Theoretical Maximum Velocity in a PCP

[rsterne]

We knew what you meant, Lloyd.... even if it's not what you typed.... *chuckle*….

I think that the concept of a large, parabolic velocity gradient radially across the bore may be overrated.... While that occurs (2:1) for laminar flow, it is almost a flat gradient until you reach within a few thou of the bore for turbulent flow.... Since the flow would be tripped to turbulent right at the valve, and once turbulent never again becomes laminar.... I think the velocity gradient would look more like the lower one in this diagram than the upper one....



Note that the velocity at the center is not a lot greater than the average flow velocity, like it is in laminar flow.... Yes, it is greater right at the center.... but the much flatter profile across the bore, radially, would mean that the pressure distribution across the pellet would be much more even.... After all, pressure is only the force on the surface (in this case the moving pellet) caused by the collision of the molecules of air against it.... The pressure gradient should follow the velocity gradient, no?....

Bob

[WhatUPSbox?]

Lloyd,
I don't have any issue with using engineering estimates to aid in understanding what is going on in the flow, I just try to keep the boundary conditions separate, since those tend not to change as the engineering estimates mature.

Bob,
I'm not sure the velocity profiles you show are applicable. The pellet obstruction flattens the velocity profile. Between the wall where the boundary condition is zero velocity, and the outer edge of the pellet where the velocity should be of the pellet, there is a complex transition. Might be worth searching to see if it has been modeled for other applications.

[Scotchmo]

We knew what you meant, Lloyd.... even if it's not what you typed.... *chuckle*….

I think that the concept of a large, parabolic velocity gradient radially across the bore may be overrated.... While that occurs (2:1) for laminar flow, it is almost a flat gradient until you reach within a few thou of the bore for turbulent flow.... Since the flow would be tripped to turbulent right at the valve, and once turbulent never again becomes laminar.... I think the velocity gradient would look more like the lower one in this diagram than the upper one....



Note that the velocity at the center is not a lot greater than the average flow velocity, like it is in laminar flow.... Yes, it is greater right at the center.... but the much flatter profile across the bore, radially, would mean that the pressure distribution across the pellet would be much more even.... After all, pressure is only the force on the surface (in this case the moving pellet) caused by the collision of the molecules of air against it.... The pressure gradient should follow the velocity gradient, no?....

Bob

"...Yes, it is greater right at the center...."

Here is that diagram with both (laminar and turbulent) scaled to have the same pellet velocity.



Hardly flat. And assuming that the pressure variation follows, we should remember that as you move toward the barrel axis center-line, the area of each subsequent pressure annulus is decreasing. The larger area outer annuli do not get the benefit of the highest pressures. If you assume that the pressure at the outer annulus is the same as the inner, does it matter? It probably does not make much difference for lower velocities (subsonic?).

[lloyd-ss]

Sorry, I just noticed the other replies as I was getting ready to post this, so some of my response might seem redundant.
------------------
What about the possibility of turbulent flow, which will have higher and lower velocities? The air that is immediately exiting the Valve > T-Port > Barrel Port, will be turbulent. That should be a valid assumption, I think. Will the air then be able to transition from turbulent flow to a laminar flow before it reaches the muzzle? Because this is a closed system, with one of the boundaries (the pellet) accelerating away from the pressure, how similar to an open flow pipe is the gun barrel?
I go back to the air piston/cylinder analogy. With a slow moving air piston, there is very little pressure or velocity differential throughout the system. As the resistance of the piston decreases and the velocity of the piston increases, the differentials chance rapidly in the matter of a few milliseconds.
If we were talking about a slow moving air piston I would be firm in a commitment that the velocity of the air immediately behind the piston was uniform, and the same as the piston. When talking about a pellet moving at 1000 fps, I am not sure at all.

[rsterne]

I agree, Lloyd.... I am not aware of any mechanism whereby turbulent flow can revert to laminar, but that doesn't mean there isn't one.... IMO the shape of the leading edge of the velocity profile in subsonic flow behind a pellet will be exactly the same shape as the base of the pellet.... Further back along the barrel, I have no idea, but would bet it is something like that shown for turbulent flow....

