Theoretical Maximum Velocity in a PCP

[TPL]

You know, there is no such limit for gas flow. There are wind tunnels exceeding many Mach.

[Sfttailrdr46]

  I know that at times I know just enough to be dangerous but this time I will ask my foolish question . The pressure equates to thrust pushing the projectile so exceeding MACH is not surprising since we have been exceeding the speed of sound for almost 80 years with various projectiles with expanding gasses just not compressed air out of our air guns or am I all wet??

[Scotchmo]

You know, there is no such limit for gas flow. There are wind tunnels exceeding many Mach.

True. But, the Speed Of Sound at that point in the tunnel may be low. What velocity in fps does Mach 5 equate to? Mach is a ratio, not a velocity.

High pressure gas becomes choked when the outlet velocity equals the speed of sound of the propelling gas. Expansion in the barrel adds additional velocity. Some of that air (at
the front) is at the highest velocity.

I don't care about the SOS of the ambient air. The velocity limit has nothing to do with that. I'm comparing the actual velocity of the projectile/gas to the SOS of the driving gas.

[Scotchmo]

okay , im following.. the gas itself never broke its own mach, rather the projectile pushed by the gas never broke the gas mach speed

Its own mach?

It can break Mach 1. And approach Mach 2, with respect to the gas in the tank. With respect to the external SOS (ambient), there is no limit on Mach ratio, so that ratio means nothing since there is no physical tie between them. The velocity limit still holds.

Foget about the SOS of the air outside. Best to look at the muzzle velocity of the gas/air VS the maximum velocity of sound of the gas in the tank. That is the ratio that matters.

[rsterne]

I think you are losing everybody, Scott, including me.... When you start dealing with Fanno Flow, and Thermodynamics, my eyes glaze over....

There is a maximum FPE that any PCP can reach in theory, and that is Force times Distance.... The force (in lbs.) is bore area times pressure, and the distance is the barrel length (in ft.).... Multiply them together, and you get the maximum FPE that can theoretically be achieved.... However, that FPE has to be split between the projectile and the gas propelling it.... With air, you are lucky to hit 50% of that, and with Helium you can hit about 70%.... Until you start exceeding those numbers, we aren't dealing with magic, or witchcraft, and we don't really need to go beyond good ol' Newtonian Physics to find an explanation of what is happening.... IMO....

Bob

[Scotchmo]

Bob,

FPE?

The topic is the "Theoretical Maximum Velocity in a PCP". You are talking about the theoretical FPE and velocity in a PARTICULAR PCP. Your estimates require caliber, barrel length, projectile mass, etc. For the topic estimate, all those variables drop out of the equation. We are left with maximum flow rate + maximum expansion.

Maximum flow velocity is the SOS of the propelling gas. Maximum theoretical expansion velocity end ups being about that same velocity.

Max velocity = max flow velocity+ max expansion velocity.

That is the velocity that a PCP will never achieve.

Newtonian Physics gives a fairly accurate estimate when modeling, if you account for the flow limit of the inlet. You won't need a fudge factor to account for what appears to be low efficiency.

But for "Theoretical Maximum Velocity in a PCP", I say:

Vmax = SOS x sqrt((2/(k-1))+1)

And I'm sticking to it (at least until proven wrong).

For discussion, we can approximate it and say "about 2xSOS".

[MJP]

And your point being?
I don't see you making any point, 2 times SOS? In what?
In side the barrel?
Gas speed at the valve throat?

Marko

[rsterne]

OK, Lloyd started this topic based on Newtonian Physics, and gave, in the first post, the formula to determine the maximum possible VELOCITY, for any SPECIFIC PCP, based on nothing more than bore area, pressure, barrel length, and projectile weight.... I added a simplified version to calculate FPE, but if you know one, and the projectile mass, you can calculate the other.... It was the thread on Lloyd's high velocity experiments where we were discussing all the ideas about why the 1650 fps "limit" wasn't a limit at all, etc.etc.etc....

Lloyd's formula, if the mass is low enough, will predict velocities well over 4000 fps, for a SPECIFIC PCP, and while you and I, for different reasons, don't think that is possible, it wasn't what the topic was really about.... I would have preferred that we kept the two topics separate, but that's just my opinion.... As I said, I know nothing about Thermodynamics, so I can't participate in that discussion.... I guess I'll just have to take your word for it, unless somebody manages to go faster (which I doubt)....

I will assume, however, that your "SOS" is the speed of sound at the pressure and temperature of the driving gas present inside the barrel?.... That being the case, using your approximation, then doubling the values in the chart below should be the "limit"....



Using 4500 psi at 20*C, that gives about 3100 fps with Nitrogen, 3200 fps with Air, and 7450 fps with Helium.... providing you can get your projectile weight down to zero, of course.... Did I get that correct?....

Now, how do I apply that to determine what a PARTICULAR PCP, running a given gas, at a given pressure, with a given barrel caliber and length, can theoretically achieve?.... I ask because that was the question Lloyd was trying to answer in this thread.... as I understood it....

Bob

[TPL]

Alternative for Newtonian Physics is Quantum Physics. Please keep Newtonian, it is well enough for this.

It's all about energy, not speed. Speed is not limited really but of course going supersonic takes a lot of energy, which we just don't have that much in airgun. You often hear SOS is a limiting factor in AG but it is not, usable energy is.

[Scotchmo]

And your point being?
I don't see you making any point, 2 times SOS? In what?
In side the barrel?
Gas speed at the valve throat?

Marko

SOS of gas inside the dump chamber. And that dump chamber must be infinite volume, or at least very large in order to maintain pressure.

Gas speed is the maximum velocity at the expansion front. That is at the base of the projectile, if there is to be one.

