Based on the cross-sectional shapes the sewers may be classified as: 1. Circular Sewers 2. Non-Circular Sewers.
1. Circular Sewers:
Sewers of circular section are most commonly used. These are best suitable for diameters up to 1.5 m.
The various advantages of circular sewers are as indicated below:
(i) A circular section gives the least perimeter for a given area, and therefore has the maximum hydraulic mean depth for running full and half full conditions. It is therefore the most efficient section at these flow conditions.
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(ii) It is the most economical section since it requires minimum quantity of material for its construction.
(iii) The section has uniform curvature and hence prevents the possibility of deposits anywhere within the section.
(iv) These can be easily manufactured.
A circular sewer may run either full or partially full.
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The hydraulic elements of circular sewers for both the conditions viz., running full and running partially full are indicated below:
(a) Circular Sewers Running Full:
Let D be the internal diameter of the sewer.
(b) Circular Sewers Running Partially Full:
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Fig. 4.2 shows a circular sewer running partially full.
Let D be the internal diameter of the sewer, d be the depth of flow, and θ be the angle subtended by the wetted perimeter at the centre of the sewer.
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Let a be the area of flow section;
p be the wetted perimeter;
r be the hydraulic mean depth, i.e., r = (a/p); and
v and the velocity of flow
It may be noted that in equations 4.19, 4.21, 4.23. 4.25 and 4.27 there is only one variable i.e., angle θ which depends on depth ratio or proportional depth d/D as indicated by equation 4.15. Thus by taking depth ratio d/D as reference variable, all other quantities can be computed. Table 4.9 shows the computed values of ratios of hydraulic elements of circular sewers running partially full.
Fig. 4.3 shows the variation of ratios of the various hydraulic elements with the depth ratio d/D. It has been observed that the value of roughness coefficient n varies with the depth of flow. As such for the ratios (v/V) and (q/Q) two sets of curves have been plotted, one for constant n (i.e., nd/n = 1) and the other for variable n.
It may be observed that if roughness coefficient n is assumed constant or independent of the variation in the depth of flow, then so long as sewers run more than half full, the velocities of flow in sewers running partially full exceed those when they are running full. The maximum velocity is obtained when the depth of flow d is about 0.81 times the diameter of the sewer and it is about 14% greater than that when running full.
Similarly maximum discharge is obtained when the depth of flow is about 0.938 times the diameter of the sewer and it is about 7% greater than that when running full. However, when the variation in the value of n with the depth of flow is taken into account, velocities equal to or greater than those for running full condition are obtained when the sewers are running partially full with depth of flow equal to or more than 0.8 times the diameter of the sewer.
Hydraulic Elements of Circular Sewers having Equal Self-Cleansing Properties at All Depths:
The various hydraulic elements of circular sewers that possess equal self-cleansing properties at all depths may be derived on the basis of the assumption that equality of tractive force implies equality of cleansing. Thus if s is tractive force per unit area when sewer is running partially full, and t is tractive force per unit area when it is running full, then for equality of self-cleansing properties at all depths, we have-
τs = τ
or w r ss = w R S
Where;
ss is slope or gradient to be provided to achieve self-cleansing velocity when the sewer is running partially full;
r is hydraulic mean depth when sewer is running partially full;
R is hydraulic mean depth when sewer is running full;
S is slope or gradient to be provided to achieve self-cleansing velocity when the sewer is running full ; and
w is unit weight of water
If vs and V are the self-cleansing velocities when sewer is running partially full and running full respectively, then from equations 4.24 and 4.13, we have-
Introducing equation 4.28 (a) in equation 4.29, we have-
If qs and Q, and as and A are the corresponding discharges and areas of flow sections then, we have-
Fig. 4.4 shows the variation of the ratios (vs/V) and (qs/Q) with the depth ratio (d/D) for (nd/n) = 1 as well as for variable (nd/n). The variation of the ratio (ss/S) with the depth ratio (d/D) is also shown in Fig. 4.4. From the curve for (ss/S) it may be observed that for (d/D) varying from 1 to 0.5, (ss/S) is nearly equal to one, thereby indicating that so long as circular sewers run more than half full they may be provided with same slopes as would be necessary to achieve self-cleansing when sewers are running full.
In other words sewers running more than half full need not be placed on steeper slopes or gradients to be as self- cleansing as sewers running full. However, when the depth of flow d drops to 0.2D, then for equal self- cleansing the slope or gradient must be two times (i.e., ss = 2S), and when the depth of flow d drops to 0.1D the slope or gradient must be four times (i.e., ss = 4S) that required for running full.