Bob

[Sfttailrdr46]

Jeez!!!! there is more than 2 of them. David

What did you expect ??  Every time I read these "Geek Squad " threads I spend several hours doing Google research to better understand what these guys are saying and sometimes I scare myself when it starts to make sense

[lloyd-ss]

Jeez!!!! there is more than 2 of them. David

What did you expect ??  Every time I read these "Geek Squad " threads I spend several hours doing Google research to better understand what these guys are saying and sometimes I scare myself when it starts to make sense

You got that right! There is so much info on the internet that I can pretend like i know something, LOL. But a serious filter is needed to separate out the "good" info.
Lloyd

[Machinist]

So.....

There are powder burners that see 40,000 psi or more, don't know what the highest pressure is.  Anyway, I was wondering what is the highest pressure that air can be compressed to? 

Would the barrel know the difference between 40,000 psi of hot, burning gas, or super pressurized air?

Steve

[Scotchmo]

So.....

There are powder burners that see 40,000 psi or more, don't know what the highest pressure is.  Anyway, I was wondering what is the highest pressure that air can be compressed to? 

Would the barrel know the difference between 40,000 psi of hot, burning gas, or super pressurized air?

Steve

The expanding gases in a powder burner are very hot (high temperatures). The stored air for an airgun is assumed to be at room temperature.

Aside for the challenges of building a 40kpsi vessel with a sufficient volume - 40kpsi would likely allow for more FPE in a subsonic rifle. But I'm not sure it would be any benefit when trying to approach the theorhetical maximum velocity of air. You would be well into the supercrictical region. Adiabatic expansion of the 40kpsi air beyond a couple of multiples would cause a significant drop in temperature. Would the temperature drop be enough to initiate condesation of the air into liquid oxygen or liquid nitrogen? Wouldn't that limit any further expansion in an adiabatic process? I don't know - just thinking of some possible outcomes.

[Nvreloader]

There are numerous PB calibers that have a MAP of 60,000+ psi values,
the max psi for a standard 25-06 is right at 62,000+ psi,
the standard 06 runs at 60,000 psi,
the standard 30-30 is right at 42,000 psi.

Standard 357 Mag pistol runs at 35,000 psi,
the lowly 22 LR runs at 29,700 psi.

Just to name a few,
but we are an Airgun site and have to deal with less than 35/4000 psi, sadly     etc.

Wouldn't it be nice, if we could get into the 20,000 psi realm?
Just think of what we do................Lloyd & Bob would be unstoppable.  LOL 

Tia,
Don

[Nvreloader]

Here comes a dumb questions, for you enablers/geeks.

I have read this post and don't remember see any info on it,

What effect does the the air column, inside the bore, ahead of the pellet/bullet,
have or come into play, with the maximum fps etc.   

This air column has weight and provides resistance when shooting the pellet, correct?

Tia,
Don

[rsterne]

I addition, with air that densely packed, the simple Boyle's law relationship between pressure and density goes out the window.... If you take air at 6,500 psi and compress it into half the volume, instead of the pressure being 13,000 psi, it is over 40,000.... When you expand it, the opposite occurs, the pressure drops extremely rapidly as it expands, reducing the power available to drive a projectile.... Simply put, 40Ksi air is not going to give twice the power of 20Ksi air.... you are working with (quickly) diminishing returns at such high pressures....

Bob

[Wildcatter]

17 pages of this.  My head exploded about 14 pages ago....

[rsterne]

Don, that air creates what is called the "nose pressure", which is only a very small factor in the overall performance.... This is because you have hundreds or thousands of psi behind the bullet, and only 15 psi in front to start with, and maybe several times that just before the bullet exits the muzzle.... The mass of the air in front of the bullet that must be pushed out of the way is tiny, so little energy is expended overcoming it....

Bob

[Machinist]

"Wouldn't it be nice, if we could get into the 20,000 psi realm?"


That kind of pressure wouldn't worry me at all...… as long as I'm in New York and the gun is in Nevada.

Steve

[DomingoT]

