Square cube law

Hi

Could someone explain the square cube law please in relation to scale down engines.

Frank

i

Basically, area varies with the square of length, and volume varies with the cube.

If you make an engine to 1/2 scale (ie, the length, height and width are 1/2 of the prototype), then the cross-sectional area of the cylinders will be 1/4 (1/2 squared) and the swept volume will be 1/8 (1/2 cubed) of the prototype.

Also referred to as 'Surface Area to Volume Ratio"

and the old truism "You can't Scale Nature"

... which is why we don't have giant insects < etc. >

MichaelG.

A nice big subject. Here goes with two examples:

1. If you double the length of a ship and keep its proportions the same, the resistance to motion through the water will increase by the square, four times, but the carrying capacity by the cube, eight times. This was Brunel's thinking behind the S.S. Great Eastern - it would carry a large cargo to Australia and return without re-coaling. They discovered coal in Oz while it was being built.
2. The power developed by an engine, without speed increase, is proportional to its volume but the heat lost through its walls is to the squared. Halve the engine size and you have an eighth of the power but a quarter of the heat loss. Therefore a small engine will be less efficient, thermally, than a large engine.

As with all such rules it is a generalization with many other factors being important. This allows many to find exceptions to the rule. However in essence it is correct.

Go on, have fun with this one!

JA

. which is why we don't have giant insects < etc. >

MichaelG.

I thought that was due to oxygen levels, used to be some big bugs about millions of years ago

b.

bodge,

May I respectfully suggest that you review the logic of your response.

Hint: ... we don't have giant insects [now, in the prevailing atmosphere]

MichaelG.

Michael G.

Thank you for being so polite

I geuss its because i cant equate " scaling nature " to the size of bugs and oxygen levels . sorry for going off topic

b.

.

bodge,

I have sent you a PM

MichaelG.

.

P.S. ... Just for clarity: It's really the 'laws of nature' that cannot be scaled.

Edited By Michael Gilligan on 09/01/2016 16:34:47

Also, because muscle and bone strength is dependent on cross sectional area and weight is proportional to volume, large animals such as elephants and rhinos have proportionally thicker legs than smaller animals.

An alternative response to the 'We don't have giant insects' debate:

We have never had huge insects because the weight goes up with the volume of the insect but the lift depends on the area of the wings. In the distant (fossil) past, there were are few bigger than modern insects, but nothing as big as an eagle or a vulture, because their lungs and feathers are more efficient. It is surmised that the big fossils were from a period when oxygen levels were higher, but this runs contra to the science of fire which says that with more % of O2 wildfires would be much more prevalent.

While the rule works against the production of power from steam (or petrol etc) it works in favour of model gliders. The smaller the plane, the easier it is to make it fly, as the weight goes down faster than the lift as you go smaller. You can even make a solid oak glider fly if it is small enough (but watch out or the cat will eat it).

There are other effects involved in the broad 'can't scale nature' debate - especially surface tension and related effects.

Regards, Tim

One obvious item that cannot be scaled is a beam. This is not square cube law but dependent on the second moment of area of the beam, that is the beam width times the beam depth cubed. This governs the deflection and load carrying capacity of the beam. It also applies to a torsion rod and makes scaling of springs difficult.

In addition moment of inertia is essentially the same as second moment of area so the same difficulties apply to flywheels.

JA

Although tiny the Somerset Coalfield is more interesting than most. Bell Pits, early engines and tramways, a canal said to be once the most profitable in England, the father of geology and of course the world famous Radstock Giant Dragonfly

I thought I saw one over Bath once, but it turned out to be a Hercules.

Another example of "you can't scale nature" is contained in some black humour from the coalfield. It relates to mammals when they accidentally fall down a mine shaft. Apparently mice walk away unharmed, dogs break their legs, men break all their bones, but horses splash. (F = mv²)

Another problem arises with SillyOldDuffer's black humour, as it is exceptionally difficult for a horse to fall down a scaled-down mine shaft. At some scales even mice would have difficulty.

Tim

Good point Tim! And if poor old Dobbin did have the misfortune to fall into a scaled down mine-shaft, he wouldn't go splash at the bottom. I'll need to experiment if my model coal mine is to be realistic. I wonder if anyone would lend me a horse...

Cheers,

Dave