Quantity of Storm Water

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 t^½
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 t^0.6
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 t^½
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 t^½
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
Character of Improvement		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,

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:

Q = 25,000
A + 125 + 15.

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 = 200M^⅝ and Talbot’s Q = 500M^¼, 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,

once in 10 years as i = 105
t + 20,

once in 15 years as i = 120
t + 20,

in which i is the intensity of rainfall in inches per hour and t is the duration of the storm in minutes.

CHAPTER IV
THE HYDRAULICS OF SEWERS

34. Principles.—The hydraulics of sewers deals with the application of the laws of hydraulics to the flow of water through conduits and open channels. In so far as its hydraulic properties are concerned the characteristics of sewage are so similar to those of water that the same physical laws are applicable to both. In general it is assumed that the energy lost due to friction between the liquid and the sides of the channel varies as some function of the velocity, usually the square, and that the total energy passing any section of the stream differs from the energy passing any other section only by the loss of energy due to friction.

The general expression for the flow of sewage would then be,

h = (f)Vⁿ,

in which h is the head or energy lost between any two sections, and V is the average velocity of flow between these sections. It is to be noted in this general expression that the quantity and rate of flow past all sections is assumed to be constant. This condition is known as steady flow. Problems are encountered in sewerage design which involve conditions of unsteady flow, and methods of solution of them have been developed based on modifications of this general expression. The average velocity of flow is computed by dividing the rate (quantity) of flow past any section by the cross-sectional area of the stream at that section. This does not represent the true velocity at any particular point in the stream, as the velocity near the center is faster than that near the sides of the channel. The distribution of velocities in a closed circular channel is somewhat in the form of a paraboloid superimposed on a cylinder.

The laws of flow are expressed as formulas the constants of which have been determined by experiment. It has been found that these constants depend on the character of the material forming the channel and the hydraulic radius. The hydraulic radius is defined as the ratio of the cross-sectional area of the stream to the length of the wetted perimeter, or line of contact between the liquid and the channel, exclusive of the horizontal line between the air and the liquid.

35. Formulas.—The loss of head due to friction caused by flow through circular pipes flowing full as expressed by Darcy is,

h = fl
d V²
2g,

in which h is the head lost due to friction in the distance l, V is the velocity of flow, g is the acceleration due to gravity, and f is a factor dependent on d and the material of which the pipe is made. A formula for f expressed by Darcy as the result of experiments on cast-iron pipe is,

f = 0.0199 + 0.00166
d,

in which d is the diameter in feet. In using the formula with this factor the units used must be feet and seconds.

Another form of the same expression is known as the Chezy formula. It is an algebraic transformation of the Darcy formula, but in the form shown here, by the use of the hydraulic radius, it is made applicable to any shape of conduit either full or partly full. The Chezy formula is,

V = C√RS,

in which R is the hydraulic radius, S the slope ratio of the hydraulic gradient, and C a factor similar to f in the Darcy formula.

Kutter’s formula was derived by the Swiss engineers, Ganguillet and Kutter, as the result of a series of experimental observations. It was introduced into the United States by Rudolph Hering and its derivation is given in Hering and Trautwine’s translation of “The Flow of Water in Open Channels by Ganguillet and Kutter.” In English units it is,

in which n is a factor expressing the character of the surface of the conduit and the other notation is as in the Chezy formula. V is the velocity in feet per second, S is the slope ratio, and R the hydraulic radius in feet. The values of n to be used in all cases are not agreed upon, but in general the values shown below are used in practice.

Values of n in Kutter’s Formula

n		Character of the Materials

0.009		Well-planed timber.
0.010		Neat cement or very smooth pipe.
0.012		Unplaned timber. Best concrete.
0.013		Smooth masonry or brickwork, or concrete sewers under ordinary conditions.
0.015		Vitrified pipe or ordinary brickwork.
0.017		Rubble masonry or rough brickwork.
0.020 0.035	}	Smooth earth.
0.030 0.050	}	Rough channels overgrown with grass.

Kutter’s formula is of general application to all classes of material and to all shapes of conduits. It is the most generally used formula in sewerage design.

The cumbersomeness of Kutter’s formula is caused somewhat by the attempt to allow for the effect of the low slopes of the Mississippi River experiments on the coefficients. The correctness of these experiments has not been well established and the slopes are so flat that the omission of the term 0.0028
S will have no appreciable effect on the value of V ordinarily used in sewer design. The difference between the value of V determined by the omission of this term and the value of V found by including it is less than 1 per cent for all slopes greater than 1 in 1,000 for 8 inch pipe (R = 0.167 feet). As the diameter of the pipe or the hydraulic radius of the channel increases up to a diameter of 13.02 feet (R = 3.28 feet), the difference becomes less and at this value of R there is no difference whether the slope is included or not. For larger pipes the difference increases slowly. For a 16 foot pipe (R = 4 feet) on a slope of 1 in 1,000 the difference is less than 0.2 per cent, and on a slope of 1 in 10,000 the difference is approximately 1 per cent. Flatter slopes than these are seldom used in sewer design, except for very large sewers where careful determinations of the hydraulic slope are necessary. It is therefore safe in sewer design to use Kutter’s formula in the modified form shown below in which the term .0028
S has been omitted.

Bazin’s formula is

in which α and β are constants for different classes of material. For cast-iron pipe α is 0.00007726 and β is 0.00000647. This formula is seldom used in sewerage design.

Exponential formulas have been developed as the result of experiments which have demonstrated that V does not vary as the one-half power of R and S but that the relation should be expressed as,

V = CR^pS^q,

in which p and q are constants and C is a factor dependent on the character of the material. The various formulas coming under this classification have been given the names of the experimenters proposing them. Examples of these formulas are: Flamant’s, in English units, for new cast-iron pipe, which is,

V = 232R^.715S^.572,

and Lampé’s for the same material which is,

V = 203.3R^.694S^.555.

These formulas are useful only for the material to which they apply, but they can be used for conduits of any shape. A. V. Saph and E. W. Schoder have shown [^[31]] that the general formula for all materials lies between the limits,

V = (93 to 142)S^{.50 to .55}R^{.63 to .69}.

Hazen and Williams’ formula is in the form,

V = 1.31CR^.63S^.54,

in which C is a factor dependent on the character of the material of the conduit. The values of C as given by Hazen and Williams are,

C	Character of Material
95	Steel pipe under future conditions. (Riveted steel.)
100	Cast iron under ordinary future conditions and brick sewers in good condition.
110	New riveted steel, and cement pipe.
120	Smooth wood or masonry conduits under ordinary conditions.
130	Masonry conduits after some time and for very smooth pipes such as glass, brass, lead, etc., when old, and for new cast-iron pipe under ordinary conditions.

This formula is of as general application as Kutter’s formula and is easier of solution, but being more recently in the field and because of the ease of the solution of Kutter’s formula by diagrams it is not in such general use. Exponential formulas are used more in waterworks than in sewerage practice.

Manning’s formula is in the form,

V = 1.486
nR^⅔S^½

in which n is the same as for Kutter’s formula. Charts for the solution of Manning’s formula are given in Eng. News-Record, Vol. 85, 1920, p. 837.

36. Solution of Formulas.—The solution of even the simplest of these formulas, such as Flamant’s, is laborious because of the exponents involved. Darcy’s and Kutter’s formulas are even more cumbersome because of the character of the coefficient. The labor involved in the solution of these formulas has resulted in the development of a number of diagrams and other short cuts. Since each formula involves three or more variables it cannot be represented by a single straight line on rectangular coordinate paper. The simplest form of diagram for the solution of three or more variables is the nomograph, an example of which is shown in Fig. 13 for the solution of Flamant’s formula. A straight-edge placed on any two points of the scales of two different vertical lines will cross the other line at a point on the scale corresponding to its correct value in the formula. Such a diagram is in common use for the solution of problems for the flow of water in cast-iron pipe.

