Cd A pe v

where Cd = Newton's dimensionless drag coefficient and A = the projected particle area in the direction of the flow. Cd varies with the Reynolds number. By substituting the equations (11.24), (11.25) and (11.26) in equation (11.23), an expression for the dynamic behavior of the particles is obtained:

dt 2

After an initial transient period the acceleration becomes zero and the velocity is constant. This velocity can be obtained from equation (11.27):

If the particles are spherical and the diameter is d, the V/A is equal to 2/3*d and equation (6.8) becomes:

vs = ( (4g (p - pe) * d) / (3Cd * pe) ) 1'2 (11.29)

Newton's drag coefficient Cd is, as mentioned, a function of the Reynolds number and of the shape of the particle. The relationship between Cd and the Reynolds number for spheres and cylinders is given in Fig. 11.6.

When the Reynolds number is below 1, the relationship between Cd and Re can be approximated by Cd = 24/Re, where Re = Reynolds number defined as:

where ^ = the viscosity

In this case (11.29) conforms with Stokes law:


Figure 11.6. Experimental variation of the drag coefficient with Reynolds number. After Fair et al. (1968).

From Fig. 11.6 it can be seen that Cd is approximately constant for turbulent flow in the range for Reynolds number between 1000 and 250,000. For this region the velocity vs is given by:

v8 = 1.82 ( ((p - pe) * d * g) / pe)1/2 (for spheres only) (11.31)

Stokes law can be modified to account for non-spherical floes by use of an "equivalent radius" and shape factor in the formulation:

Vs = settling velocity, length/time g = acceleration due to gravity, length/time^

R = equivalent radius (based on a sphere of equivalent volume),length pp = density of the cell, mass/length3

pw = water density, mass/length3 v = kinematic viscosity Fs = shape factor

The shape factor has a value < 1.0 and accounts for all factors, reducing the settling velocity.

Most nitrogen components are unfortunately readily dissolved in water, which implies that precipitation cannot be used as an easy solution to the problem of nitrogen removal, in contrast to phosphorus, which is widely removed from waste water by the use of chemical precipitation. Nitrogen removal by use of precipitation may, however, be carried out by the following two processes:

Mg2"1" + NH4++ HPO4- <=> Mg (NH4) P04 (s) + H+ (11.33)

Dissolved proteins + pr├ęcipitants = insoluble proteins (11-34)

Process (11.33) looks at the first glance as a very attractive solution, as phosphate and ammonium are precipitated simultaneously. The stoichiometric ratio between the two components in municipal waste water is however not favorable for the precipitation. The concentration of phosphate Is about 10 mg /1 or 0.3 mmol I I, while ammonium is normally present as ammonium-N in a concentration of about 30 mg /1 or about 2 mmol/l.

This implies that phosphate must be added to assure a proper precipitation. This makes the process much more expensive, although the product magnesiumammonium-phosphate is an appreciated fertilizer. The process has therefore not been used for removal of nitrogen from municipal waste water except in pilot plant experiments in Bari, Italy. It might be more attractive to utilize the process, where the stoichiometric ratio between N and P is more favorable, but the ratio in most industrial waste waters is even more unfavorable than in municipal waste water.

The precipitation of proteins by use of various pr├ęcipitants such as ligno-sulfonic acid, iron (III) chloride, calcium hydroxide, glucose-tri-sulfate or just pH-adjustment has been widely used. The precipitation is carried out at the isoelectric point of the proteins, where the destabilization is most easily performed. As waste water often contains a wide range of proteins, it is not possible to adjust the pH for all proteins at the same time, so that precipitation of proteins never can become 100 % effective.

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