In elementary compressible-flow theory (Chap. 9), compressed air will exhaust from a small hole in a tank at the mass flow rate m_ ≈C, where is the air density in the tank and C is a constant. If o is the initial density in a tank of volume v, derive a formula for the density change (t) after the hole is opened. Apply your formula to the following case: a spherical tank of diameter 50 cm, with initial pressure 300 kPa and temperature 100°C, and a hole whose initial exhaust rate is 0.01 kg/s. Find the time required for the tank density to drop by 50 percent.

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CV i Volumev Density ρ (t)