Fig. 13.—Diagram for the Solution of Flamant’s Formula for the Flow of Water in Cast-iron Pipe.

Fig. 14 has been prepared to simplify the solution of Hazen and Williams’ formula. The scales of slope for different classes of material are shown on vertical lines to the left of the slope line. For use these scales must be projected horizontally on the slope line. The scales for other factors are shown on independent reference lines.

For example let it be required to find the loss of head in a 12 inch pipe carrying 1 cubic foot per second when the coefficient of roughness is 100. A straight-edge placed at 1.0 cubic feet per second on the quantity scale, and 12 inches on the diameter scale crosses the slope line at .00092 opposite the slope scale for c = 100. It crosses the velocity line at 1.31 feet per second.

Kutter’s formula is the most commonly used for sewer design and has been generally accepted as a standard in spite of its cumbersomeness. Fig. 15 is a graphical solution of Kutter’s formula for small pipes, and Fig. 16 for larger pipes. The diagrams are drawn on the nomographic principle and give solutions for a wide range of materials, but they are specially prepared for the solution of problems in which n = .015. In their preparation the effect of the slope on the coefficient has been neglected. Fig. 17 is drawn on ordinary rectangular coordinate paper and can be used only for the solution of problems in which n = .015. Both diagrams are given for practice in the use of the different types.

Fig. 14.—Diagram for the Solution of Hazen and Williams’ Formula.

Fig. 15.—Diagram for the Solution of Kutter’s Formula.
For values of n between 0.010 and 0.020. Specially arranged for n = 0.015. Values of Q from 0.1 to 10 second-feet.

Fig. 16.—Diagram for the Solution of Kutter’s Formula.
For values of n between 0.010 and 0.020. Specially arranged for n = 0.015. Values of Q from 10 to 1,000 second-feet.

Fig. 17.—Diagram for the Solution of Kutter’s Formula.

Fig. 18.—Conversion Factors for Kutter’s Formula.

In Figs. 15 and 16 the diameter scales are varied for different values of the roughness coefficient n. The velocity scale is shown only for a value of n = .015. The velocity for other values of n can be determined by the method given in the following paragraphs.

37. Use of Diagrams.—There are five factors in Kutter’s formula: n, Q, V, d (or R), and S. If any three of these are given the other two can be determined, except when the three given are Q, V, and d. These three are related in the form Q = AV, which is independent of slope or the character of the material. There are only nine different combinations possible with these five factors, which will be met in the solution of Kutter’s formula. The solution of the problems by means of the diagrams is simple when the data given include n = .015. For other given values of n the solution is more complicated. Results of the solution of types of each of the nine problems are given in Table 17 and the explanatory text below.

If n is given and is equal to .015, the solution is simple.

For example in Table 17 case 1, example 1; to be solved on Fig. 15. Place a straight-edge at 1.0 on the Q line and at 6 inches on the diameter line for n = .015. The slope and the velocity will be found at the intersection of the straight-edge with these respective scales.

All problems in which n is given as .015 and the solution for which falls within the limits of Fig. 15 or 16 should be solved by placing a straight-edge on the two known scales and reading the two unknown results at the intersection of the straight-edge and the remaining scales.

For example in case 1, example 2 find the intersection of the horizontal line representing Q = 100 with the sloping diameter line representing d = 48 inches. The vertical slope line passing through this point represents S = .0065 and the sloping velocity line passing through this point represents 8.5 feet per second.

In general problems in which n = .015, can be solved on Fig. 17 by finding the intersection of the two lines representing the given data, and reading the values of the remaining variables represented by the other two lines passing through this point.

TABLE 17

Solutions of Problems by Kutter’s Formula

Case	Example	Given					Found
Case	Example	n	Q	V	d	S	n	Q	V	d	S
1	1	0.015	1.0	2.5	6				5.0		0.0575
1	2	.015	100.0						8.5		.0065
1	3	.020	1.0		6				5.0		.13
1	4	.020	100.0		48				8.5		.0125
2	1	.015	5.0			0.0003			1.2	28
2	2	.010	5.0			.0003			1.7	23.5
3	1	.015			18	.002		4.0	2.25
3	2	.018			18	.0008		2.0	1.1
4	1	.015	2.0	2.5						12	.00475
4	2	.011	2.0	2.5						12	.0022
5	1	.015		5.0	36			35.0			.0038
6	1	.018		5.0		.001		185.0		80
7	1		3.0		18	.002	0.019		1.7
7	2		50.0		36	.005	.012		7.0
8	1		6.0	2.5		.003	.018			21
9	1			4.2	66	.00059	.011	100.0

If n is given and is not equal to .015 the solution is not so simple. In Fig. 15 and 16 the diagram is so drawn that the position of the diameter scales for all values of n is fixed on the vertical “diameter” line. The scales of diameter change for each value of n. These scales of diameter are shown for each value of n from .010 to .020 on vertical lines to the left of the “diameter” line. For use, the proper diameter scale for any given value of n must be projected horizontally upon the vertical “diameter” line. The velocity can be determined on Fig. 15 and 16, only when the diameter has been determined and then only when the diameter scale for n equal .015 is used, since the only scale shown for velocity is for n = .015.

For example, in Case 1, Example 3 there are given n = .020, Q, and d. Find the intersection of the vertical line for n = .020 with the sloping diameter line for d = 6 inches. Project the intersection horizontally to the right to the vertical “diameter” line. Place a straight-edge at this point and at Q = 1.0 on the quantity scale. The required value of S is read at the intersection of the straight-edge and the slope scale and is equal to 0.13. The intersection of the straight-edge in this position with the velocity scale is not the required value of the velocity since the velocity scale is made out for n = .015 and not .020. It is necessary to change the position of the straight-edge so that it may lie on Q equal 1.0 and on d equal 6 inches for n equal .015. The value of V is shown in this position as 5 feet per second.

The reverse process for Fig. 15 and 16 is illustrated by Case 4, Example 2 in which n = .011 and Q and V are also given. When Q and V are given the value of d is fixed independent of all other factors. Therefore the value of d can be read from the scale with n = .015 and is found to be 12 inches. Now find the value of d = 12 inches on the scale for n = .011 and project on to the “diameter” line. Place the straight-edge at this point and at Q = 2. The required slope is read as .0022.

Fig. 17 is prepared for the solution of problems in which n = .015 only. For problems in which n has some other value it is necessary to transform the data to equivalent conditions in which n = .015. This is done by means of the conversion factors shown in Fig. 18. The given slope or velocity is multiplied by the proper factor to convert from or to the value of n = .015.

For example in Case 1, Example 4 there are given n = .020, Q, and d. With Q and d given the value of V can be read from Fig. 17 without conversion. The corresponding value of S for n = .015 is .0065. It is now necessary to use the transformation diagram Fig. 18. The hydraulic radius of the given pipe is one foot. On Fig. 18 at the intersection of the slope line for R = 1.0 foot and n = .020 the value of the factor is read as 1.92. Since the given n is for rougher material than that represented by n = .015 the required slope must be greater than for n = .015 to give the same velocity. It is therefore necessary to multiply .0065 × 1.92 and the required slope is .0125.

In Case 6, Example 1 there are given n = .018, d, and S. The remaining factors are to be solved by Fig. 17. Solve first as though n = .015 in order to find an approximate value of d or R. In this case it is evident that d is greater than 57 inches. The value of R is therefore about 1.25. Referring to Fig. 18 the conversion factor for the slope for n = .018 is about 1.52. Since the given slope for n = .018 is .001, for an equal velocity and for n = .015 the slope should be less. Therefore in reading Fig. 17 it is necessary to use a slope of .001
1.52 = .00066. The diameter is found to be about 80 inches. Since this is nearer to the correct diameter the value of the conversion factor must be corrected for this approximation. The hydraulic radius for an 80 inch pipe is 1.67 feet, and the conversion factor from Fig. 18 is about 1.48. The slope for n = .015 should be therefore .001
1.48 = .000675 and from Fig. 17 the required diameter and quantity are read as 80 inches and 185 second-feet, respectively.

