ස 7 8 9 10 side of the area. Letv represent the uniform rate of flow over the surface, in feet per minute. From the inlet as a center and with v, 2v, etc., to 10v as radii, draw a series of arcs across the area in question. (These arcs may be called run-off contours, and the areas between them run-off areas.) Then during one minute the rain which has been falling upon the surface of this area within the arc v will all have entered the inlet; that within the second run-off area, between arcs v and 20, will have flowed into the area between arc v and the inlet; and that from the tenth area, between arcs 90 and 100, will be between 8v and 9v. During the second minute, the rain which fell upon the second area will enter the inlet, that which fell upon the third area will reach the first, etc. FIG. 8.-DIAGRAM OF RUN-OFF CON- Although the intensity of rainfall was the same on all the areas, and although the velocity of flow has been the same for all parts of the surface, the areas from which the run-off reaches the inlet during successive minutes, and consequently the amounts of such run-off, will differ because of the different areas included between successive pairs of arcs. In the illustration, there will be considerable increase in volume per minute reaching the inlet during the first five minutes, a slight decrease during the next three minutes, and a rapid decrease during the final two minutes. This condition is upon the assumption that the rainfall continued one minute and then stopped entirely. Assume now that the rain continues at the same intensity for ten minutes. Then each of the run-off areas, 1, 2, 3, etc., will continue to receive and pass on towards the inlet the same amount of run-off per minute. During the second minute the inlet will receive the run-off from areas 1 and 2. During the third minute that from areas 1, 2, 3, and during the tenth minute, the sum of the run-off from all 10 areas. If there is no additional area draining to this inlet, the total amount reaching it after the tenth minute will remain constant so long as the intensity of rainfall remains constant, and will be equal to Ar, in which A is the total area. Given the same area, let us now assume that the rainfall during the first minute totals .or of an inch, during the second minute totals .02 of an inch, and gradually increases to .1 of an inch during the tenth minute. During the first minute the rainfall reaching the inlet will be that equivalent to .or inch on area No. 1. During the second minute the rainfall reaching the inlet from area No. I will be that caused by the .02 inch, but that reaching it from area No. 2 will be caused by the rain of the previous minute, or .01 inch. Similarly, the run-off reaching the inlet during the tenth minute will be that produced by rainfall of 0.1 inch on area No. 1, .09 inch on area No. 2, .08 inch on area No. 3, etc., and .or inch on area No. 10. We may now assume an area in a business district where the entire surface is covered with roofs, court yards and street paving, all of which is impervious, and that it yields practically 100 per cent of the run-off. All of the area draining to one inlet may be assumed to have (as such areas very frequently do have) a slope of less than 2 feet per ico, which for practical' purposes may be considered uniform throughout. (If all roof water is carried directly to the sewer, the total area of the roofs would be deducted from the area considered, so far as this calculation is concerned. If the roof water leaders discharge upon the ground surface, their run-off is to be included, but we may neglect the effect of the elevation of the roof upon the time of run-off, since the time required for reaching the surface from an ordinary roof is in most cases less than one minute.) The probability is that run-off from this area does not follow a direct route to the inlet, but flows to and along the nearest street gutter. In drawing the run-off contours on such an area, therefore, having determined or assumed a certain rate of flow in feet per minute, we may locate points on the oneminute contour by laying off this distance from the inlet out to the gutters of the radiating streets; other points are determined by finding points on the area that slopes toward each gutter. from which the distance to that gutter and down the gutter to the inlet equals the same one-minute distance. Then, connecting these points, we have the one-minute contour. In the same way, taking twice this distance, we may locate the two-minute contour, etc. Next, assume varying rates of flow for different sections of the area. We may then use the same method for locating the minute-contours, except that we will use for each section of the area the rate of flow determined for that section. For instance, if it be assumed that the run-off from yards to the gutter will be at the rate of 0.4 foot per second, while the flow down the gutter to the inlet will be at the rate of 2 feet per second, then, if the center of the block be 100 feet from the gutter, it would take four minutes and ten seconds for water from the center of the block to reach the gutter, and during the fifty seconds remaining of a five-minute interval it would flow 100 feet in the gutter. Consequently, one point in the five-minute contour for this block would be in the center of the block opposite a point in the gutter 100 feet from the inlet. The above analysis of run-off forms the basis of the calculation of run-off by the rational method. One more assumption of the problem needs to be qualified for practical use. It has been assumed so far that the entire area is impervious and that the run-off is therefore 100 per cent of the rainfall. As we have seen, under certain conditions this percentage may not exceed 20 or 30. To allow for this (since it is run-off and not rainfall with which we are concerned) we may assume for each area a factor of imperviousness, I, and multiply the area by this. Then if this product AI be multiplied by the rainfall rate R, AIR will represent the run-off. In order to avoid the tedious calculation of run-off by the rational method, many engineers use formulas which take into account only a few of the variable conditions. In fact, it is only within the past ten or fifteen years that the rational method has come into anything like general use. Most of these formulas were based upon measurements of rainfall and sewer flow in certain cities or even smaller areas, and, since they are largely empirical, will only by chance give correct results for any other areas. The best known of these formulas are as follows: B is the mean breadth of the drainage area in miles; c is a constant varying from 0.37 to 1.95 in Craig's formula, and from 0.75 for paved streets to 0.31 for macadamized streets in the Bürkli-Ziegler and McMath formulas; d is the diameter of the sewer in inches; D is the diameter in feet; L is the extreme length of the drainage area in miles; M is the drainage area in square miles; N is the length in feet in which the sewer falls one foot; Qis the cubic feet per second reaching the sewer; R is the average rate of rainfall in inches per hour during the heaviest rainfall (one inch per hour, all flowing off, is practically the same as one cubic foot per second per acre); In Kirkwood's and Adam's formulas R=1; in McMath's, R= 2.75. S is the general fall of the arca per thousand; t is the duration in minutes of the intensity (b-ct), in which b is 2.1 for Rochester, N. Y., and c equals .0205. Of the above formulas, those most frequently used are probably the McMath and the Bürkli-Ziegler, which are the same except for the exponent of the fraction That of McMath Α S was based upon measurements made at St. Louis, and Mr. McMath himself has stated that he considers this formula adapted to large areas only and that it was derived in an entirely empirical manner from St. Louis data only. This formula is no longer used in St. Louis, but the rational method was adopted some years ago. The use of these formulas is still quite general but is not to be recommended. It is probably due largely to disinclination to incur the labor involved in the rational calculation, and a feeling that the factors of imperviousness, intensity of rainfall, etc., must be estimated upon such unsatisfactory data that the result is not sufficiently reliable to warrant the labor involved. It is true that there is some uncertainty in the data available for use in such calculations, but many investigations have been made during recent years which have gradually reduced such uncertainty, and more general use of the method will undoubtedly increase the reliability of the results obtained. At least, this method may be expected to give much more reliable results than the formulas, especially under conditions to which the formulas do not apply with any exactness, which conditions are themselves indefinite or unknown. As an illustration of the unreliability of the formulas, it is found that the several formulas give the following quantities of run-off from a 10-acre area with a 1 per cent slope: Hawskley, 12; Adams, 15; BürkliZiegler, 27; McMath, 21; Parmley, 39; Hering, 20. An area of 57.1 acres of St. Louis with a slope of 1 per cent gave 122 cubic feet per second by the rational method and only |