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" <gdmo@planete.net> 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