If n is not given but must be solved for, the solution on Fig. 15 and 16 is relatively simple. The desired value of n is read at the intersection of the sloping diameter line representing the known diameter and the horizontal projection of the intersection of the straight-edge with the vertical “diameter” line.

For example in Case 7, Example 1 there are given Q, d, and S. Lay the straight-edge on the given values of Q = 3 and S = .002. At the point where the straight-edge crosses the vertical “diameter” line project a horizontal line to the sloping diameter line for d = 18 inches. The vertical line passing through this point represents a value of n = .019. In order to find the value of V lay the straight-edge on Q = 3 and d = 18 inches for n = .015. The value of V is read as 1.7.

A slightly different condition is illustrated in the solution of Case 8, Example 1 in which Q, V and S are given. Determine first the value of d as though n = .015. Then proceed to determine n as in the preceding examples.

The solution for an unknown value of n on Fig. 17 is not so simple. It must be determined by working backwards from the conversion factor.

For example in Case 7, Example 2 there are given Q, d, and S. The value of V is read directly as though n = .015 as 7 feet per second. The value of S read for n = .015 is .0075. But the given slope is .005. Since the given slope is flatter than that for n = .015 the conversion factor is less than unity and is therefore .005
.0075 = 0.67. With this value of the conversion factor and the value of R given as 0.75 the value of n is read from Fig. 18 as slightly greater than .012.

38. Flow in Circular Pipes Partly Full.—The preceding examples have involved the flow in circular pipes completely filled. The same methods of solution can be used for pipes flowing partly full except that the hydraulic radius of the wetted section is used instead of the diameter of the pipe. Diagrams are used to save labor in finding the hydraulic radius and the other hydraulic elements of conduits flowing partly full.

The hydraulic elements of a conduit for any depth of flow are: (a) The hydraulic radius, (b) the area, (c) the velocity of flow, and (d) the quantity or rate of discharge. The velocity and quantity when partly full as expressed in terms of the velocity and quantity when full as calculated by Kutter’s formula will vary slightly with different diameters, slopes and coefficients of roughness. The other elements are constant for all conditions for the same type of cross-section. The hydraulic elements for all depths of a circular section for two different diameters and slopes are shown in Fig. 19. The differences between the velocity and quantity under the different conditions are shown to be slight, and in practice allowance is seldom made for this discrepancy.

In the solution of a problem involving part full flow in a circular conduit the method followed is to solve the problem as though it were for full flow conditions and then to convert to partial flow conditions by means of Fig. 19, or to convert from partial flow conditions to full flow conditions and solve as in the preceding section.

For example let it be required to determine the quantity of flow in a 12–inch diameter pipe with n = .015 when on a slope of .005 and the depth of flow is 3 inches. First find the quantity for full flow. From Fig. 15 this is 2.0 cubic feet per second. The depth of flow of 3 inches is one-fourth or 0.25 of the full depth of 12 inches. From Fig. 19, running horizontally on the 0.25 depth line to meet the quantity curve, the proportionate quantity at this depth is found to be on the 0.13 vertical line, and the quantity of flow is therefore 2 × 0.13 = 0.26 cubic feet per second.

Fig. 19.—Hydraulic Elements of Circular Sections.

d = 12′ 0″	s = .0004	n = .015
d = 1′ 0″	s = .01	n = .013

Another problem, involving the reversal of this process is illustrated by the following example:

Let it be required to determine the diameter and full capacity of a vitrified pipe sewer on a grade of 0.002 if the velocity of flow is 3.0 feet per second when the sewer is discharging at 30 per cent of its full capacity, the depth of flow being 12 inches. From Fig. 19 the depth of flow when the sewer is carrying 30 per cent of its full capacity is 0.38 of its full depth. Since the partial depth is 12 inches the full diameter is 12
.038 = 31.6 inches. The velocity of flow at 38 per cent depth is 86 per cent of the full velocity. Since the velocity given is 3.0 feet per second, the full velocity is 3.0
.86 = 3.5 feet per second. With a full velocity of 3.5 feet per second and a diameter of 31.6 inches from Fig. 16 the full capacity of the sewer is 18 cubic feet per second.

39. Sections Other than Circular.—The ordinary shape used for small sewers is circular. The difficulty of constructing large sewers in a circular shape, special conditions of construction such as small head room, soft foundations, etc., or widely fluctuating conditions of flow have led to the development of other shapes. For conduits flowing full at all times a circular section will carry more water with the same loss of head than any other section under the same conditions. In any section the smaller the flow the slower the velocity, an undesirable condition. The ideal section for fluctuating flows would be one that would give the same velocity of flow for all quantities. Such a section is yet to be developed. Sections have been developed that will give relatively higher velocities for small quantities of flow than are given by a circular section. The best known of these sections is the egg shape, the proportions and hydraulic elements of which are shown in Fig. 20. Other shapes that have the same property, but which were not developed for the same purpose are the rectangular, the U-shape, and the section with a cunette. The egg-shaped section has been more widely used than any other special section. It is, however, more difficult and expensive to build under certain conditions, and has a smaller capacity when full than a circular sewer of the same area of cross-section. Various sections are illustrated in Fig. 22 and 23.

The U-shaped section is suitable where the cover is small, or close under obstructions where a flat top is desirable and the fluctuations of flow are so great as to make advantageous a special shape to increase the velocity of low flows. The proportions of a U-shaped section are shown in Fig. 23 (6). Other sections used for the same purpose are the semicircular and special forms of the rectangular section.

The proportions and the hydraulic elements of the square-shaped section are shown in Fig. 21. This is useful under low heads where a flat roof is required to carry heavy loads, and the fluctuations of flow are not large.

Sections with cunettes have not been standardized. A cunette is a small channel in the bottom of a sewer to concentrate the low flows, as shown in Fig. 22 (7). A cunette can be used in any shape of sewer.

Fig. 20.—Hydraulic Elements of an Egg-shaped Section.
d = 6′ 0″ s = .00065 n = .015

Fig. 21.—Hydraulic Elements of a Square Section.
d = 10′ 0″ s = .0004 n = .015

Sections developed mainly because of the greater ease of construction under certain conditions are the basket handle, the gothic, the catenary, and the horse shoe. Some of these shapes are shown in Fig. 22 and 23. They are suitable for large sewers on soft foundations, where it is desirable to build the sewer in three portions, such as invert, side walls, and arch. They are also suitable for construction in tunnels where the shape of the sewer conforms to the shape of the timbering, or in open cut work where the shape of the forms are easier to support.

Problems of flow in all sections can be solved by determining the hydraulic radius involved, and substituting directly in the desired formula, or by the use of one of the diagrams after converting to the equivalent circular diameter. The determination of the hydraulic radius of these special sections is laborious, and hence other less difficult methods are followed. Problems are more commonly solved by converting the given data into an equivalent circular sewer, solving for the elements of this circular sewer and then reconverting into the original terms, or by working in the other direction. The hydraulic elements of various sections when full are given in Table 18.

TABLE 18

Hydraulic Elements of Sewer Sections. Sewers Flowing Full.

