Читать книгу Sewerage and Sewage Treatment - Harold E. Babbitt - Страница 17
Оглавление5. 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 | H 1 | d1 − H1 | V | S | S 1 | L | Elevation | |||
| D | d | d 1 | 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.
