# How can I calculate the neutral buoyant weight of an airship at altitude?

I love airships. I often like to do research on the old designs from the 20s and 30s. Trying to figure out how they work and how they're built etc.

Lately I've been working on trying to calculate how much ballast would have to be dropped to change the altitude by a defined amount. Let me give an example.

I have an airship that is floating happily along at a steady 500ft altitude. The airship, and everything on it weighs exactly 10,000lbs. There's mountains or something ahead so they decided to climb to 7,000ft. To do that they need to become lighter which means dropping ballast. How would I go about calculating the amount of weight I need to loose to become neutrally buoyant at 7,000ft.

I get pretty confused because obviously the air pressure will only be about 79kPa at the new altitude. Which means my helium will expand. But how will that affect the amount of lift it generates? As 1 cubic meter of helium will create less lift, but its expanded so I have more cubic meters. Also I know that the temperature of the air will be different at the new altitude and that would affect things.

I hope my question is clear. And thanks in advance for the help!

• You should read this, it deals with at least part of your question regarding the volume displaced, as the airship expands at higher altitudes. en.wikipedia.org/wiki/Archimedes'_principle – user140606 Feb 7 '17 at 21:19
• Be careful. Your helium will only expand as much as you let it expand. It may want to expand at the higher altitude, but the airship balloon is designed to keep the helium at roughly the same volume at all times. – Luke Pritchett Feb 7 '17 at 21:21
• @luke right, I guess I should have specified that my example airship has ballonets to allow for helium expansion. It's pressure limit is 10,000ft so we're ok as long as we make our calculations properly :) – tyler mackenzie Feb 7 '17 at 21:27
• The lifting ability of a fixed mass of unpressurized helium will not change with the surrounding temperature and pressure, as long as the helium is in equilibrium with the surrounding air. – DJohnM Feb 7 '17 at 23:50
• @DJohnM Is that to say that, in an ideal situation, as long as the helium has room to expand and it changes temperature with the surrounding air. then it will be neutrally buoyant at any altitude? – tyler mackenzie Feb 7 '17 at 23:57

## 1 Answer

Well my question was technically answered by @DJohnM in the comments, but it's been two days and he hasn't posted an answer for me to accept, so I thought I'd post an answer here for anyone else who was wondering about this question.

As long as the helium is given room to expand without being pressurized and the helium keeps the same temperature as the surrounding air. The airship will have the exact same amount of lift at any altitude. Dropping ballast or venting helium is only a means applying an upward or downward thrust. To go from 500ft up to 7000ft, we drop 100 kg of ballast, now the airship will continue to rise with 100kg of thrust until the buoyancy is levelled out again by venting helium. If we didn't vent helium the airship would continue to rise indefinitely.

Actually it would rise until it reached the expansion limit of it's helium bags. As it rises the helium expands, eventually there will be no room left to expand and either helium is vented or the bag bursts.

Now for some numbers:

Altitude:         500 ft (152.4 m)       7,000 ft (2133.6 m)
Air Pressure:     99.5 kPa               78.2 kPa
Temperature:      19 °C                  6.1 °C
Air Density:      1.186 kg/m³            0.975 kg/m³
Helium Density:   0.164 kg/m³            0.135 kg/m³
Lift:             1.023 kg/m³            0.840 kg/m³
Volume:           10,000 m³              12,166 m³
Total Lift:       10,225 kg              10,225 kg


So as you can see, The air pressure and temperature drop as we climb higher, which results in reduced lift. But the helium also expands, so the reduced lift and expanded helium cancel out. same lift at any altitude.

Some things will mess with the buoyancy. As we climb, the temperature drops, the helium in the airship will cool slower than the surrounding air. so once we reach 7000ft and vent helium to level out, we have warm helium. as that helium cools with will provide less lift and we will start to sink again unless ballast is dropped. Also we've probably been burning fuel this whole time, getting lighter as more fuel is burned. ballast will have to be dropped to maintain our desired altitude.