# What are the physical laws for upscaling an RC model to 1:1?

What are the laws of physics for upscaling an RC model from a smaller to larger scale or even to a full sized 1:1 aircraft? I guess the basic thing may be lift/weight ratio which was nicely described here:

... when you scale an airfoil up, the lift it produces increases with its area, which grows with the second power of size, but its weight increases with volume, which grows with the third power of size.

The above means the lift/weight ratio is worse for bigger vehicles which I guess makes them fly worse (less range, higher fuel consumption).

Are there any more laws to be taken into account for such upscaling? I mean in terms of aerodynamics or power source (battery/fuel) etc. Does the air behave differently with scale? Do different laws apply for different aircraft types (plane, helicopter, quadcopter etc.)?

• Well, when you scale it up more, you can do less and less of the "tricks" RC planes can do. – SMS von der Tann Oct 12 '16 at 22:51
• When i see "law" i first think of regulation. Physics maybe the thing. Just my humble opinion though :) good question anyway – vasin1987 Oct 13 '16 at 3:26
• – Peter Kämpf Oct 13 '16 at 11:28
• And be sure to look at this one, too – Peter Kämpf Oct 13 '16 at 11:30
• aerotelemetry.com/press-design-news-airplanes-aviator – Daniel Oct 14 '16 at 15:50

This is not a full answer: much is mentioned by Michel Touw, but I'd like to correct a common misconception.

The mass growing as the third power of linear dimensions applies only to solid objects. But the airplane is not solid; rather, it's a thin-skinned frame. (Unless it's a solid foam RC model). Therefore, its mass should grow with the area, as the square of length.

However, this is, of course, not quite true. In reality, you need thicker skin, thicker spars, etc., and these elements will get heavier 'properly' with the cube of length - and even more, because they'll need to support themselves as well, with their added mass.

In the end, the mass of airplanes grows between the second and third power, and much closer to second.

Let's take B734 and B772 for example, which differ close to twice in linear size. Their empty weight (roughly 33 and 140 tons, depending on modification) differs by 4.25 times, which is approx. 22.1. MTOW has a similar ratio: 68 vs 300 tons, 22.15.

A lot of things happen, depending from what perspective you look at it. I'll first start with some scaling laws. Note that ~ means 'scales with' or 'is proportional to'.

Basic scaling laws:

(1) length   ~    length
(2) density  ~    1 (is constant)


From (1) we can derive that:

(3) area     ~    length^2
(4) volume   ~    length^3


From (2) and (4) we can derive that:

(5) mass     ~    length^3


We also know that accelerations on an object don't change (example: gravity):

(6) acceleration ~ 1


Since force = mass * acceleration, from(5) and (6):

(7) force  ~    length^3


Since acceleration is contant (and here it may get a little weird):

(8) time   ~    sqrt(length)


From (1) and (8):

(9) velocity  ~  sqrt(length)


The idea from time may more comprehensible when you look at vibrating stick: a very long stick will have a longer oscillation period than a short one.

Aerodynamics

Now for the aerodynamics. As the length scale increases, to keep everything the same (proportional) the plane will also have to fly faster (by the sqrt of the length scale). This can also be seen from the lift equation:

L = C_L * 0.5 * rho * V^2 * S
L = W, for which holds that W and L (forces)  ~ length^3
length^3 ~ 1 (from C_L) * 1 (from 0.5) * 1 (from rho (density)) * V^2 * length^2
V^2 ~ length
V ~ sqrt(length)


The increase in both length and velocity increase something called the Reynoldsnumber, which is equal to:

Re = (rho*V*x)/mu  where:
rho = density of medium (air)
V = velocity
x = characteristic distance
mu = viscosity of the medium


As the Reynoldsnumber increase, in general the drag gets lower for a given amount of lift. This also allows (for most airfoils) a larger maximum lift coëfficiënt (the lift per area generated at a certain velocity and density).

Also your Mach number increases, since velocity increases.

M = V/a where
V is the velocity of the aircraft
a is the speed of sound


As M > 0.3 you have to take this effect into account. It decreases your lift coëfficiënt. However 0.3M is around 100 m/s, and your craft probably won't fly at that speed. At 0.3M the error is about 5%.

These are the 2 main aerodynamic factors that come into play. Note that both numbers are dimensionless.

Other factors

You can also think of other factors such as the range on your transmitter, structural changes in proportions. Controlability: as time increases with length, your craft will feel less agile, requiring longer approaches which may be out of sight.

But I seemed you wanted to know mainly about the aerodynamics. With the basic scaling laws, you will be able to derive what happens with most other factors you encounter.

• Before reading this answer I wondered if the viscosity of the air would be a factor. I don't think I saw anything about properties of the air in your answer. Are the (fixed) properties of the air a factor in scaling? – Todd Wilcox Oct 13 '16 at 13:11
• No, the fixed properties of air don't change the scaling. Just like density doesn't scale. The velocity of the flow however, does affect the scaling as can be seen in the answer. – Michel Touw Oct 13 '16 at 14:14
• Could you maybe use some mathjax to make the equations a little more readable? – Jay Carr Oct 13 '16 at 17:46
• @ToddWilcox, viscosity is a factor. It's part of the Reynolds number. The Re number is the ratio of the mass vs viscous properties of the fluid. And because Re for airplanes is large (thousands to millions), it basically shows that aerodynamic forces are produced mostly by pushing masses of air around rather than by 'shearing' it. And thus you see air density (rho) in every equation. – Zeus Oct 17 '16 at 1:00

You may want to have a look at the free NASA publication which explains the relationship between models and actual aircraft.

http://www.nasa.gov/connect/ebooks/aero_modeling_flight_detail.html