A large balloon is to be inflated from radius a₁ to a₂ using a piston pump that delivers air at a constant mass flow rate w. If the wall tension is proportional to how much the radius a exceeds a₀, then the balloon pressure P and ambient pressure P₀ will be related as

where χ is a measure of the elasticity of the balloon material. (This is analogous to the Young–Laplace equation, the constant surface tension in that relationship being replaced here by a variable wall tension.) For simplicity, assume that χ is constant and that the air temperature in the balloon remains at the ambient value T₀.

(a) Explain qualitatively why a larger χ will increase the time t_p that is needed for the inflation.

(b) Derive an expression that could be used to predict t_p if χ and a₀ were known.

Transcribed Image Text:

P - Po = 2x (a - ao) a (P12.14-1)