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Determination of a. Integrating Equation (9.8) from t = 0 and C = 0 to t = t and C = C produces

In this equation, because [Cos] is a constant, only one pair of values of t and C is needed to determine KLa. In practice, however, there can be several pair of these values obtained from an experiment. One way to determine KLa is to plot the straight line of the relationship between t and In([Cos] - [C])/[Cos]. The slope of this line determines KLa. A more practical and easy method, however, is to average the t's and the (ln([Cos] - [C])/[Cos])'s to obtain a single pair of value. This pair is then used to solve for KLa.

The averaging for the t's and (ln[Cos] - [C])/[Cos])'s have the same number of addends, their sums may be simply equated. This is shown below.

Solving for From KLa,

Zm=n m=1tm

From this equation, KLa may be calculated, which can then be corrected to obtain (KLa)20 using the temperature relation. (KLa)20 is one of the factors needed to calculate a.

The organisms in wastewater respire, so oxygen utilization must be incorporated. Calling the respiration rate by r, Equation (9.11) is modified to d[C] dt

This equation may be rewritten as d [C ]

= (KLa)w([Co,w] - [C]) - r = (KLa)w([^] - [C]) (9.23)

where (K'La )w is an apparent overall mass transfer coefficient. It encompasses both (KLa)w, the true overall mass transfer coefficient, and r.

The second part of Equation (9.23), [d(C)/dt] = (K'La)w([Cosw] - [C]), is similar in form to Equation (9.8). It can therefore be manipulated to obtain an equation similar to Equation (9.21). This is shown below.

(K fLa )w may now be solved as

Xm=1

Having solved (K'La)w and for n values of [C], Equation (9.23) may be written as m=n m=n m=n

(KLa)w£(£[Cos] - [C])m - X fm = (K'La)w£(£[Cos] - [C])m (9.26)

from which (KLa)w may be solved once f is determined in a separate test. For a constant f, Xm=i fm = nr. Thus,

, (K'La)wXm ==nmCos] - [C])m + nf (KLa )w = --(9.27)

(KLa)w may now be corrected to obtain (KLa)w 20 using the Arrhenius temperature relation. Once (KLa)w 20 and (KLa)20 are known, a can be computed.

The actual laboratory experimentation involves deaerating the sample, first. This is done by consuming the dissolved oxygen using sodium sulfate (Na2SO3) with cobalt chloride (CoCl2) added as a catalyst. The sulfite converts to sulfate when reacted with the dissolved oxygen. From the stoichiometry of the reaction, 7.9 mg/L of the sulfate is needed per mg/L of the dissolved oxygen. Ten to 20% excess is normally used. Cobalt chloride has been used in concentration of 1.5 mg/L to act as catalyst. As soon as the sample is completely deoxygenated, reaeration is allowed to take place using the type of aeration system to be employed in the prototype such as bubble-diffusion, turbine, surface-aeration, cascade, perforated plate, and spray tower. The increase in dissolved oxygen concentration with respect to time is monitored. The data obtained are then used to calculate the aeration parameters. In the cases of cascades, perforated plate towers, and spray towers, the time may be taken as the time it takes the mass of water or droplets to fall through the height. The concentration at the top of the cascade or tower would have to be zero; that at the bottom would have to be whatever is measured.

Example 9.8 A settling column 4 m in height is used to determine the a of a wastewater. The wastewater is to be aerated using a fine-bubble diffuser in the prototype aeration tank. The laboratory diffuser releases air at the bottom of the tank. The result of the unsteady state aeration test is shown below. Assume P = 0.926, f = 1.0 mg/L • h and the plant is 304.79 m above mean sea level. For practical purposes, assume mass density of water = 1000 kg/m . Assume an ambient temperature of 25°C. Calculate a.

Tap Water at 6.5 °C Wastewater at 25 °C

Time (min)

[C] (mg/L)

Time (min)

[C] (mg/L)

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