XO = C(M − N)S
in which X = flow of the stream expressed in second-feet; O = grams of free oxygen in one cubic foot of water; S = rate of sewage discharge in second-feet; M = grams of oxygen required to consume the organic matter in one cubic foot of diluted sewage as determined by the permanganate test with 4 hours boiling; N = grams of oxygen available in the nitrites and nitrates in one cubic foot of diluted sewage; C = ratio between the amount of oxygen in the stream and that required to prevent putrefaction. Where C is equal to or greater than one, satisfactory conditions have been attained.
In using this formula it is necessary to make analyses of trial mixtures of sewage and water until the correct mixture has been found.
Hazen’s formula is:[[134]]
D = x
S = 4m
O,
in which D = dilution ratio; x = volume of water; S = volume of sewage; m = result of the oxygen consumed test expressed in p.p.m. after 5 minutes, boiling with potassium permanganate; O = amount of dissolved oxygen in the diluting water expressed in p.p.m.
For comparison with Rideal’s formula the factor of 7 should be used instead of 4 to allow for the increased time of boiling.
Since the amount of oxygen needed is dependent on the amount of organic matter in the sewage rather than the total volume of the sewage, and since the amount of organic matter is closely proportional to the population, the amount of diluting water has sometimes been expressed in terms of the population. Hering’s recommendation for the quantity of diluting water necessary for Chicago sewage was 3.3 cubic feet of water per second per thousand population. Experience has proven this to be too small. Between a minimum limit of 2 second-feet and a maximum of 8 second-feet of diluting water per thousand population the success of dilution is uncertain. Above this limit success is practically assured and below this limit failure can be expected.
Even with these carefully devised formulas and empirical guides, the factors of reaëration, dilution, sedimentation, temperature, etc., may have so great an effect as to vitiate the conclusions. As shown in Table 75 dilution in winter is far more successful than in summer. The lower temperatures so reduce the activity of the putrefying organisms that consumption of oxygen is greatly retarded.
225. Governmental Control.—A comprehensive discussion of the legal principles governing the pollution of inland waters is contained in “A Review of the Laws Forbidding the Pollution of Inland Waters,” by E. B. Goodell, published by the United States Geological Survey in 1905, as Water Supply Paper No. 152.