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Problem 3.99 - Problem*3.99[Difficulty 4 Given Spherical...

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Problem *3.99 [Difficulty: 4] Given: Spherical balloon filled with helium lifted a payload of mass M=230 kg. At altitude, helium and air were in thermal equilibrium. Balloon diameter is 120 m and specific gravity of the skin material is 1.28. Find: The altitude to which the balloon rose. 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) t z D M Assumptions: (1) Static, incompressible fluid (2) Static equilibrium at 49 km altitude (3) Ideal gas behavior Taking a free body diagram of the balloon and payload: Σ F z F buoy M He g M s g Mg = 0 = Substituting for the buoyant force and knowing that mass is density times volume: ρ air g V b ρ He g V b ρ s g V s 0 = ρ air V b ρ He V b ρ s V s M 0 = The volume of the balloon: V b π 6 D 3 = The volume of the skin: V s π D 2 t = Substituting these into the force equation:
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