Combining air pressures - Time to reach equilibrium

[oujapuja]

I need to produce a relationship between the radius of a hole between compartment A (Pressure P1, Volume V1) and compartment B (P2, V2) and the time it takes for these pressures to reach an equilibrium point. I am assuming the gases begin at the same temperature at this point. I'm struggling for a starting point and can't see where the time relationship comes from. I am also wondering how much detail I need to go into to get an accurate relationship and would be grateful for any hints or tips on how to tackle this, Thanks.

 

[tiny-tim]

welcome to pf!

hi oujapuja! welcome to pf!

find how speed relates to pressure difference, and how speed relates to the amount of matter in each compartment

 

[oujapuja]

hi oujapuja! welcome to pf!

find how speed relates to pressure difference, and how speed relates to the amount of matter in each compartment

Thanks for your reply tim. I haven't had to do calculations like this for a long time so I'm very rusty, so thanks for the advice.

I've used Bernoulli's equation to link the pressure difference to flow velocity and have an expression for mass flow rate. I've also worked out an equation for the additional mass added to the lower pressure compartment using the ideal gas law (I think!).

My thinking is if I integrate my mass flow expression for the change in mass and change in time, using my limits as the additional mass I have calculated, this will yield the time taken? If I can deal with the following problem (hopelessly optimistic of me).

The other issue I am struggling with is the varying pressure difference in my mass flow expression and am unsure of how to deal with it. Tried a few replacements using the ideal gas law, but couldn't seem to get anywhere.

I'm also wondering how much effect the hole will have on the flow and I wonder whether I need to use a friction factor, somewhere (again, very rusty, sorry!).

Any tips again would be great. I'm quite keen on solving this, so thanks for not just solving it for me!

 

[tiny-tim]

hi houjapuja!

it's a little difficult to advise when you haven't shown your equations

for the hole, just use Bernoulli's equation, and forget friction etc

i think you'll have to make an assumption about temperature … i'd assume the temperature stays the same

 

[oujapuja]

From Bernoulli's equation, I have the following for velocity U:

U=\sqrt{\frac{2*(P1-P2)}{ρ}}

My mass flow equation is then:

mass flow rate = A*\sqrt{2*ρ*(P1-P2)} (A - Cross Sectional Area of hole)

Using: Mass in compartment B at equilibrium = original mass + additional mass (Ma)
Using the ideal gas law to find masses, I arrived at the following:

Ma = \frac{V2}{V1+V2}*\frac{P1*V1}{R*T1} - \frac{V1}{V1+V2}*\frac{P2*V2}{R*T2}

Again, I'm unsure if this is correct.

As I mentioned, I was thinking it would be correct to integrate my mass flow expression with mass limits (0, Ma) and time Limits (0, t), aiming to find t.

If all this is correct then the problem for me lies in the flow velocity as this will be changing with time. Am I right in trying to substitute the pressure difference in terms of mass?

 

[tiny-tim]

hi oujapuja!

If all this is correct then the problem for me lies in the flow velocity as this will be changing with time. Am I right in trying to substitute the pressure difference in terms of mass?

looks ok (but what is ρ in your first equation?)

your Vs and Ts are constants,

so you have equations for M_a and dM_a/dt which you should be able to solve

 

[oujapuja]

Sorry for the late reply!

I didn't even think of the density - should there be two density values (one for each side) in my velocity equation?

I know that the equation for Ma is a constant, calculated with inital pressure values. But what I am still unsure on is the value of (P1-P2) in my mass flow expression - won't this change with changing mass in each side hence affecting the mass flow expression? How can I incorporate this into my equation? I've tried replacing them with various expressions from the ideal gas law etc. but the equation turns into a horrible mess.