10. Water Quality Management. (Stark and Nicholes, 1972) Four cities discharge waste water into the same stream. City 1 is upstream, followed downstream by city 2, then city 3, then city 4. Measured alongside the stream, the cities are approximately 15 miles apart. A measure of the amount of pollutants in waste water is the BOD (biochemical oxygen demand), which is the weight of oxygen required to stabilize the waste constituent in water.

A higher BOD indicates worse water quality. TIle Environmental Protection Agency

(EPA) sets a maximum allowable BOD loading, expressed in lb BOD per gallon. The removal of pollutants from waste water takes place in two forms: (1) natural decomposition activity stimulated by the oxygen in the air, and (2) treatment plants at the points of discharge before the waste reaches the stream. The objective is to determine the most economical efficiency of each of the four plants that will reduce BOD to acceptable levels.

The maximum possible plant efficiency is 99%.

To demonstrate the computations involved in the process, consider the following definitions for plant 1:

QI = Stream flow (gal/hour) on the IS-mile reach 1-2 leading to city 2 PI = BOD discharge rate (in lb/hr)

~r

~n eer.

'eion s1-

:le-

2.3 Selected lP Applications 67 XI = efficiency of plant 1 (:::;.99)

bi = maximum allowable BOD loading in reach 1-2 (in lb BOD/gal)

To satisfy the BOD loading requirement in reach 1-2, we must have PI(l - xd :::; b1QI In a similar manner, the BOD loading constraint for reach 2-3 takes the form

(

BOD diSCharge) ( BOD diSCharge)

1 - Tl2. +. :::; b Q ( ) rate In reach 1-2 rate In reach 2-3 2 Z or The coefficient Tl2 «1) represents the fraction of waste removed in reach 1-2 by decomposition.

For reach 2-3, the constraint is Determine the most economical efficiency for the four plants using the following data (the fraction of BOD removed by decomposition is 6% for all four reaches):