Calculate Outflow Velocity with High Pressure Tank & Water Vessel

[MattRob]

Okay, so there's a system with a high pressure tank, connected to a water vessel:

The high pressure air tank is (a), and the water tank is (b).
The first choke point has the opening area c₁, the second c₂, and the outflow nozzle has the opening area c₃.

Let's say the air pressure is pressure p, and the area onto which it distributes force onto the water is A_w.

V_e is the velocity at which it flows out the tank. What will this velocity be?

I have a vague idea, that give area A_w, I can just calculate how much force is being exerted on the water by pressure p, and apply that much force to area c₃. Is that correct?
V_e = (p * A_w) / c₃

The only problem with that though is that should be describing an acceleration, not a velocity...
But to determine a velocity, I need a mass. A mass requires a volume, though, not an area, but all I have is an area (c₃).

So... How do I find V_e?

 

[Studiot]

Velocity times crosssectional area = volume rate of flow.

What quantity do you know that transforms volume into mass?

 

[Michio Cuckoo]

Use Bernoulli's principle. its basically a form of the law of conservation of energy.

 

[MattRob]

Velocity times crosssectional area = volume rate of flow.

What quantity do you know that transforms volume into mass?

What velocity? Outflow velocity?

How am I going to know the volume rate of flow if I don't know the outflow velocity?

All I know is
1) the pressure of tank a (of atmospheric air)
2) the crossectional area that that pressure is being applied on the water mass
3) the crossectional area of the different chokepoints c1, c2, and c3.

What I'm trying to find is the mass flow rate out the back, how much mass in how much time, and at what velocity? (In other words, how much momentum, the entire system is a propulsion system, it's goal is to generate thrust, the question is how much thrust will it generate given those known variables?)

 

[Studiot]

Hello Matt,

Your diagram was so much neater than my rough sketches and the text so crisp that I (and probably other respondents) thought this was a homework type question. So the replies were rather cyptic.

I don't know if you are familiar with Bernoulli's theorem?

OK so you have a water reservoir pressurised to pressure P_b by a compressed air reservoir.

I think you can neglect gravity in this application.
So that the pressure in the water will be the same throughout the reservoir.
This will equal P_a initially and at least for some time after the valve C3 is opened.

This situation will be independent of C1 and C2.

So the pressure at C3 on initial opening the valve will be P_a.

The water jets out and a little way beyond C3 the pressure will be atmospheric.

At the moment of opening the valve we can make the assumption that the water reservoir is large enough to take the general flow velocity within the reservoir itself to be zero.

This gives us all the information we need to set up and solve the Bernoulli equation for the velocity just outside C3.



You can then use mass (flow rate) = Volume(flow rate) x density
and thence to your momentum calculation.

Do you need help with setting up the Bernoulli equation and is this homework?

For a more refined model you could note that the actual velocity will be less by what is known as a discharge coefficient. I suggest 0.8 at a first stab.