Limitation on Depth of Flow in Sewers:
From considerations of ventilation in wastewater flow sewers should not be designed to run full. All sewers are to be designed to flow 0.8 full at ultimate peak flow.
From the foregoing discussion it is evident that in a circular sewer if the depth of flow becomes less than half full, both velocity as well as discharge reduce considerably with the reduction in the depth of flow. As such circular sewers are suitable only where variation in discharge of sewage is not large.
The circular sewers are, therefore, suitable for separate sewerage system in which since sewers carry only dry weather flow there is no large variation in discharge. On the other hand in a combined sewerage system since sewers carry both storm water (or rain water) flow and dry weather flow there is considerable variation in the discharge.
This is so because the storm water (or rain water) flow is usually much more than the dry weather flow which during summer may not be even 5 to 10 percent of the combined discharge. Thus if circular sewers are provided for a combined sewerage system, they will mostly run with very low depths. Therefore for combined sewerage system non-circular sewers such as egg-shaped sewers (or ovoid sewers) are more suitable.
2. Non-Circular Sewers:
The non-circular sewers may be classified according to the cross-sectional shapes as follows:
(i) Egg-shaped or ovoid sewers
(ii) Sewers of other sections
These are briefly discussed below:
(i) Egg-Shaped or Ovoid Sewers:
There are various forms of egg-shaped or ovoid sewers out of which the following two forms are most common-
(a) Standard or Metropolitan egg-shaped section
(b) New egg-shaped section
Both these forms of egg-shaped sewers are shown in Fig. 4.5, which also indicates the geometrical parameters of these sewer sections.
The egg-shaped sewers may be used both in the combined sewerage system where the dry weather flow is small as compared with the total capacity of the sewer, as well as in the separate sewerage system for a town or city where the present population is only a small proportion of the ultimate development as it allows for increased future flows.
The main advantage of an egg-shaped sewer is that for small flows the depth is greater and the velocity is somewhat higher than in a circular sewer of equivalent capacity. Thus egg-shaped sewers have better self-cleansing property at small flows as compared to circular sewers.
However, egg-shaped sewers have the following disadvantages:
(i) The egg-shaped sewers are less stable than the circular sewers, because the small end of the egg is down and it has to support the weight of the upper broader section.
(ii) It is more difficult to construct these sewers.
(iii) These sewers are expensive as more material is required and the cost of construction is also high.
The hydraulic elements of both the forms of egg-shaped sewers for running full as well as partially full conditions are indicated below:
Hydraulic Elements of Standard or Metropolitan Egg-Shaped Sewer:
(a) Sewer Running Full:
When running full, the area of flow section of the sewer is composed of:
(i) Semi-circular crown (or roof) portion ABCGA ;
(ii) Intermediate (or central) portion AGCDHFA ; and
(iii) Invert portion FHDEF
(b) Sewer Running 2/3 Full:
In this case sewer will run up to AGC, and hence area of flow section is composed of:
(i) Central portion AGCDHFA; and
(ii) invert portion FHDEF.
Area of central portion AGCDHFA = 2.79b2
Area of invert portion FHDEF = 0.23b2
∴ Total area of flow section
a = (2.79b2 + 0.23 62) = 3.02b2 … (4.35)
Wetted perimeter may be determined as follows:
Portions AF and CD =3.86b
Portion FED = 0.93b
∴ Total wetted perimeter
p = (3.86b + 0.93b) = 4.79b … (4.36)
Hydraulic mean depth
r = a / p = 3.02b2 / 4.79b = 0.63b … (4.37)
(c) Sewer Running 1/2 Full:
On the same lines it can be shown that-
a = 2.03b2 … (4.38)
p = 3.92b … (4.39)
r = 0.52b … (4.40)
(d) Sewer Running 1/3 Full:
On the same lines it can be shown that
a = 1.17b2 … (4.41)
p =2.91b … (4.42)
r =0.40b … (4.43)
Hydraulic Elements of New Egg-Shaped Sewer:
(a) Sewer Running Full:
When running full, the area of flow section of the sewer is composed of:
(i) Semi-circular crown (or roof) portion ABCGA;
(ii) Intermediate (or central) portion AGCDHFA; and
(iii) Invert portion FHDEF
Wetted perimeter of the sewer may be determined as follows:
Hydraulic mean depth;
(b) Sewer Running 2/3 Full:
In this case sewer will run upto AGC, and hence area of flow section is composed of:
(i) Central AGCDHFA; and
(ii) Invert portion FHDEF.
