Genghis Mater, who has Engineers for a father, husband, and son (me) maintains that an Engineer can be identified as somebody who when all other avenues have been explored will finally reach for the instructions.
In that spirit, I thought I'd get an aerodynamics textbook off the shelf. This, as is traditional, divides the take-off into three segments, as follows:-
Ground roll
S1 = integral from 0 to Vr of [W / 2g (T - D ) ] dVē
Assuming that's correct, then for most of that ground roll, lift isn't a significant player, but drag is. So, for most of the ground roll, so long as the flap setting used isn't significantly increasing drag it's not a player.
But the boundary condition, Vr (rotation speed) is clearly a function of CL.max and hence increased flaps reduces take-off roll until the increase in Cdo (or Cdi) doesn't become sufficient to overwhelm that.
Rotation segment
The book cops out this time, suggesting that the S1 formula should be used over 1 second segments, based upon a known and assumed constant rotation rate. So, on that basis, the same argument applies.
Initial climb segment
The initial part of this is the curved path from parallel with the runway to the steady climb gradient, this is defined by...
S2 = Vē * (T - D) / (delta-n x g x W)
So, what are the players here...
V - clearly the slower you can go (which is going to be a factor of Vs) the shorter the run, and since this is a squared term it's probably the most significant.
(T - D), no surprises there, as before you want to keep drag down. But, since we're at the left hand side of the drag curve it's mostly induced drag that you want to reduce, Cdo shouldn't be too big a player. Flaps *probably* help here by shifting the main spanwise centre of lift inboard, reducing tiplosses.
delta-N, the increase in g during rotation implies that high-lift devices are helping again, since the more g you pull in the rotation, the more S2 is reduced.
I have to say, seeing W on the bottom of that equation is rather anti-intuitive, higher weight should surely be increasing S2. However, thinking through it, greater W will probably reduce delta-n by the same factor, so I don't think that one should get too worked up about it, and anyhow it's not flaps dependent.
Which all seems to come down to the principle of use flaps to keep speed down, and they reduce take-off distance, until you hit the law of diminishing returns and increases in CL.max are more than compensated for by much greater increases in Cd
And on the 747 thing
I'll start by admitting that I have absolutely no knowledge of the type at-all, I'm just playing with equations here. Climb rate in a steady climb is given by...
RoC = V . (T - D ) / W
We can obviously assume that the weight is the same between 10° and 20° flaps, so presumably the better climb rate you are getting at flaps 20 is down to one or more of...
(1) Your best climb speed is faster at flaps 20, or
(2) Total drag is reduced at flaps 20 (seems a little unlikely to me), or
(3) In some way the engine is put into a higher thrust regime at flaps 20 (seems even more unlikely), or
(and I'm sticking my neck out a little here)...
(4) The pressure errors are altered in flaps 20 such that the static tends to indicate a higher rate of climb on the VSI ?
I must admit, I find it hard to hang my hat on any of these. Type-knowledge anybody?
G