Problem 3.100 - Problem*3.100[Difficulty 3 Given A...

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Problem *3.100 [Difficulty: 3] Given: A pressurized balloon is to be designed to lift a payload of mass M to an altitude of 40 km, where p = 3.0 mbar and T = -25 deg C. The balloon skin has a specific gravity of 1.28 and thickness 0.015 mm. The gage pressure of the helium is 0.45 mbar. The allowable tensile stress in the balloon is 62 MN/m 2 t M D Find: (a) The maximum balloon diameter (b) The maximum payload mass Solution: We will apply the hydrostatics equations to this system. Governing Equations: F buoy ρ g V d = (Buoyant force is equal to mass of displaced fluid) p ρ R T = (Ideal gas equation of state) π D 2 p /4 π Dt σ Assumptions: (1) Static, incompressible fluid (2) Static equilibrium at 40 km altitude (3) Ideal gas behavior The diameter of the balloon is limited by the allowable tensile stress in the skin: Σ F π 4 D 2 p π D t σ = 0 = Solving this expression for the diameter: D max 4t σ p = F buoyant M b g Mg z D max 4 0.015 × 10 3 × m 62 × 10 6 × N m 2 1 0.45 10 3 bar × bar m 2 10 5 N × = D max 82.7m = To find the maximum allowable payload we perform a force balance on the system:
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