What happens at extreme pressures?

[rsterne]

There are lots of PCPs that run 3000 psi, and a few up to 4500.... Is there any point in running even higher pressures?.... What is to be gained, and is that gain linear?.... Some of you are quite familiar with Boyle's law which relates pressure inversely to volume for an ideal gas.... However, air is not an ideal gas, and when the pressure goes over 3000 psi, we need to take into account van der Waal's corrections for it's behavior.... I was not even aware of this until a few months ago when PakProtector mentioned it, while explaining why the 550 CI "60 minute" CF tanks we use to hold air at 4500 psi don't hold 97 CF of air at that pressure, like Boyle's law would predict, but in fact they only hold 88 CF.... I've done up a graph which shows how some of our common gasses don't follow Boyle's law....



The graph shows the relationship between pressure and volume as compared to an ideal gas (black line).... I couldn't find any data for air, so I took the geometric average between the lines for Oxygen and Nitrogen in the ratio of 21:78, ignoring the other 1% constituents, to come up with the green line for air.... Note that it crosses what we would expect using Boyle's Law at just under 3500 psi, and in fact below that you can actually cram more air into a given space than Boyle's Law would predict for a given pressure.... At 1500-2000 psi, the air would actually be about 8% more dense than an ideal gas.... Above 3500 psi, however, it becomes less dense, in other words the pressure rises faster than you would expect.... At 6000 psi, it is only about 79% as dense as Boyle's law would predict.... If you took a litre of air at 3000 psi and compressed it to 6000 psi, instead of occupying 500 cc, it would occupy about 650 cc.... In other words, even though you doubled the pressure, you didn't double the density....or halve the volume.... 

This has led me to question what happens in a PCP at extreme pressures.... When we fire a PCP, a small pulse of high pressure air is released by the valve, accelerating the pellet as it expands down the barrel.... For the first part of that process, the valve is open, and the pressure drop is relatively small.... Then the valve shuts, and the second phase begins, likely an adiabatic expansion, or nearly so, as the pellet continues to uncover increasing amounts of barrel volume before it exits.... All of our theory (including Lloyd's spreadsheet) assume Boyle's law is in effect, so that as the volume doubles, the pressure is now half.... If we are working with 3000 psi air, and the residual muzzle pressure is, say, 500 psi, we are working (in air) on that part of the curve above from about 104% to 108% and back to 103% again.... Although not completely flat, it's probably close enough for a model, particularly since the beginning and ending point are nearly identical densities....

Now let's look at what would happen if we started at twice the pressure, 6000 psi, and with a large reservoir, so that the pressure remained close to 6000 through the "valve open" stage.... Once the valve shuts and the air begins to expand, we're at about 79% density, and let's this time assume our muzzle pressure is 1000 psi (also twice the pressure)... At that point, we're at 106% density.... Assuming we are running the same volumes as our 3000 psi gun above, our slug of 6000 psi air, which will have a given number of molecules in it, when expanded to the same full barrel volume, will drop more in pressure than predicted by Boyle's law.... While we were expecting the muzzle pressure to be 1000 psi, in fact it will be less, roughly 745 psi if my calculations are correct.... In other words, the pressure during the expansion phase will drop faster than we expected....

Does this make any significant difference to the shot?.... Probably not, as the last part of the expansion near the muzzle is pretty ineffective in producing acceleration.... However, I would like to raise one more question....

Compared to 3000 psi air, 6000 psi air is only about (79% / 104%) = 76% as dense as the doubling of density we expected by Boyle's law.... Does this make ANY difference to the acceleration of the pellet, because fewer molecules than expected are hitting it.... or is the pressure the only thing that matters?.... In other words, will 6000 psi air produce twice the acceleration that 3000 psi air would or not?.... I'm betting that pressure is the important factor.... Anyone?....

Bob

[Rocker1]

You make my head hurt Bob;  David

[QVTom]

In other words, will 6000 psi air produce twice the acceleration that 3000 psi air would or not?.... I'm betting that pressure is the important factor.... Anyone?....

I'm no physicist......  Isn't the pressure just the force of the molecules pushing away from each other? If it takes fewer molecules than the ideal gas law suggests to make 6000 psi of force isn't the retained energy still 6000 psi?  Therefore I assume you are correct, the force (calibrated measured energy) is what counts.

Do we have a physicist in the house?

Tom

[rsterne]

Pretty sure pressure is the over-riding factor, but I'd like to know if the decrease in density has any effect.... and like Tom, I'm no physicist....

Bob

[DOKF]

Bob,

I am a physical chemist.  Van der Waals corrections to the ideal gas law take into account the intermolecular attractions disallowed by the ideal gas law, as well as the actual molecular volume as opposed to the "swept volume" of a rapidly moving gas molecule.

Helium and Hydrogen have the least intermolecular attractions, so should be closest to ideal  As your calculations show, He has an almost linear volume response to increased pressure, while the others have a more complex behaviour. 

