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Old 05-19-2020, 07:28 AM
brossa is offline
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A buoyancy question - esp. for gases in gases


We had a big bonfire last night, and as I was watching the flames shoot into the sky and the smoke rise for dozens of feet, I got to thinking about buoyancy. For big objects with defined volumes, the buoyancy force on the object in a fluid is the same as the mass of the fluid displaced by the object. OK, I get that; at least for relatively 'big' objects with nice hard edges separating the 'object' from the 'fluid'.

But ISTM that this requires a certain 'bulk' of material. A single atom of helium doesn't feel a buoyancy force when it is mixed with air; two atoms of helium don't know they are a 'bubble' of helium gas, etc. But if you get 'enough' helium, or hot air, together in one place, even without a confining envelope, it rises until/while it mixes with the surrounding air. I have conceptual questions about this process:

How much 'stuff' is enough for buoyancy forces to be meaningful? I would guess that the mean free path traveled by a molecule has something to do with the scale of such things. No individual molecule knows that it's part of a collection, it only reacts to banging into other molecules and the force of gravity.

What shape would an unconfined blob of less-dense gas assume as it rose through the air, losing volume as its edges diffuse away? Are there currents inside the blob? In a mass of helium, for example, for a given helium atom at the center moving in a random direction, there would be more possible collisions with air molecules along the bottom of the mass than along the top. Looked at another way, the mean free path downward, would be shorter than the mean free path upward. That would make the whole mass rise, I think, and the speed of the rise would depend on what? The mean velocity of the gas molecules in the bubble vs. that of the surrounding air?

Help me to better understand the microscopic behavior here, I guess...
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Old 05-19-2020, 09:45 AM
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I think that if the region of lighter gas is significantly larger than the mean free path of a molecule, it can act like a "bubble", and rise as a whole. Instead of a membrane of latex or whatever acting as the skin, the "skin" is a fuzzy layer approximately one MFP thick. At least, until it diffuses completely into the surrounding gas.

I think. But if someone has a better answer, listen to them.
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Old 05-19-2020, 10:46 AM
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We come across situations like this when we model many different combustion processes. Basically imagine a pipe with air flowing inside it; at a certain point you inject fuel into the center of this pipe. How long does it take for the fuel to mix well with the air ?

The physics of the question does seem simple but in real life, many different models and experiments need to be conducted and most of the final results are proprietary and many times just data fits.

Relevant to the OP, please read the Wikipedia article on Prandtl Mixing Length. https://en.m.wikipedia.org/wiki/Mixi...ical_intuition

I did not find hydrogen-air prandtl mixing lengths in the public domain but there are smart dopers on this board who may help you.
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Old 05-19-2020, 07:47 PM
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Uh-oh, when I see 'Navier-Stokes' I know I'm in over my head.
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Old 05-19-2020, 08:40 PM
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Well a lot of basic fluid dynamics problems are not solved by simple physics and many solutions have empirical terms built into it. The basic physics is sound and easy, but you need tons of experimental data to make reasonable predictions.

Take for example the simple fluttering of a flag in the wind. If you do a search on any research publications, you will see 100s of papers addressing some nuance on this.
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Old 05-19-2020, 11:51 PM
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Quote:
Originally Posted by brossa View Post
Uh-oh, when I see 'Navier-Stokes' I know I'm in over my head.
The Navier-Stokes equations in their conceptual form are actually pretty easy to understand (by pretty easy I mean you can basically cover it in a semester-long class after you’ve covered partial differential equations, thermodynamics, and basic fluid mechanics). Actually using them in simulation, on the other hand, is ferociously complicated because of the breakdown between continuous fluid and particle kinematics, the resulting inherent nonlinearity of turbulence, and then adding the complex chemical kinetics and thermodynamics of combustion phenomena, requiring quasi-empirical models to describe every different type of interaction. Because of the vast number of interactions on an atomic or molecular level it will never be computationally possible to directly simulate a flow field at the fundamental level without a model that is exactly as complex as the scenario it is simulating which is actually thermodynamically prohibitive in any digital computation machine.

However, the question of the o.p. can be answered fairly simply if inexactly in terms of the diffusion coefficients and molecular weights of the respective gases. A free (and chemically inert) gas will essentially rise at a constant rate until diffusion mixes it thoroughly with the surrounding gas relative to the buoyant force, except as attenuated by kinetic losses through the exchange of momentum with the fluid above it. One can witness this in a discrete form with air exhausted by a SCUBA diver, in which the bubbles rise and divide, diffusing with the much denser water although the dramatic pressure differentials in water also playing a much more significant factor than a lightweight gas in atmosphere.