Section	Area in Terms Vertical Diameter Squared D²	Hydraulic Radius in terms of Vertical Dia. D	Vert. Dia. D in Terms of Dia. d of Equivalent Circular Section	Source
Circular	0.7854	0.250	1.000
Egg	0.5150	.1931	1.295	Eng. Record, Vol. 72: 608
Ovoid	0.5650	.2070	1.208	Eng. Record, Vol. 72: 608
Semi-elliptical	0.8176	.2487	1.041	Eng. News, Vol. 71: 552
Catenary	0.6625	.2237	1.1175	Eng. Record, Vol. 72: 608
Horseshoe	0.8472	.2536	0.985	Eng. Record, Vol. 72: 608
Basket handle	0.8313	.2553	0.979	Eng. Record, Vol. 72: 608
Rectangular	1.3125	.2865	0.7968	Hydraulic Dgms. and Tbls. Garrett
Square (3 sides wet)	1.0000	.333	0.7500	Eng. Record, Vol. 72: 608
Square (4 sides wet)	1.0000	.250	1.0000	Eng. Record, Vol. 72: 608

1. Standard Egg-shaped Section, North Shore Intercepter, Chicago, Illinois.

2. Rectangular Section, Omaha, Nebraska, Eng. Contracting, Vol. 46, p. 49.

3. Trench in firm ground. 4. Trench in Rock.
Note.—Underdrains and Wedges to be used only when Ordered by the Engineer.

7. Brick and Concrete Sewer showing cunette.

5. Soft Foundation. 6. Wet ground.

8. Brick and Concrete Sewer, Evanston, Ill., Eng. Contracting, Vol. 46, p. 227.

Fig. 22.

1. Tunnel Sections.				2. Open Cut Sections.
Type A.	Type B.	Type C.	Type D.
Where Rock is more than 16′ above Springing Line.	Where Rock is more than 7′ and less than 16′ above Springing Line on both Sides.	Where Rock is between Springing Line and 7′ above Springing Line on both Sides.	Where Rock drops below Springing Line on either Side.	16′ 6″ Sewer. 25′ Fill	Where Rock is above Springing Line

Mill Creek Sewer, St. Louis, Eng. Record, Vol. 70, pp. 434, 435.

3. Circular Concrete Section in Soft and Hard Ground, Eng. Record, Vol. 59, p. 570.

4. Semi-Elliptical Section, Louisville, Ky., Eng. News, Vol. 62, p. 416.

5. Reinforced Concrete Sewer, Harlem Creek, St. Louis, Eng. News, Vol. 60, p. 131.

6. U-Shaped Section, San Francisco, Eng. News, Vol. 73, p. 310.

Fig. 23.

Equivalent sections are sections of the same capacity for the same slope and coefficient of roughness. They have not necessarily the same dimensions, shape, nor area. The diameter of the equivalent circular section in terms of the diameter of each special section shown is given in Table 18. The inside height of a sewer is spoken of as its diameter.

For example let it be required to determine the rate of flow in a 54–inch egg-shaped sewer on a slope of 0.001 when n = .015. First convert to the equivalent circle. From Table 18 the diameter of the equivalent circle is 1
1.295 times the diameter of the egg-shaped sewer, which becomes in this case 43 inches. From Fig. 16 the capacity of a circular sewer of this diameter with S = 0.001 and n = .015 is 28 cubic feet per second, which by definition is the flow in the egg-shaped sewer.

As an example of the reverse process let it be required to find the velocity of flow in an egg-shaped sewer flowing full and equivalent to a 48–inch circular sewer. Both sewers are on a slope of 0.005 and have a roughness coefficient of n = .015. It is first necessary to find the quantity of flow in the circular sewer, which by definition is the quantity of flow in the equivalent egg-shaped sewer. The velocity of flow in the egg-shaped sewer is found by dividing this quantity by the area of the egg-shaped section. As read from the diagram the quantity of flow is 90 cubic feet per second. From Table 18 the area of the egg-shaped sewer is 0.51D² where D is the diameter of the egg-shaped sewer, and D = 1.295d where d is the diameter of the equivalent circular sewer. Therefore the area equals (0.51) × (1.295 × 4)² = 13.5 square feet and the velocity of flow is 90
13.5 = 6.7 feet per second. This is slightly less than the velocity in the circular section.

Some lines for egg-shaped sewers have been shown on Fig. 17 by which solutions can be made directly. For other shapes, and for sizes of egg-shaped sewers not found on Fig. 17 the preceding method or the original formula must be used for solution. Problems in partial flow in special sections are solved similarly to partial flow in circular sections, by converting first to the conditions of full flow or by working in the opposite direction.

40. Non-uniform Flow.—In the preceding articles it is assumed that the mean velocity and the rate of flow past all sections are constant. This condition is known as steady, uniform flow. In this article it will be assumed that conditions of steady non-uniform flow exist, that is, the rate of flow past all sections is constant, but the velocity of flow past these sections is different for different sections. Under such conditions the surface of the stream is not parallel to the invert of the channel. If the velocity of flow is increasing down stream the surface curve is known as the drop-down curve. If the velocity of flow is decreasing down stream the surface curve is known as the backwater curve. The hydraulic jump represents a condition of non-uniform flow in which the velocity of flow decreases down stream in such a manner that the surface of the stream stands normal to the invert of the channel at the point where the change in velocity occurs. Above and below this point conditions of uniform flow may exist.

Conditions of non-uniform flow exist at the outlet of all sewers, except under the unusual conditions where the depth of flow in the sewer under conditions of steady, uniform flow with the given rate of discharge would raise the surface of water in the sewer, at the point of discharge, to the same elevation as the surface of the body of water into which discharge is taking place. By an application of the principles of non-uniform flow to the design of outfall sewers, smaller sewers, steeper grades, greater depth of cover, and other advantages can be obtained.

The backwater curve is caused by an obstruction in the sewer, by a flattening of the slope of the invert, or by allowing the sewer to discharge into a body of water whose surface elevation would be above the surface of the water in the sewer, at the point of discharge, under conditions of steady, uniform flow with the given rate of discharge.

The drop-down curve is caused by a sudden steepening of the slope of the invert; by allowing a free discharge; or by allowing a discharge into a body of water whose surface elevation would be below the surface of the water in the sewer, at the point of discharge, under conditions of steady, uniform flow with the given rate of discharge. The last described condition is common at the outlet of many sewers, hence the common occurrence of the drop-down curve.

The hydraulic jump is a phenomenon which is seldom considered in sewer design. If not guarded against it may cause trouble at overflow weirs and at other control devices, in grit chambers, and at unexpected places. The causes of the hydraulic jump are sufficiently well understood to permit designs that will avoid its occurrence, but if it is allowed to occur the exact place of the occurrence of the jump and its height are difficult, if not impossible, to determine under the present state of knowledge concerning them. The hydraulic jump will occur when a high velocity of flow is interrupted by an obstruction in the channel, by a change in grade of the invert, or the approach of the velocity to the “critical” velocity. The “critical” velocity is equal to √(gh), where h is the depth of flow and g is the acceleration due to gravity. The velocity in the channel above the jump must be greater than √(gh₁), where h₁ is the depth of flow in the channel above the jump. The velocity in the channel below the jump must be greater than √(gh₂), where h₂ is the depth of flow below the jump. The jump will not take place unless the slope of the invert of the channel is greater than g
C²,in which C is the coefficient in the Chezy formula. With this information it is possible to avoid the jump by slowing down the velocity by the installation of drop manholes, flight sewers, or by other expedients.

The shape of the drop-down curve can be expressed, in some cases, by mathematical formulas of more or less simplicity, dependent on the shape of the conduit. The formula for a circular conduit is complicated. Due to the assumptions which must be made in the deduction of these formulas, the results obtained by their use are of no greater value than those obtained by approximate methods. A method for the determination of the drop-down curve is given by C. D. Hill.[^[32]] In this method it is necessary that the rate of flow past all sections shall be the same; that the depth of submergence at the outlet shall be known; and that the depth of flow at some unknown distance up the stream shall be assumed. The shape and material of construction of the sewer and the slope of the invert should also be known. The problem is then to determine the distance between cross-sections, one where the depth of flow is known, and the other where the depth of flow has been assumed. This distance can be expressed as follows:

L = (d₂ − d₁) − (H₁ − H₂)
S − S₁ = d′ − H′
S′,

in which L = the distance between cross-sections; d₁ = the depth of flow at the lower section; d₂ = the depth of flow at the upper section; H₁ = the velocity head at the lower section; H₂ = the velocity head at the upper section; S = the hydraulic slope of the stream surface; S₁ = the slope of the invert of the sewer.

