And I post this without checking the link. The Breitlinger Orbiter overcame this very thing by being a hybrid of hot air balloon and helium balloon. Helium: for the long term. Hot air: to be able to mitigate the effects of altitude variations mentioned above.
You’re right, that does seem pretty insignificant; that’s why my intuition failed me. I wouldn’t be shocked to learn that there’s 0.001 PSI pressure difference on the two sides of my front door. So by my calculations, if we took 26 ft[sup]3[/sup] of sea-level air and put it into one of those 380,000-ft[sup]3[/sup] balloons, sealed, inside a vacuum, it would puff up just as much.
Cool.
I like the comparison. Those high-altitude balloons look barely inflated when they take off. Just a skinny bag of mylar with enough helium to get it aloft.
At 100k feet they look like real balloons, though. Ain’t the standard atmosphere a blast?
Normal air pressure drops about 30 parts per million for each foot of altitude. If we take a 60 high foot hot air baloon where the air pressure at the bottom is 15.000 PSI, then the air pressure at the top would be about 14.973 PSI. Assuming hot air has 2/3 the density of cold air, the hot air pressure drops about 20 PPM for each foot of altitude. The air pressure of the two gasses are the same at the bottom, 15.000 psi. But the pressure of the hot gasses at the top is 14.982 PSI. A pressure difference of .009 PSI (1.3 PSF) exists at the top.
I agree with your math, but this seems like a larger density difference than I would have expected. Can the flame really keep all that air that hot?
My 2/3 figure was a WAG as I don’t know the temperature inside the balloon. If the temp outside the balloon is 290 kelvin (65 f) and the hot-air temp is 435 k (325f), then since the temperature has a ratio of 2/3 then the density would have ratio of 2/3.
325 does seem a bit hot now that I’ve done more math. Perhaps the ratio is 0.70 or 0.75.
Going back to this link, it says that a 200,000 cubic ft hot-air balloon can lift 1050 lb (350 lb balloon + 700 lb gondola). So the lift is 0.005 lb/ft[sup]3[/sup], indicating that the average density of the hot air is about 8% higher than the surrounding air. That would be a roughly 25 C (45 F) difference in temperature.