air flow physics formulae

[Jotsy]

sorry if this has been discussed before. if it has, please link me cos my search skills seem to be really lacking lately.

anyways, my physics is a little rusty and i was wondering if there were any formulae for the expansion rate of compressed air? basically, with air compressed to X psi in Y amount of space, with a sudden opening Z mm^2 wide released into normal air:

- how fast will the air empty?
- how fast will it accelerate a ball up to 300fps? (assuming zero friction if you want)
- how much space (barrel length) will it need?

another question would be if it would be possible to accelerate the ball up to a certain speed and let the ball get to 300fps on its own inertia.

by the time the ball hits 300fps at the end of the effective barrel, if pressure behind the ball (in the barrel) and pressure in front of the ball (outside air) was equal (in theory using up the most minimum amount of air to accelerate the ball up to 300fps), would this affect the travel of the ball in anyway?

[Pyroboy597]

I do not know the actual formula, but applying friction would be easy as long as you could get the coefficient of the inside of a barrel against the outside of a paintball.

[LorneCash]

I hate Fluid Mechanics!

Unless you're a fluid mechanics PHD with some sophistocated software there is no way you would ever get numbers that make any sense for this type of calculation. It would be much easier to do it by observation/experimentation. If you look in Tom's data thread that's what he did. It's a shame that the pictures are no longer there. I would have liked to see them too.


Here's a little better explanation if you dare to take a look...



The motion of a non-turbulent, Newtonian fluid is governed by the Navier-Stokes equation.

The equation can also be used to model turbulent flow, where the fluid parameters are interpreted as time-averaged values. The time-derivative of the fluid velocity in the Navier-Stokes equation is the material derivative.

The material derivative is distinct from a normal derivative because it includes a convection term, a very important term in fluid mechanics. This unique derivative will be denoted by a "dot" placed above the variable it operates on.

here's the link to the Navier-Stokes equations




A non-turbulent, perfect, compressible, and barotropic fluid undergoing steady motion is governed by the Bernoulli Equation.

where g is the gravity acceleration constant (9.81 m/s2; 32.2 ft/s2), V is the velocity of the fluid, and z is the height above an arbitrary datum. C remains constant along any streamline in the flow, but varies from streamline to streamline. If the flow is irrotational, then C has the same value for all streamlines. The function is the "pressure per density" in the fluid, and follows from the barotropic equation of state, p = p(r).

here's the link to the Bernoulli Equation

[LorneCash]

Also, inertia will never speed anything up. Here's the dictionary's definition:

Inertia - The tendency of a body to resist acceleration; the tendency of a body at rest to remain at rest or of a body in straight line motion to stay in motion in a straight line unless acted on by an outside force.

In other words, inertia means it's harder to stop something that weighs more.

[Jotsy]

thanks lorne, thats the kind of stuff i wanted to know. i'll try and decipher it later

[thefool]

dont know if this is the right place to ask this but i hope it is. How would you determain the volume of gas needed(at different pressures) to accelerate the .68 diamerter sac to 300 fps, if you ignore friction casue it would make everything much more complicated.

[LorneCash]

I Hate Fluid Mechanics... Revisited

How would you determine the volume of gas needed(at different pressures) to accelerate the .68 diamerter sac to 300 fps, if you ignore friction casue it would make everything much more complicated.

Some things to consider:
Force = Mass * Acceleration
Pressure = Force / Area
Area = PI * diameter^2 / 4
diameter = .68in
Mass we can find easy bet it is significantly different for different paintballs

Pressure is going to be a problem because you may know your initial pressure but it changes instantaniously as the ball moves down the barrel. Also there is a column of air in the barrel that you are compressing and have to push out of the way. If that wasn't enough, there will come a piont when the pressure behind the ball is less than the pressure in front of the ball. Hopefully that is the exact point at which your barrel porting starts. If not it's actually slowing down you ball. One other thing to consider is there is a small gap between the barrel and the ball and air can slip through that gap. The size of that gap also greatly affects the pressure behind the ball. Anyone with a barrel kit with multiple bore sizes can tell you that, but in order to calculate it's affect you would need some extreemly expensive FEA (Finite Element Analysis) Software with a fluid dynamics/ fluid flow module. Then you have to know the Renolds nuber of your air and how turbulent it is because it's clearly not a laminar flow... For more info on this see my first post in this thread. And lucky for us you said we could ignore the friction between the ball and the barrel but that's not even where most of the friction is... Have you heard of wind resistance AKA friction with the air it's passing through. That's also a Major consideration and can't be ignored.

Even if you knew all that that is still only going to get you the acceleration now you need the velocity.

Velocity is the time derivative of acceleration and you need to know some calculus to figure that out.