In order to solve such problems with a satisfactory degree of accuracy the difference between d₁ and d₂ should be taken sufficiently small to divide the entire length of the sewer to be investigated into a large number of sections. The solution of the problem requires the determination of the wetted area, the hydraulic radius, and other hydraulic elements at many sections. The labor involved can be simplified by the use of diagrams, such as Fig. 19, or by specially prepared diagrams such as those accompanying the original article by C. D. Hill. The solution of the problem can be simplified by tabulating the computations as follows:

Drop-down Curve Computation Sheet

Uniform discharge. Varying depth

D = Q = A = V = Q A = S₁ = L = d₁ − H₁) S₁
1	2	3	4	5	6	7	8	9	10	11	12	13
Depth			R	H	H₁	d₁ − H₁	V	S	S₁	L	Elevation
D	d	d₁	R	H	H₁	d₁ − H₁	V	S	S₁	L	Sewer	W. L.

At the head of the computation sheet should be recorded the diameter of the sewer in feet, the assumed volume of flow, the area of the full cross-section of the sewer, the velocity of the assumed volume flowing through the full bore of the sewer, and the gradient or slope of the invert. In the 1st column enter the assumed depth in decimal parts of the diameter for each cross-section; in the 2nd column enter the same depth in feet; in the 3rd column enter the difference in feet between the successive cross-sections; in the 4th column enter the hydraulic radius corresponding to the depth at each cross-section; in the 8th column enter the velocity, equal to the volume divided by the wetted area, for each cross-section; in the 5th column enter the corresponding velocity head; in the 6th column enter the difference between the velocity heads at successive cross-sections; in the 7th column enter the difference between the quantities in the third and in the sixth columns; in the 9th column enter the hydraulic slope corresponding to the velocity and hydraulic radius of each cross-section; in the 10th column enter the difference between the hydraulic slope and the slope or gradient of the sewer; in the 11th column enter the computed distance between successive cross-sections; in the 12th column enter the elevation of the bottom of the sewer at each cross-section; and in the 13th column enter the corresponding elevation of the surface of the water.

The table should be filled in until the distance to the required section is determined, or if the distance is known, it should be filled in until the depth of flow with the assumed rate of discharge has been checked.

If only the depth of flow at some section is known and it is required to know the maximum rate of flow with a free discharge, or a discharge with a submergence at the outlet less than the depth of flow with the maximum rate of discharge, it is necessary to make a preliminary estimate of the maximum rate of flow in order to fill in the quantity Q at the head of the table. The procedure should be as follows:

1st. Assume a depth of flow at the outlet. 2nd. Compute the area (A) and the hydraulic radius (R) at the known section and at the outlet. 3rd. Determine the area and the hydraulic radius half way between these two sections as the mean of the areas and the hydraulic radii of the two sections. 4th. Determine the rate of flow through the sewer from the condition that the difference in head at the two sections is the head lost due to friction caused by the average velocity of flow between the sections (equals lV²
C²R) plus the gain in velocity head (equals V₂² − V₁²
2g), which then combined and transposed result in the expression:

in which Q = rate of flow; A = the area determined in the 3rd step; A₁ = the area at the upper cross-section; A₂ = the area at the lower cross-section; C = the coefficient in the Chezy formula; g = the acceleration due to gravity; h = the difference in elevation of the surface of the stream at the two cross-sections; l = the distance between the cross-sections; R = the hydraulic radius determined in the third step.

5th. Continue this process by assuming different depths at the outlet until the maximum rate of discharge has been found by trial.

With this rate of discharge and depth of flow at the outlet, the depth of flow at the known section can be checked. If appreciably in error a correction should be made by the assumption of a different depth of flow at the outlet. The approximate character of the method is scarcely worthy of the refinement in the results which will be obtained by checking back for the depth of flow at the known section. It will be sufficiently accurate to assume the rate of flow obtained by trial from the preceding expression, as the maximum rate of discharge from the sewer.

CHAPTER V
DESIGN OF SEWERAGE SYSTEMS

41. The Plan.—Good practice demands that a comprehensive plan for a sewerage system be provided for the needs of a community for the entire extent of its probable future growth, and that sewers be constructed as needed in accordance with this plan.

Sewerage systems may be laid out on any one of three systems: separate, storm, or combined. A separate system of sewers is one in which only sanitary sewage or industrial wastes or both are allowed to flow. Storm sewers carry only surface drainage, exclusive of sanitary sewage. Combined sewers carry both sanitary and storm sewage. The use of a combined or a separate system of sewerage is a question of expediency. Portions of the same system may be either separate, combined, or storm sewers.

Some conditions favorable to the adoption of the separate system are where:

a. The sanitary sewage must be concentrated at one outlet, such as at a treatment plant, and other outlets are available for the storm drainage.

b. The topography is flat necessitating deep excavation and steeper grades for the larger combined sewers.

c. The sanitary sewers must be placed materially deeper than the necessary depth for the storm-water drains.

d. The sewers are to be laid in rock, necessitating more difficult excavation for the larger combined sewers.

e. An existing sewerage system can be used to convey the dry weather flow, but is not large enough for the storm sewage.

f. The city finances are such that the greater cost of the combined system cannot be met and sanitary drainage is imperative.

g. The district to be sewered is an old residential section where property values are not increasing and the assessment must be kept down.

Some additional points given in a report by Alvord and Burdick to the city of Billings, Montana, are:

The separate system of sewerage should be used, where:

1st. Storm water does not require extensive underground removal, or where it can be concentrated in a few shallow underground channels.

2nd. Drainage areas are short and steep facilitating rapid flow of water over street surfaces to the natural water courses.

3rd. The sanitary sewage must be pumped.

4th. Sewers are being built in advance of the city’s development to encourage its growth.

5th. The existing sewer is laid at grades unsuitable for sanitary sewage, it can be used as a storm sewer.

A combined system must be relatively larger than a separate storm sewer as the latter may overflow on exceptional occasions, but the former never.

A combined system of sewerage should be used where:

1st. It is evident that storm and sanitary sewerage must be provided soon.

2nd. Both sanitary and storm sewage must be pumped.

3rd. The district is densely built up.

42. Preliminary Map.—The first step in the design of a sewerage system is the preparation of a map of the district to be served within the limits of its probable growth. The map should be on a scale of at least 200 feet to the inch in the built up sections or other areas where it is anticipated that sewers may be built, and where much detail is to be shown a scale as large as 40 feet to the inch may have to be used. The adoption of so large a scale will usually necessitate the division of the city or sewer district into sections. A key map should be drawn to such a scale that the various sections represented by separate drawings can all be shown upon it. In preparing the enlarged portions of the map it is not necessary to include these portions of the city in which it is improbable that sewers will be constructed, such as parks and cemeteries.

The contour interval should depend on the character of the district and the slope of the land. In those sections drawn to a scale of 200 feet to the inch for slopes over 5 per cent, the contour interval need not be closer than 10 feet. For slopes between 1 and 5 per cent the contour interval should be 5 feet. For flatter slopes the interval should not exceed 2 feet, and a one foot interval is sometimes desirable. In general the horizontal distances between contours should not exceed 400 feet and they should be close enough to show important features of the natural drainage. Elevations should also be given at street intersections, and at abrupt changes in grade. For portions of the map on a smaller scale the contours need be sufficiently close to show only the drainage lines and the general slope of the land.

