolved respiratory gases to and from the outside of the aggregate, and for movement of dissolved nutrient sources and metabolic products. Several methods have been employed to estimate fractal dimensions of floes in three dimensions. One approach is to measure light scattering of suspended floes [17,18]. The two-slopes method calculates fractal dimensions A from the slope of the cumulative size distribution for maximum length / and the slope of the cumulative solid volume F [17] ... 

Confocal optical microscopy can be used to take a sequence of randomly chosen images through a bacterial floe. Methodologies for calculation of three-dimensional fractal dimensions have been described for this approach [18-20]. One method determines the fractal dimension of each section 7)f using a two-point correlation function C(r) [20] ... 

Another approach to determining the fractal dimension of three-dimensional floes has been to view them individually by light microscopy to measure the largest floe length [21]. Fractal dimension can be determined from the scaling relationship [21] ... 

Fractal dimensions in two dimensions A can be determined from the floes projected surface area, since aggregate projected surface area A is related to aggregate length / [21] ... 

Floe structure changes under different conditions, for example freezing and thawing produced more rounded pores [47]. Floe structure affects settleability and filterability. Some floes settle rapidly and others more slowly. For floes from two different sewage treatment plants, floes with fractal dimensions (determined using a two-point correlation function Section 8.2.4) of 1.9 and 1.8 settled more slowly than floes with fractal dimensions of 2.2 and 2.1 [47]. Differences in fractal dimension reflected different species compositions. Though the fractal dimension values reflect heterogeneity... 

Additionally, fractal dimension has been used in analysis of images of activated sludge to distinguish between floes and filaments [49]. It is now also used to quantify biofllms on surfaces [50,51]. 

The variation of the fractal dimension of kaolinite floes with pH was found using the method of dynamic scaling. Weitz et al (9) showed that for a rapidly aggregating system (a condition fulfilled by kaolinite) that... 

It should be mentioned that, since the 1990s, Gmachowski (1990, 1995, 1998, 2005) has presented a series of papers which combined functional relationships derived from porous-sphere-models with experimental data. He deduced relations for the fractal prefactor kc and the hydrodynamic diameter R, which are frequently cited in the hterature (e.g. Vanni 2000 Tang et al. 2002 Gmy and Cugniet 2004 Lee and Kramer 2004). However, his experiments mainly concern coarse flocculated polystyrene latices (floe size > 100 pm), which are probably not comparable with the aggregate types discussed here (e.g. in Gmachowski s papers the fractal prefactors increase monotonically with fractal dimension, which is in striking contrast to the behaviour found in Fig. 4.8). 

Figure 12 Fractal floes the dimensions and extremes of fractal floes (shown in 2D).

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