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28. The Rational Method.—The water which falls during a storm must be removed rapidly in order to prevent the flooding of streets and basements, and other damages. The quantity of water to be cared for is dependent upon: the rate of rainfall, the character and slope of the surface, and the area to be drained. All methods for the determination of storm-water run-off, whether rational or empirical, depend upon these factors.

The so-called Rational Method can be expressed algebraically, as,

Q = AIR,

in which Q =rate of run-off in cubic feet per second;A =area to be drained expressed in acres;I =percentage imperviousness of the area;R =maximum average rate of rainfall over the entire drainage area, expressed in inches per hour, which may occur during the time of concentration.

The area to be drained is determined by a survey. A discussion of R and I follows in the next two sections. An example of the use of the Rational Method is given on page 95.

29. Rate of Rainfall.—Rainfall observations have been made over a long period of time by United States Weather Bureau observers and others. Continuous records are available in a few places in this country showing rainfall observations covering more than a century. Such records have been the bases for a number of empirical formulas for expressing the probable maximum rate of rainfall in inches per hour, having given the duration of the storm. Table 13 is a collection of these formulas with a statement as to the conditions under which each formula is applicable. The formula most suitable to the problem in hand should be selected for its solution.^[22]

TABLE 13
Rainfall Formulas
Name of Originator	Conditions for which Formula is Suitable	Formula
E. S. Dorr		i = 150 t + 30
A. N. Talbot	Maximum storms in Eastern United States	i = 360 t + 30
A. N. Talbot	Ordinary storms in Eastern United States	i = 105 t + 15
Emil Kuichling	Heavy rainfall near New York City	i = 120 t + 20, etc.
L. J. Le Conte	For San Francisco. See T. A. S. C. E. v. 54, p. 198	i = 7 t½
Sherman	Maximum for Boston, Mass.	i = 25.12 t.687
Sherman	Extraordinary for Boston, Mass.	i = 18 t ½
Webster	Ordinary for Philadelphia, Pa.	i = 12 t0.6
Hendrick	Ordinary storms for Baltimore. Eng. & Cont., Aug. 9. 1911	i = 105 t + 10
J. de Bruyn-Kops	Ordinary storms for Savannah, Ga.	i = 163 t + 27
C. D. Hill	For Chicago, Ill.	i = 120 t + 15
Metcalf and Eddy	Louisville, Ky. Am. Sew. Prac., Vol I.	i = 14 t½
W. W. Horner	St. Louis, Mo. Eng. News, Sept. 29, 1910	i = 56 (t + 5).85
R. A. Brackenbuy	For Spokane, Wash. Eng. Record, Aug. 10, 1912	i = 23.92 t + 2.15 + 0.154
Metcalf and Eddy	New Orleans	i = 19 t½
Metcalf and Eddy	For Denver, Colo.	i = 84 t + 4
Kenneth Allen	Central Park, N. Y. 51–Year Record. Eng. News-Record, April 7, 1921, p. 588	i = 400 2t + 40[23]

30. Time of Concentration.—By the time of concentration is meant the longest time without unreasonable delay that will be required for a drop of water^[24] to flow from the upper limit of a drainage area to the outlet. Assuming a rainfall to start suddenly and to continue at a constant rate and to be evenly distributed over a drainage area of 100 per cent imperviousness and even slope towards one point, the rate of run-off would increase constantly until the drop of water from the upper limit of the area reached the outlet, after which the rate of run-off would remain constant. In nature the rate of rainfall is not constant. The shorter the duration of a storm the greater the intensity of rainfall. Therefore the maximum run-off during a storm will occur at the moment when the upper limit of the area has commenced to contribute. From that time on the rate of run-off will decrease.

The time of concentration can be measured fairly well by observing the moment of the commencement of a rainfall, and the time of maximum run-off from an area on which the rain is falling. A prediction of the time of concentration is more or less guess work. As the result of measurements some engineers assume the time of concentration on a city block built up with impervious roofs and walks, and on a moderate slope, is about 5 to 10 minutes. This is used as a basis for the judgment of the time of concentration on other areas. For relatively large drainage areas such a method cannot be used. The procedure is to measure the length of flow through the drainage channels of the area, to assume the velocity of the flood crest through these channels and thus to determine the time of concentration. Table 14 shows the flood crest velocities in various streams of the Ohio River Basin under flood conditions. The velocity over the surface of the ground may be approximated by the use of the formula^[25]

V = 2,000I√S,

in which V =the velocity of flow over the surface of the ground in feet per minute;I =the percentage imperviousness of the ground;S =the slope of the ground.

For areas up to 100 acres where natural drainage channels are not existent this formula will give more satisfactory results than guesses based on the time of concentration of certain known areas.

Having determined the time of concentration, the rate of rainfall R to be used in the Rational Method is found by substitution in some one of the rainfall formulas given in Table 13.

