If you have a circular headgate, how can you use that as an orifice to measure flow? It’s easy if the gate is fully open – orifice equations from the USBR Water Measurement Manual can be used. The area of a circle is PI*r^2.
Here are the orifice equations for a rectangular headgate – the same equations are used for a circular headgate, or really a headgate of any shape:
A = the area of the orifice
g = gravity, 32.2 ft/sec^2
h1 = the upstream depth in feet
h2 = the downstream depth in feet
The first equation with the coefficient of 0.61 is for a gate on a wall. The second with the
coefficient of 0.70 is for a circular gate at the end of a pipe.
The difference between h1 and h2 can also be measured down from the top of a
wall with a level top.
Now, how do you get the area of a partly open circular headgate? There is a long equation with a good description at: https://mathworld.wolfram.com/Circle-CircleIntersection.html However, it sure is easier to get it from a table:
There you go. Based on the change of stem height, you can calculate how much the headgate is open. Convert that to a percentage. For example. A 2.0′ diameter headgate open 0.5′ is 25% open. Then use the table’s 50% row to where it intersects the 2.00 foot diameter column to get an area of 1.913 square feet.