How Does Pressure in a Sealed Container Change Over Multiple Cycles?

[jstefanop]

Im tryng to solve an equation for how pressure changes over time in a container. It goes something like this

There is a 10 liter sealed container(V1) with x amount of initial pressure (P1), and a 1 liter container attached to it (V2) with one valve that opens between them, and then another valve that opens to atmosphere. The initial pressure in the 1 liter attachment is atmospheric (P2).

For each cycle the valve opens between the pressureised 10 liter container and the one liter at 1 atm, the pressure then equalizes in both containers. Then the inner valve is closed and the outer valve is opened on the 1 liter container and pressure is vented to atmosphere (so the 1 liter container returns to 1atm) then the outside valve is closed and the cycle continues.

I need to find out after how many cycles does the large 10 liter container reach a certain pressure above 1 atm...

I can figure out what the pressure is in the container after they equalize each time which is
P= (P1*V1 +P2*V2)/(V1+V2)

So let's say the initial pressure is of P1 is 2atm after the first cycle the pressure would be P = (2*10l +1l*1atm)/(1l+10l) = 1.91 atm

then the second cycle would be P = (1.91*10l +1l*1atm)/(1l+10l) = 1.83 atm

etc etc

so what would be the equation t describe this pressure drop over a certain number of cycles? i.e. P(c) = ?

 

[Bystander]

P= (P1*V1 +P2*V2)/(V1+V2)

Okay.

say the initial pressure is of P1 is 2 x atm

Now try it.

 

[Merlin3189]

I'll have to accept your physics, because I don't know how to do that, but juggling your equation round and using V₁, V₂, and P₂ as constants, I get a formula for Pn after n cycles, as

( (P_n ) -1 ) = $(\frac {10}{11})^n$ ((P₀) -1 )

Edit - added excess brackets just to make clear

 

[Merlin3189]

Or in a more general form

( P_n - P_x ) = ( P₀ - P_x ) $( \frac{V1}{V1 + V2})^n$

Where P_x is the sink pressure, P₀ is the container starting pressure and P_n is the container pressure after n cycles.

Edit - BTW this looks so nice to me, that I think your physics must be correct!
Edit2 - and it looks as if you could start with container pressure below atmospheric as well.