Since the discussion is hovering about "pressure"...
More pressure, more max velocity? This is a wonderful example of how intuition doesn't always take you to the right place.
Several posts earlier, I posted the max theoretical velocity a pellet can achieve, as
2 * Speed of sound in reservoir / (gamma -1)
This formula is an unconstrained limit that can never ever by surpassed. It is also very hard to come close to, at least on Earth.
If  you look at this formula carefully,  you may find it disturbing.  The reason is that the pressure doesn't appear in it. You may argue that the pressure appears indirectly, because for a "real" gas, gamma depends on the pressure. But even if gamma didn't change with pressure ("ideal" gas), you would expect the formula for the max velocity of an air gun to have the pressure in it. But it doesn't. Why?
Imagine a barrel with a volume of compressed gas at the breech end, and a pellet keeping the gas in place. Now you release the pellet, and the pellet accelerates down the barrel. Assume the pellet weighs almost nothing, and the outside environment is vacuum.  Assume there is no friction, the gas has no viscosity, and there is no heat transfer. Under these conditions, the pellet is pushed down the barrel by an expansion wave. Behind the pellet, there is a column of gas that is expanding, with the front end expanding more quickly than the rear end.  The pellet will travel, therefore, with the velocity of the front of this expanding wave. This velocity is given by the formula above. The gas cannot move any faster (the front of the expanding wave is the fastest it can go, that is why it is the FRONT of the wave!), therefore, the pellet cannot be accelerated to a velocity higher than the one I quote.
Why is it that the front of this wave will NOT move any faster if you RAISE the pressure of the volume of gas BEFORE you release the pellet? The higher the pressure (keeping the same temperature), the more dense the gas is. Nature has arranged things in a way that no matter how much you raise the pressure, the kinetic energy tied to the higher density behind the wave front compensates  for any increment in velocity the front end of the wave could have due to the higher pressure. The speed of the front end of the wave reaches a limit - that limit is the formula I indicated above.
If you impose constraints, such as the barrel having a given length, the pellet a given weight, etc., things change. The formula for the max theoretical velocity "under those constraints" becomes a bit more complicated and depends explicitly on the pressure.

[rsterne]

So according to your formula for the maximum velocity, which is typically about Mach 5, the pressure, caliber and barrel length are all unimportant (they aren't after all, in the formula).... All that matters is that the pellet have zero mass and surf along the front of the expansion wave, and the whole thing be surrounded by a vacuum.... This is the maximum velocity that can be achieved by a "gas gun", and that depends on the speed of sound for the gas used.... Hydrogen, having the highest speed of sound, can produce the highest velocity, which is why it (or Helium) is used in "light gas guns" to reach near orbital velocities for testing micrometeorite impacts, etc.... They additionally use compression of that gas to raise the temperature and pressure, using a piston typically driven by a "controlled explosion"....

Let's backtrack to the title of the thread, where we are considering Pre-Charged Pneumatic airguns.... Typically we are using these on Earth, surrounded by an atmosphere and gravity, and therefore constrained by the practicalities of materials science which limits pressure, caliber and barrel length in some way.... Respectfully, pressure, caliber and barrel length all enter into the potential energy the PCP can produce.... and for a given FPE the velocity is then inversely proportional to the square root of the mass of the pellet.... If you get the pellet light enough, and with minimum bore drag, it is virtually "surfing" down the barrel on the leading edge of the expanding column of gas supplied by the reservoir.... That gas itself has mass as well, and absorbs some energy from the system....

Given the above constraints, can you give us a more realistic formula or number for the maximum velocity of a PCP ?.... because we all know that Mach 5 is simply out of the question.... Further, if you personally wanted to test that maximum, how would you build a PCP to produce the maximum possible velocity using compressed air at room temperature as your power source, and no vacuum applied to the barrel?....

Bob

[lloyd-ss]

Domingo,
 I have taken a first pass at reading your paper, Internal Ballistics of PCP Airguns, and found it quite fascinating. Several more readings will be needed to become comfortable with the information. A few of the stand-out points for myself were the mass flow discussion regarding the controlling/limiting of mass flow  through the restriction points at either end of the transfer port; the concept of uniform air density in the barrel behind the pellet; the quantifying method for dealing with the acceleration of the air mass behind the pellet; and the temperature gradients and "hot spots" in the barrel and ports. I had not considered that there could be hot spots within the system, but it does indeed make sense.   
A lot to absorb! Thank you for sharing this information.
Lloyd

[ss23]

If the Generals could calculate and predict were the shells would land and hit, they wouldn't need forward Artillery spotters!

[DomingoT]

@ Bob,
It isn't that pellet mass and barrel length don't matter. They must be viewed as "constraints", such that the velocity is "bounded" by the theoretical limit. In a way, the theoretical limit is like the speed of light, which has its greatest value in vacuum.  It doesn't mean the medium the light travels through doesn't matter, it just means the vacuum value is a reference that can never be surpassed. A theoretical limit is valuable because a transgression would mean you violated at least one the laws of nature.
I am looking into getting a more general formula in a friendly format