Theoretical Maximum Velocity in a PCP

[rsterne]

Your calculations for the maximum FPE agree exactly with Lloyd's.... For the .50 cal (area 0.196 in^2) with a 1 ft. barrel with 3000 psi you get 588 FPE and Lloyd, with 3500 psi, got 686 FPE.... the only difference being the pressure used....

The reason for the "50%?" remark at the end of the first post is that REAL PCPs have many losses, so can never approach the theoretical maximum, there was no "missed 0.5 factor".... For a start, the bullet is not the only mass being accelerated, the air itself has significant mass.... That 1 ft. long .50 cal barrel-full of 3000 psi air would weigh 142 grains.... in a 24" barrel, twice that, or about the same weight as the bullet.... In addition, the maximum FPE would require constant pressure over the entire 1 ft. of barrel travel, which would require an infinite reservoir volume.... When the reservoir is of limited volume, you get a pressure drop during the shot, and then the pressure drops even more from adiabatic cooling.... It also ignores barrel friction and other losses.... In fact, only the very best of our PCPs even approach 50% of the theoretical FPE.... most fall far short....

Bob

[Scotchmo]

...THUS: a 1/2 inch diameter slug, with 3ksi pushing it, REGARDLESS of mass, will achieve ~580 ft lbs, regardless of mass, regardless of mass... see caveman terms.
...

That sort of works up to a point. But the air itself has mass. When trying for very high velocities, the air mass required to propel the pellet becomes a significant part of the equation. And there are limitations in the rate of flow and rate of expansion for air at room temperature.

[subscriber]

...THUS: a 1/2 inch diameter slug, with 3ksi pushing it, REGARDLESS of mass, will achieve ~580 ft lbs, regardless of mass, regardless of mass... see caveman terms.
...

That sort of works up to a point. But the air itself has mass. When trying for very high velocities, the air mass required to propel the pellet becomes a significant part of the equation. And there are limitations in the rate of flow and rate of expansion for air at room temperature.

Noah,

To understand why projectile mass matters; return to reality:   With a real airgun, you are stuck with a finite air supply of, where the pressure drops as the projectile accelerates down the barrel.  Now; lighter projectiles mean higher acceleration.  Higher acceleration to higher velocities mean much more of the system energy is spent accelerating the air in the barrel up to that higher velocity; because Kinetic Energy = V x V x 1/2 X mass. 

So, in real PCPs, heavier projectiles are more efficient and carry more energy than light ones, because less energy is wasted accelerating the air column in the barrel behind them.  End of story.

[rsterne]

A perfect example is my 6mm PCP.... With a 73.4 gr. lead bullet, and tuned for maximum power, it shoots at 1029 fps, which is 172.6 FPE....With no changes, other than using an airsoft BB which only weighs 1.8 gr. I get 2092 fps, but that is only 17.5 FPE.... Incidently, the theoretical maximum FPE for this gun is (0.243 x 0.243 x PI/4) x 4200 x (28 / 12) = 454 FPE.... A large percentage of the lost FPE (only 38% ended up in the lead bullet, and less than 4% in the plastic one) went into accelerating the 250 cc of 4200 psi air released by the valve, which weighed 78.6 grams, or 1212 grains.... and yes, most of that left the muzzle long after Elvis had left the building....

Bob

[WhatUPSbox?]

Just curious if anyone with access to welding gases has taken shots with argon (38% denser than air) at the same pressure to see the drop off of FPE, might be able to get an idea of what the average velocity of the gas column is.

[asdf1234]

Okay, good, I didn't see the 0.5 on the equation posted, didn't double check the exact numbers.

Re air mass, I remember years ago some science on paintball markers with ported barrels, the porting was found to substantively reduce pressure and turbulence in front of the ball, and acceleration actually increased, even compared to the exact barrel prior to drilling holes through it.

The point on e=df is just to bring it down to earth for considering that its all linear.

Extrapolating to eliminate air acceleration in favor of max M.E. per charge e just means increasing mass, perhaps to the point of being a harpoon gun...
Can someone make a heavy "harpoon" and measure M.E. compared to lead?
I'm curious in seeing it taken to the limit.

Barrel length is also linearly equivalent to force... could double length.
Also: Bumping pressure to 10ksi should be pretty easy within the gun.

[rsterne]

Eventually, long barrels become too awkward to deal with.... High powered PCPs are typically running 28" to 34" barrels....

Using higher pressures, while technically possible, means heavier (stronger) guns, and difficulty finding a way to fill them.... This thread is not so much about how fast you can go, but what those limits are, based on barrel volume times pressure.... If you haven't checked the math already, you can use the barrel volume, since a .50 cal barrel a foot long has the same potential FPE as a .25 cal barrel 4 feet long.... so you can simply use this formula for the theoretical maximum FPE....

