Airship Lift and Ceiling

What is modelled

An airship floats because the air it displaces weighs more than the helium inside and the structure around it. As the ship climbs, the air gets thinner and the helium expands. Air-filled ballonets let the gas expand without bursting the hull. Up to the pressure height the lift stays the same, because the hull is only getting fuller; above it the helium must be vented and the lift falls with the air density until it equals the weight, at the static ceiling.

Equations

• Standard atmosphere T = T0 − 6.5 h up to 11 km, then constant; pressure from dp/dh = −ρ g.
• Gross lift L = Vg (ρair − ρgas) with ρ = p M / (R T) and the gas at T + ΔT.
• Gas volume Vg = V f0 (p0 / p)((T + ΔT) / (T0 + ΔT)), limited by the hull volume.
• Static ceiling where L(h) = m g.

Checks run in the page

• The pressure profile from hydrostatic integration against the closed form.
• Archimedes lift against the ideal-gas form.
• Pressure height by bisection against a fine scan.
• The weight-lift balance at the static ceiling.

Honest simplifications

The gas pressure equals the outside pressure (no superpressure), the gas temperature differs from the air by a fixed amount, the ballonets are ideal, and wind, aerodynamic lift, engine thrust, fuel burn and structural limits are ignored. A real airship needs trim, ballast and valve rules, so this shows only the static buoyancy picture.

• Press Hot day, impure gas to see the lift fall when the air is thin and the helium is diluted.
• Press Heavy freighter to find a ship that can barely leave the ground.
• Press Big hull, high flight and watch the ceiling climb above ten kilometres.