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Fluid Mechanics(Mass Balances)2

The tank has a volume V=1 m^3 and contains air that is maintained at a constant temperature by being in thermal equilibrium with its surroundings.
If the initial absolute pressure is po=1 bar, how long will it take for the pressure to fall to a final pressure of 0.0001 bar if the air is evacuated at a constant rate of Q= 0.001 m^3/s, at the pressure prevailing in the tank at any time?(independent of pressure)

Let's say the pressure in the tank is P(t).

We can find the pressure at some infinitesimal later point in time as:

$P(t+\Delta t)=P(t)$1-0.001\Delta t$$

Rearranging this we get

$\frac{P(t+\Delta t)-P(t)}{\Delta t}=-0.001P(t)$

Now taking the limit as the time interval goes to zero, we get

$\frac{dP(t)}{dt}=-0.001P(t)$

This differential equation has a very simple solution:

$P(t)=P_0e^{-0.001t}$

We know that $P_0$ is 1 bar, so we simply plug in that value, along with 0.0001 bar for P(t), and solve for t.

$0.0001=e^{-0.001t}$

$\ln (0.0001)=-0.001t$

$t=9210.3$

So it will take a little over 2.5 hours for the pressure to reach 0.1 mbar.

thanks ^_^