Maximum FPE = Barrel Volume (cu.in) x Pressure (psi) / 12 (in/ft)

In reality, approaching half that is difficult, so I often use a formula to predict a "lofty goal" for PCPs, with the proviso that the weight of the bullet is half the FPE goal (so the velocity is 950 fps)....

Lofty Goal FPE = Barrel Volume (cu.in) x Pressure (psi) / 24

I have never quite made it, I usually need about a 10% longer barrel, or 10% higher pressure, to get there.... If you use a heavier bullet, getting to, or even surpassing, that 50% goal is easier to accomplish (but still rarely done)…. Most production PCPs don't even reach half of the "lofty goal" FPE....

Bob

[asdf1234]

Eventually, long barrels become too awkward to deal with.... High powered PCPs are typically running 28" to 34" barrels....

Using higher pressures, while technically possible, means heavier (stronger) guns, and difficulty finding a way to fill them....

A simple pressure booster is just a couple rods and tubes, it can be machined from steel and integrated into the action.
We're talking about a handful of in^3?

It might be a 4 lbs piston & volume assembly... although I suppose you might as well have it charge via battery at that point.

[rsterne]

I suppose, if what you are talking about is what is possible vs practical.... then just about anything is possible.... Don't forget, if you only have a reservoir consisting of a "handful of in^3", you will get a significant pressure drop during the shot.... particularly if you are in search of high FPE, which means lots of barrel volume for the reservoir air to expand into.... For example, for a .50 cal with a 24" barrel, starting with 10,000 psi, to maintain 9,000 psi at the muzzle, you will need a reservoir of about 111 CI (1820 cc).... The VanDerWaals effect rears its ugly head above 3000 psi, and by the time you get to 10,000 psi, dropping to 9,000 only reduces the air density by 4% instead of 10%.... This requires reservoirs to be more than twice as large as predicted by Boyle's Law for a given pressure drop.... This also affects shot count, of course, something we already see when using 4500 psi instead of 3000.... the shot count is noticeably less than you would calculate using Boyle's law....

May I ask if you have ever built a PCP, or a pressure booster?.... If not, then doing that to understand what is "simple" or even practical might be a good idea.... If you have, then please share some of your projects and results, we would all love to see them....

A much easier way to greatly increase the FPE of a given PCP is to use Helium instead of air.... Since the primary power loss is from accelerating the gas, using a gas that is 1/7th the density has a huge advantage.... For a PCP shooting a bullet at about 950 fps, Helium will usually drive that same bullet Supersonic, with about a 50% increase in FPE.... The gains in FPE are even greater for lighter bullets, because the losses with air are higher, so the FPE gains shifting to Helium are larger....

Bob

[Scotchmo]

...
Also: Bumping pressure to 10ksi should be pretty easy within the gun.

10ksi air?

At some point, compressability of a supercritical fluid becomes nearly flat, almost non-existent.

Why not 100ksi vs 50ksi? No benefit, because at those pressures, it would make zero difference in FPE.

For air, I wouldn't go above 6000psi (for a number of reasons).

https://www.researchgate.net/figure/The-temperature-pressure-phase-diagram-for-nitrogen_fig1_315888614


It sounds like you might be talking about compressing air on the fly after pulling the trigger), with a booster. If so, then you are really talking about an air powered spring gun. Temperatures of the propelling gas would be higher than room temperature, so velocity limits would also be higher.

[rsterne]

Not sure I understand, Scott.... The critical pressure for air is 550 psi.... above that we are dealing with a supercritical fluid anyways, providing the temperature is above -140*C.... Yes, eventually only a small change in density requires a huge change in pressure.... no doubt about that.... and VanDerWaals equations deal with that.... Here is a very good calculator showing the relationship between pressure and density for air, up to 1000 bar (14,500 psi)....

https://www.peacesoftware.de/einigewerte/luft_e.html

As an example, a 100% increase in pressure from 500 bar to 1000 bar only requires a 37% increase in air density....

Bob

[Scotchmo]

Not sure I understand, Scott.... The critical pressure for air is 550 psi.... above that we are dealing with a supercritical fluid anyways, providing the temperature is above -140*C.... Yes, eventually only a small change in density requires a huge change in pressure.... no doubt about that.... and VanDerWaals equations deal with that.... Here is a very good calculator showing the relationship between pressure and density for air, up to 1000 bar (14,500 psi)....

https://www.peacesoftware.de/einigewerte/luft_e.html

As an example, a 100% increase in pressure from 500 bar to 1000 bar only requires a 37% increase in air density....

Bob

"As an example, a 100% increase in pressure from 500 bar to 1000 bar only requires a 37% increase in air density...."

