Factors of Scale

[Zipper730]

I'm curious if there are any rules of thumb (or scientific formula) that would allow a person to deduce the effect that scaling up an aircraft has on weight (provided the basic shape remains more or less the same) as well as the effects on weight increasing maximum load-factor has on a design.

 

[Shortround6]

For a solid object the weight goes up with the cube of the increase.
Double the length (or caliber of a shell) and the weight goes up 8 times.

Of course for hollow objects this becomes less accurate real quick.

Also aircraft, for the most part, don't scale up in perfect proportion. Adding 20% to a fuselage length rarely means adding 20% to it's width and/or 20% it's depth/height.

 

[swampyankee]

To expand what Shortroud6 said: if everything is scaled in proportion, including things like skin gauge and tubing thicknesses, weight scales by the cube of the size, areas by the square, and moments of inertia by the fourth. Some things simply don't scale, like people.

Light aircraft are frequently structurally inefficient, as aluminum only comes in specific sizes, and thicknesses of skins and bulkheads may be dictated by criteria like not having a dropped screwdriver punch a hole in the floor or handling denting a leading edge. It's also not likely to be cost effective to do things like electrochemically mill skins or use titanium instead of steel or vacuum-melt 18-8PH instead of 4130 or advanced joining techniques vs rivets, so an unmodified scaling up in size would be heavier than designing to size, but an unmodified scaling down could be too fragile.

 

[Zipper730]

For a solid object the weight goes up with the cube of the increase.

That I'm aware of, but that doesn't seem to apply with aircraft design as the structures are mostly hollow. I was curious if there were any rules of thumb.

 

[swampyankee]

That I'm aware of, but that doesn't seem to apply with aircraft design as the structures are mostly hollow. I was curious if there were any rules of thumb.

It does, if everything is scaled in the same proportion. Usually it's not if the scaling is large, that is is probably quite close for 90 to 110 percent of original, but not so good for 50 or 200 percent, because of things like minimum sheet metal gauges, manufacturing expenses, and things that don't scale, like instruments and people.

 

[wuzak]

For a solid object the weight goes up with the cube of the increase.
Double the length (or caliber of a shell) and the weight goes up 8 times.

Of course for hollow objects this becomes less accurate real quick.

Also aircraft, for the most part, don't scale up in perfect proportion. Adding 20% to a fuselage length rarely means adding 20% to it's width and/or 20% it's depth/height.

I believe lift increases with the square of wing area, so if the scale is 2 the lift increases by 4. But weight increases by 8.

 

[wuzak]

That I'm aware of, but that doesn't seem to apply with aircraft design as the structures are mostly hollow. I was curious if there were any rules of thumb.

Clearly there isn't.

 

[GregP]

The North American B-45 was:
75 ft long, 89 foot span, 91,000 lbs max weight

The Boeing B-52 was:
159 feet long (about twice), 185 foot span (about twice), 488,000 max gross (about 5.3 times, not quite cube or it would be 8 times). But, it is close to cube.

Is it representative? Not really. It doesn't work so well with the B-46 and B-47. They had close to the same length and span, but the B-47 grossed 2.4 times the B-46 and max weight. So, the answer CAN be all over the place.

But, the B-47 and B-52 are very close in design. Both have long, skinny, swept wings with podded engines and long, relatively narrow fuselages. The B-52 is about 1.5 times the length and span of the B-47. 1.5 squared is 2.25 and the B-52 at max gross is 2.12 times as heavy as the B-47 at max gross. So, for very similar designs, the cube rule holds for the B-47 / B-52. How does it do with other, very close aircraft.

How about the Rockwell B-1 and the Soviet Tu-160? VERY close in design.
Tu-160 is about 1.21 as long, 1.32 times the span. Split the difference and call it 1.25. 1.25 squared is 1.56, and the Tu-160 at max gross is 1.27 times the weight of the B-1. So, it seems close enough to say it looks to be slightly under cubed, but in the ballpark for at least these two very similar designs.

You might try a few yourself. Interesting question.

 

[Zipper730]

The North American B-45 was:
75 ft long, 89 foot span, 91,000 lbs max weight

The Boeing B-52 was:
159 feet long (about twice), 185 foot span (about twice), 488,000 max gross (about 5.3 times, not quite cube or it would be 8 times). But, it is close to cube.

Well, technically the B-52A was 420000 if I recall. This seems only valid if I was to make comparisons between aircraft of similar structural strengths.

The B-45 and B-52 were stressed for similar g-loadings (B-45: 4.5 ultimate; B-52: 3-4.5 ultimate). A fighter would be stressed for higher figures.

You might try a few yourself. Interesting question.

I'll start looking through statistics

 

[Timppa]

I believe lift increases with the square of wing area, so if the scale is 2 the lift increases by 4.

Think again, lift increases with..

 

[swampyankee]

I believe lift increases with the square of wing area, so if the scale is 2 the lift increases by 4. But weight increases by 8.

Leaving out issues like Reynolds number effects, and holding airspeed constant, lift is linear with wing area or to the square of the scaling factor.

 

[Zipper730]

Using the P-40E and P-47 as an estimate it seems to work okay as a function of empty weight

 

[GregP]

Hi Zipper,

I'm thinking of aircraft with a similar planform, general setup, and construction techniques.

