Since the density of air, ρair is initially equivalent to the initial density in the tank, ρo, and all other physical properties are the same, ambient pressure must be equivalent to initial tank pressure:
Perform a mass balance across the tank of the form in-out=accumulation:
Where c is a given constant, ρ is the density of the tank at time t, Q is the volumetric flowrate leaving the tank, and V is the v olume of the tank.
Rearranging, noting that all physical properties are constant except for pressure:
Cancelling out like terms:
Rearranging for integration:
Integrating from the initial pressure, Po, to an arbitrary pressure, P:
Rearranging for pressure:
To find the lowest attainable pressure, P*, let time approach infinity:
To determine the time it takes to fall from halfway from its initial value to P*:
Substituting P* into the integrated expression:
Substituting numeric values yields:
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