Continuing the discussion from xkcd thread:
Is it possible to make a tiny glass sphere strong enough to hold a vacuum? Would it have enough buoyancy to float in water? In air?
What about other materials, like titanium, carbon, or some composite?
What if it’s not a true vacuum? How much internal pressure would be needed to keep this tiny sphere from collapsing, and would it still float?
I can’t answer the first question - how to manufacture a tiny hollow glass sphere filled with a vacuum, but glass is extremely strong in compression so yes, it would be strong enough to resist atmospheric pressure.
In order to be buoyant in air, an evacuated vessel would need to have a mass slightly less than the mass of the air it displaces - sorry if that’s obvious, but that is how buoyancy works and it’s not a lot to work with.
Calculations at a really small scale yield results that are difficult to envisage, so just to begin, consider a sphere that has a volume of 1 cubic metre; the mass of that amount of air at sea level etc is about 1.25kg - so you’d have to make a sphere of some impermeable material with excellent compressive strength, out of less than 1.25kg of that material.
There will be scaling effects that bring this closer to being possible at smaller scales, but I believe it never quite gets to the break-even point where vacuum could be used practically to create net positive buoyancy in air.
Here are a few previous threads on the topic:
Perhaps I’m on the entirely wrong track here, but I’d guess that the pressure on the glass sphere, and hence the strength it needs to have, increase with the square of the diameter of the sphere, but its volume (and hence buoyancy) increases with the cube. So as you make your sphere progressively smaller, you’re actually making it harder to reach the point where it floats in air compared to a larger sphere.
I suspect buoyancy is not how it is proposed to work. Very small heavier than air objects can remain in the atmosphere a long time due to drag, diffusion, and updrafts. This is why dust and tiny spiders and so on can remain in the atmosphere for extended periods despite being heavier than air.
Even if it was possible, the process to create a “bubble” of vacuum is hazy at best.
Aren’t light bulbs glass spheres containing a vacuum? A vacuum is a very small force of a maximum of 1 atmosphere at the most. Even a high vacuum is limited to this because the definition of a vacuum is little or no molecules. Getting little amount of molecules is easy, getting none is not possible.
Unless I am missing something here I believe the answer is yes. I suspect I am missing something, I need coffee.
I think light bulbs typically contain inert gas because (for a tungsten bulb) this slows down the loss of atoms boiling off the filament (or something like that).
But the question isn’t whether you can make a rigid container that can be evacuated (you can - we make vacuum chambers out of glass) - the question is whether you can do it using an amount of material that weighs less than the evacuated air.
@Mangetout utterly nailed it in the first post and more succinctly here in the second.
Those prior threads he cites go into great detail. Worth reading. There are others I think.
Vacuum is only slightly lighter than hydrogen. But hydrogen at ambient pressure of your outside atmosphere at whatever altitude you want to cruise is incredibly powerful at supporting your sphere from the inside. Conversely vacuum provides zero support.
You can make your hydrogen spheres out of the thinnest lightest imaginable plastic sheet, if they’re filled with hydrogen (or any gas really) to the same pressure as outside. But if it’s vacuum inside and atmosphere outside, you need very stiff, relatively thick walls to keep the sphere from being crushed. The extra lightness of the vacuum doesn’t come close to repaying the extra heaviness of the container.
It’s a losing game all the way around.
More cites:
It’s not hard to calculate the necessary dimensions. For the spherical shell of glass with outer radius r_1 and inner radius r_2 to have the same overall mass as a spherical volume of air of radius r_1, we must have
where \rho_g is the density of glass and \rho_a is that of air. Solving for this, we get
Or to put it another way, the wall thickness must be \rho_a / (3 \rho_g) times the radius of the sphere to make it neutrally buoyant.
This number is very very small — about 180 parts per million, assuming air at room temperature and soda-lime glass. A glass sphere with a radius of 1 mm would have to have walls about 180 nanometers thick — less than a wavelength of light. I don’t want to say this is impossible to manufacture, but it would be quite tricky.
And even if you could construct it, it probably couldn’t withstand an external pressure of 1 atm. If the formula on this page is to be believed, such a sphere would buckle at an external pressure of 0.046 atm. And that’s if it was perfectly constructed with no microscopic flaws; otherwise, it couldn’t even withstand that. Note that this same logic would apply to a glass sphere of any size, microscopic or not, since the ratio t/r is fixed by the mass requirement.
(Floating in water is much easier, because the density of water is much closer to the density of glass. In that case it works out that the wall thickness needs to be about 15% of the radius, and it can withstand about 32,000 atm.)
Beautiful. Thank you. The money quote is right here:
So the glass is ~1/20th of the needed strength enough to survive. Assuming perfect glass. IRL, we’d need to design in a safety factor or nearly every sphere would fail as soon as it was made.
And while we’re at it, folks sometimes suggest things like rigid hoops with a membrane stretched across them, but those are even less efficient. A continuous sphere is the best you can do.
I assume the same would be true of anything where the membrane (in tension) is supported with internal radial spokes in axial compression (like a circus tent, but in the form of a sphere).
Yup. This was covered in excruciating detail in some of those earlier threads. No matter what you do, no matter how nifty your structure is, you can’t beat a sphere. Every part of a sphere is in compression. This will always beat any other structure.
Oh, thank you for the formulas! Very helpful.
So, we have the thickness of the material as
t = \frac{1}3r\rho_a/\rho
given the radius of sphere, the density of air, and the material’s density, respectively.
And the maximum pressure, from your cite, is
P = \frac{2}{\sqrt{3(1-\nu^2)}} (\frac{t}r)^2 E
given the material’s Poisson ratio, material’s thickness, sphere’s radius, and the material’s Young’s modulus, respectively. (This is a best case scenario with a perfectly manufactured cow sphere.)
Combining, this gives the maximum pressure a neutrally buoyant sphere can withstand,
P = \frac{2}{9\sqrt{3(1-\nu^2)}} (\frac{\rho_a}\rho)^2 E
Note that this does not depend on the radius or thickness of the sphere–it’s is purely a function air density and properties of the material.
I’d like to throw in values for various materials, but out of time at the moment.
E is of order 100 GPa or 10^6 atm.
\rho_{air} is order 1 \mathrm{kg/m^3}
For structural materials \rho ranges from maybe 300 \mathrm{kg/m^3} for some woods to 20000 \mathrm{kg/m^3} for tungsten.
The rest of the numbers don’t matter.
It looks like an oak balloon might support almost an atmosphere.
Edit: plugging in the numbers that “don’t matter” knocks it down another order of magnitude
(Hand wave) The engineers will take care of it. I’m just the idea guy.
Yeah, but… Hear me out…
A carbon fibre cylinder with titanium domes on the ends. If there’s any risk of it buckling we could listen out for that and just back it off.
Mythbusters got a lead balloon to float. And it wasn’t even a sphere. It was a bloated cube. A blube.
I forget the size of the blube, around 10 feet or so. And the thickness of the lead was 0.002". That gives a blube diameter to shell thickness ratio of 60,000 to 1.
What about a glass sphere with a radius of one micron, or even 100 nanometers? The math you cited is beyond my ability to follow but I’d be surprised to hear that there is no advantage to making the sphere smaller. Can’t the shell thickness be proportionally less at smaller radii?
ETA: and of course the original cite was for a sphere filled with dry nitrogen so the shell only has to be strong enough to remained superpressurized at constant altitude.