TABLE 14
Flood Crest Velocities in Ohio River Basin in March, 1913
From Table 12. U. S. G. S., Water Supply Paper. No. 334
River	Stations	Distance between Stations in Miles	Distance to Mouth of River, Miles	Distance of Lower Station below Starting-point, Miles	Velocity between Stations, Miles per Hour	Velocity from Pittsburgh, Miles per Hour	Time between Stations in Hours
Ohio	Pittsburgh, Pa., to Wheeling, W. Va.	90	967	90	9.0	9.0	10.0
Ohio	Wheeling, W. Va., to Marietta, Ohio	82	877	172	5.9	7.2	14
Ohio	Marietta, Ohio, to Parkersburg, W. Va.	12	795	184	0.9	4.8	14
Ohio	Parkersburg to Point Pleasant, W. Va.	80	783	264	6.7	5.3	12
Ohio	Point Pleasant to Huntington, W. Va.	44	703	308	11.0	5.7	4
Ohio	Huntington to Catlettsburg, W. Va.	9	659	317	0.8	4.1	11
Ohio	Catlettsburg, W. Va., to Portsmouth, Ohio	38	650	355		5.0
Ohio	Portsmouth Ohio, to Maysville, Ky.	52	612	407	5.2	5.0	10
Ohio	Maysville, Ky., to Cincinnati, Ohio	61	560	468	6.8	5.2	9
Ohio	Cincinnati, Ohio, to Louisville, Ky.	136	499	604	11.4	5.9	12
Ohio	Louisville, Ky., to Evansville, Ind.	183	363	787	1.9	5.3	96
Ohio	Evansville, Ind., to Mt. Vernon Ind.	36	180	823	9.0	5.3	4
Ohio	Mt. Vernon, Ind., to Paducah, Ky.	101	144	924	2.1	4.6	48
Ohio	Paducah, Ky. to Cairo, Ill.	43	43	967	2.9	4.2	15
Monongahela	Fairmont, W. Va., to Lock No. 2 Pa. (Upper)	107	119	107	6.7		16
Little Kanawha	Creston, W. Va., to Dam. No. 4 W. Va. (Upper)	16	48	16	16.0		1
New	Radford, W. Va., to Hinton, W. Va.	78	139	78	3.0		26
Kanawha	Kanawha Falls, W. Va. to Charleston, W. Va.	37	95	37	2.6		14
Scioto	Columbus, Ohio, to Chillicothe, Ohio	52	110	52	4.7		11
Miami	Dayton, Ohio, to Hamilton, Ohio	44	77	44	14.7		3
Kentucky	Highbridge, Ky., to Frankfort, Ky.	52	117	52	5.2		10
Cumberland	Celina, Tenn. to Nashville, Tenn.	190	383	190	2.9		64.5
Tennessee	Knoxville to Chattanooga, Tenn.	183	635	183	3.2		57
Note.—The velocities shown are the velocities of the crest of the flood wave and are not the average velocity of the flow of the river. The velocity of the crest of the flood wave should be used in determining the time of concentration. The flood crest velocity is slower then that of the river because of the storage in the river basin.

31. Character of Surface.—The proportion of total rainfall which will reach the sewers depends on the relative porosity, or imperviousness, and the slope of the surface. Absolutely impervious surfaces such as asphalt pavements or roofs of buildings will give nearly 100 per cent run-off regardless of the slope, after the surfaces have become thoroughly wet. For unpaved streets, lawns, and gardens the steeper the slope the greater the per cent of run-off. When the ground is already water soaked or is frozen the per cent of run-off is high, and in the event of a warm rain on snow covered or frozen ground, the run-off may be greater than the rainfall. The run-off during the flood of March, 1913, at Columbus, Ohio, was over 100 per cent of the rainfall. Table 15^[26] shows the relative imperviousness of various types of surfaces when dry and on low slopes. The estimates for relative imperviousness used in the design of the Cincinnati intercepter are given in Table 16.

TABLE 15
Values of Relative Imperviousness
Roof surfaces assumed to be water-tight	0.70–	0.95
Asphalt pavements in good order	.85–	.90
Stone, brick, and wood-block pavements with tightly cemented joints	.75–	.85
The same with open or uncemented joints	.50–	.70
Inferior block pavements with open joints	.40–	.50
Macadamized roadways	.25–	.60
Gravel roadways and walks	.15–	.30
Unpaved surfaces, railroad yards, and vacant lots	.10–	.30
Parks, gardens, lawns, and meadows, depending on surface slope and character of subsoil	.05–	.25
Wooded areas or forest land, depending on surface slope and character of subsoil	.01–	.20
Most densely populated or built up portion of a city	.70–	.90

