Floe packing

Obviously, Equation 12b has to be modified somewhat for different floe packings. However, in this approximate treatment, we only wish to investigate trends, and for that purpose a simple cubic arrangement suffices. 

We have assumed that a random packed structure is more likely at high rates with a distribution of floe sizes. The volume fraction of floes will depend upon the floe packing fraction, giving rise to a floe diameter 2af with particles per floe ... 

Floes pack in a space-filling, Euclidean fashion hence, at the floe level of structure, the material can be considered as an orthodox amorphous substance. Within the floes, however, particles pack in a non-Euclidean, fractal fashion. For such a structural arrangement, the volume fraction of particles in a floe (O ) is equivalent to the volume fraction of particles in the entire system (O), namely = O. This well-known relation of polymer physics (11) has been experimentally shown to also apply to colloidal aggregates above their gelation threshold (12). 

As stated above, Afd is related to the contact pair potential Afg(0). In a floe, each particle is in close contact with z other particles. If A/a is small, the z lens-shaped overlap volumes (see Figure A) surrounding each particle do not overlap with each other, and Afd equals zAf (0)/2 where Afg(0) is given by Equation 8. For higher values of A/a, the lenses overlap partly, and Afd < zAfs(0)/2. Above a certain value of A/a (which depends on the packing of the particles in the floe), there is no polymer left within the interstices of the floe and all the solvent in the floe is within a distance A from the surface of at least one particle. Then the volume of solvent which is transferred towards the solution when a particle is added to the floe is readily calculated. 

The high degree of packing of the organisms within a volume can lead to the formation of floes (suspended aggregates), where millions of cells cluster to form particles with dimensions in the order of millimetres [29]. Models for uptake by such ecosystems also assume sphericity, and start from a continuity equation accounting for the consumption of the species throughout the floe ... 

The term D is called the fractal index and represents the packing change with distance from the centre of the floe. Computer simulation and experiments allow the value of D to be related to the mechanism of aggregation. Typical values are for ... 

The lower the value of D the more open the packing. Suppose we now imagine a dispersion which has been driven to instability by, say, changing temperature. We can visualise sites for the nucleation of floes occurring randomly throughout the whole volume of the dispersion. The total number of primary particles remains unchanged so we can determine the volume fraction of floes ... 

There is a concentration limit to the applicability of the fractal approach. As simultaneous nucleation occurs the floes will grow and as they grow they may eventually touch. Further growth would require the interpenetration of the floes and when this occurs the idea of a fractal floe is lost because the structure changes. We can roughly estimate the limits on this by assuming the floes reach a packing fraction before interpene-... 

Sediment Volume. If the dispersion is unstable, the sediment bed will be quite deep and sedimenting particles will stick together where they first strike the sediment bed, thus forming an open structure with considerable occluded liquid. If the dispersion is stable to reagglomeration, the particles will move freely past one another to avoid contact as long as possible. The result is a thin sediment bed with maximum solids packing and minimum occluded liquid (12). Since dispersed particles setde more slowly than floes, centrifugation maybe needed to force sedimentation of small particles within a reasonable analysis time. 

The UASB type of digester consists of a bottom layer of packed sludge, a sludge blanket, and an upper liquid layer. Wastewater flows upward through a sludge bed that is covered by a floating blanket of active bacterial floes. Settler screens separate the sludge floes from the treated water, and gas is... 

In principle, it is possible to calculate Xg from the Gmin values given in Fig. 6, using equation (5). If the floes are assumed to be close-packed structures, then n must be between 8 (random-close packing) and 12 (hexagonal close-packing). However, equation (5) is based on a monodisperse suspension. 

Figure 8. Idealized fat crystal network under extension. Particles (a) are packed in a fractal fashion within floes ( ). A force (F) acting on the network causes the links between floes to yield, and the original length of the system in the direction of the applied force (L) to increase (A/.). Thus, the inter-floc separation distance (I), also increases.

Flocculation comes from the Latin word flocculate meaning loose and woolly. Flocculated systems result in rapid rate of settling because each individual unit is composed of many particles and is therefore larger. However, due to the loose packing of floes they are easily dispersible on shaking. Deflocculated systems on the other hand are made up of smaller particles whose settling rate is slower, but the settled particles tend to form an irreversible compact and are difficult to redisperse. This phenomenon is called caking. For coarse suspensions, a deflocculated suspension will have better uniformity of dose but poorer stability... 

Flow of any concentrated suspension will become impossible when the solid particles can form a continuous three-dimensional network of contacts throughout the sample. This so-called maximum packing fraction 4> depends mainly on the particle size distribution and the particle shape. Broader particle size distributions result in lower values of 4>m, because the smaller particles can fill the gaps between the bigger ones, and a deviation from spherical shape results in lower values of 4>m due to steric hindrance of packing. Also flocculation will result in a decrease in the value of 4>in, because the individual floes are only loosely packed. 

In this treatment we will consider the case where spherical particles are distributed in an aperiodic, fractal fashion within floes. These floes form chains which pack in a space-filling (close-packed), periodic fashion, forming a three-dimensional network. The volume fraction of particles in the network ( I>) is given by ... 

In a completely deflocculated system the particles are not associated pressure on the individual particles can lead in this layer to close packing of the particles to such an extent that the secondary energy barriers are overcome and the particles become irreversibly bound together. In flocculated systems (where the repulsive barriers have been reduced) particles settle as floes and not as individual particles. The supernatant clears but, because of the random arrangement of the particles in the floes, the sediment is not closely packed and caking does not readily occur. 

Fig. 6.11. Typical packing structures in concentrated suspensions (a) simple network of chains (b) denser floes, cormected by chains (c) dense packing of spheres surroimded by a thick immobile layer of solvent or absorbed species (redrawn from Sormtag and Strenge [6]).

The systematic classification of the feature shapes revealed 12 basic forms that are named after well-known objects, see Table 2.1.1. Subdivisions occur and further geometric types await detection. The basic shapes may occur in pure form or mixed and merged. Importantly, they relate to the crystal packing. For example, if a cleavage plane ends on the surface, parallel floes will occur, or flat lying molecules on a surface will lead to random volcanoes, etc. An exhaustive collection of these feature types and subtypes can be found in Ref. [18]. 

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