Fig. 10.—Daily and Hourly Variations of Sewage Flow.
The fluctuations of flow in commercial and industrial districts are so different from those in residential districts that the formulas given should not be used in the design of sewers other than those draining residential areas. It is reasonable to suppose that fluctuations in rates of flow from industrial districts are dependent upon the character of the tributary industries. A study of these industries will give valuable light on the maximum and minimum rates at which sewage will be delivered to the sewers.
Hourly, daily, and seasonal fluctuations in rates of sewage flow are of interest in the design of pumping stations to give knowledge of the rates at which the pumps must operate at various periods. The fluctuations in rates of sewage flow during various hours and days in different cities and districts are shown in Fig. 10. Fluctuations in rate of flow of sewage lag behind fluctuations in rate of water consumption, the time being dependent on the distance through which the wave of change must travel in the sewer.
26. Effect of Ground Water.—Sewers are seldom laid with water-tight joints. Since they usually lie below the ground water level it is inevitable that a certain amount of ground water will enter. Various units have been suggested for the expression of the inflow of ground water in an attempt to include all of the many factors. Some of these units are: gallons per acre drained by the sewer per day, gallons per mile of pipe per day, gallons per inch diameter per mile of pipe per day, etc. Since the ground water enters pipe sewers at the joints, the longer the joints the greater the probability of the entrance of ground water. The last unit is therefore the most logical but the accuracy of the result is scarcely worthy of such refinement and the unit usually adopted is gallons per mile of pipe per day.
No definite figure can be given for the amount of ground water to be expected in sewers since the character of the soil and the ground water pressure must be considered. Relatively normal infiltration may be found from 5,000 to 80,000 gallons per mile of pipe per day. The minimum is seldom reached in wet ground and the maximum is frequently exceeded. Table 12 shows the amount of ground water measured in various sewers as given by Brooks.[21]
27. Résumé of Method for Determination of Quantity of Dry weather Sewage.—The steps in the determination of the quantity of sewage are: determine the period in the future for which the sewers are to be designed; estimate the population and tributary area at the end of this period; estimate the rate of water consumption and assume the sewage flow to equal the water consumption; determine the maximum and minimum rates of sewage flow; and finally, estimate the maximum rate of ground water seepage and add it to the maximum rate of sewage flow to give the total quantity of sewage to be carried by the proposed sewers.
| TABLE 12 | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| Data on the Infiltration of Ground Water into Sewers | |||||||||
| Abstracted from paper by J. N. Brooks in Transactions Am. Society of Civil Engineers, Vol. 76, p. 1909. | |||||||||
| Place | Shape | Diameter or Dimensions in Inches | Material | Wet Trench, Per Cent of Total Length | Avg. Head of Ground Water, Fee | Character of Subgrade | Gallons per 24 Hours | ||
| Per Foot of Joint | Per Inch Diameter Per Mile of Pipe | Per Mile of Pipe | |||||||
| Boston, Mass. | Circ. | 8 to 36 | V.P. | 2.6 | 1,818 | 40,000 | |||
| East Orange, N. J. | 10 | Q. | 22,400 | ||||||
| East Orange, N. J. | 8 to 24 | V.P. | 0.8 | 540 | 8,650 | ||||
| Joint trunk sewer, New Jersey | G. & Q. | 25,000 | |||||||
| Rogers Park, Ill. | 6 | 0.3 | 207 | 1,240 | |||||
| Altoona, Pa. | 30 | 5.0 | 2,890 | 86,592 | |||||
| Concord, Mass. | 18 | 8 | 43,000 | ||||||
| Malden, Mass. | Circ. | V.P. | 60 | 50,000 | |||||
| Westboro, Mass. | 15 | V.P. | 100 | 88,100 | 1,320,300 | ||||
| Fond du Lac, Wis. | Circ. | 24 | V.P. | 100 | 5 | C. | 1.5 | 1,010 | 24,370 |
| East Orange, N. J. | Circ. | 10 to 24 | V.P. | 100 | 4.7 | 2,540 | 43,250 | ||
| Ocean Grove, N. J. | Circ. | 4 to 12 | V.P. | 100 | 3 | S.C. | 2.7 | 1,890 | 15,126 |
