## V

Figure 11.4 Destabilization by flocculation.

Lawler et al. (1983) has presented a mathematical model describing changes in the particle size (PSD) immediately below the solid/liquid interface in gravity thickening based upon Brownian motion, fluid shear, and differential sedimentation. Although the model predicted trends for the coagulation and differential sedimentation for changes in time, solids concentration, particle stability, and the subsidence velocity at the interface, the model was limited because the subsidence velocity cannot be predicted and a simplified approach to the hydrodynamics of differential sedimentation was used. Several mathematical models were developed by Babenkov (1983) to describe the relation between the density and the size of the floes formed during coagulation. The characteristics of the final floes depended on the size and density of the micro-flocs. Aluminum sulfate and cationic polymers provided efficient coagulation for coal processing waste waters.

In many cases agitation is used to accelerate the aggregation of colloidal particles. When particles follow a fluid motion they have different velocities, so that opportunities exist for interparticle contacts. When a contact between particles is caused by fluid motion the process is sometimes called ortho-kinetic flocculation (Overbeck, 1962).

The following equation describes the rate of change in the concentration of particles:

dt 3

where h = collision efficiency factor

G = velocity gradient N = concentration of particles (number/vol.) d = diameter of particles t = time.

G can be calculated (Camp and Stein, 1943) and (Camp, 1955) from p

P = the power input to the fluid V = the volume m = the viscosity of the fluid

Agitation will not increase the aggregation rate of particles smaller than about 1 /; diameter, whereas particles with a diameter of 1 y or more will grow as a result of fluid motion. Since 1 p particles do not settle well, a flocculation tank to allow aggregation must be included in a treatment system which uses sedimentation tanks at a later stage to separate solids from water. Flocculation tanks are designed to provide interparticle contact by orthokinetic flocculation. Design data include selection of velocity gradients, reactor configuration, reactor data and detention time necessary to produce sufficient aggregation. It is difficult to base the design on equations because such parameters as h and P are almost impossible to measure, and even the velocity gradient G can be difficult to determine. It is therefore necessary to provide information for design based on laboratory and pilot plant experiments. However, the interpretation of such an experiment is only possible using a mathematical description of the orthokinetic flocculation.

Suspended matter is removed from water by various separation processes, including sedimentation or settling. The principles of this process will be presented here, while other possible separation processes will be touched upon in Section 11.3, dealing with the design problems. Precipitates and coagulates might settle. Settling rates depend on the difference in density between the suspended matter and the water, the size and shape of the matter, the viscosity of water, the turbulence and velocity of the flow field. In addition, the physiological state of the phytoplankton cells also plays an important role.

In most cases it is not difficult to describe the sedimentation itself, but it is far more difficult to account for the influence of the hydrological flow pattern. Therefore theoretical approaches based upon physical considerations should almost always be accompanied by measurements of sedimentation rates, either directly or indirectly. This latter determination is often carried out by use of tracers, for instance by use of isotopes.

Removal by settling is most often described as a first order reaction: dm

dr where m is the concentration of suspended matter and s is the rate of removal by sedimentation, s is thereby also the ratio between the settling rate, Vs, and the depth D:

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