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Problem: Model a leaking pressurized cylinder volume filled with an ideal gas (density ρ and molecular weight w) charged to a pressure P at temperature T. The rigid cylinder of fixed volume V has a small leak (adiabatic – T doesn’t change significantly) of area a. The leak rate will diminish with time since as material escapes, the driving pressure decreases. What function describes the pressure decay as a function of time, t?

Solution: The model in the attached pdf uses the ideal gas law and Bernoulli flow equation to find that

dP/dt = -constant * SQRT(P)

I'm not sure what the solution to this is but see that it will decay more slowly than the exponential solution I expected to find.

Question: Is the above differential solution correct, and if so, what is P(t)? If it is incorrect, what is a correct approach?

Solution: The model in the attached pdf uses the ideal gas law and Bernoulli flow equation to find that

dP/dt = -constant * SQRT(P)

I'm not sure what the solution to this is but see that it will decay more slowly than the exponential solution I expected to find.

Question: Is the above differential solution correct, and if so, what is P(t)? If it is incorrect, what is a correct approach?