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How long does it take to heat up a hot air balloon?

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Not long at all would be the short answer. The time to heat a balloon enough to fly depends on the ambient temperature (surrounding air) and how much heat you have to apply. read more

if you heat the balloon and it is freely expanding (increase in volume) ,then there wont be a pressure increment, the nature will try to keep balloon pressure constant. but as soon as the volume expansion is restricted (Depend of material used to make the balloon) or the volume cant be further increased, the pressure will start to increase. read more

Hot air balloons do this with a burner positioned under an open balloon envelope. As the air in the balloon cools, the pilot can reheat it by firing the burner. Modern hot air balloons heat the air by burning propane, the same substance commonly used in outdoor cooking grills. read more

Most balloons can fly for about 1 hour depending on the outside temperature and the weight carried. On a cold day with only the pilot flying, a 2-3 hour flight would be possible. read more

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Further Research

I’ll Fly Away
www.illflyaway.net

Launching and Landing
science.howstuffworks.com

Propane and Rising Balloons
science.howstuffworks.com