Compressability, and ideal properties depend on induced dipole forces (Van der Waals Forces), which in turn depend on the size of the electron cloud on the molecule.  H₂ and He have the smallest electron clouds, CO₂ the largest, with N₂ and O₂ somewhere in between.

Pressure is the common metric because of the ease of use, but as you've seen, it may not be the most reliable.  Of more use would be "n" from PV = nRT, the number of molecules.  Gas density is already a factor in the ideal gas law where P/RT = n/V

As your plots show, the common gases develop linear volume ~ pressure relationships after a certain pressure.  Unfortunately many canisters and associated equipment may not be able to withstand the pressures required for V P linearity.

What your plots don't show is what happens when the gases condense to liquids.  There will be a disconnect at the phase transition, as is observed for CO₂.  All gases will liquefy at some pressure and temperature as does CO₂, probably the least ideal of the common gases.

Another consideration would be pressure regulation, which also perturbs the ideal gas law.

[PakProtector]

As your plots show, the common gases develop linear volume ~ pressure relationships after a certain pressure.  Unfortunately many canisters and associated equipment may not be able to withstand the pressures required for V P linearity.

The plots show what looks like a linear error developing in the deviation from ideal.

On the rifle's use of high pressure I think the lower density will be of greatest use where it is at its largest...as in, there is more density error where there is more use for high presser low density working fluid.
cheers,
Douglas

[DOKF]

As your plots show, the common gases develop linear volume ~ pressure relationships after a certain pressure.  Unfortunately many canisters and associated equipment may not be able to withstand the pressures required for V P linearity.

The plots show what looks like a linear error developing in the deviation from ideal.

i prefer to think the ideal is a deviation from reality, with significant intermolecular attractions as pressure increases.  The Ideal Gas Law really only works at small deviations from STP under adiabatic conditions.  Pressures here are far from STP, making the Ideal Gas Law more a Rule of Thumb.

Regards,

Kim

[rsterne]

I would agree that Helium shows a linear trend, but divergent from an Ideal gas (the horizontal black line) which is the reference for the graph.... I more correctly should have titled the graph "n" (the number of molecules) vs. Pressure (for constant volume), my bad.... but at least I explained the deviation correctly (as to direction).... I am well aware that state changes screw this up terribly, which is why I chose 20*C and the pressure ranges I did.... and didn't use CO2....

Any thoughts about which is important in terms of the force on a pellet/bullet?.... I think it's pressure and the valve of "n" (which relates to the density) doesn't really matter....

BTW, I have a BSc. in Organic Chemistry, I took Physical Chemistry 3 times, got a D - F - D.... *LOL*.... I hate Calculus, that was my downfall.... By the way, if you see Shroedinger, please shoot his cat for me.... 

Bob

[DOKF]

I would agree that Helium shows a linear trend, but divergent from an Ideal gas (the horizontal black line) which is the reference for the graph.... I more correctly should have titled the graph "n" (the number of molecules) vs. Pressure (for constant volume), my bad.... but at least I explained the deviation correctly (as to direction).... I am well aware that state changes screw this up terribly, which is why I chose 20*C and the pressure ranges I did.... and didn't use CO2....

Any thoughts about which is important in terms of the force on a pellet/bullet?.... I think it's pressure and the valve of "n" (which relates to the density) doesn't really matter....

BTW, I have a BSc. in Organic Chemistry, I took Physical Chemistry 3 times, got a D - F - D.... *LOL*.... I hate Calculus, that was my downfall.... By the way, if you see Shroedinger, please shoot his cat for me.... 

Bob

Bob,

I thought I detected a science background in your posts.   

I agree with you that P and n are the important factors.  As phrased, the pressure is set at 6,000 and the volume fixed, so the apparent volume shouldn't be a consideration.  The energy delivered is then directly related to the pressure per # of molecules released, provided the  pressure change is small for every decant.  Remember that RT is an energy term (commonly used in thermodynamics equations), thus the energy of each decant is then PV/n where V is the actual (fixed) not apparent (corrected) volume.

At some high pressure, the volume correction term will actually serve to limit pressure increases, as seen with CO₂ where the max pressure is determined by the vapour pressure of CO₂(l).

Of course then you will have to deal with the inefficiencies of energy transfer when the air slug hits the projectile.  But the losses should be diminish relatively with increased P.

best,

Kim

PS  I spent some time in Sparwood during my undergrad, working the mines.  Spent every spare moment fishing the local waters; West Slope Cutts, Dolly Varden's (Bull), and whitefish.  Love smoked whitefish!  Occasionally see some from northern AB in the local stores.  Smoked goldeye and mooneye are also favs.

[Sfttailrdr46]

  The geek squad strikes again and now you have a chemist added to the mix . My electronics background of forty years ago just lets me understand enough to get massive headaches and sort of follow the train of thought. Kim welcome to the GTA you are in good company with the geek squad . They drag some of us along kicking and screaming and force us to learn things we did not know we needed to shoot AG's