The following may be shown on the preliminary map: the elevation of lots and cellars; the character of the built up districts, whether cheap frame residences, flat-roof buildings, manufacturing plants, etc.; property lines; width of streets between property lines and between curb lines; the width and character of the sidewalks and pavements; street car and railroad tracks; existing underground structures such as sewers, water pipes, telephone conduits, etc.; the location of important structures which may have a bearing on the design of the sewers such as bridges, railroad tunnels, deep cuts, culverts, etc.; and the location of possible sewer outlets and the sites for sewage disposal plants.

Fig. 24 shows a preliminary map for a section of a city, on which the necessary information has been entered. The map is made from survey notes. All streets are paved with brick. The alleys are unpaved. The entire section is built up with high-class detached residences averaging one to each lot. The lots vary from 1 to 3 feet above the elevation of the street.

43. Layout of the Separate System.—Upon completion of the preliminary map a tentative plan of the system is laid out. The lines of the sewer pipe are drawn in pencil, usually along the center line of the street or alley in such a manner that a sewer will be provided within 50 feet or less of every lot. The location of the sewers should be such as to give the most desirable combination of low cost, short house connections, proper depth for cellar drainage, and avoidance of paved streets. Some dispute arises among engineers as to the advisability of placing pipes in alleys, although there is less opposition to so placing sewers than any other utility conduit. The principal advantage in placing sewers in alleys is to avoid disturbing the pavement of the street, but if both street and alley are paved it is usually more economical to place the sewer in the street as the house connections will be shorter. On boulevards and other wide streets such as Meridian Avenue in Fig. 24, the sewers are placed in the parking on each side of the street, rather than to disturb the pavement and lay long house connections to the center of the street.

All pipes should be made to slope, where possible, in the direction of the natural slope of the ground. The preliminary layout of the system is shown in Fig. 24. The lowest point in the portion of the system shown is in the alley between Alabama and Tennessee Streets. The flow in all pipes is towards this point, and only one pipe drains away from any junction, except that more than one pipe may drain from a terminal manhole on a summit.

44. Location and Numbering of Manholes.—Manholes are next located on the pipes of this tentative layout. Good practice calls for the location of a manhole at every change in direction, grade, elevation, or size of pipe, except in sewers 60 inches in diameter or larger. The manholes should not be more than 300 to 500 feet apart, and preferably as close as 200 to 300 feet. In sewers too small for a man to enter the distance is fixed by the length of sewer rods which can be worked successfully. In the larger sewers the distances are sometimes made greater but inadvisedly so, since quick means of escape should be provided for workmen from a sudden rise of water in the sewer, or the effect of an asphyxiating gas. In the preliminary layout the manholes are located at pipe intersections, changes in direction, and not over 300 to 500 feet apart on long straight runs at convenient points such as opposite street intersections where other sewers may enter.

No standard system of manhole numbering has been adopted. A system which avoids confusion and is subject to unlimited extension is to number the manholes consecutively upwards from the outlet, beginning a new series of numbers prefixed by some index number or letter for each branch or lateral. This system has been followed with the manholes on Fig. 24.

Fig. 24.—Typical Map Used in the Design of a Separate Sewer System.

Fig. 25.—Typical Map Used in the Design of a Storm Sewer System.

45. Drainage Areas.—The quantity of dry weather sewage is determined by the population rather than the topography. Lot lines and street intersections or other artificial lines marking the boundaries between districts are therefore taken as watershed lines for sanitary sewers. The quantity of sewage to be carried and the available slope are the determining factors in fixing the diameter of the sewer. Since there may be no change in diameter or slope between manholes the quantity of sewage delivered by a sewer into any manhole will determine the diameter of the sewer between it and the next manhole above. In order to determine the additional amount contributed between manholes a line is drawn around the drainage area tributary to each manhole. This line generally follows property lines and the center lines of streets or alleys, its position being such that it includes all the area draining into one manhole, and excludes all areas draining elsewhere. An entire lot is usually assumed to lie within the drainage area into which the building on the lot drains. In laying out these areas it is best to commence at the upper end of a lateral and work down to a junction. Then start again at the upper end of another lateral entering this junction, and continue thus until the map has been covered.

The areas are given the same numbers as the manholes into which they drain. The dividing lines for the drainage areas on Fig. 24 are shown as dot and dash lines, and the areas enclosed are appropriately numbered. If more than one sewer drains into the same manhole the area should be subdivided so that each subdivision encloses only the area contributing through one sewer. Such a condition is shown at manhole C2. The areas are designated by subletters or symbols corresponding to the symbol used for the sewer into which they drain. For example, the two areas contributing to manhole C2 are lettered C2_K and C2_D. The sewer from manhole C3 to C2 receives no addition, it being assumed that all the lots adjacent to it drain into the sewer on the alley. There is therefore no area C2. Likewise there is no area A1_C.

46. Quantity of Sewage.—The remaining work in the computation of the quantity of sewage is best kept in order by a tabulation. Table 19 shows the computations for the sewers discharging from the east into manhole No. 142. The computation should begin at the upper end of a lateral, continue to a junction, and then start again at the upper end of another lateral entering this junction. Each line in the table should be filled in completely from left to right before proceeding with the computations on the next line. In the illustrative solution in Table 19, computations for quantity have not been made between manholes where it was apparent that there would be an insufficient additional quantity to necessitate a change in the size of the pipe.

In making these computations the assumptions of quantity and other factors given below indicate the sort of assumptions which must be made, based on such studies as are given in Chapter III. The density of population was taken as 20 persons per acre, the assumption being based on the census and the character of the district. The average sanitary sewage flow was taken as 100 gallons per capita per day. The per cent which the maximum dry weather flow is of the average was taken as M = 500
P^⅕, in which P is the population in thousands. The per cent is not to exceed 500 nor to be less than 150. The rate of infiltration of ground water was assumed as 50,000 gallons per mile of pipe per day.

In the first line of Table 19, the entries in columns (1) to (6) are self-explanatory. There are no entries in columns (7) to (10), as no additional sewage is contributed between manholes 3.5 and 3.4. In column (11), 2250 persons are recorded as the number tributary to manhole No. 3.5 in the district to the north and west. These people contribute an average of 100 gallons per person per day, or a total of 0.346 second foot. This quantity is entered in column (13). The figure in column (14) is obtained from the expression M = 500
P^⅕. Column (15) is .01 of the product of columns (13) and (14). Column (16) is the product of the length of pipe between manholes 3.5 and 3.4, and the ground water unit reduced to cubic feet per second. Column (17) is the sum of column (16), and all of the ground water tributary to manhole 3.5, which is not recorded in the table. Column (18) is the sum of columns (15) and (17).

No new principle is represented in the second and third lines.

In the fourth line the first 10 columns need no further explanation. The (11th) column is the sum of the (10th) column, and the (11th) column in the third line. It represents the total number of persons tributary to manhole 3.4 on lateral No. 8. Column (13) in the fourth line is the sum of column (13) in the third line and the (12th) column in the fourth line, and the (15th) column in the fourth line is the product of the 2 preceding columns in the fourth line. Note that in no case is the figure in column (15) the sum of any previous figures in column (15). With this introduction the student should be able to check the remaining figures in the table, and should compute the quantity of sewage entering manhole No. 142 from the west, making reasonable assumptions for the tributary quantities from beyond the limits of the map.

