Hot Airships

Wednesday, July 9th, 2008

I was recently reminded of the SF trope of establishing a parallel universe by sticking a few airships in the sky, and that got me to thinking about everyone’s favorite hydrogen- and helium-filled leviathans — and why hot air works so well for balloons, yet we never hear about hot-air dirigibles.

It turns out, they exist. The Airship Alberto, for instance, uses hot air for lift, which means it can deflate and fold for storage, like an ordinary balloon, without a large hangar and ground crew:

The Alberto — named after Alberto Santos-Dumont, of course — has a semi-rigid, umbrella-like skeleton, unlike a balloon, which gives it the structural rigidity to handle higher air speeds and tighter maneuvering — and with vectored thrust from the rear-mounted propeller, it is quite maneuverable.

Of course, the real advantage of the Alberto‘s hot-air design is that it’s one-tenth as expensive as a helium dirigible. (They expect the sale price of the aircraft to be between $100,000 and $200,000, the price of a small airplane or mid-sized sailboat.)

So why use helium at all? Because a given volume of hot air can lift only about one-third as much as the same volume of helium — so a hot-air airship needs to be roughly 1.5 times as tall, wide, and long as a comparable helium-filled airship.

If helium is that much better than hot air, how much better is hydrogen than helium? Not that much better, it turns out. Amateur chemists might note that hydrogen has an atomic weight of 1, versus helium’s atomic weight of 4, and assume that it would have four times the lift. After a little thought, they might realize that hydrogen (H2) has a molecular weight of 2, versus helium’s 4, implying half the density. The key is that half the density does not mean twice the buoyancy:

The density at sea-level and 0°C for air and each of the gases is:

Thus helium is almost twice as dense as hydrogen. However, buoyancy depends upon the difference of the densities (?gas) – (?air) rather than upon their ratios. Thus the difference in buoyancies is about 8%, as seen from the buoyancy equation:

  • Buoyant mass (or effective mass) = mass × (1 – ?air/?gas)
  • Therefore the buoyant mass for one liter of hydrogen in air as:
    • 0.08988 g * (1 – (1.292 / 0.08988) ) = -1.202 g
  • And the buoyant mass for one liter of helium in air as:
    • 0.1786 g * (1 – (1.292 / 0.1786) ) = -1.113 g

The negative signs indicate that these gases tend to rise in air.

Thus hydrogen’s additional buoyancy compared to helium is:

  • 1.202 / 1.113 = 1.080, or approximately 8.0%

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