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Inverted siphons (also called depressed sewers) allow stormwater or wastewater sewers to pass under obstructions such as rivers. Our calculation allows up to five parallel siphons to go under the river. Unlike the main sewer pipe, the siphon pipes flow under pressure and must have flow velocities greater than 3 ft/s (0.9 m/s) to keep material suspended; therefore, several siphons having smaller diameters than the main sewer may be required. Our calculation computes the siphon diameters, velocities, and inlet chamber wall heights and siphon invert elevations.

Overall Diagram:

Plan view of inlet chamber (3 siphons):

Section A-A (exploded scale):

For ease of fabrication, all siphon inverts can be located at the elevation of the lowest siphon invert.

Register to fully enable the "Click to Calculate" button in the calculation below:

Units: cm=centimeter, cfs=cubic feet per second, ft=feet, gpm=US gallons per minute,
gph=US gallons per hour, gpd=US gallons per day, m=meters, MGD=Millions of US gallons per
day, s=second

Links on this page: Introduction Equations Variables Manning n coefficients

Glossary Error messages and validity
References

**Introduction **

Stormwater and wastewater sewers often encounter obstructions such as rivers, other pipes,
subways, tunnels, or valleys. To pass these obstructions, a common method is for the sewer
pipe to drop sharply, then run horizontal under the obstruction, and finally rise to the
desired elevation. The piping going under the obstruction is traditionally called an
"inverted siphon", but since the pipe is not actually acting as a siphon, a
better term is "depressed sewer" (Metcalf and Eddy, 1981).

Unlike the main sewer pipe, the siphon pipe(s) flow under pressure. Special care must
be taken in inverted siphon design since losses are greater for pressurized flow, and the
velocity in each siphon pipe must be at least 3 ft/s (0.9 m/s) for sewage or 4 ft/s (1.2
m/s) for storm water (Metcalf and Eddy, 1981). Therefore, even if there is only one main
sewer pipe, several siphons may be required. If minor losses due to bends or elbows in the
siphon are significant compared to the siphon length, include the equivalent length of the
elbows. Increase the siphon length (L_{s}) so that L_{s} is the physical
length of a siphon plus the equivalent length of minor losses due
to elbows in siphon.

**Equations and Methodology**
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Equations are primarily from Metcalf and Eddy (1981) but are supplemented by equations in
Chow (1959) and Viessman and Hammer (1998). Note that Manning's equation is
empirical, and its form in the following equations requires use of meters and seconds for
the units.

Compute the maximum flow in the main sewer pipe using Manning's equation for full pipe flow:

Compute the diameter of each siphon, D_{i}, or the flow through each siphon, Q_{i},
using Manning's equation for full pipe flow through each siphon:

Compute the wall heights, y_{j} (relative to main invert), in the inlet
box. The walls separate the siphons from each other. The wall heights are the
same height as the water depths, y_{j}, in the main pipe corresponding to the
discharge through the siphons. Here, Q_{j=1} is the discharge through siphon
1, Q_{j=2} is the discharge through siphons 1 and 2, and so on. Manning's
equation for a partially full main pipe is used, but is solved backwards (numerically) in
order to compute y_{j}. We allow up to five siphons (four walls).

Compute the siphon invert elevations in the inlet chamber. According to Metcalf
and Eddy (1981), there is no loss in the inlet box for flow going from the main culvert to
the first siphon since the flow travels in a straight path. However, for siphons 2
through n the flow must turn 90^{o} to go over the chamber wall (a head loss of
1.5 velocity heads) and has an additional head loss of one velocity head as the flow
enters siphon i. Therefore, for i=2 to n siphons and j=2 to n-1 walls:

where E_{i} is relative to the invert of the main pipe. Note that for the
first siphon, H_{i}=0, and for the last siphon y_{j} is replaced by D_{m}.
Often, all siphon inverts are located at the same elevation (the elevation of the lowest
siphon) for ease of construction.

**
Variables**
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y

**Manning n Coefficients**
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Manning n values are from Metcalf and Eddy (1981), AISI (1980), and footnoted items in references for pipes in good condition.

Pipe Material |
Manning n |
Pipe Material |
Manning n |

Uncoated cast iron | 0.013 | Coated cast iron | 0.012 |

Commercial wrought iron - black | 0.013 | Commercial wrought iron - galvanized | 0.014 |

Smooth brass and glass | 0.010 | Smooth lockbar and welded "OD" | 0.011 |

Riveted and spiral steel pipe | 0.015 | Corrugated Metal | 0.022* |

Common clay drainage tile | 0.012 | Vitrified sewer pipe | 0.013 |

Brick in cement mortar, brick sewers | 0.013 | Glazed brickwork | 0.012 |

Cement mortar surfaces | 0.012 | Neat cement surfaces | 0.011 |

Wood stave pipe | 0.011 | Concrete pipe | 0.013 |

Corrugated Polyethylene (PE) with smooth inner walls^{ a,b} |
0.009-0.015 | ||

Corrugated Polyethylene (PE) with corrugated inner walls^{
c} |
0.018-0.025 | ||

Polyvinyl Chloride (PVC) with smooth inner walls^{ d,e} |
0.009-0.011 |

* Corrugated metal pipe n value can vary significantly with pipe diameter and type of corrugations (values can range from 0.012 to 0.033) - AISI (1980).

