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Introduction
Discharge (flowrate) 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 dishcarge 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
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 flowrate.
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
"D must be >0."  Check your input.  Diameter, D, was entered as 0.0 or negative.  No computations.
"h must be > 0."  Check your input.  Water depth, h, was entered as 0.0 or negative.  h=0 is an empty pipe.  No computations.
"h/D must be <= 0.5."  For the discharge equation to be valid, the pipe cannot be more than half full.  No computations.
"Must have 0<h/D<=0.5."  The pipe cannot be empty and it cannot be more than half full.  No computations.
"T must be > 0."  Check your input.  Water top width, T, was entered as 0.0 or negative.  T=0 is an empty pipe.  No computations.
"T must be <= D."  Water top width, T, cannot exceed pipe diameter, D.  No computations.

"Need h/D >0.095 for accuracy."  ISO 4371 recommends that h/D be greater than 0.095 for the discharge equation to be accurate.  Computations completed.
"Need h>0.05 m for accuracy."  ISO 4371 recommends that water depth, h, be greater than 0.05m for the discharge equation to be accurate.   Computations completed.
"Need T>0.3 m for accuracy."  ISO 4371 recommends that water top width, T, be greater than 0.3m for the discharge equation to be accurate.   Computations completed.

"NaN" or "Infinity".  Variable was computed as positive infinity or negative infinity.  Occurs if a very large number is divided by a very small number resulting in a value that is out of the range of precision of the Java compiler.  All of our programs are written in double precision which has a range of  ±1.79769313486231570x10³⁰⁸ and  ±4.9406564584126544x10^-324.

References
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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