224. Quantity of Diluting Water Needed.—In a large majority of the problems of disposal of sewage by dilution it is not necessary to add sufficient diluting water to oxidize completely all organic matter present. Ordinarily it is sufficient to prevent putrefactive conditions until the flow of the stream, lake, or tidal current, has reached some large body of diluting water or where putrefaction is no longer a nuisance. It is never desirable to allow the oxygen content of a stream to be exhausted as putrescible conditions will exist locally before exhaustion is complete. The exact point to which oxygen can be reduced in safety is in some dispute. Black and Phelps have assumed 70 per cent of saturation as the allowable limit; Fuller has placed it at 30 per cent; Kinnicutt, Winslow, and Pratt have placed it at 50 per cent. Since the reaction between the oxygen and the organic matter is quantitative, others have placed the limit in terms of parts per million of oxygen. Wisner,[[132]] has recommended a minimum of 2.5 p.p.m. as the limit for the sustenance of fish life, which is not far from Fuller’s limit for hot-weather conditions.
Formulas of various types have been devised to express the rate of absorption of oxygen with a given quantity of diluting water which is mixed with a given quantity and quality of sewage. The quantity of sewage is sometimes expressed in terms of the tributary population or in other ways. Knowing the rate at which oxygen is exhausted and the velocity of flow of the stream, the point at which the oxygen will be reduced to the limit allowed is easily determined. The accuracy of none of these formulas has been proven, and their use, without an understanding of the effect of local conditions, may lead to error. They may be used as a check on the bio-chemical oxygen demand determinations, which should be conclusive.
The following formula, based on the work of Black and Phelps, is a guide to the amount of sewage which can be added to a stream without causing a nuisance. It is:
in which C = per cent of sewage allowed in the water; O′ = per cent of saturation or the p.p.m. of oxygen in the mixture at the time of dilution; O = per cent of saturation or the p.p.m. of oxygen in the stream after period of flow to point beyond which no nuisance can be expected; t = time in hours required for the stream to flow to this point; k = constant determined by test determinations of the factors in the following expression:
in which O′1 = per cent of saturation or the p.p.m. of oxygen in the diluting water before mixing with the sewage;
In the solution of these formulas it is desired to determine the permissible amount of sewage to discharge into a given quantity of diluting water. This value is expressed by C in the first equation. In solving this equation:
O′ is determined by laboratory tests and should represent the conditions to be expected during various seasons of the year; O is determined by judgment. It may be 30 per cent or 50 per cent or more as previously explained; t is determined by float tests or other measurements of the stream flow; k is determined by laboratory tests in which mixtures of various strengths are incubated for various periods of time. Different values of k will be obtained for different characteristics of the sewage; but for the same sewage the value of k should be unchanged for different periods of incubation.
Rideal devised the formula:[[133]]