There is a twin-engine, delta-wing Mirage 2000 fighter and a twin-engine, delta-wing Mirage IV bomber, both by Dassault. I'd bet they are decently close to the cube rule in weight.

Another two that might be close, but not the same are the Fw 190 and the F8F, but there isn't much size difference, so there is no point to the comparison. I would not suspect the P-47 and the Fw 190 because the P-47 had a turbocharger setup and very complex construction whereas the Fw 190 was only supercharged and was simply built.

We might have a good look at it using the F4D Skyray and the Vulcan bomber, though. Hard to say without doing it and it's not really my premise.

 

[swampyankee]

A more generic look will tend to show that wing loading does tend to increase with size, which tends to be compensated for by increasingly complex high-lift devices. One doesn't put triple-slotted Fowlers and leading edge flaps on something the size of a Foxjet, but would on a 747.

 

[Zipper730]

We might have a good look at it using the F4D Skyray and the Vulcan bomber, though. Hard to say without doing it and not my premise.

I would have figured that the Vulcan was rated for 3.5g normal and the F4D was probably rated for 6.0-6.5 or so normal. Wouldn't everything only scale right if the aircraft had similar strength?

 

[Zipper730]

A more generic look will tend to show that wing loading does tend to increase with size, which tends to be compensated for by increasingly complex high-lift devices. One doesn't put triple-slotted Fowlers and leading edge flaps on something the size of a Foxjet, but would on a 747.

I'm not sure if this has any validity or not as it relates to aircraft weight or what you said about wing-loading.

Admittedly I figured power loading and wing-loading were in direct proportionality. I'm not sure how frontal area scales up with drag though

 

[GregP]

A Cessna 152 and a Cessna 152 Aerobat have exactly the same max weight. The standard C-152 has a standard empty weight of 1,101 lbs and the C-152A has a standard empty weight of 1,125 lbs.

The difference is 2.17%. Close to meaningless and some of that weight is aerobatic harnesses that are not in the standard airplane.

 

[Builder 2010]

The one thing that doesn't scale well when you go smaller, i.e., making a scale flying model, is the speed at which is flies. Air molecules are a fixed size so the amount of them passing over and under the airfoil doesn't scale. To get lift in a small model wing, especially if it is heavily loaded, requires faster air flow across the surface. This is specifically why small birds have to flap their wings so much faster than large birds. The B-17 cruised at 187 mph. In scale RC model meets some of the judges are wanting the planes to fly at scale speeds too. So for the 1:16 B-17 I built the scale speed would be a bit under 12mph. Instead the model flew at about 80mph. At 12mph the wing has literally no lift at all. When model design aircraft are tested in wind tunnels, the air velocity is increased to simulate the Reynolds number of the air over the actual wing.

 

[Zipper730]

The one thing that doesn't scale well when you go smaller, i.e., making a scale flying model, is the speed at which is flies. Air molecules are a fixed size so the amount of them passing over and under the airfoil doesn't scale.

Now that's interesting, especially with Reynolds numbers I'd have figured lift figures would be better when smaller...

To get lift in a small model wing, especially if it is heavily loaded, requires faster air flow across the surface. This is specifically why small birds have to flap their wings so much faster than large birds.

That's interesting, I'd have figured that if the bird's weight went up to the cube of size, that the wing area would go up to the square root and it'd be actually worse. With scaling not necessarily working in that order, the issue is how much the bird's body scales up versus how big the wings get.

The B-17 cruised at 187 mph.

According to billrunnels , who was a B-17 bombardier said that the cruise speed was around 150 mph indicated. At 22500', this comes out to 205-210 mph; at 25000', this comes out to 213-221 mph; at 27500', this comes out to. 226-231 mph based on the barometer set to 29.92" as is current custom (not sure about the 1940's), and air temperatures ranging from -40 to -55F at 22500, and ranges of -40 to -70 F at altitudes of 25000-27500 F.

If I was to use 187 mph as a cruise-figure in indicated airspeed, that would correlate to

• 22500' @ -40F/29.92" = 262 mph
• 22500' @ -55F/29.92" = 257 mph
• 25000' @ -40F/29.92" = 276 mph
• 25000' @ -55F/29.92" = 272 mph
• 25000' @ -70F/29.92" = 266 mph
• 27500' @ -40F/29.92" = 288 mph
• 27500' @ -55F/29.92" = 287 mph
• 27500' @ -70F/29.92" = 282 mph

These figures seem higher than the normal cruise speeds listed though at 27500 they seem right up on the red-line for the later B-17's.

In scale RC model meets some of the judges are wanting the planes to fly at scale speeds too. So for the 1:16 B-17 I built the scale speed would be a bit under 12mph. Instead the model flew at about 80mph. At 12mph the wing has literally no lift at all. When model design aircraft are tested in wind tunnels, the air velocity is increased to simulate the Reynolds number of the air over the actual wing.

Now that is interesting, though I vaguely remember something about stalling being easier to trigger at low Reynolds numbers (which is strange, I'd figure more turbulent flow...)

 

[pbehn]

Clearly there isn't.

There is, and it is a man with 10 hand thumbs and another 10 foot thumbs knows all the rules.