This is a question that comes up quite often, and I usually tell people that it depends on the water depth over the pipe entry.
This means that we either use the weir or the orifice formula.
Or if the pipe is flowing full, as in a syphonic system, we can use the pipe flowing full formulas, Colebrooke-White etc, and work out the pipe size using the hydraulic grade.
However this is not necessarily always the case.
What if water enters a downpipe from lower level roofs, balconies etc?
A plumbing stack has this situation all the time.
So now we can start to have some real fun.
When water flows down a vertical pipe (assuming it is not designed as syphonic) the pipe does not flow full.
There is also air being pushed down, and air trying to rise up.
This is why plumbing stacks have vents all over the place, to relieve any unwanted air pressures.
Storm water pipes have entries all over the place which do the same thing.
Water starts out clinging to the sides, and as the flow gets greater it starts to spiral, and getting greater still starts to oscillate, forming plugs. Rapid oscillations develop unwanted noise and vibrations.
It has been determined (refer Standard Plumbing Engineering Design by Louis S Nielsen) that a flow which occupies about 1/4 to 1/3 of the cross sectional area of the downpipe flows without a problem.
The next thing then is to find out the terminal velocity so we can work out the flow.
We don’t need to go into that, suffice to say, that I have reduced all the formulas from the above text into something simple.
This is what can be referred to as a recommended maximum flow based on the above concepts, and equates to a flow that occupies 7/24 of the DP area.
So Q = 0.00004 * dia^2.666 where Q is in L/s, and dia in mm.
This equates to an inground pipe of the same size with a grade of about 1:100.
So, there you go, just don’t make any inground pipe smaller than the DP.
Want to impress your clients?
Turns out this is not an exact science, but just for the sake of BS..ing at design meetings, terminal velocity is about 5m/s and it takes about 2 stories to get there.