What is the theoretical max flight height of a hot air ballon?

A few days ago I saw a mini hot air balloon consisting of only a burner and a cloth. As it ascended towards the sky it got me thinking; if that thing had infinite fuel, how high would it be able to fly? And wouldn’t it ascend faster as the atmosphere gets colder?

A hot air balloon works by heating the air inside the canopy, which causes it to expand. Because the canopy is open at the bottom, the some of the air molecules escape out of the bottom. What’s left is fewer molecules occupying the same volume of air – that means lower density, and lower weight!

The large canopy displaces a volume of cool air in the atmosphere. That gives you a bouyant force, equal to the weight of that cool air, which tends to lift the balloon up. When the bouyant force is greater than the weight of the balloon, including the canopy, the burner, and the hot air inside it, the balloon goes up.

So, as you heat the air inside the canopy, you cause it to expand more and more, which makes it less and less dense, giving you less and less weight to lift. But, as you go up, the density of the atmosphere starts to go down, which means the weight of the displaced volume of air goes down. That means your bouyant force goes down, too.

(You’re right that the temperature of the air goes down as you go higher, but it’s the density that is critical. Density is inversely related to temperature, and directly related to pressure, and the pressure goes down more quickly than the temperature, so density goes down with altitude.)

You can imagine, in the limit, you have a super strong, super light material to make your balloon canopy from. Then, instead of heating the air to make it less dense, you could seal off the canopy, and pump the air out. Remember, your canopy is super strong, so you can pump all the air out, and it will still hold its shape. So now, the weight of the air inside the canopy is 0 lb. Your canopy is also super light – let’s say it weighs 0 lb! Then you can get rid of everything else – no burner, no basket, and the total weight of the balloon is 0 lb.

Now, as long as you have even a tiny bouyant force, it will be greater than the weight of the balloon. So displacing any air at all will cause you to move upward. Now, at very high altitudes, there’s not much air to displace. But even in space, there’s a couple of molecules. So you’ll continue to rise away from Earth infinitely, or at least until you start to get to some gas molecules that are more strongly attracted to another body. Then, you’ll start moving away from that body. Plus, the solar wind will start pushing you around.

Of course, there’s no such thing as a weightless balloon canopy, and it certainly couldn’t hold its shape under a complete vacuum. The real maximum height a balloon can reach is determined by its volume, its weight, the weight of the air in the canopy, and the density of the atmosphere it flies in.

The weight of the air in the canopy is related to how much heat energy you can put in, but you’ll lose some heat energy to the surrounding atmosphere through the canopy. Plus, there’s a limit to how hot the air can get before it melts or burns through the canopy, and then you’re really in trouble!

As a point of reference, the record altitude for a hot air balloon flight is almost 69,000 ft, set in 2005 by Vijaypat Singhania.

Another point of reference – closed balloons filled with helium or hydrogen are used to reach altitudes of as much as 120,000 ft. Hydrogen and helium are lighter than air, so they are bouyant, just like the hot air in a hot air balloon. But because the balloon is closed, as it goes up into lower pressure air, the trapped hydrogen/helium stretch the bag, giving it more volume, and thus displacing more air.

I hope that helps you out!

Mini Hot Air Balloon Burning(9th minute)

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