Circular culvert free-flow discharge calculation

Water free-flowing from end of pipe. End depth method. ISO 4371 equation.


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Discharge, Q: 
Water Depth, h: 
Culvert Diameter, D: 
Ratio, h/D:   
Top Width, T: 
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Units in circular culvert free-discharge water calculator: cm=centimeter, cfs=cubic foot per second, cms=cubic meter per second, ft=foot, gal=gallon (U.S.), m=meter, min=minute, s=second, yr=year

Free-Flowing Water Circular Culvert (Pipe) Drawings and Equations

Culvert discharge                  Flow diagram

Culvert equations and cross-section

Introduction to Circular Culvert Free-Discharge Calculation using End Depth Method
Discharge (flow rate) can be computed for water flowing out of a circular culvert that is up to half full. The end depth method is based on the International Organization of Standards publication 4371 dated 1984 (ISO, 1984). Knowing the culvert diameter and the depth of flow where the water drops off (i.e. at the end of the culvert), the discharge can be computed. If it is more easy or accurate to measure the water top width at the drop-off (instead of water depth), the top width can be used with diameter to compute the discharge. For the equation to be used properly, there are several criteria that should be met:
• Clear water.
• Circular cross-section for a distance at least 20h upstream of the drop-off.
• Culvert should be horizontal although a slight downward slope (up to 1/2000) is okay.
• Culvert walls should be smooth such as would be provided by a neat cement finish.
• Flow should be sub-critical upstream of the drop-off.
• The drop, d, should be greater than h.
• The channel width, T, should be greater than 0.3 m (11.8 inch).
• The end depth, h, should be greater than 0.05 m (1.97 inch).
• The ratio h/D should be between 0.095 and 0.5, and cannot exceed 0.5.

Variables for Circular Water Pipe Free-Discharge Equations [L] indicates length units. [T] indicates time units.
d=Drop elevation [L]. Elevation difference between bottom of channel and tail water. Must be at least h for the discharge equation to be reliable.
D=Culvert diameter [L]. Note that h/D must be between 0.095 and 0.5 for the discharge equation to be reliable. The calculation only functions for h/D≤0.5.
g=Acceleration due to gravity=9.8066 m/s2.
h=Measured water depth at end of channel (i.e. at the drop-off) [L]. Must be greater than 0.05 m (1.97 inch) for the discharge equation to be reliable. Note that h/D must be between 0.095 and 0.5 for the discharge equation to be reliable. The calculation only functions for h/D≤0.5.
L=Straight length upstream of drop-off [L]. Must be at least 20h for the flow to be well-established.
Q=Discharge [L3/T]. Also known as flow rate.
T=Top width of water in the culvert [L]. Must be greater than 0.3 m (11.8 inch) for the discharge equation to be reliable.

Error Messages for Circular Culvert Free-Flow Water Calculation
Messages indicating input values are out of the acceptable ranges (the seemingly redundant messages are checked at different phases of program execution):
"D must be > 0",  "h must be > 0",  "h/D must be ≤ 0.5",  "Must have 0< h/D ≤ 0.5",  "T must be > 0",  "T must be ≤ D",  "Enter 1e-5 ≤ D ≤ 1000 m",  "Enter 1e-7 ≤ h/D ≤ 0.5",  "Enter 1e-5 ≤ h ≤ 1000 m".

The calculation finished, but per ISO (1984), equation accuracy is reduced when the following messages appear:
"Need h/D > 0.095 for accuracy",  "Need h > 0.05 m for accuracy",  "Need T > 0.3 m for accuracy".

References for Circular Culvert Free-Discharge Equations
International Organization of Standards (ISO 4371). 1984. Measurement of liquid flow in open channels by weirs and flumes - End depth method for estimation of flow in non-rectangular channels with a free overfall (approximate method).  Reference number: ISO 4371-1984(E).


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