**Glossary**
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Inlet chamber - usually concrete manhole where main culvert branches into several siphon
pipes.

Invert - inside bottom of pipe.

Main - culvert through which flow occurs before and after the siphon.

Siphon - pipe or pipes flowing full and under pressure which go underneath the
obstruction. Not siphons by the true definition. True siphons flow uphill then
back down. Siphons used here go down then back up.

**Error Messages and Validity **
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calculation

** Initial input checks.** The following messages are generated
from improper input values:

*"Need 1e-9<Main n<1e9", "Need 1e-9<Siphon n<1e9".*
The Mannings n values for the main culvert and siphons must be between these limits.

*"Need 1e-9<D _{1}<1e9 m", "Need 1e-9<D_{2}<1e9
m", "Need 1e-9<D_{3}<1e9 m", "Need 1e-9<D_{4}<1e9
m", "Need 1e-9<D_{5}<1e9 m".* If siphon diameters
are input, they must be between these limits.

*"Need 1e-9<Q _{1}<1e9 m^{3}/s", "Need 1e-9<Q_{2}<1e9
m^{3}/s", "Need 1e-9<Q_{3}<1e9 m^{3}/s",
"Need 1e-9<Q_{4}<1e9 m^{3}/s".* If siphon flows
are input, the flows must be between these limits.

** Run-time errors.** The following messages may be
generated after performing some calculations:

*"Need siphon Q>0".* If diameters are being computed, the
flowrate through the last siphon is automatically computed such that the sum of the flow
through all siphons is equal to the discharge through the main culvert. If the
siphon flows input by the user exceed the discharge in the main culvert, then the flow in
the last siphon will be negative, which will generate the error message. You should
reduce the flows in the siphons so that there is positive flow in the last siphon.
Or, you could reduce the number of culverts.

*"Siphons under-designed".* Shown only if siphon flows are being
computed and Qs/Qm<0.95. You need to increase the siphon diameters. This
message will not be generated if diameters are being computed - because diameters are
computed so that the total flow through the siphons is exactly equal to the discharge
through the main culvert.

*"Siphons over-designed".* Shown only if siphon flows are being
computed and Qs/Qm>1.05. Since wall heights cannot be computed for flows grossly
exceeding that of the main culvert, the calculation stops. You need to decrease the
siphon diameters. This message will not be generated if diameters are being computed
- because diameters are computed so that the total flow through the siphons is exactly
equal to the discharge through the main culvert.

**References and Bibliography
**
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calculation

AISI (American Iron and Steel Institute). 1980. Modern Sewer Design.

^{a} Barfuss, Steven and J. Paul Tullis. Friction factor test on high
density polyethylene pipe. Hydraulics Report No. 208. Utah Water Research
Laboratory, Utah State University. Logan, Utah. 1988.

^{c} Barfuss, Steven and J. Paul Tullis. Friction factor test on high
density polyethylene pipe. Hydraulics Report No. 208. Utah Water Research
Laboratory, Utah State University. Logan, Utah. 1994.

^{e} Bishop, R.R. and R.W. Jeppson. Hydraulic characteristics of PVC
sewer pipe in sanitary sewers. Utah State University. Logan, Utah.
September 1975.

Chow, V. T. 1959. Open-Channel Hydraulics. McGraw-Hill, Inc. (the classic text)

Hammer, M. J. and M. J. Hammer, Jr. 1996. Water and Wastewater Technology. Prentice Hall, 3ed.

Metcalf and Eddy, Inc. 1981. G. Tchobanoglous, editor. Wastewater Engineering: Collection and Pumping of Wastewater. McGraw-Hill, Inc. (Note that there are some errors in the invert elevations computed on p. 175.)

^{d} Neale, L.C. and R.E. Price. Flow characteristics of PVC sewer pipe.
Journal of the Sanitary Engineering Division, Div. Proc 90SA3, ASCE. pp.
109-129. 1964.

^{b} Tullis, J. Paul, R.K. Watkins, and S. L. Barfuss. Innovative new
drainage pipe. Proceedings of the International Conference on Pipeline Design and
Installation, ASCE. March 25-27, 1990.

Viessman, W. and M. J. Hammer. 1998. Water Supply and Pollution Control. Addison-Wesley, 6ed.

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(modified January 10, 2013 - added text about including minor losses due to bends/elbows)