TABLE 19

Computations for Quantity of Sewage For a Separate Sewerage System

On Street	From Street	To Street	From Manhole	To Manhole	Length Feet	Mark of Added Areas	Area, Acres	Population per Acre	Number of Persons	Total Persons Tributary	Avg. Sanitary Flow, C.F.S.	Cumulative Avg. Sanitary Flow, C.F.S.	Per cent Max. Sanitary is of Average	Total Max. Sanitary, C.F.S.	Increment of Ground Water, C.F.S.	Cumulative Ground Water, C.F.S.	Total Flow, C.F.S.	Line Number
Nebraska St.	Map margin	Alley S. Grant St.	3.5	3.4	338					2250	0.0000	0.346	425	1.47	0.005	0.0187	1.66	1
Alley S. of Grant St.	Railroad	E. of Missouri St.	8.3	8.2	328	8.2	2.7	20	54	54	.0084	.0084	500	0.041	.0048	.0048	0.046	2
Alley S. of Grant St.	E. of Missouri St.	E. of Kansas St.	8.2	8.1	355	8.1	3.41	20	68	122	.0106	.0190	500	0.095	.0052	.010	0.105	3
Alley S. of Grant St.	E. of Kansas St.	Nebraska St.	8.1	3.4	340	3.4₈	2.68	20	54	176	.0084	.0274	500	0.137	.0050	.015	0.152	4
Nebraska St.	Alley S. of Grant St.	Alley S. of Meridian	3.4	3.3	380					2428	.0000	.373	423	1.58	.0058	.208	1.79	5
						7.1
Alley S. of Meridian	Railroad	Nebraska St.	7.2	3.3	800	3.3₇	7.14	20	142	142	.0221	.0221	500	0.111	.0117	.0117	0.123	6
Nebraska St.	Alley S. of Meridian	Alley S. of Smith Av.	3.3	3.2	304					2568	.0000	.395	414	1.63	.0045	.224	1.85	7
						6.1
Alley S. of Smith Ave.	Railroad	Nebraska St.	6.2	3.2	609	3.2₆	3.82	20	76	76	.0119	.0119	500	0.060	.0089	.0089	0.069	8
Nebraska St.	Alley S. of Smith Ave.	S. of Cordovez St.	3.2	3.1	300					2644	.0000	.407	414	1.68	.0044	.237	1.92	9
S. of Cordovez St.	Railroad	Nebraska St.	4.1	3.1	410	3.1₄	3.10	20	62	62	.0096	.0096	500	0.048	.006	.006	0.054	10
S. of Cordovez St.	Map margin	Nebraska St.	5.1	3.1	380	3.1₅	2.69	20	54	54	.0084	.0084	500	0.042	.0056	.0056	0.048	11
Nebraska St.	S. of Cordovez St.	Long St.	3.1	148	172					2760	.0000	.425	409	1.74	.0025	.251	1.99	12
Long St.	Map margin	Nebraska St.	149	148	380	148	1.53	20	31	31	.0048	.0048	500	0.024	.0056	.0056	0.030	13
Long St.	Nebraska St.	N. Carolina St.	148	147	492					2791	.0000	.430	409	1.76	.0072	.264	2.02	14
Long St.	N. Carolina St.	Georgia St.	147	146	430					2791	1.000 [^[33]]	.430	409	1.76	.0064	1.27	3.03	15
Long St.	Georgia St.	Harris St.	146	145	419	146	0.81	20	16	2807	.0025	.433	407	1.76	.0061	1.28	3.04	16
						2.1
Long St.	Harris St.	Tennessee St.	145	143	725	143–145	6.6	20	132	2936	.0205	.454	403	1.83	.024	1.30	3.13	17

Column No. (1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)	(14)	(15)	(16)	(17)	(18)

TABLE 20

Computations for Slope and Diameter of Pipes for a Separate Sewerage System

On Street	From Street	To Street	From Manhole	To Manhole	Length Feet	El. of Surface		Total Flow, C.F.S.	Slope	Dia. of Pipe, Inches	Velocity when Full, Ft. per Second	Capacity when Full, Second-Feet	El. of Invert		Line Number
On Street	From Street	To Street	From Manhole	To Manhole	Length Feet	Upper Manhole	Lower Manhole	Total Flow, C.F.S.	Slope	Dia. of Pipe, Inches	Velocity when Full, Ft. per Second	Capacity when Full, Second-Feet	Upper Manhole	Lower Manhole	Line Number
Nebraska St.	Map margin	Alley S. Grant St.	3.5	3.4	338	105.8	102.4	1.66	0.0108	10	3.25	1.78	97.80	94.40	1
Alley S. of Grant St.	Railroad	E. of Missouri St.	8.3	8.2	328	113.5	112.0	0.046	.00575	8	2.00	0.71	105.50	103.62	2
Alley S. of Grant St.	E. of Missouri St.	E. of Kansas St.	8.2	8.1	355	112.0	107.7	0.105	.0110	8	2.78	0.98	103.61	99.70	3
Alley S. of Grant St.	E. of Kansas St.	Nebraska St.	8.1	3.4	340	107.7	102.4	0.152	.0156	8	3.27	1.18	99.69	94.40	4
Nebraska St.	Alley S. of Grant St.	Alley S. of Meridian	3.4	3.3	380	102.4	100.7	1.79	.00385	12	2.28	1.79	94.07	92.61	5
Alley S. of Meridian	Railroad	Kansas St.	7.2	7.1	400	111.8	107.0		.0120	8	2.90	1.03	103.80	99.00	6
Alley S. of Meridian	Kansas St.	Nebraska St.	7.1	3.3	400	107.0	100.7	0.123	.0157	8	3.28	1.18	98.99	92.70	7
Nebraska St.	Alley S. of Meridian	Alley S. of Smith Av.	3.3	3.2	304	100.7	99.3	1.85	.0042	12	2.36	1.85	92.37	91.09	8
Alley S. of Smith Ave.	Railroad	East of Kansas St.	6.2	6.1	305	109.3	105.3		.0131	8	3.00	1.08	101.30	97.30	9
Alley S. of Smith Ave.	East of Kansas St.	Nebraska St.	6.1	3.2	304	105.3	99.3	0.069	.0197	8	3.70	1.32	97.29	91.30	10
Nebraska St.	Alley S. of Smith Ave.	S. of Cordovez St.	3.2	3.1	300	99.3	101.1	1.92	.00213	15	2.00	2.45	90.84	90.20	11
S. of Cordovez St.	Railroad	Nebraska St.	4.1	3.1	410	100.8	101.1		.00574	8	2.00	0.71	92.80	90.62	12
S. of Cordovez St.	Map margin	Nebraska St.	5.1	3.1	380	104.6	101.1	0.054	.00854	8	2.46	0.87	96.60	93.10	13
Nebraska St.	S. of Cordovez St.	Long St.	3.1	148	172	101.1	98.7	1.99	.00213	15	2.00	2.45	90.04	89.87	14
Long St.	Map margin	Nebraska St.	149	148	380	103.8	98.7	0.030	.0134	8	3.04	1.08	95.80	90.70	15
Long St.	Nebraska St.	N. Carolina St.	148	147	492	98.7	103.8	2.02	.00213	15	2.00	2.45	89.86	88.94	16
Long St.	N. Carolina St.	Georgia St.	147	146	430	103.8	99.1	3.03	.0016	18	2.00	3.50	88.69	88.00	17
Long St.	Georgia St.	Harris St.	146	145	419	99.1	96.9	3.04	.0016	18	2.00	3.50	87.99	87.32	18
Alley S. of Janis St.	End of Janis St.	Harris St.	2.2	2.1	350	105.2	98.1		.0203	8	3.78	1.35	97.20	90.10	19
Harris St.	Alley N. of Janis St.	Long St.	2.1	145	135	98.1	96.9		.0088	8	2.53	0.89	90.09	88.90	20
Long St.	Harris St.	Kentucky St.	145	144	258	96.9	94.4		.00353	18	2.98	5.20	87.31	86.40	21
Long St.	Kentucky St.	Tennessee St.	144	143	282	94.4	93.6		.00635	18	4.00	7.00	86.39	84.60	22
Tarbell Ave.	Harris St.	Long St.	1.1	143	417	98.7	92.6		.0146	8	3.18	1.14	90.70	84.60	23
Long St.	Tennessee St.	Alley W. of Tenn. St.	143	142	185	92.6	92.3	3.13	.0016	18	2.00	3.50	83.77	83.47	24

Column No. (1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)	(14)	(15)

47. Surface Profile.—A profile of the surface of the ground along the proposed lines of the sewers should be drawn after the completion of the computations for quantity. An example of a profile is shown in Fig. 26 for the line between manholes No. 3.5 and No. 147. The vertical scale should be at least 10 times the horizontal. A horizontal scale of 1 inch to 200 feet can be used where not much detail is to be shown, but a scale of one 1 to 100 feet is more common and more satisfactory and even one inch to 10 feet has been used. The information to be given and the method of showing it are illustrated on Fig. 26. The profile should show the character of the material to be passed through and the location of underground obstacles which may be encountered. The method of obtaining this information is taken up in Chapter II. The collection of the information should be completed as far as possible previous to design, and borings and other investigations made as soon as the tentative routes for the sewers have been selected.

