Arthur Lupton's Lead Balloon

All units are SI

Data

Acceleration due to gravity = G = ca 10
Density of atmosphere = Da = ca 1.2 for air
Density of gas inside balloon = Db = ca 0.083 for Hydrogen
Difference in densities = Da –Db = Dd = ca 1.117 for air/Hydrogen
Density of balloon skin = Ds = 11.4 × 103 for Lead
Tensile/creep strength of skin = S = ca 1.72 x 106 for the creep strength of Lead

Unknowns

Excess pressure inside balloon = Px
Radius of balloon = R
Thickness of balloon skin = Ts
Payload of balloon = L

Constraining conditions

Condition (1) that when balloon is fully inflated, the excess internal pressure must support the skin of the upper hemisphere:
            πR2.Px = 2πR2.Ts.Ds.G
whence:
            Px &ge 2Ts.Ds.G                       eqn 30.1

Condition (2) for the balloon not to sink, when buoyancy must at least be balanced by weight:
            4/3πR3.Dd &ge 4πR2.Ts.Ds.G
whence:
            R &ge 3Ts.Ds.G /Dd                       eqn 04.3

Condition (3) for the balloon's skin to hold firm, when the creep strength (which is the critical measure in the case of Lead) must equal or exceed the tensile stress on the skin:
            2πR.S.Ts &ge πR2. Px
whence:
            R &le 2S.Ts/Px                       eqn 04.5C

Evaluations

Eliminate PX between 30.1 and 04.5C:
            R &le S/Ds.G
whence:
            R &le 15.088 metres
and substitute this value for R in 04.3 to evaluate TS :
            Ts &le 49.278 micrometres.

Calculate payload of balloon (L) for R = 15.088m and TS = 49.278 m:
            L= 4/3 πR3. Dd – 4πR2. Ts.Ds.G
            = 4/3 πR2(R.Dd – 3Ts.Ds.G)
whence:
L = 0.2098kg