In the case of heated gas rising from a bonfire, it should be recognized that many of the emissions are actually heavier than air and are only rising because of their randomized kinetic energy (e.g. ‘heat’) that causes the diffusion rate to be higher. Of course, as it rises through the air it exchanges momentum with the surrounding particles (including suspended water vapor) and comes to a temperature equilibrium, so all of that, plus turbulence, wind, et cetera would have to be accounted for to give a precise simulation of “unconfined blob of less-dense gas assume as it rose through the air, losing volume as its edges diffuse away”.

So, the best we can do with these kinds of simulations is present a crudely approximate model with large uncertainties. Don’t even get me started on heat amplification form interacting shock waves; that still gives me nightmares. And people wonder why we can’t have a single-state-to-orbit spaceplane...

Stranger
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Old 05-20-2020, 05:10 PM
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I think one atom of helium DOES know that it is lighter than surrounding air.

Well, let's be more careful, before I get silly. I think lighter gases have a tendency to move upward relative to heavier gasses, though this effect constantly competes with diffusion and, perhaps, advection to keep the gasses mixed. I think this is the principle by which hydrogen and helium tend to leave the Earth's atmosphere. The opposite effect of heavier atoms or molecules sinking is what's exploited in centrifuges to concentrate certain isotopes of radioactive elements, as gas (such as uranium hexafluoride), as the different isotopes will have somewhat different densities. But these effects aren't huge, which is why it takes very fast expensive centrifuges such a long time to do their job.
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Old 05-20-2020, 05:25 PM
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Luckily for us, oxygen and nitrogen in the atmosphere don't separate, with a layer of pure oxygen below a layer of pure nitrogen, despite oxygen molecules being heavier than nitrogen molecules. This is because gravity is not strong enough to overcome the random turbulence in the lower atmosphere that keeps them mixed. If you get high enough, above the turbopause at an altitude of about 100 km, the space between molecules is far enough that turbulence becomes insignificant enough that the atmosphere is indeed stratified by molecular weight.
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Old 05-20-2020, 06:22 PM
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Markn+ - in the heterosphere above the turbopause, are things really stratified by molecular weight - ie there is a layer of N2 alone below a layer of O2 alone below a layer of O alone below a layer of He alone below a layer of H alone? Or are there just bands where the primary gas is a single species? So the bottom has N2/O2 and some O, He, H, then when you climb a bit the N2/O2 drops out and O/He/H dominates (but mostly O, and less total O than in the lower band), etc? With the partial pressures of each gas falling off with altitude, but at different rates?

Napier - I thought that H and He leave Earth because their mass is such that at a given temperature they are much faster than other gas molecules, so they are the only molecules that have much of a chance of hitting escape velocity from random thermal collisions. But I could be wrong. I also thought that gas centrifuges only enriched heavier gases at the bottom, not really separating them out in layers like with immiscible liquids. But this goes to my question as to the microscale behavior of buoyancy: how does a single atom experience a buoyancy force?

Stranger On A Train - you said that increased heat causes the diffusion rate to be higher, which is why the heated gas from a fire rises. Can you clarify why the increased diffusion rate would make the packet of gas rise?
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Old 05-20-2020, 07:15 PM
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All gases naturally intermix. The density of any gas in the atmosphere decreases with height, and the decrease is more rapid for heavier gases. So higher up, you'll find a greater proportion of lightweight gases, compared to heavier gases. They're not separated, though.

And heat causes gas at a given pressure to be at a lower density, and it also causes faster diffusion, but these are two mostly-separate effects.
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Old 05-20-2020, 09:18 PM
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A simplified view of what happens on a molecular level has similaries to buoyancy for larger bodies. If you have a gas consisting entirely of one type of molecule, say inside a vertical tube of constant temperature, there will be a density gradient, with higher density near the ground. At thermodynamic equilibrium because of this density gradient, a molecule will have more collisions with molecules below it than from above, pushing it upward with a force that is on average counteracted by the weight of the molecule for no net average vertical force. If the mass of one molecule is suddenly reduced, keeping all else the same, the weight no longer balances on average with the imbalance in collisions, so it will tend to move upward. The difference in force being the difference in weight between the new lighter molecule and the one it replaced, while in large bodies the net vertical force is the difference in weight of the body and the fluid it displaced.

On the molecular level, there is much fluctuation since molecule position and velocity is always changing due to collisions, which is why I said "on average" and "tend to move upward". The above summary was a simplified view, since the dynamics of the collisions will change for the lighter molecule due to it taking on a higher velocity than a heavier molecule, and a molecule with different mass will also differ in size.
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Old 05-20-2020, 09:40 PM
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Also related to the buoyancy effect is the counterintuitive Granular convection or the Brazil nut effect. If you take a box of cereal and shake it up, the bigger particles (heavier) will move to the top.

https://en.m.wikipedia.org/wiki/Granular_convection
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