I don’t understand the question, sorry

I think I do understand it but I don't know how to answer it, sorry.

Water flows through a near horizontal pipe in a different way to the way it flows through a near vertical pipe, especially when gravity-fed. In a horizontal pipe it flows along the bottom, in a vertical pipe it swirls around the outside. But I've no idea what happens as the pipe fills up.

I'd guess that it flows down through a vertical pipe more easily since the flow is gaining potenetial energy as it falls, which will cause it to speed up and or get warm in some measure. But I don't know what the equations are, let alone what the solution is.

Not sure quite what you mean by a 75mm horizontal supply, but perhaps can dodge the issue by another route.
Most of the rain you collect will fall – I hope - at less than 50mm per hour. That’s heavy rain. Suppose you are collecting from 100 square metres of roof, your flow rates will thus mostly be less than 5 cubic metres per hour, or about 1.5 litres/sec. Crudely calculated, a downpipe with a drop of 2.5 metres could theoretically give a velocity of no more than 5 metres per second (See calculation below), or rather less in practice due to friction and whatnot. If the pipe is 50mm dia the flow rate could be max 14l/sec; if 25mm dia it would be 3.5 l/sec. You could maybe adjust the figures for your actual situation to get an idea of what size pipe might do. (Edit:These flowrates are corrected - see post below)

All this assumes the pipe is filled with water, perhaps from a hopper, but however in such a way that a vortex with air at the centre does not occur.

As Will says, avoiding blockages is a big factor. My own system was bodged together 10 years ago mostly with what I had lying about, and where I’ve improved it since, it has usually been to minimise blockage. I drain a 300m^2 barn roof (which sheds lumps of moss and fragments of clay tile) down about 50m of cheap 50mm conduit, connectors improvised with the help of gaffer or self-amalgamating tape, etc. The odd imperfect joint leak is not an issue at the low pressures involved.

Calculation: The water (of mass m) in the vertical pipe has kinetic energy due to its speed, by the well-known formula: KE = 1/2 m V^2. That energy comes from the potential energy the water had due to it being higher when at the top of the pipe. The average height difference is half the length h of the vertical pipe, so PE = 1/2 h m g ( g being gravity of 9.8 metres/sec^2). Suspect you wish you hadn't asked....
Anyway, since the KE gained equals the PE lost, the halves and the 'm's cancel out leaving:
V^2 = hg Edit: ...at first, but increasing to V^2 =2hg. Flow rates above corrected. See comments below.

Mike - nice! But I think you need another factor 2 in the derivation of Bernoulli's equation.
0.5V^2 = hg
https://en.m.wikipedia.org/wiki/Bernoulli%27s_principle#Incompressible_flow_equation

TBH downpipes do not flow in this ideal way, because the entrained air messes it up by increasing the velocity and decreasing the density. The Froude number is your friend here, you make it low enough that the air can escape up the centre of the pipe, (or high enough that the air is dragged down the pipe - but then you have surging at intermediate flows)

Well, if its an overflow under heavy flow conditions where the water level is just above the top of the outlet I think the flow rates in https://www.dutypoint.com/systems/55-friction-head-loss give an accurate capacity if the pipe is clear. What are the consequences for overflow failure?

So what you need is a flow rate for the incoming pipe - is it from a gutter - how big is the roof - how big is the gutter and what fall - what area roof is it. Does the incoming pipe run full (unlikely) - what length and fall is the incoming pipe. Only one of these parameters will give the capacity your overflow requires. So for example tiny gutters on a big roof would overflow the gutters.

have a look at this page https://www.gutter-techuk.co.uk/flow-calculations it gives the capacity of a typical roof and gutter make up. An 25mm full & vertical overflow pipe could cope with that roof/gutter combination. It may gurgle/slurp/surge during intermediate flows but that may not be important to you.

Mmmm....! No.. :-)

As you said, it's an energy balance. When each particle of water falls down the pipe, the head energy it has lost is mgh, equal to the kinetic energy gained 0.5mv^2. It can't be created or destroyed, etc. There's no averages or h/2 involved, the whole of h is lost.

If you like this kind of stuff, consider the pipe near to the top, and near the bottom, the volume flowing through both ends is identical and therefore so is the velocity at both ends and everywhere inbetween (if the pipe is running full of water). So the elevation energy is different top to bottom, but the kinetic energy is the same at both ends, how can that balance?

Hint: 0.5v^2 = hg + P/rho.

I'd italicise the P if I knew how to!

The answer is why we put air admittance valves on downpipes to stop them running full of water and sucking a vacuum at the top...

The air allows the water to accelerate into a thinner film as it falls, so occupy less of the pipe section, the air fills in the rest and smooths out the pressure.

In the real world, pipes have friction, so an empirical fourth term was added to the energy balance, by a Mr Darcy. (Not that one!)
https://en.m.wikipedia.org/wiki/Darcy–Weisbach_equation

Posted By: mike7It won't be very effective if the space above the outlet is small. Better to site the outlet lower in the tank and extend the elbow upward a reasonable distance with a hopper or funnel to create in effect a long weir for excess water to flow over at the intended max height.

A fancier option could be to put the outlet at the max height and turn your elbow downward and extend it as far as you like. This will siphon water out once the level rises above the outlet, but you'll want a small airhole at outlet height to break the siphon when the level drops below it.

The hopper or funnel is a great idea. I don't think the upturned option is recommended though, as the outlet is supposed to skim any debris off the top of the tank, plus remove the warmer water from the tank.

I can't make out the wording of the black box - I assume it's some sort of leaf/debris filter. I think position a hopper at the level on your diagram where the word 'overflow' is on the diagram. Make the hopper deep then connect one (or more) pipes to that. The hopper could be a 75mm pipe with say a 32mm adaptor at the bottom. It'll slup in heavy flow conditions but I reckon it will do the job. The 'hopper' could be sealed or open I think.

That's a proper hack.

It looks like the Wisy Filter allows the debris to fall through - in which case you may want to catch the larger bits so they don't clog the smaller downpipes. Use the link I recommended above to get the full pipe flow rate - for a vertical pipe the flow rate will be when the head loss equals the height. good luck.