The discussion so far has simply dealt with material balances. For example, they predict the concentration of nitrogen in the effluent, alkalinity requirement and production, and carbon requirement. None, however, would be able to compute the size of the reactor and how fast or how long the sewage should be treated; and, although the concentrations of the nitrogen in the effluent have been predicted, the actual resulting value would depend on how long the reaction was allowed to take place. The subject that deals with studying how fast reactions proceed to completion is called reaction kinetics. As a result of knowing how long a reaction proceeds, the size of the reactor can then be determined. In general, there are three types of kinetics involved in the removal of nitrogen: nitrification kinetics, denitrification kinetics, and carbon kinetics.
The kinetics of the microbial process may be conveniently divided into kinetics of growth and kinetics of food (or substrate) utilization. The kinetics of growth was derived previously in the Background Prerequisites chapter, in connection with the derivation of the Reynolds transport theorem. The result was
where [X ] is the concentration of mixed population of microorganisms utilizing the organic waste; t is the time of reaction; is the maximum specific growth rate of the mixed population in units of per unit time;  is the concentration of substrate; Ks is called the half-velocity constant; and kd is the rate of decay.
Substrate kinetics may be established by noting that as organisms grow, substrates are consumed. Therefore, the rate of decrease of the concentration of the substrate is proportional to the rate of increase of the concentration of the organisms. The first term on the right-hand side of Equation (15.71) is the rate of increase of microorganism that corresponds to the rate of decrease of the substrate. Thus, the rate of decrease of the substrate is
U is a proportionality constant called the specific substrate utilization rate; it is the reciprocal of Y, the specific yield of organisms. The specific substrate utilization rate simply means the amount of substrate consumed per unit amount of organisms produced.
Primary effluent Qo> S0, Xq
Sludge recycle Qr, Xu, S
Sludge wasting Qw> Xu> S
Secondary clarifier S
Secondary clarifier S
Sludge underflow Qu> Xy, S
FIGURE 15.1 Schematic of the activated sludge process.
15.10.2 Material Balance around the Activated Sludge Process
Biological nitrogen removal is basically a part of the general activated sludge process. Figure 15.1 shows the schematic of the process. Primary effluent is introduced into the reactor where air is supplied for consumption by the aerobic microorganisms. In the reactor, the solubles and colloids are transformed into microbial masses. The effluent of the reactor then goes to the secondary clarifier where the microbial masses are settled and separated from the clarified water. The clarified water is then discharged as effluent to some receiving stream. The settled microbial masses form the sludge. Some of the settled sludges are waste and some are recycled to the reactor. When once again exposed to the air in the reactor, the organisms become reinvigorated or activated, thus the term activated sludge. The wasting of the sludge is necessary so as not to cause buildup of mass in the reactor.
Develop the mathematics of the activated sludge process by performing a material balance around the control volume as indicated by the dashed lines. The symbols depicted in the figure mean as follows:
Qo = influent flow to the process from the effluent of the primary clarifier
So = substrate remaining after settling in the primary clarifier and introduced into the reactor
Xo = influent mass concentration of microorganisms
Qr = recycle flow from the secondary clarifier
S = substrate remaining after consumption by organisms in the reactor
X = mass concentration of organisms developed in the reactor
Xe = mass concentration of organisms exiting from the secondary clarifier
Qw = rate of flow of sludge wasting
Qu = underflow rate of flow from the secondary clarifier
Xu = underflow mass concentration of organisms from the secondary clarifier.
The volume of the control volume is comprised of the volume of the reactor, volume of the secondary clarifier^ and the volumes of the associated pipings. The total rate of increase is d I V [ X ] dV/dt ; the local rate of increase is d|V [ X] dV/dt ; and the convective rate of increase is $A[X]V• n dA. Thus, by the Reynolds transport theorem, dIV[X]dV dIV[X] dV , » V - V +I [X]v • ndA (15.73)
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