Monod published the first kinetic studies on the limitation of bacterial growth by substrate concentration. For these experiments he used an aerobic batch reactor, glucose as substrate, and a culture of Escherichia coli (Monod 1942, 1950). His inspiration came from Michaelis-Menten kinetics (see Section 3.2.1.4), which are only valid for a system consisting of one substrate and one enzyme. Although the biodegradation of one substrate by a pure culture of bacteria and, of course, by a mixed culture is a more complicated process, the standard form of the growth kinetics of bacteria shows the same mathematical structure if the growth is not limited by the concentration of dissolved oxygen:

and:

where v is the specific growth rate (d-1), vmax is the maximal specific growth rate (d-1), S is the substrate concentration (g L-1), KS is the saturation coefficient (g L-1), rx is the growth rate (g L-1 d-1 MLSS) and X is the concentration of bacteria (g L-1 MLSS).

Monod's first experiments were carried out in batch reactors. He described his measurements of S and X with respect to time by using the bacterial balance:

dX S

and the substrate balance:

where the substrate consumption rate is:

S vo

and YX/S is the true yield coefficient.

The initial conditions of Eqs. (6.3) and (6.4) are:

The simplest solution can be obtained for a relatively high X0 and a relatively low S0, resulting in a nearly constant X = X0. Only Eq. (6.4) must be solved, which is performed by the separation of variables:

with:

YX/S

The results of an experiment can be tested as a function of S = f(t) and the coefficients KS and rS,max can be determined by transformation of Eq. (6.7) into:

In order to confirm that the results of the batch experiment correspond with the model used to develop Eq. (6.4), we should plot, e.g.:

ln So/S t versus

and the points should form a straight line with a positive slope rSmax/KS and a negative ordinate intercept -1/KS.

For substrate and oxygen limitation, Eq. (6.1) must be expanded: S c'

The region where oxygen limitation is significant can be estimated using K' = 0.2 mg L-1 O2 at T = 20 °C (Putnaerglis 1987). For SP KS the point of limitation may be given as u = 0.9 pmax (90% of maximal growth rate):

For c'6 1.8 mg L-1 we do not expect O2 limitation (T = 20 °C, domestic sewage, no influences of mass transfer at the surface or inside of flocs formed by bacteria).

With KS = 50-120; 85 mg L-1 BOD5 at T = 20°C (Sundstrom and Klei 1979), the region of substrate limitation can be estimated roughly to be:

If both concentrations S and c' decrease to low values during a batch process, the kinetic evaluation is complicated. Therefore, kinetics should be studied considering only under substrate limitation or oxygen limitation. But if we want to study oxygen limitation during a batch process without aeration, the substrate concentration (BOD5, COD, DOC) decreases only about a few milligrams per liter if c' decreases from 8 mg L-1 to 0 mg L-1. This kind of batch experiment can be carried out even for low S values.

If we avoid substrate limitation c'

can be tested.

122 | 6 Aerobic Wastewater Treatment in Activated Sludge Systems For X; const, the oxygen balance is:

dc Fmax C

and a corresponding solution to Eq. (6.9) is obtained which can be linearized as shown before.

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