# Would atmosphere floating inside an airship be counted towards the mass of the airship?

Airships are filled with hydrogen. However as pointed out by Peter Kampf in the answer to the question below when an airship is on the ground only a portion of its hydrogen cells are filled with hydrogen.

During WW1 Zeppelins began their missions with Hydrogen cells at 33% full. How did this provide enough lift to get off the ground?

This chart details the hydrogen density in a given airship before takeoff.

LZ104's cells are only filled to 42% at sea level. Does this mean that below the hydrogen cells sits an empty volume of air equal to roughly 58% of the volume of the cells? If so is this air counted towards the mass of the airship?

I ask because water in a ship's hull makes it heavy and sink. Why wouldn't air in an airship make it heavy and fall to earth? Or would it?

• Water in a ship's hull displaces air which would help to let it float. If that water would displace something of equal density (cargo, for example) it would not make the ship heavier. Commented Oct 15, 2020 at 9:42
• From a physics perspective all matter has mass. So yes, the air counts as mass.
– GdD
Commented Oct 15, 2020 at 9:48
• Or in other words: if there's water in a ship's hull that means there isn't air in the ship's hull and if there's not enough air then it sinks. But some water is okay. In fact since ships have toilets, you know that it's okay for them to have water. Commented Oct 15, 2020 at 15:04
• Yes towards inertial mass, no towards gravitational mass. Commented Oct 15, 2020 at 15:41
• @GdD: That was a shorthand for "yes, it has a weight, but it is counteracted by the buoyancy". Commented Oct 15, 2020 at 19:57

Would atmosphere floating inside an airship be counted towards the mass of the airship?

Yes, but not for buoyancy.

Does this mean that below the hydrogen cells sits an empty volume of air equal to roughly 58% of the volume of the cells?

Yes, exactly.

When calculating the lift produced by the interior of the airship, the gas with lower density compared to outside air (hydrogen) produces lift while the internal air of equal density to outside air has no influence. It simply cancels out. While hydrogen displaces air of higher density, creating lift in the process, this internal air displaces outer air of equal density. So for lift that internal air can be disregarded.

However, when maneuvering this air does count. It increases the inertia of the ship, both when changing direction and when accelerating or decelerating. To this air the airship designers even counted the outside air which is influenced by the motion of the ship. An airship in motion continuously displaces air at its bow and lets it flow together again at the stern. The air is thus given an impulse which - as the product of mass and speed - can be described as an additional mass of the airship at this speed. In the case of an elongated body, this additional mass in the direction of travel is small in relation to the mass of the displaced air, but is of the same order of magnitude in case of lateral or vertical movements. In case of rotational movements, the inertial forces and moments of the gases within the airship envelope are added. To make it easier to calculate the forces on the airship hull, the concept of virtual mass was introduced. This virtual mass depends on temperature, altitude and speed. Although this technique allows a simple mathematical description of the effect near an operating point, it can only be generally applied with additional effort. If you are interested in the actual values: NACA-Report 184 contains tables and correction factors for the calculation of virtual mass.

I ask because water in a ship's hull makes it heavy and sink.

Because this water would displace air which is lighter than water and makes the ship float. This comparison only works when air would be replacing hydrogen in the gas bags. See the empty volume in the airship as akin to an external tank which is added to a ship and submerged. Its weight would be carried by the water. The air-filled volume in an airship is needed to have space for the gas bags to expand into without disturbing the aerodynamic shape of the hull.

Why wouldn't air in an airship make it heavy and fall to earth? Or would it?

Because this air displaces air of equal density. It cancels out in the buoyancy calculation.

• Thanks Peter that is fascinating. So when an airship has a large diameter it produces tremendous lift. However the hull is also full of air which makes the ship more challenging to turn left or right. Very interesting. Commented Oct 15, 2020 at 15:52

It all comes down to buoyancy. One would actually be heavier if weighed in a vacuum chamber. Water entering a ship does not make it sink, just as air in a dirigible does not affect its weight (if it is being weighed in an air medium).

What makes a ship sink is anything denser than water, like steel. To float, the volume of steel must be lightened by adding a volume of something less dense than water (steel + air = water density). The displacement of the ship is the amount of water volume that is of equal density (there for weight) of the entire ship (including superstructure). If the ship floods, it's density becomes greater than water (steel + water > water density).

So it is with airships: hydrogen + frame + engines + fuel = density of air for a given volume (weight) of air they displace. In an air medium, air in the ship does not count.

It may be noted that both airships and submarines can rise or sink by trading light volume for heavy (submarine air and water, air ship hydrogen and air), but lift can also be used when under power for the same purpose.

• Fun fact: an airliner flying at 35,000 feet would actually have to account for the higher density of pressurized air in the cabin as extra weight. Commented Oct 15, 2020 at 6:12
• How would the airliner actually account for it? Commented Oct 15, 2020 at 14:38
• @MichaelHall The density is easilly got from the equation of state rho = p / (R * T). The added mass density is only (p - p_outer) / (R * T). It is a similar problem to weighing the same object in air and in vacuum - sometimes the difference is negligible, sometimes it is not. Commented Oct 15, 2020 at 14:58
• @RobertDiGiovanni That's a fascinating point about pressurization. So if somehow airship cabins were pressurized the extra air would have been counted towards mass. I hadn't thought of that. Commented Oct 15, 2020 at 16:21
• A380 pressurized to 8000ft would be ~1.2tonne heaver than with 35000ft density given 2100m^3 pressurized volume. That's about 2x the mass of the fuel that is burned during takeoff and climb to 1500ft. Commented Oct 15, 2020 at 18:24

As the airship ascends, the air is displaced by the expanding hydrogen.

What this means practically is that LZ104 filled to 42% can fly at a higher altitude than LZ40 filled to 70%, because its H2 can expand from 42% to 100% (2.5x) rather than from 70% to 100% (about 1.5x) before it runs out of lift.

As you can see in the "max altitude" figures in that table, LZ104 can reach 8200m vs LZ40's 3900m.

An extreme example is those high altitude balloon launches where you can see a tiny bubble of helium at the top of a huge envelope.