Advanced materials question

Is is possible yet to build a sphere enclosing a vacuum, which is lighter than air and won’t collapse at sea level pressure?

Depends. Do you mean that the whole system (sphere + vacuum inside) is lighter than the equivalent volume of air? If so, then you can build your sphere out of pretty much anything as long as you make it large enough, since the volume inside the sphere increases faster than its surface area. An extreme example of this would be building a sphere out of 5 cm-thick steel plates; if it contained vacuum and was bigger than about 300 m across, it would be lighter than air.

Yes, but will your 300 m steel sphere support the 2.9 billion kilograms being applied to it?

It’s is not so much the material but the size.
Any sphere, enclosing a vacuum, and is lighter that air will be impossible to make unless on a nano size scale, if then.

A sphere, enclosing a vacuum that is lighter than air, is a pressure vessel easily designed to not collapse at sea level pressure.

I think you are both wrong. The ration betweem the surface of a sphere and the volume of a sphere is constant regardless of size. The pressure per-sq in on the surface of the sphere will be the same whether it is macro or nano.

Buckminster Fuller adressed this. He was of the opinion that a geodesic sphere of large enough size could be fitted with one-way vents. It would be heated by the sun. The air inside would expand and exit via the vents. The sphere would then be lighter than the surrounding air. Fuller made some suggestions about using such floating spheres as living spaces.

I dunno about you, but one-third the radius looks pretty dependant on size, to me.

D’oh :smack:

Simple geometric relationship.

Also simple physics.
When the strength of the sphere is concerned the thickness increases by the square of the linear dimensions.
Thus a very tiny, nano, sphere could be evacuated and the containing sphere could be very thin.
I’ll leave the calculations for others to toy with.

Helium and a balloon won’t work for ya?

It is not just a surface area or volume issue. Preventing a vacuum vessle from imploding requires a lot of stiffness. This scales really badly. Even a pressure vessle scales non-liniarly due to Pascal’s law. If it is possible to make a “vacuum ballon” it would be much easier at nano to micro scale.

A spherical pressure vessel has to be much thicker than a vacuum vessel of the same size.
P.V. failure is predictable knowing the material, sphere size, wall thickness, etc.
In the case of a vacuum vessel the implosion forces are balanced all around and it is random imperfections in the wall that result in failure.
Hence the smaller the sphere the thinner the wall can be to contain the vacuum!

Whether or not such a sphere can be made is dependent on material properties and not on scale at all.

Take a certain material with stiffness (Young’s modulus) of E and stress limit of S and construct a hollow sphere out of it. The classic thin-wall vessel equation gives a pressure limit of:

P = 2S (t/r)

where t is the thickness and r is the radius of the sphere.

Since this is an external pressure vessel, buckling may be an issue. In that case, my reference says the critical buckling pressure is somewhere between

P = 0.37E(t/r)[sup]2[/sup]
P = 1.21E(t/r)[sup]2[/sup]

depending on manufacturing precision.

Both of these equations include the geometric factor (t/r). Thus, whether failure is governed by crush or by buckling, for a given material and given pressure (atmospheric in this case), there will be a certain minimum (t/r) ratio.

And that (t/r) ratio will determine the answer to the question. A constant (t/r) ratio will give a constant shell volume to internal volume ratio, which in turn will give a constant shell weight to displaced air weight ratio (assuming a particular material, which we have been).

[As an aside, it could be argued that if this problem is buckling limited, there may be a particular scale where manufacturing precision is optimized, in which case there is some scale dependence. That’s a practical issue rather than a theoretical one, though.]

[As another aside, I swear this question came up in GQ before, but I can’t find the thread. C’est la vie.]