| Ocean Grove, N. J. | Circ. | 4 to 12 | V.P. | 100 | 4 | S.C. | 7.9 | 5,480 | 43,764 |
| East Orange, N. J. | Rect. | 24 × 36 | Brick | 100 | 570,000 | ||||
| Westboro, Mass. | Brick | 415,850 | |||||||
| Altoona, Pa. | Rect. | 33 × 44 | B. & C. | 5,390 | 264,000 | ||||
| Columbus, Ohio. | H.S. | 42 × 42 | Concrete | 120 | 6,340 | ||||
| Bronx Valley, N. Y. | Circ. | 44 to 72 | Concrete | G. | 123 | 7,266 | |||
| Cincinnati, Ohio. | Estimated in design. Data not from Brooks | 67,500 | |||||||
| Milwaukee, Wis. | Residential districts, gals. per acre per day. Not taken from Brooks | 1460 to 2200 | |||||||
| Abbreviations: H.S. = horseshoe shaped; B. & C = Brick and concrete; V.P. = vitrified pipe; G. = gravel; Q. = quicksand; S. C. = sand clay; C. = clay. | |||||||||
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,
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]
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
but when T is greater than t,
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
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,
Other formulas have been devised which are more applicable to drainage areas of more than 1,000 acres.[30] Such areas are met in the design of sewers to enclose existing stream channels draining large areas. Kuichling’s formulas, published in 1901 in the report of the New York State Barge Canal, were devised for areas greater than 100 square miles. The following modification of these formulas for ordinary storms on smaller areas was published for the first time in American Sewerage Practice, Volume I, by Metcalf and Eddy:
Fig. 12.—Comparison of Empirical Run-off Formulas.
It is to be noted that the only factor taken into consideration is the area of the watershed. It is obvious that other factors such as the rate of rainfall, slope, imperviousness, etc., will have a marked effect on the run-off.
There are other run-off formulas devised for particular conditions, some of which are of as general applicability as those quoted. Two formulas which are frequently quoted are: Fanning’s, Q = 200M⅝ and Talbot’s Q = 500M¼, in which M is the area of the watershed in square miles. A comprehensive treatment of the subject is given in American Sewerage Practice, Vol. I, by Metcalf and Eddy.
A comparison of the results obtained by the application of a few formulas to the same conditions is shown graphically in Fig. 12. It is to be noted that the divergence between the smallest and largest results is over 100 per cent. As these formulas are not all applicable to the same conditions, the differences shown are due partially to an extension of some of them beyond the limits for which they were prepared.
33. Extent and Intensity of Storms.—In the design of storm sewers it is necessary to decide how heavy a storm must be provided for. The very heaviest storms occur infrequently. To build a sewer capable of caring for all storms would involve a prohibitive expense over the investment necessary to care for the ordinary heavy storms encountered annually or once in a decade. This extra investment would lie idle for a long period entailing a considerable interest charge for which no return is easily seen. The alternative is to construct only for such heavy storms as are of ordinary occurrence and to allow the sewers to overflow on exceptional occasions. The result will be a more frequent use of the sewerage system to its capacity, a saving in the cost of the system, and an occasional flooding of the district in excessive storms. The amount of damage caused by inundations must be balanced against the extra cost of a sewerage system to avoid the damage. A municipality which does not provide adequate storm drainage is liable, under certain circumstances, for damages occasioned by this neglect. It is not liable if no drainage exists, nor is it liable if the storm is of such unusual character as to be classed legally as an act of God.
Kuichling’s studies of the probabilities of the occurrence of heavy storms are published in Transactions of the American Society of Civil Engineers, Vol. 54, 1905, p. 192. Information on the extent of rain storms is given by Francis in Vol. 7, 1878, p. 224, of the same publication. Kuichling expresses the intensity of storms which will occur,