TABLE 16
Coefficients of Imperviousness Used in the Design of the Cincinnati Sewers
Character of Improvement	Typical Commercial Area, 30.4 A. None Undeveloped. Sand and Gravel	Combined Tenement and Industrial. 35.6 A., 55 per Acre. Clay, Sand and Gravel	Residential, 291.1 A. 20 per Acre, Middle Class, Detached Dwellings, Yellow and Blue Clay Overlying Beds of Shale and Sandstone
Area in 1000’s Square Feet	Per Cent Total Area	I, Estimated	Equivalent Imp. Area, 1000’s Square Feet	Area in 1000’s Square Feet	Per Cent Total Area	I, Estimated	Per Cent of Total Area	I, Estimated
Roofs:
	Public and commercial	881.2	66.5	0.90	793.0	66.8	4.3	0.40	4.8	0.40
	Residences					289.2	18.6	.90	13.1	.90
	Barns and sheds					79.2	5.1	.75	1.4	.75
Interior Walks:
	Brick	7.5	0.6	.40	3.0	35.6	2.3	.40	0.6	.40
	Cement	10.0	0.7	.75	7.5	22.6	1.5	.75	2.6	.75
Street Walks:
	Brick	6.1	0.5	.40	2.4	48.2	3.1	.40	1.0	.40
	Cement	139.3	10.5	.75	104.5	78.1	5.0	.75	3.4	.75
Street Pavements:
	Asphalt, brick, wood block	145.5	11.0	.85	123.7				5.0	.85
	Granite block	111.4	8.4	.75	83.6				1.0	.75
	Macadam and cobble	23.2	1.8	.40	9.3	238.6	15.4	.40	4.8	.40
	Granite and poor macadam								0.4	.20
Unimproved yards and lawns:					692.4	44.7	.15
	Tributary to paved gutters								57.1	.15
	Not tributary to paved gutters								7.9	.10
	Total	1324.2	100.0		1127.0	1550.7	100.0		100.0
Impervious coefficient for the district	85.1	44.4	35.9

C. E. Gregory^[27] states that I, in the expression Q = AIR is a function of the time of concentration or the duration of the storm. If t represents the time of concentration and T represents the duration of the storm, then when T is less than t

I = 0.175t^⅓,

but when T is greater than t,

I = 0.175 t(T^{4 3} − (T − t)^{4 3}).

Gregory condenses Kuichling’s rules with regard to the per cent run-off, as follows:

1. The per cent of rainfall discharged from any given drainage area is nearly constant for heavy rains lasting equal periods of time.

2. This per cent varies directly with the area of impervious surface.

3. This per cent increases rapidly and directly or uniformly with the duration of the maximum intensity of the rainfall until a period is reached which is equal to the time required for the concentration of the drainage waters from the entire area at the point of observation, but if the rainfall continues at the same intensity for a longer period this per cent will continue to increase at a much smaller rate.

4. This per cent becomes larger when a moderate rain has immediately preceded a heavy shower on a partially permeable territory.

Gregory’s formulas have not been generally accepted and are not widely used in practice. Marston stated:^[28]

All that engineers are at present, warranted in doing is to make some deduction from 100 per cent run-off... the deduction... being at present left to the engineer in view of his general knowledge and his familiarity with local conditions.

Burger states^[29] in the same connection:

In its application there will usually be as many results (differing widely from each other) as the number of men using it.

In spite of these objections the Rational Method is in more favor with engineers than any other method.

32. Empirical Formulas.—The difficulty of determining run-off with accuracy has led to the production by engineers of many empirical formulas for their own use. Some of these formulas have attracted wide attention and have been used extensively, in some cases under conditions to which they are not applicable. In general these formulas are expressions for the run-off in terms of the area drained, the relative imperviousness, the slope of the land, and the rate of rainfall.

The Burkli-Ziegler formula, devised by a Swiss engineer for Swiss conditions and introduced into the United States by Rudolph Hering, was one of the earliest of the empirical formulas to attract attention in this country. It has been used extensively in the form

in whichQ =the run-off in cubic feet per second;i =the maximum rate of rainfall in inches per hour over the entire area. This is determined only by experience in the particular locality, and is usually taken at from 1 to 3 inches per hour;S =the slope of the ground surface in feet per thousand,A =the area in acres;C =an expression for the character of the ground surface, or relative imperviousness. In this form of the expression C is recommended as 0.7.

The McMath formula was developed for St. Louis conditions and was first published in Transactions of the American Society of Civil Engineers, Vol. 16, 1887, p. 183. Using the same notation as above, the formula is,

McMath recommended the use of C equal to 0.75, i as 2.75 inches per hour, and S equal to 15. The formula has been extended for use with all values of C, i, S, and A ordinarily met in sewerage practice. Fig. 11 is presented as an aid to the rapid solution of the formula.

Fig. 11.—Diagram for the Solution of McMath’s Formula,