So really please everyone don't ask any more questions like this. There is no point in taking the time to figure out the mathematical side when it would be much quicker and more accurate to just do some tests. If you really really want to know, build some kind of a test chamber calculate it's volume fill it to a pressure and release it on a paintball in a barrel and let me know. Trust me it's much simpler than trying to do it with equations.

[muskratjim]

hey lorne what is it like to be so smart lol

[thefool]

thats not really the question i was asking, you over complicated it. All i really wanted to know how to do if figure out say with a mag, what size the dump chamber would have to be at a given pressure. Obviosly the guys at adg did it.

[LorneCash]

thats not really the question i was asking, you over complicated it. All i really wanted to know how to do if figure out say with a mag, what size the dump chamber would have to be at a given pressure. Obviosly the guys at adg did it.

Well if you're familiar with your valve you know that there's a brass Ram in the back with two springs behind it. The answer is that you don't have to know how big the chamber must be you simply make it adjustable. That's exactly what you are doing when you adjust your velocity. You are changing the size of the "dump" chamber. Assuming you have a constant pressure comming into the gun the springs will always compress the same ammount. All you are doing is adjusting the size of that chamber while the springs are compressed. Doing that mathematically is near impossible. Calculating it based on the internal volume and the compression of the springs is not necessary because knowing that number won't help you. It changes drastically with temperature and pressure and somewhat with what type of gas you are using. You can see that if you know the Ideal gas law:

Pressure * Volume = n * R * Temperature

where:
n = quantity of gas normally expressed in moles
R = universal gas constant = 8.3145 J/mol K

Basically you make your device adjustable and just shoot it over a chronograph and adjust it. But still the diameter and roundness of your paint and smoothness of your barrel have a HUGE part in how large that volume needs to be, probably even more so than temperature or pressure... See my last post again for more on that. And trust me, there is no way the guys at AGD spent the time to actually figure what that volume was. there's no need.

[athomas]

One of the things that gets overlooked, is the restriction within the powertube. The powertube inside diameter restricts air flow so that you don't have a constant pressure behind the ball. Another restriction is where the bolt stem is held out the front of the powertube. The bolt spring holds the bolt stem back against the air flowing out the powertube much like that of a regulator.

[thefool]

well i looked into a bit and corrent me if im wrong but if you wanted to get a very rough idea of the volume and pressure youd need to accelerate a ball in say 6" (average working length of a barrel) all youd have to do would pv1/.4(1-(v1/v2)^.4)
where p is initial pressure
v1 is initial volume
v2 is expanded volume (all the space the air will take up by the time the ball is 6" down the barrel)
and .4 is a constant the name of which slips me right now.

so supposing your operation pressure is 300, you would need an input volume of 1.35 inches to accelerate the ball to 330 fps

Tell me if something seems wrong, i somehow dont belive that agd did it completly based off trial and error.

[LorneCash]

well i looked into a bit and corrent me if im wrong but if you wanted to get a very rough idea of the volume and pressure youd need to accelerate a ball in say 6" (average working length of a barrel) all youd have to do would pv1/.4(1-(v1/v2)^.4)
where p is initial pressure
v1 is initial volume
v2 is expanded volume (all the space the air will take up by the time the ball is 6" down the barrel)
and .4 is a constant the name of which slips me right now.

so supposing your operation pressure is 300, you would need an input volume of 1.35 inches to accelerate the ball to 330 fps

Tell me if something seems wrong, i somehow dont belive that agd did it completly based off trial and error.

There's no "=" is your equation so I have no Idea what you are solving for. Without that it is impossible for me to explain exactly how wrong you are...lol I need to know that so I can tell what assumptions you are making about that system. Basically you can't use a steady state equation like that because like I explained before both the pressure and volume are changing with time. The equation has to be a differential equation. And really you got to trust me, there is no way AGD did anything more than trial and error on this.

[MANN]

First let me say that nothing is impossible, only improbable. Everything can be solved with math. I dont know for sure but I am sure that Tom did a little math on his valve before he sent it to the CNC machine. As sure as I am that he did some math I am positive that he did a little testing also before he sent his design to the CNC machine

Second off let me say that I am an Engineering student not an English student so please forgive my horible english/spelling/writing skills( I went into engineering because it required no spelling and no foregin language).I do MATH. I am also a Junior Mech Eng student so if I'm wrong well oh well. They dont test us on paintball guns. the following is my opinion. Please check my math(I am drinking and studing for a final friday so I am a little scatterbrained.)(I know I should keep the drinking untill after the final, however this is the only way to learn about dimensionless numbers )

Paintball guns accelerate the balls using a combiniation of air pressure, and bolt movement. The bolt moves the paintball past the breech of the feedneck, and the pressure is what gives it the "boot". You could technically calculate what pressure you need to get the ball moving X fps using the following formulas Pressure=Force * Area. Pressure being the psi placed upon the ball, Force being calculated, and area being the area of the ball. Once Force is calcualted you can use the formula Force= mass* acceleration. Force being calculated from the previous formula, and mass being measured with a scale(not pounds. make sure you convert right.) and acceleration would be how fast the ball would go.

