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Volume of a cylinder...

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Old October 17th, 2010, 21:42   #1
LUTNIT
 
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Volume of a cylinder...

I always just used the listed recommended barrel length for differently ported cylinders without calculating the volume. I finally sat down and tried calculating the volume of a cylinder to see what the max barrel length is and I got a really weird answer no matter what I do. I used a vernier caliper to get the measurements and then rounded them to the nearest 0.25mm to account for differences in tolerances and such between brands.

Cylinder = Modify stainless non-ported regular cylinder (not bore up)
Cylinder head = stock TM from an M733
Piston head = Modify polycarb

Cylinder length = 72mm
Cylinder head depth = 7.25mm (only the section that sits inside the cylinder)
Piston head depth = 6mm (since the piston head is still inside the cylinder when fully drawn back)

So functional cylinder depth is 58.75mm

Inner diameter of cylinder = 23.5mm
Therefore radius = 11.75mm

Volume of cylinder = 3.1415 x 11.75^2 x 58.75 = 25 482mm^3

Inner radius of barrel = 3.015mm (Prometheus 6.03mm)

Area of cross section = 3.1415 x 3.015^2 = 28.56mm^2

So volume of the cylinder divided by the area of a cross section of the inside of an inner barrel should equal max barrel length in mm's. Here the problem...

25 482 / 28.56 = 892.23mm

The max barrel length for a normal non-ported cylinder is listed as 500-590mm but somehow I get almost double that. Is my math screwed or is this theoretical calculation method inappropriate to airsoft due to air leaks and such? I have always heard of people using volume calculations to try to best match the cylinder to the inner barrel but after doing this I wonder how its possible.
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Old October 17th, 2010, 21:50   #2
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You need a higher volume in the cylinder than in the barrel because we operate our guns at higher than atmospheric pressure. In order to accelerate the pellet, we need to have a higher pressure on one side of the pellet to overcome the atmospheric pressure on the other side.

The air compresses in the cylinder so we need exceed the barrel volume with the cylinder volume.
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Old October 17th, 2010, 21:52   #3
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Is there supposed to be some amount of compression involved?

edit: too late...lol
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Old October 17th, 2010, 21:53   #4
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Well, that would be a good calculation if you were using a non-compressible fluid.

Edit: Also too late.
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Old October 17th, 2010, 21:57   #5
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Hmmmm..... I was thinking that but so many people who talk about cylinder matching online talk about using this basic formula to find if the barrel you chose needs a differently ported cylinder that I thought it might work.

Adding compression into the mix of course changes everything and I guess people who use the straight up calculations are doing it wrong to get the answers they want or are on crack.
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Old October 17th, 2010, 22:06   #6
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I would figure the best way to go about it would be to use something more along the lines of a pressure balance instead of a volume balance. Use your volumes to determine and compare your internal pressure to 101KPa. If the internal pressure drops below 101kPa before the end of the barrel, your BB will be deccelerating.
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Old October 17th, 2010, 23:10   #7
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When calculating a cylinder for your barrel use 1.7 x the volume of the barrel as a base number for the volume of the cylinder. For a stronger spring you should go higher to account for higher compression. 1.7 is just a starting guideline for a stock spring of 280-300fps with .2g bbs.
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Old October 18th, 2010, 03:03   #8
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assuming a 0.2g pellet, with muzzle velocity of 400fps

400fps = 120m/s, avg speed = 60m/s assuming constant acceleration
barrel time = 0.55m/60m/s = 0.0091666s
acceleration = 120m / 0.009166s = 13,091 m/s^2

force = mass * accelleration
force = 0.0002 * 13,091m/s^2 = 2.62N

barrel area = 0.0000283m^2

pressure = force/area = 92.66kN/M^2 = 13.44psig (pressure over atmospheric)

That's kind of neat because that corresponds to an atmospheric pressure of 1.91atm which is roughly in the ballpark of the 1.7 factor that venture is quoting.
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Old October 18th, 2010, 20:11   #9
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Madmax, I want to have your baby!

Thanks everyone.
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Old October 19th, 2010, 23:13   #10
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You can't have my baby. My wife would be pissed and I wouldn't have an heir to continue my airsoft work.

That calculation is pretty rough. Assuming a constant acceleration is a bit of a whopper of an assumption given that the mainspring will assert a force related to the piston position. I think my calculation stands as a reasonably good "beer coaster" look into what's going in our AEGs, but if you really want a good theoretical explanation, I think one would have to creak open a differential equation textbook and hammer away. I've probably got one somewhere, but I don't have good table data on the force-deflection behavior of our AEG springs and I think our AEGs may have some funny phase of operation that would require a few sequences of equations to properly model. Maybe someone here languishing away in university attaining a math degree might be so inclined to give this one a try. I'd even settle for a pimply Eng Sci looking to codify the world.

In the first phase we'd see a non linear acceleration of the piston as the spring starts from it's fully compressed length. If you have a spring with two more more pitch angles (loose and tighter coils) then you may observe two spring constants if the lower pitch coils start fully compressed against each other. Coils that are crunched together aren't compressible. They may be pushing outwards, but they can't be compressed any closer together. In this situation, your spring behaves like it's got a solid section so it'll have a higher constant. Once these tighter coils open up, suddenly more winds "appear" so the springs constant reduces. I suspect that a spring designer could use this feature to reduce the pick up force that a sector would have to exert to make the first piston tooth engagement to reduce the wear on this crappy initial engagement. This may or may not be effectively done as airsoft designers often tend to be avid hobbyists, but not the greatest at engineering (e.g. poor mat'l selections et al). This first phase is the most simple one as all you've got is a rigid mass (the piston) being accelerated along a stiffer spring constant then a less stiff one. I think it may be ok to neglect pneumatic effects because the air isn't compressing because it's flying out the cylinder side opening.

The second phase is a more complicated one. Once the piston passes the cylinder opening, air begins to compress in the cylinder. I suppose that the pressure will build until it overcomes the detenting effect that the hopup has against a breeched ball. In this phase the initially moving mass of the piston is pushed by a single constant spring (I'm assuming that the tight coils have opened by now) and air pressure starts to rise non linearly in relation to piston movement. The pressure-distance relation would be inverse as each halving of cylinder volume would result in a doubling of pressure. Pressure will rise until the bb pushes past the hopup and things get even more messy.

The third phase is ultra messy because the bb is free to accelerate. Basically you've got two moving masses, piston and bb, with the piston pushed one way by the mainspring, and a squishy nonlinear "spring" pneumatically coupling the piston face to the spring. This is pretty much where I want to get off because I didn't do all that well in differential equations which we ended up calling difficult equations.

Still there are some approximations which might also prove to be fatal. If we were to be concerned with the moving mass of the piston, we should also consider the moving spring which is heavy steel. I suppose it could be approximated by assuming it's centre of gravity would move half the distance and speed of the piston, but it adds yet another term to the chain of crap. We also have to neglect air leakage around the bb as well as friction in the cylinder and barrel because that might be very difficult to model. A constant friction is easy to work with, but in my experience, orings do funny things when they're compressed. Under low pressure, an oringed assembly might slide nice, but when pressurized, round cross sectioned orings squish into an oval cross section so their sides flare more firmly against their sealing surfaces. While this prevents leaks, it increases dynamic friction.

If I really really wanted to know how everything was happening, I'd probably skip most of the hardcore math and put together a P90 build and swap out barrels with a fixed cylinder set. Measure muzzle velocity at varying barrel lengths and make a fudge factor chart to cheat around the whole messy ball of assumptions and math.
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