48. Slope and Diameter of Sewers.—After the quantity of sewage to be carried has been determined, and the profile of the ground surface has been drawn, it is possible to determine the slope and diameter of the sewer. A table such as No. 20 is made up somewhat similar to No. 19, or which may be an extension of Table 19 since the first 6 columns in both tables are the same. The elevation of the surface at the upper and lower manholes is read from the profile.

The depth of the sewer below the ground surface is first determined. Sewers should be sufficiently deep to drain cellars of ordinary depth. In residential districts cellars are seldom more than 5 feet below the ground surface. To this depth must be added the drop necessary for the grade of the house sewer. Six-inch pipe laid on a minimum grade of 1.67 per cent is a common size and slope restriction for house drains or sewers. An additional 12 inches should be allowed for the bends in the pipe and the depth of the pipe under the cellar floor. Where the elevation of the street and lots is about the same, and the street is not over 80 feet in width between property lines, a minimum depth of 8 feet to the invert of sewers, 24 inches or less in diameter is satisfactory. This is on the assumption that the axes of the house drain and the sewer intersect. For larger pipes the depth should be increased so that when the street sewer is flowing full, sewage will not back up into the cellars or for any great distance into the tributary pipes.

Fig. 26.—Typical Profile Used in the Design of a Separate Sewer System.

The grade or slope at which a sewer shall be may be fixed by: the slope of the ground surface; the minimum permissible self-cleansing velocity; a combination of diameter, velocity, and quantity; or the maximum permissible velocity of flow. Sewers are laid either parallel to the ground surface where the slope is sufficient or where possible without coming too near the surface they are laid on a flatter grade to avoid unnecessary excavation. The minimum permissible slope is fixed by the minimum permissible velocity.

The velocity of flow in a sewer should be sufficient to prevent the sedimentation of sludge and light mineral matter. Such a velocity is in the neighborhood of 1 foot per second. Since sewers seldom flow full this velocity should be available under ordinary conditions of dry weather flow. The minimum velocity when full should therefore be about 2 feet per second. Under this condition, the velocity of 1 foot per second is not reached until the sewer is less than 18 per cent full. The velocity in small sewers should be made somewhat faster than in large sewers since the velocity of flow for small depths in small pipes is less than for the same proportionate depth in large pipes. The maximum permissible velocity of flow is fixed at about 10 feet per second in order to avoid excessive erosion of the invert. If the sewer is carefully laid this limit may be exceeded in sanitary sewers.

The method for determining the grade and diameter of sewers is best explained through an illustrative problem which is worked out in Table 20 for the profile shown on Fig. 26. The figures are inserted in the table from left to right in each line, one line being completed before the next one is commenced. The headings in the first 6 columns are self-explanatory. The elevations of the surface at the upper and lower manholes are read from the profile. The total flow is read from column (18) in Table 19. The slope of the ground surface is then computed, and with the quantity, slope, and coefficient of roughness, the diameter of the pipe and the velocity of flow are read from Fig. 15.

The following conditions may arise:

(1) The diameter required is less than 8 inches. Use a diameter of 8 inches as experience has shown that the use of smaller diameters is unsatisfactory.

(2) The velocity of flow when the sewer is full is less than 2 feet per second. Increase the slope until the velocity when full is 2 feet per second.

(3) The diameter of the pipe required is not one of the commercial sizes shown in Fig. 15. Use the next largest commercial size.

(4) The slope of the ground surface is steeper than necessary to maintain the required minimum velocity and the upper end of the sewer is deeper than the required minimum depth. Place the sewer on the minimum permissible grade, or upon such a grade that its lower end will be at the minimum permissible depth.

(5) The slope of the ground surface is so steep as to make the velocity of flow greater than the maximum rate permissible. Reduce the grade by deepening the sewer at the upper manhole and using a drop manhole at this point.

It is not permissible to use a pipe larger than that called for by the above conditions. This is attempted sometimes in order to reduce the grade and thereby save excavation, under the rule of a minimum velocity of 2 feet per second when full. It is better to use the smaller pipe on the flat grade as the quantity of sewage is insufficient to fill the larger sewer and the minimum permissible velocity is more quickly reached.

Having determined the slope, the diameter, and the capacity of the pipe to be used, these values are entered in the table. The elevations of the invert of the pipe at the upper and lower manholes are next computed and entered in the table. This method is followed until all of the diameters, slopes, and elevations have been determined.

The slopes are computed from center to center of manholes, but an extra allowance of 0.01 of a foot is allowed by some designers for the increased loss in head in passing through the manhole. When it becomes necessary to increase the diameter of the sewer the top of the outgoing sewer is placed at the same elevation or below the top of the lowest incoming sewer. No extra allowance is made to compensate for loss in head in the manhole in this case. This case is illustrated in columns (14) and (15) in lines (16) and (17) of Table 20. All of the conditions listed above are illustrated in Table 20, except the condition for a velocity greater than 10 feet per second.

The first condition is met at the head of practically every lateral, and is illustrated in the second line.

The second condition is also illustrated in the second line. The slope of the ground surface is 0.0046, which gives a velocity of only 1.8 feet per second in an 8–inch pipe. The slope is therefore increased to 0.00575, on which the full velocity is 2 feet per second.

The third condition is met in the first line. The diameter called for to carry 1.66 cubic feet per second on a slope of 0.0108 is slightly less than 10 inches. A 10–inch pipe is therefore used and its full capacity and velocity are recorded.

The fourth condition is illustrated in the fourteenth line. The cut at manhole No. 3.1 is 11.1 feet. The slope of the ground is 0.014, much steeper than is necessary to maintain the minimum velocity in a 15–inch pipe. The pipe is therefore placed on the minimum permissible slope, and excavation is saved. The student should check the figures in Table 20 and be sure that they are understood before an attempt is made to make a design independently.

49. The Sewer Profile.—The profile is next completed as shown in Fig. 26, the pipe line being drawn in as the computations are made. The cut is recorded to the nearest ⅒th of a foot at each manhole, or change in grade. It should not be given elsewhere as it invites controversy with the contractor. The cut is the difference of the elevation of the invert of the lowest pipe in the trench at the point in question, and the surface of the ground.

The stationing should be shown to the nearest ⅒th of a foot. It should commence at 0 + 00 at the outlet and increase up the sewer. The station of any point on the sewer may show the distance from it to the outlet, or a new system of stationing may be commenced at important junctions or at each junction.

Elevations of the surface of the ground should be shown to the nearest ⅒th of a foot, and the invert elevation to the nearest 1
100th of a foot.

Only the main line sewer is shown in profile in Fig. 26. The profiles of the laterals computed in Table 20, have not been shown. The approximate location of all house inlets are shown on the profile and located exactly, and are made a matter of record during construction.