Re: [indoor construction] VP or VD

From: markdrela <drela_at_mit.edu>
Date: Wed, 25 Oct 2006 14:58:40 -0000

--- In Indoor_Construction_at_yahoogroups.com, "John Barker"
<john.barker783_at_...> wrote:
>
> If the torque coefficient and the air density
> are taken as constant for our purposes
> it can be said that Q is proportional
> to n^2.D^5 or that n^2 is proportional to Q/D^5.

True. But this assumes that the prop *shape* stays the same.
So it assumes that the chords increase proportionately with D,
which isn't the case for a VD prop.

If we treat the chord and diameter variation separately,
and assume that
 * prop is turning at its pitch speed (i.e. blades are unstalled),
the approximate scaling relations are

 T ~ sqrt[1 + (pi D/P)^2] * chord * D^2 * cl * V * RPM
 Q ~ sqrt[1 + (pi D/P)^2] * chord * D^2 * cl * V^2

If we further assume that
 * V = constant at the min-power speed of the airplane,
 * cl = constant at its best cl/cd point
 * P/D doesn't change appreciably (roughly true for a good VD prop)
we get

 T ~ D^2 * RPM
 Q ~ D^2

or equivalently

 RPM ~ T / D^2
 Q ~ D^2

For my VD example, D was increased from 40cm to 56cm while holding
thr thrust T fixed. These simple rules then predict that

1) RPM should decrease by a factor of (40/56)^2 = 0.51
2) Torque should increase by a factor of (56/40)^2 = 1.96

This was almost exactly what the full prop calculation method gave.
So the simple rules above work very well for design decisions. But as
I said before, it's important to increase blade angle together with
the diameter to keep the blade airfoils happy.
Received on Wed Oct 25 2006 - 08:03:58 CEST