Area of central portion AGCDHFA = 2.85b2
Area of invert portion FHDEF = 0.05b2
∴ Total area of flow section
a = (2.85b2 + 0.05b2) = 2.90b2 … (4.47)
Wetted perimeter may be determined as follows:
Portions AF and CD = 4.32b
Portion FED =0.38b
∴ Total wetted perimeter
p = (4.32b + 0.38b) = 4.70b … (4.48)
Hydraulic Mean Depth:
(c) Sewer Running 1/2 Full:
On the same lines it can be shown that:
a = 1.93b2 … (4.50)
p =3.85b … (4.51)
r = 0.50b … (4.52)
(d) Sewer Running 1/3 Full:
On the same lines it can be shown that-
a =1.05b2 … (4.53)
p =2.78b … (4.54)
r =0.38b … (4.55)
Fig. 4.6 shows the variation of ratios of the various hydraulic elements of standard egg-shaped (or ovoid) sewers with the depth ratio (d/D’) computed by Crimp and Bruge’s formula, where D’ is the height of the egg-shaped sewer which is equal to 3b, and it is the depth of flow when the sewer is running full. D’ is also known as vertical diameter of the egg-shaped sewer.
Hydraulically Equivalent Circular Sewers:
The computations of various hydraulic elements such as area of flow section, wetted perimeter, hydraulic mean depth, etc., of egg-shaped sewers involve complicated mathematical calculations. Therefore while designing egg-shaped sewers, it is usual practice to first calculate the approximate diameter of a hydraulically equivalent circular sewer which would give the same discharge while running full when laid at the same gradient, and then to convert it into dimensions of an egg-shaped section having an equal area.
For standard egg-shaped sewer running full, area of flow section is given by equation 4.32 as-
A = 4.59b2
If Do is the maximum width (or horizontal diameter) of the egg-shaped sewer then since Do = 2b, we have-
The hydraulic mean depth of an egg-shaped sewer is same as that of a hydraulically equivalent circular sewer when running full, but it is higher for smaller depths of flow. Hence at smaller depths of flow the velocity in egg-shaped sewers is higher than that in the hydraulically equivalent circular sewers.
The proportionate velocities at smaller depths of flow in circular as well as in standard egg-shaped sewers are given in Table 4.10. It may be observed that as the depth of flow becomes less than 0.25 times the full depth, the proportionate velocity in egg-shaped sewer becomes higher than that in hydraulically equivalent circular section.
(ii) Sewers of Other Sections:
In case of soft soils there is difficulty of providing foundation for sharply-curved circular or egg-shaped sewer. In such cases sewers of other shapes such as semi-elliptical, parabolic, horse-shoe, rectangular or U- shaped in which the bottom is comparatively flat may be adopted.
A brief description of each of these sewer sections is given below:
(a) Semi-Elliptical Section:
It is made up of three circular arcs or it may be a true semi-ellipse. The main advantage of this section is that the shape of the arch more nearly coincides with the line of resistance under the conditions of actual working. Due to this the arch portion can be made thin. The whole section depends to a large extent upon the stability of the invert. In general the hydraulic properties of the semi- elliptical section are good. For sewers having width more than 2 m this section is more suitable.
(b) Parabolic Section or Delta Section:
This section has a large carrying capacity as compared with a circular section of the same height. For equal capacity and height the width of the trench required for this section is more than that required for the semi-elliptical section. The normal flow level in this section is lower than in a circular section. This section is suitable for handling relatively small quantities.
(c) Horse-Shoe Section:
This section has a semi-circular arch above the springing line. The side walls below the springing line may be vertical or inclined inward. The invert may be horizontal, circular or parabolic arc. The shape of this section is such that a large external load can be supported without aid from the backfilling. In soft soil, considerable stress is produced in the bottom; and unless the bottom is of reinforced concrete, cracks are likely to develop at the centre.
(d) Rectangular Section:
This section has excellent hydraulic properties until it is filled. All four sides of rectangular section should be of reinforced concrete. The advantages of this type of section are simplicity of form work, economy of masonry and space in the trench, and easier construction. The rectangular section is used where headroom is limited.
(e) U-Shaped Section:
It has fairly good hydraulic properties until it becomes filled. This section may have the form of a true U or it may consist of a smaller section that is set into the bottom of a larger sewer. U-shaped section is suitable for sewers about 1 m wide and over 1 m in depth. In proportion to the area this section requires considerable masonry, however the construction is easier.