After some algebraic manipulation,

The same mathematical operations may be applied to the substrate. These will not be repeated but the result will be written at once, which is

-V IF = TTTb][ X] V = -{0 + (-GoE 5o ] + (Qo - Qw)[ 5 ] + Qw [ 5 ])}

Note that the total derivative and Eulerian derivative are prefixed by a negative sign. This is because they represent a rate of decrease of the mass of the substrate as opposed to the rate of increase of the mass of the microorganisms. Equation (15.78) may be manipulated to produce

Equating the right-hand sides of Eqs. (15.77) and (15.79) and rearranging, Qw[] + (Qo - Qw)[] - Qo[] Qo y

[X] V is the mass of biomass (organisms) contained in the volume of the reactor at any time. The is also called the mixed liquor volatile suspended solids. The volatile suspended is used to estimate the biomass in the reactor. Qw[Xu] + (Qo -Qw)[Xe] - Qo[Xo] is the net rate of biomass wasting. Since the mass is wasted to prevent buildup of solids, it represents the mass that has accumulated over some interval of time in the reactor. Hence, if the quantity of biomass in the reactor is divided by the rate of biomass wasting, the ratio obtained is the time it takes to accumulate the biomass solids inside the reactor. This ratio 6c = [ X ] V/Qw[Xu] + (Qo - Qw)[Xe] - Qo[Xo] is called by various names: mean cell residence time (MCRT), sludge retention time (SRT), and sludge age. This ratio is different from the ratio obtained by dividing the quantity of water in the reactor by the net rate of discharge from the reactor. This latter ratio, V/Qo = 6, is called the nominal hydraulic retention time (NHRT). The word nominal is used here because 6 is not the actual detention time of the tank. The actual detention time is V/(Qo + Qr), where Qr is the recycled or recirculated flow. Using 6c and 6 in Equation (15.81)

Y, kd, ¡xm, etc. are called kinetic constants.

Note: All example values of kinetic constants reported in succeeding discussions are at 20°C.

The kinetics of nitrification are a function of several factors the most important of which include pH, temperature, and the concentrations of ammonia and dissolved oxygen. Experience has shown that the optimum pH of nitrification lies between 7.2 and 8.8. Outside this range, the rate becomes limited. As shown in the nitrification reaction, acidity is produced. If this acidity is not buffered by addition of sufficient alkalinity, the pH could control the process and the kinetics become pH limited. We have not addressed the mathematics of this issue.

Temperature affects the half-velocity constant Ks of the Monod equation. It also affects maximum growth rates. Growth rates for Nitrosomonas and Nitrobacter are affected differently by change in temperature. At elevated temperatures Nitrosomonas growth rates are accelerated while those of Nitrobacter are depressed. Nitrosomo-nas is, however, slow growing compared to Nitrobacter that the kinetics is controlled by Nitrosomonas.

The value of the concentration of dissolved oxygen in nitrification does not promote unhampered growth of organisms. Aeration equipment are not that efficient; they cannot provide an unlimited amount of dissolved oxygen. In fact, the reason why sludge is wasted is to avoid exhaustion of the dissolved oxygen as a result of too large a concentration of organisms consuming oxygen in the reactor. Thus, the nitrification process is always limited by dissolved oxygen.

Expressing the effects of these predominantly limiting factors on the specific growth rate "n of the nitrifiers, the following Monod equation is obtained:

" =(¡íídlSNS;;:|;ii)( ¡d^|lO¡k_)( c'» > a583)

where finm is the maximum ¡in; [SNH ] is the [S] concentration of NH4-N.

Note: Because TKN hydrolyzes to produce NH+, [SNHJ may also represent [TKN]); KsNH^ is Ks for nitrification; [SO2] is the concentration of dissolved oxygen; KsO is the half-velocity constant for oxygen; CpH and is a fractional correction factor due to the effect of pH.

KsNH , ¡inm, and CpH have been determined experimentally and are given, respectively, by4 (Mandt and Bell, 1982):

The values of KsOi ranges from 0.25-2.46 mg/L. T is in °C, KsN^ is in mg/L, and Vnm is per day.

Applied to nitrification, Equation (15.77) in conjunction with Equation (15.83) may be written as

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