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Anar22 · Nov 8, 2011, 07:16 AM

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)

jcaron2 · Nov 8, 2011, 09:19 AM

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.

Anar22 · Nov 8, 2011, 09:47 AM

thanks ^_^