# What is the sack-to-ship ratio?

In the world I'm building, the day lasts 9 of their years. This means that they are almost always on the move, living in blimp-like cities and houses. I've been designing these and I am having trouble with the exact ratio for the balloon size to the size of the cargo. What is the sack-to-ship ratio? Weight ratios will work as well.

The net lift for the gases is how much the volume of air they displace weighs, minus the weight of the lifting gas itself. The following values are from this site and are per 1,000 cubic feet of volume.

| Weight of Lifting Gas |  Weight of Air |  Net Lift
+-----------------------+----------------+------------
Hydrogen |  5.31 lbs             |  76.36 lbs     |  71.05 lbs
Helium   |  10.54 lbs            |  76.36 lbs     |  65.82 lbs

So, if you have 71 lbs of cargo, you need 1,000 cubic feet of hydrogen to lift it. The volume which the cargo takes up is irrelevant.

You'll need more than to just lift cargo, of course, you have to lift the rest of the ship including the gas bag, deck, rigging, etc.

This is assuming the atmosphere is the same density as on Earth. The specific values are with respect to Earth's gravity, but the ratio will remain for any (reasonable) value of gravity.

If your planet isn't populated exclusively by Americans, Myanmas, or Liberians then they probably use the glorious metric system.

In which case the table looks more like this for a volume of one cubic meter:

| Weight of Lifting Gas |  Weight of Air |  Net Lift
+-----------------------+----------------+------------
Hydrogen |  0.090 kg             |  1.292 kg      |  1.202 kg
Helium   |  0.178 kg             |  1.292 kg      |  1.114 kg

If you got some cargo from a backward country and they said it weighed 71 lbs, once you properly weighed it at 32.2 kgs you would know you need about 26.8 cubic meters of helium to lift it.

In your specific case, if you wanted to lift a typical mobile home weighing 6758.53 kilograms (14,900 lbs):

$$6\,758.53\ \mathrm{kg} \times {{1\ \mathrm{m^3}} \over {1.114\ \mathrm{kg}}} \approx 6\,070\ \mathrm{m^3}$$

You'd need 6,070 cubic meters (~214,000 cubic feet) of hydrogen. This is about two and a half Olympic swimming pools in volume. Or, more specifically, a sphere 22.6 meters (~74 ft) in diameter (a bit over 8 stories tall).

• Don't forget that these gases are rare in terrestrial planet atmospheres. Since ownership of the gas = survival, this could be a source for quite a bit of drama. It also means these gas bags could be by far the most valuable thing on the planet to its inhabitants. Commented Jul 21, 2015 at 18:24
• it is, a good ship is very valuable. Also think of how one madman could blow up an entire city Commented Jul 21, 2015 at 18:30
• @Jim2B Helium certainly is and it's getting worse. But, they can make large volumes of gaseous hydrogen with water electrolysis. Of course, there is that boom factor there... Commented Jul 21, 2015 at 18:50
• You would probably want to increase the gas bag diameter by something on the order of 30%, assuming the gas bag and rigging weigh about as much as the trailer. Commented Jul 21, 2015 at 20:42
• could you add the actual equation for figuring this out Commented Jul 21, 2015 at 20:57

If you want to forego the use of lifting gas, and don't mind going big, look at Buckminister Fuller's "Cloud 9"

Fuller's insight was the volume of enclosed air in a geodesic dome increased by the cube/square law; it increased by the power of 3 as the dome doubled in area. At some point the amount of air inside the dome vastly outweighed the dome itself, and a temperature differential of as little as 1 degree f could cause the dome to take off like a hot air balloon.

A 100-foot-diameter, tensegrity-trussed, geodesic sphere weighing three tons encloses seven tons of air. The air-to-structural-weight ratio is two to one. When we double the size so that the geodesic sphere is 200 feet in diameter, the weight of the structure increases to seven tons while the weight of the air increase to fifty-six tons – the air-to-structure ratio changes as eight to one. When we double the size again to a 400-foot geodesic sphere – the size of several geodesic domes now operating – the weight of the air inside increases to about 500 tons while the weight of the structure increases to fifteen tons. The air-weight-to-structure-weight ratio is now thirty-three to one. When we get to geodesic sphere one-half mile in diameter, the weight of the structure itself becomes of relatively negligible magnitude, for the ratio is approximately a thousand to one.

Even larger domes work better, since you have a huge "reserve" of lifting power so long as the interior of the dome is warmer than the outside air. The waste heat of human activity and machinery inside the dome will actually help keep it aloft at night.

Like any hot air balloon, you are adrift in the wind (you could add engines and propellers like a Dirigible airship), and the balloon will work better in cold climates where the temperature differential is more pronounced.

The main issue here is are your people technologically capable of creating and maintaining such a structure? Once the idea of a geodesic or similar lightweight structure is developed, it should not take too long for someone to make the same deductions that Fuller did.

• But the dome materials must be stronger as the dome gets bigger. Commented Oct 4, 2015 at 3:07
• The beauty of a geodesic dome of this nature is much of the strength can be from tension members, which are far lighter than compression members. Much of the domes strength could be found in a cable net "pulling" against the forces of the expanding gasses within. Commented Oct 4, 2015 at 5:08
• While I do agree the bottom half is compression that doesn't change the fact that it's still subject to the square-cube law. You can't just scale it up with the same strength! Commented Oct 4, 2015 at 5:11
• The entire Fuller "Cloud 9" is a sphere (two geodesic domes joined at the base, if you will), so the entire structure will be in tension, which was Fuller's point in this thought exercise. Since the temperature differentials needed for flight could be as little as 1 degree farenhight, the amount of stress will be rather minimal as well. Cloud 9 structures could have been built in the 1960's, using then available technology, using modern materials like carbon fibre would make this much easier today. Commented Oct 4, 2015 at 5:18
• If it's purely in tension then rope is a reasonable model--but what's going to happen if you make a dome out of rope??? If it's a lighter-than-air craft I think you can get away with the only compression element being a hoop around the middle, though. Commented Oct 4, 2015 at 22:38