Helium Balloon: Buoyancy, Latex Wall Pressure and Slow Leakage

What is modelled

A party balloon filled with helium in a room. Helium is about seven times lighter than air, so the air it displaces weighs more than the balloon, the latex, the string and the gas inside; the difference is the lift. But the latex wall is not gas-tight: helium atoms are small and dissolve in and diffuse through the rubber, while a little oxygen and nitrogen creeps in. The balloon slowly shrinks and loses lift until it sinks, long before it is empty.

Equations

• Gas law and buoyancy: n R T = p V, lift = ρair V − (mgas + mlatex + mstring + mclips), with ρair = p Mair/(R T). Pure helium at 20 °C lifts about 1.04 g per litre.
• Stretched latex wall (neo-Hookean membrane, stretch λ = R/R₀): overpressure P = (2 μ t0/R0)(1/λ − 1/λ7), which has a maximum at λ = 71/6 = 1.38; the gas pressure inside is patm + P. The wall thins as t = t0/λ2.
• Leakage (solution-diffusion): dni/dt = Pi A/t (pi,out − pi,in) with permeabilities in the ratio He : O₂ : N₂ = 25 : 17 : 6 barrer and a temperature factor exp[(Ea/R)(1/293 − 1/T)], E_a = 25 kJ/mol. For a fixed shell the helium decays as n0 e−t/τ with τ = V t/(P A R T).

Checks run in the page

• Densities of air and helium at 20 °C against tables.
• The wall pressure from the derivative of the strain energy (finite difference) against the closed form, and the location of the pressure peak.
• Helium leakage through a rigid shell for 6 hours against the exponential law.
• Step-size independence of the time-to-sink, and the initial lift written two ways.

Honest simplifications

• Permeabilities are typical literature values for natural rubber, not measurements of your balloon; real balloons also differ in thickness from place to place, in the seal at the knot, and many are treated with a coating ("Hi-Float" style) that is represented here only as a permeability multiplier.
• The wall is a single neo-Hookean membrane with a constant shear modulus (0.4 MPa) and relaxed radius 4.5 cm; real latex stiffens at large stretch, creeps and the balloon is not a perfect sphere. Temperature of the gas is the room temperature at all times.
• Motion is a simple drag model with a ceiling stop; there is no draught, no static electricity and no spin. The balloon on its 1.5 m string just floats at the end of it.
• The time compression applies to the leakage only; the motion always runs in real time.

What to try

• Press Release and watch it rise to the ceiling, then raise the time compression: after hours the radius shrinks and it sinks while a good part of the helium is still inside.
• Add paperclips with the slider to see how few grams ground it; the readout shows the exact amount.
• Compare "Plain 11-inch latex" with the coated balloon, and a warm room with a cold one: permeability rises with temperature.
• Look at the third graph: the wall pressure rises, peaks and then falls as the balloon is blown up.