Gas Flow Equation: Calculating Force & Velocity

[Havaren]

First off hello and I hope I have this right

Homework Statement

My problem- I have a cylinder of a specific value. One end of the cylinder has a piston and the other end is restricted by a nozzle that is attached to a tube. The information I need is how the flow out the smaller nozzle is effected by the movement of the piston.

Homework Equations

The equations I know I will need is the force the spring applies on the piston and the velocity of the piston as it moves forward. I know in life that friction form the pistons and the cylinder wall would be an issue as well, but for this exercise I get to ignore that.

I don't know if it is important but the gas in question is ambient air. So if needed I will be using the basic average air mixture at sea level at average tempature.

The Attempt at a Solution

This is where I get stuck. I do not have a strong physics background as of yet. From my research and questions I have gotten answered by my local professors I will need to understand the continuity principle of fluid. I also believe that I will need to know bernoulli's principle equation. Or perhaps the main equation found here http://www.air-dispersion.com/usource.html

Also I have not provided actual values for the problem because at this stage I am looking for the formulas and understanding on how to apply said formulas.

Anyway, I hope I did this correctly and I hope you guys can assist me. Thanks in advance!

 

[BvU]

Hello Havaren and welcome to PF,

I like detectives and sometimes I like to play detective too. But it distracts from effectively helping people posting on PF if I have to untangle their story. So, for the moment, instead of an answer you get more questions in return:

I have a cylinder of a specific value

value for what ? radius, height, wall thickness ? Don't let us guess!

the force the spring applies on the piston

what spring ? How does it fit in the problem ? What's at the other end ?

the velocity of the piston as it moves forward

Forward is towards the nozzle I assume (Sherlock Holmesing away...)
Can you proofread and perhaps elucidate with a clear diagram ? I added an example

Omitting relevant equations altogether is not such a good idea. No idea if you are proficient in rocket science or just starting this way. Fortunately you give a clue further on: no strong physics background yet. (It will come, don't worry). But just from a few shreds of your account, I can think of the ideal gas law, some spring stuff, some pressure stuff, expressions for work, would get us going. For now, a standstill.

 

[Havaren]

Ach! Sorry I wasn't very clear. Of course the post made sense to me XD

Ok,the actual values are not as important right now as the formula setup because I will be using the formula to calculate values for several different volumes of cylinders, different piston weights and different springs.
For now however I will go with some basic values
Spring constant (k)= 5N/cm
Cylinder interior radius (r) = 11.90mm
Cylinder height (H_c) =49.00mm
Spring draw length (H_s) =49.00mm
Piston weight = 15g
Nozzle interior radius = 1.95mm

As for diagrams you linked one that is pretty much prefect but I will try uploading the one I have as well.

As for formulas that I know I will be using so far I have spring force formula and hooks law probably to get the speed and pressure applied to the piston. (Correct me if I am wrong on this, but it seems like it is the right approach to me)
What I so far on this is (kx_i^2 -kx_f^2)/2 where I is the spring constant, and x is distance (initial and final) ( ok looking at that now, I think I might be missing something in my notes. I will re-review this section and see if I can't find what it is, if anything)

I will of course be using the basic geometric volume for the cylinder.
V=Pi*r^2*HI am still unsure as to other formulas I would be using. I know that I will need a formula for applying the acceleration of the piston to the pressure created in the cylinder and how that affects the out flow.

Here is a relatively good animated cycle of the object I am looking at. Please ignore the fact that there is an open forward end, I'm looking for theoretical maximums. Also ignore the fact the nozzle that is show is moving as it is moving on top of a fixed nozzle.

https://www.physicsforums.com/attachments/67534

 

[BvU]

Great if I could only see the attachment instead of getting a 404!

 

[Havaren]

thank you internets... let's try again shall we?

 

[BvU]

Love the picture. Should have known: when I googled some of the words a lot of gunsmithery pictures popped up. So now PF is into the arms race. Next thing you know there will be conspicuously inconspicuous unmarked vans in your street...

Well, let's pretend this is a toy instead of a machine gun (that doesn't mean such toys are not unlawful in this country, but in some less developed civilizations they might still be in use... when young -sigh- I had a pea shooter that had a lot of similarity with this contraption. No fancy batteries, just squeeze hard. The things were outlawed shortly after)

The equations I know I will need is the force the spring applies on the piston and the velocity of the piston as it moves forward.

For springs we usually assume a linear relationship between displacement from equilibrium and force exercised: F = -kX. Minus sign is because it counteracts the displacement: pull to the right, spring pulls to the left. Energy required: W = ¹/₂ k X². Reason I bring in energy is that this is what you want to transfer from your battery (electric energy -- don't forget to bring spare batteries!) via the spring (mechanical potential energy) to the white disk/sphere thingies (kinetic energy) as efficiently as possible.

If I am telling you things you know already, that is because of your formulation (need force spring applies...). Correct me if necessary. [edit]your wording can well be interpreted that you have these equations at hand and know them already but are reluctant to type them out.

So piston has moved back, sucked air in the chamber

(haven't found the hole in the picture, nor how it's opened and closed at the right moments...)
Reason I bring this up is it might well be that the valve role is played by the pea itself! That way the chamber is filled with air allright, but not to atmospheric pressure by the time the piston is at its extreme left position. All kinds of pros and perhaps a few cons:
Piston can really acquire speed when released, which helps deliver the energy in a short burst
No complicated valve mechanism needed, so no friction and more reliability
No pea-restrainer/holder needed, less friction again.​

and compressed the spring to its minimum.

Next phase in the drama: end of rak is reached, pinion free of rack and piston compresses air in cylinder which can simultaneously escape through nozzle, where there is a white pea sitting in the way. Nozzle accelerates pea, tube keeps air wave size restrained. Pea magazine opening prevents energy loss of air burst by replenishing vacuum.

There is an awful lot going on at the same time here. I am tempted to bribe a colleague who does CFD for a living to set up a Multi 10k$ simulation, but I guess your pocket money can't foot the bill.

So you will have to make a lot of assumptions:
Piston moves left with nozzle open for a fraction a of its trip with a anywhere between 0.2 and 0.8
Piston moves left the rest of the way pulling a slight vacuum (adiabatic expansion formulas)
Pinion is freed
Piston compresses air in chamber (formulas under adiabatic compression) while pea sits still.
pea is not moving until pressure > b times atmospheric with b slightly > 1
pea off nozzle, piston exercises force on air, pressure is sustaining nozzle flow and imparting energy on pea until it leaves the tube they call barrel.

Quite a complicated design, but you knew that already.

Start with adiabatic compression formulas. Pretend pea is held in place until all spring energy is converted to pressure (this is past the spring equilibrium point because at that point the piston has considerable kinetic energy). Then apply Bernouilly and puff the pea out. Muzzle speed might be of the order of air speed if things are nicely tuned (any gunsmith listening in ?)

Look forward to your next homework assignment. Must be the one where two half-balls of U235 must be collided with sufficient momentum.

 

[Havaren]

This is actually a really common toy called an air for gun or marker. Depending where you are in the world of course.

For the air flow, when the piston is on the back stroke (when it is being forced to compress the spring) the nozzle has no obstructions as it is pulled back at the same time. The mechanism that pulls back the nozzle allows it forward before the piston is released however, pushing the nozzle forward into a gasket unit, effectively creating a seal. So by that mechanic, the air flows in the nozzle in this particular model ( there are some with ported cylinders, and I will address those at some point, but I am focusing on this particular issue at the moment.)

Thank you for everything so far. I will sit down with the numbers I have so far and see if I can't plug them out and get some useful data.