Aggregates mass scaling

Another important property of fractal aggregates is that their mass scales with their size raised to the fractal dimension power. For a naturally occurring aggregate, apparently random to the eye, size is difficult to define in terms of the typically ragged border and anisotropic shape. Thus, a convenient measure of size that can be precisely quantified is the radius of gyration, R defined as... 

Figure AA Dependence of the mean square radius of gyration on the aggregate mass for a collection of reaction-limited cluster aggregation mechanism (RLCA) aggregates. Here, the scaling relationship Rl is verified.

Based on the solution of the velocity field, the force and torque acting on the aggregates can be calculated and the hydrodynamic equivalent diameters for translation (xh,t) and rotation (xi -) can be derived. They scale with the aggregate mass via a power law that reflects the stmctural properties of the aggregate. That means, while for hep aggregates the aggregate mass (N) is proportional to the third power of Xh, there is a fractal-tike relationship for DLCA aggregates with a hydrodynamic dimension 4i close to the fiactal dimension (Fig. 4.17, left cf. discussion to Kirkwood-Riseman theory on pp. 164). This fractal relationship... 

Biomass containment in continuously operated bioreactors is an essential prerequisite for the feasibility of practical industrial-scale dye biodegradation. Biofilm airlift reactors have demonstrated excellent performance for their ability to control mixing, interphase mass transfer and biofilm detachment rate. Further studies are required to further exploit the potential of this type of reactors with either aggregated cells or biofilm supported on granular carriers. 

Large quantities of particles are handled on the industrial scale, and it is frequently necessary to define the system as a whole. Thus, in place of particle size, it is necessary to know the distribution of particle sizes in the mixture and to be able to define a mean size which in some way represents the behaviour of the particulate mass as a whole. Important operations relating to systems of particles include storage in hoppers, flow through orifices and pipes, and metering of flows. It is frequently necessary to reduce the size of particles, or alternatively to form them into aggregates or sinters. Sometimes it may be necessary to mix two or more solids, and there may be a requirement to separate a mixture into its components or according to the sizes of the particles. 

General Remarks. These equations for the dynamics of a cell mass are in the form of a nonlinear partial integro-differential equation. The nonlocal structure and nonlinearity of the transport term allows for short length scale variation on the order of a cell diamater that models the cellular nature of the aggregate. Equations of a similar form arise in the work of... 

D. A. Weitz and M. Y. Lin, Dynamic scaling of cluster-mass distributions in kinetic colloid aggregation, Phys. Rev. Lett. 57 2037 (1986). 

---