> a normal helicopter massively outperformed a small quadcopter.
not really. The power loading (power divided by surface of the propeller's circle) is the key factor what matters for efficiency (until of course Reynolds number - air viscosity factor - starts to change - happens when we come to Kolibri scale)
In general, helicopter with 3m long blades (6m diameter) on one 500kw turbine or something like 16 of 30kw motors with 1.5m diameter propeller (with the same number of blades as helicopter) each is pretty much equivalent. That is pretty theoretical starting point. After that we can optimize these systems in different ways. 16-copter allows to use somewhat bigger, yet less-bladed, like 2-3 bladed propellers which are more efficient than multi-blades of helicopter (and you can't really have 2 bladed rotor on helicopter). You can use ducts on 1.5m propellers - not a case for helicopter. So all this allows to increase efficiency.
Ya there are definitely many ways of looking at this. I found the link about rotor power halving for each doubling of diameter (where it says: "So in this process the power is halved to (2m)(v/2)2/2 = mv2/(2x2) = P/2."):
Downwash definitely figures into this, as well as using fewer blades.
But, the key point is that we intuitively know that smaller craft require exponentially less power than larger craft. We could build a large craft with a very wide rotor to approach the efficiency of small craft, but I think it will turn out to be easier to use multiple small craft and scale how much weight we lift linearly.
"the key point is that we intuitively know that smaller craft require exponentially less power than larger craft."
This is where your intuition has let you down. Think about a normal helicopter's lifting capability and range as you go between models and full size aircraft. Or why you see bees walking.
not really. The power loading (power divided by surface of the propeller's circle) is the key factor what matters for efficiency (until of course Reynolds number - air viscosity factor - starts to change - happens when we come to Kolibri scale)
In general, helicopter with 3m long blades (6m diameter) on one 500kw turbine or something like 16 of 30kw motors with 1.5m diameter propeller (with the same number of blades as helicopter) each is pretty much equivalent. That is pretty theoretical starting point. After that we can optimize these systems in different ways. 16-copter allows to use somewhat bigger, yet less-bladed, like 2-3 bladed propellers which are more efficient than multi-blades of helicopter (and you can't really have 2 bladed rotor on helicopter). You can use ducts on 1.5m propellers - not a case for helicopter. So all this allows to increase efficiency.