It looks like you understand the problem -

As you increase the pressure of a supercritical fluid, the change in density/pressure becomes less and less. At some point, the increase in pressure produces no more usable energy in the air. It behaves more like a liquid as far as increasing pressure.

I guess 10ksi is doable (though not really easy).  At that pressure, any additional gains are only 1/2 what you would expect from an ideal gas.

[Rob M]

so 6000 psi is an ideal working pressure if you can compress it .. very interesting .

[rsterne]

In my opinion, 6000 psi is the maximum pressure I would consider.... but the losses in shot count are already significant.... In practical terms, with current portable bottles at 4500 psi, anything much over 3600 psi leave you with difficulty filling in the field.... unless you use a portable compressor or booster.... Air boosters, while they can be made capable of over 4500 psi, waste far more air than they can cram into a tank or gun....

Bob

[Rob M]

yes.. i was thinking in terms of msp/ssp designs ... the one working design over 6000 psi was 10 yrs ago on the green forum by a brillant designer named Ron.. It was a dual stage MSP pistol.. The main issue as i recall was melting seals on the way to 6000

[asdf1234]

In my opinion, 6000 psi is the maximum pressure I would consider.... but the losses in shot count are already significant.... In practical terms, with current portable bottles at 4500 psi, anything much over 3600 psi leave you with difficulty filling in the field.... unless you use a portable compressor or booster.... Air boosters, while they can be made capable of over 4500 psi, waste far more air than they can cram into a tank or gun....

Bob

The thread is discussing several variables simultaneously... so everything I've said could be right or wrong depending on the project goals.

Efficiency, practicality, capacity, fps, energy, etc.

But for max-fps I'm glad we decided on small-caliber miniature electric 10ksi booster. 

And you can feed it helium. 

[rsterne]

Can't wait until you build one.... 

Bob

[Scotchmo]

Eventually, long barrels become too awkward to deal with.... High powered PCPs are typically running 28" to 34" barrels....

Using higher pressures, while technically possible, means heavier (stronger) guns, and difficulty finding a way to fill them....

A simple pressure booster is just a couple rods and tubes, it can be machined from steel and integrated into the action.
We're talking about a handful of in^3?

It might be a 4 lbs piston & volume assembly... although I suppose you might as well have it charge via battery at that point.

Topic of thread:
"Theoretical Maximum Velocity in a PCP"

If it involves adiabatic compression and expansion, it is not a pre-charged gun. You are really talking about a piston gun, not a PCP. Many of the limiting factors for a true PCP will not apply to a piston gun.

[rsterne]

I think Noah is talking about using a (4 lb.) booster pump, mounted on/in the gun, to raise the pressure in the reservoir to 10 Ksi…. He referred to only having to compress a "handful of in^3s", which is why I pointed out the need for a large reservoir to prevent severe pressure loss during the shot.... Perhaps he can explain whether that is what he means (in which case it is a PCP)…. or if the compression piston is driven by expanding air DURING the shot, in which case you are correct, it is an "air driven boosted springer", using adiabatic compression of the firing air, and not a PCP at all.... They operate under quite different conditions....

Bob

[asdf1234]

I think Noah is talking about using a (4 lb.) booster pump, mounted on/in the gun, to raise the pressure in the reservoir to 10 Ksi…. He referred to only having to compress a "handful of in^3s", which is why I pointed out the need for a large reservoir to prevent severe pressure loss during the shot.... Perhaps he can explain whether that is what he means (in which case it is a PCP)…. or if the compression piston is driven by expanding air DURING the shot, in which case you are correct, it is an "air driven boosted springer", using adiabatic compression of the firing air, and not a PCP at all.... They operate under quite different conditions....

Bob

I think you've got it.
I would just try to work backwards from terminal ballistics goals.

If the goal is FPS you should go as lightweight as possible, whereas if the goal is energy you should go as large bore as possible, yes?
These are more or less opposites, for lets say mass or chamber volume, so its easy to say I'm right or wrong, depending on the project goal.
Originally most people seemed to be discussing energy, but after I mentioned that y'all were overthinking it, the conversation switched back to fps, then someone brought up practicality...
... so there's no consensus on what to solve for.

If the premise of the thread is FPS and definitely NOT energy, efficiency, or practicality, then the math is easier.
Practicality is pretty abstract, and efficiency is more complicated math.

Ignoring those two variables, yes, I was referring to a booster to bump to 10ksi, in the gun or off the gun, prior to the shot... so no reason to charge with air, just plug it in and let it waste air to charge itself to 10ksi.
If you want to solve for energy, then it would get very large and heavy, sure.

Helium and 10ksi are each probably good choices for both energy and fps, but probably not for either practicality and/or efficiency.