Do atmospheric physics prevent hot air balloons from ascending over 60,000ft?

Question

I was reading the altitude records for hot air balloons on Wikipedia, and noted that the max hot air balloon altitude was about 60,000ft. It didn't really say if there was a reason why. I know that closed envelope balloons frequently climb much higher.

I was wondering... If you have a pressurized cabin (for keeping the operator conscious), and you mix oxygen into the fuel to heat up the air, is there really any reason that a balloon cannot ascend higher? Does the thinning of the air prevent it from holding heat as well? Do other atmospheric events come to play in not allowing a hot air balloon to ascend higher? I don't understand hot air balloons that well and could be missing a major component =)

Answer 1

Hmmm...

• Your lift goes by the volume of the gas bag times the difference in density: $V (\rho_{ext} - \rho_{int}) g$. As the total density goes down, you have to get a greater and greater fractional change in density to keep going up.

• To a reasonable approximation the density is inversely proportional to the temperature ($PV = nRT$ implies $\rho \propto n/V \propto 1/T$) and proportional to the pressure which falls steadily with altitude ($\rho \propto P(h)$. This means that to go higher you need greater and greater differences in temperature between the gas in your balloon and the exterior air. But heat losses will go up linearly or faster in $\Delta T$.

Which doesn't suggest a way to find some magical upper limit, but does mean that at some point going higher will be very difficult without a bigger balloon. Which of course means a bigger heat source and more fuel which means stronger everything and therefore more parasitic weight unless you go to exotic materials meaning even more expense for what is basically a lark. Isn't this fun.