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 mbarand T = -25 deg C. The balloon skin has a specific gravity of 1.28 and thickness 0.015 mm. The gage pressure ofthe helium is 0.45 mbar. The allowable tensile stress in the balloon is 62 MN/m2t MD Find:(a) The maximum balloon diameter(b) The maximum payload massSolution:We will apply the hydrostatics equations to this system.Governing Equations:Fbuoyρg⋅Vd⋅=(Buoyant force is equal to massof displaced fluid)pρR⋅T⋅=(Ideal gas equation of state)πD2∆p/4πDtσAssumptions:(1) Static, incompressible fluid(2) Static equilibrium at 40 km altitude(3) Ideal gas behaviorThe diameter of the balloon is limited by the allowable tensile stress in the skin:ΣFπ4D2⋅∆p⋅πD⋅t⋅σ⋅−=0=Solving this expression for the diameter:Dmax4t⋅σ⋅∆p=FbuoyantMbg Mg z Dmax40.015×103−×m⋅62×106×Nm2⋅10.45 103−⋅bar⋅×bar m2⋅105N⋅×=Dmax82.7m=To find the maximum allowable payload we perform a force balance on the system:
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