From: frank@bigdog.engr.arizona.edu (Frank Manning)
Newsgroups: sci.math
Subject: Re: a plane twice bigger has a engine more powerful
Date: 11 Nov 1996 01:51:03 GMT
In article <01bbcf46$b8ab1220$330296c2@club-internet.club-internet.fr>
"gdm" writes:
> This is a problem that I didnot succeed to solve when I was a student. It
> was a long time ago. A toy-plane flies well. I want to build a plane twice
> bigger. I thought candidly that the engine had to be 2*2*2=8 times more
> powerful. I was wrong: the engine had to be 8*root(2) more powerful but I
> had never understood why and I regret to have not dared to ask my teacher.
> Please explain me the exact reason because, for 20 year ago, I think
> sometimes at this problem.
Intersting problem. First we need to clarify what "twice bigger" really
means. I'll assume that means we double the linear dimensions of the
airplane. We'll use wingspan as a characteristic dimension.
We also need know the weight of the larger airplane. We'll assume the
"density" of the two airplanes is equal. In other words, the weight is
proportional to the cube of some characteristic dimension -- wingspan,
for example.
Actually, it's not obvious that this is a reasonable assumption. But
having worked this problem before, I'll leave it as an exercise to the
reader (hint -- a 1/10 scale model of an existing full-scale airplane
seems to fly OK if the weight ratio is 10^3).
Now we need to know a little fluid mechanics. Actually, this is more of
a sci.physics problem than sci.math, but anyway...
It helps to know that the lift-to-drag ratio (L/D) depends only on the
angle of attack and external shape of the airplane. This isn't
precisely true, since L/D is a weak function of Reynolds number, but
we'll ignore that effect. We also ignore Mach number effects.
We assume the two airplanes have the same external shape. Both
airplanes are assumed to fly straight-and-level at the same angle of
attack, which means (L/D)1 = (L/D)2.
Drag = Lift / (L/D)
Thrust = Drag (for nonaccelerated, straight-and-level flight)
Power = Thrust * Velocity
= Drag * Velocity
Power = Velocity * Lift / (L/D)
Power1 Velocity1 * Lift1 / (L/D)1
------ = --------------------------
Power2 Velocity2 * Lift2 / (L/D)2
But (L/D)1 = (L/D)2
Power1/Power2 = (Velocity1/Velocity2) * (Lift1/Lift2)
Lift = (1/2) * Air_Density * Velocity^2 * Area * CL
(Drag is similar, except use CD instead of CL.)
Velocity = sqrt(2 * Lift / Air_Density * Area * CL)
Both airplanes are flying at the same angle of attack, which means CL is
the same for each:
Velocity1 sqrt(2 * Lift1 / (Air_Density * Area1 * CL))
--------- = --------------------------------------------
Velocity2 sqrt(2 * Lift2 / (Air_Density * Area2 * CL))
sqrt(2 / (Air_Density * CL)) * sqrt(Lift1 / Area1)
= --------------------------------------------------
sqrt(2 / (Air_Density * CL)) * sqrt(Lift1 / Area1)
(Lift1 / Area1)
= sqrt(-------------)
(Lift2 / Area2)
Power1 Lift1 * sqrt(Lift1/Area1)
------ = ------------------------- = Power_Ratio
Power2 Lift2 * sqrt(Lift2/Area2)
Lift1^(3/2) * Area2^(1/2)
Power_Ratio = -------------------------
Lift2^(3/2) * Area1^(1/2)
In straight-and-level flight, Lift = Weight. From the airplane density
relationship:
Lift1 = ka * Span1^3
Lift2 = ka * Span2^3
Span2 = 2 * Span1
Lift2 = ka * (2 * Span1)^3
Lift2 = ka * Span1 * 8
There is a similar relationship between the two wing areas:
Area1 = kb * Span1^2
Area2 = kb * Span2^2
Area2 = kb * (2 * Span1)^2
Area2 = kb * Span1 * 4
To summarize:
Area2 = 4 * Area1
Lift2 = 8 * Lift1
Lift1 ^(3/2) * (4*Area1)^(1/2)
Power_Ratio = ---------------------------------
(8*Lift1)^(3/2) * Area1 ^(1/2)
+-------------------------------+
Power2 = 2^(7/2)*Power1 = | Power2 = 8 * sqrt(2) * Power1 |
+-------------------------------+
where
Air_Density: Self explanatory
Area: Wing area
CL: Lift coefficient
CD: Drag coefficient
Drag: Aerodynamic drag
ka, kb: Proportionality constants
(L/D): Lift-to-drag ratio, also CL/CD
Lift: Aerodynamic lift
Power: Thrust power = Thrust * Velocity
Span: Wingspan
Thrust: Aerodynamic force produced by propulsion system
Velocity: True airspeed
Subscript meanings:
Span1: Span of small airplane
Span2: Span of large airplane
etc.
-- Frank Manning
-- Chair
-- American Institute of Aeronautics & Astronautics, Tucson Section