The next formula you will need to use is the formula ((PV)/T)state 1=((PV)/T)state 2 this will get you close to figuring out your problem. Where T=temp(can remove from formula assuming that the temp inside your valve is the same as outside your valve.) to have pv=pv. state 1 is when the "gas" ie air is inside your valve and state two is after you open your valve(fire your gun) in this case you know the volume of the chamber inside your gun(I hope. if not find it) and you know the pressure inside your gun( I think the regulator inside the automag regulates it to 60-70 psi maybe?? depending on how much you adjust your velocity knob on the back of your gun). All you have to figure out is the volume between the "holding chamber" , the effective barrel length, and your ball to have the second volume(good luck) ( I would test with water and a graduated cylinder, but prob won't work) Solve the equation to figure out how much pressure is required to get the ball moving.

The next formula is the const acceleration formula. It is the intergral of the velocity with respect to a velocity and between two points is equal to the intergral of acceleration with respect to distance between two points. Or v^2=vo^2=2a(s-so). To have a ball moving 300 fps the acceleration should be about (300fps^2=0 fps^2 * 2a (6in-0in)) which is 7500 ft/s^2. This is the acceleration required to move an object 6 inches and from a velocity of 0fps to 300fps.

Lets try an example. We are going to use an automag. (Because it is my favorite) First we need to work it backward. To acheive a ball accel of 7500 fps ^2 you would need the force to be (7500fps^2(2286mps^2)*mass(lets say 5 grams just a guess))(convert fps to meters per second) (mass needs to be in kg so .005kg) so the answer would be 11.43N(newtons) F=ma. Next you would have to find the area of a paintball .68" would be (.0002343m^2) use this area in the Pressure = force/area to calculate the required pressure to get the ball moving. so basicly 48783Pa = 7.07psi.(note check my math. I am switching back and forth between english and SI units and I might miss something)

I know what you are thinking. This is a very low pressure . Let me continue please.

So lets say you have a chamber in your automag (I am guessing .3313 in^2, this means the chamber is .75" diameter, and .75" long) at 60 psi. when released the volume changes to the previous volume plus the internals of the barrel. if you have a 6" effective barrel and we neglect the volume that is around the bolt and ball. than the area changes to 2.5 in"^2.
solving for the second pressure equals 7.9512

Compare it to the pressure required 7.07psi. and you are probally thinking damn he just solved the worlds problem. One problem with all of the above is that we neglected alot of stuff first we have to consider leaking!! air is going out the feed tube, out screw holes, around the ball, and out every orifice it can find that is less restricting than down the barrel. Second is that the coef of friction for the paintball and the barrel. We also have to figure if all of my assumptions are correct. ie mass of ball, pressure inside a mag, air chamber inside of mag, etc..

I hope this helps, and also explains why Tom is King and is able to sell his great gun to us. After all of that I would try to back it up experimentally. Btw if anyone from agd reads this I am very intrested in working for you.

For an experiment hook up an air compressor, and set the output pressure to 10 psi and try to get a tight seal around a barrel with a ball in it, and give it a blast. Use a crono to find the velocity, and this will give you an idea what pressure is required. Another experiment is to use a blowgun and do the same thing.

Good Luck and I am VERY SORRY if i missed something above

Josh

PS If anyone has already taken Mech eng Fluids and can help me with dimensionless number conversions please email me [email protected]

[thefool]

several things here i think might be unaccounted for, as loarn cash previosly stated ball to barrel frinction exists, but if that is to be ignored you have to consider how the air is being realesed, i need these calculations for a gun im designing and the pressure relase is as flat as i could make it with some dropoff at the end. also im not sure if this applies but all the expantion in a paintball gun orrurs under adiabatic conditions which adds even more complexity.

Also when you are finding acceleration how are you doing it for 6 inchs? or *inchs? you need something that will be a justable to different lengths.

That is why i think the best way to do this is instead of spending hours trying to figure out teh higher level math involved in this example the best way to do things is make a prototype where a lot of factors are ajustable and after taking the measurements from other guns find how much air is used.

For instance an ion haas a ~.9cu dump stock and operates at about 180 psi. using real world data from others i think will be just as effective and much less time consuming.

Just for fun though, the way i tryed doing it was the formula for expantion of gases under adiabatic conditions and got numbers that were about 10x off so i think i misplaced a decimle piont somewhere but i dont know where and i dont feel like typeing what i did right now