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I wonder how a function $$f=f(h, m_{\mathrm empty}, V_{max}, T_{h}), $$ with the top height $h$, the empty mass $m$, the maximum volume $V_{max}$, and the temperature of the heater $T_{h}$ would look like for a balloon with a remote controlled heating element and desired top height?

balloon

(bolune=balloon=helium balloon)

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Hi user1078642, and welcome to Physics Stack Exchange! This question isn't very clear as it is; in particular, how exactly does the heating element work? What is the nature of the gas leak? Basically, you would need to define more precisely the model you are having us use to figure this out. –  David Z Apr 4 '12 at 15:45
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Are you allow new air into the balloon (as in a regular hot air balloon) or are you assuming a sealed system as you have drawn?

A regular hot air balloon where you heat outside air will run as long as you have fuel.

A sealed system will require you to constantly increase the temperature to maintain a constant pressure and so a constant volume as the amount of gas present decreases due to leakage. Ultimately it will be limited by the temperature the envelope can survive or the breakdown of the gas molecules.

I suppose in theory if you had a envelope that was completely heat proof and you were able to heat the gas inside to a plasma you could hold the balloon up for a long time!

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helium balloon - like this –  user1078642 Apr 4 '12 at 15:33
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