Air at atmospheric temperature T ₀ is being pumped and stored in an underground cavern initially containing air at temperature T ₀ and atmospheric pressure P ₀ . Model the cavern as an open thermodynamic mass system with insulated boundary. The mass flow rate of the inflowing air (M) is constant. Show analytically that the cavern temperature rises from To to (c _p /c _V )To during a time of order m ₀ / M, where m ₀ is the initial mass inventory of the cavern.

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