This equation:
Vmax = SOS x sqrt((2/(k-1))+1)

is a reduction of the following equation (circled in red), with V* (throat velocity) approaching M, the result being V. Throat velocity cannot exceed M.



Fanno flow is the flow (and expansion) of a gas in a constant area duct, with friction. That's basically what we have with an airgun barrel. The same velocity limits will apply.

[Scotchmo]

OK, Lloyd started this topic based on Newtonian Physics, and gave, in the first post, the formula to determine the maximum possible VELOCITY, for any SPECIFIC PCP, based on nothing more than bore area, pressure, barrel length, and projectile weight.... I added a simplified version to calculate FPE, but if you know one, and the projectile mass, you can calculate the other.... It was the thread on Lloyd's high velocity experiments where we were discussing all the ideas about why the 1650 fps "limit" wasn't a limit at all, etc.etc.etc....

Lloyd's formula, if the mass is low enough, will predict velocities well over 4000 fps, for a SPECIFIC PCP, and while you and I, for different reasons, don't think that is possible, it wasn't what the topic was really about.... I would have preferred that we kept the two topics separate, but that's just my opinion.... As I said, I know nothing about Thermodynamics, so I can't participate in that discussion.... I guess I'll just have to take your word for it, unless somebody manages to go faster (which I doubt)....

I will assume, however, that your "SOS" is the speed of sound at the pressure and temperature of the driving gas present inside the barrel?.... That being the case, using your approximation, then doubling the values in the chart below should be the "limit"....



Using 4500 psi at 20*C, that gives about 3100 fps with Nitrogen, 3200 fps with Air, and 7450 fps with Helium.... providing you can get your projectile weight down to zero, of course.... Did I get that correct?....

Now, how do I apply that to determine what a PARTICULAR PCP, running a given gas, at a given pressure, with a given barrel caliber and length, can theoretically achieve?.... I ask because that was the question Lloyd was trying to answer in this thread.... as I understood it....

Bob

"I will assume, however, that your "SOS" is the speed of sound at the pressure and temperature of the driving gas present inside the barrel?"

Answer: No, not the barrel, the dump chamber.

"Using 4500 psi at 20*C, that gives about 3100 fps with Nitrogen, 3200 fps with Air, and 7450 fps with Helium.... providing you can get your projectile weight down to zero, of course.... Did I get that correct?...."

Answer: Yes. The equations yields 2970fps. "about 2xSOS" (+/-)

"Now, how do I apply that to determine what a PARTICULAR PCP, running a given gas, at a given pressure, with a given barrel caliber and length, can theoretically achieve?."

Answer: In your spreadsheet, choke the flow at the throat when it reaches Mach 1. And get rid of the efficiency fudge factor.

[Scotchmo]

"There is a maximum FPE that any PCP can reach in theory, and that is Force times Distance...."

Bob,
That is an incorrect assumption.

That is true only until flow becomes choked. With the choke occurring at Mach 1,

< Mach 1:
energy gain = Force x Distance

> Mach 1 (choked):
energy gain = Power x Time

There is a limit on the power that a "throat" can produce.

Once that power limit is reached, the Force downstream from the point where projectile reaches Mach 1 (choke point), will fall off. Energy gain in the system is then proportional to time, not distance.

[Rob M]

"There is a maximum FPE that any PCP can reach in theory, and that is Force times Distance...."

Bob,
That is an incorrect assumption.

That is true only until flow becomes choked. With the choke occurring at Mach 1,

< Mach 1:
energy gain = Force x Distance

> Mach 1 (choked):
energy gain = Power x Time

There is a limit on the power that a "throat" can produce.

Once that power limit is reached, the Force downstream from the point where projectile reaches Mach 1 (choke point), will fall off. Energy gain in the system is then proportional to time, not distance.

COME'on Bob, everyone knows this ! (-'

[rsterne]

Scott, the THEROETICAL MAXIMUM is as I stated.... you can't get more FPE out than you put in.... HOWEVER, various factors, such as a smaller than infinite reservoir, the fact that the gas actually has mass, friction between the bullet and/or gas and the barrel, and a host of others can REDUCE that.... Your equations, which while elegant theory are unproven (in this context), are just part of that group of things that mean we can never get to the maximums Lloyd predicted in his original post in this thread.... and he freely admits that, and so do I.... If they are correct (and I'm not smart enough to understand them), then would not the velocity (eg. 2970 fps on 4500 psi air@ 20*C) only be achievable with a bullet of zero mass?.... In that case the FPE achieved by the projectile would also be.... zero.... all the energy would have to go into accelerating the gas.... As to your answer to my question....

"Now, how do I apply that to determine what a PARTICULAR PCP, running a given gas, at a given pressure, with a given barrel caliber and length, can theoretically achieve?."

Answer: In your spreadsheet, choke the flow at the throat when it reaches Mach 1. And get rid of the efficiency fudge factor.

I haven't a clue how to do that.... Perhaps you can show us?.... In the meantime, I am getting perfectly satisfactory results by using a "fudge factor" of 50% with air and 70% with Helium, of the theoretical maximum predicted by FPE = barrel volume (in^3) x pressure (psi) / 12 (in/ft).... Exceeding that is difficult, and doing it with a bullet of half the FPE (in gr.) is something I haven't yet seen.... Is it a "limit", no (the limit is the FPE predicted by the equation, with no fudge factor).... but it's a darn good indication of a "lofty goal" that if you accomplish it, you can pat yourself on the back....

Please don't misunderstand me, I'm not saying your method is incorrect, or even flawed.... I'm just saying I don't understand it, and I therefore can't use it....

Bob

[Scotchmo]

Scott, the THEROETICAL MAXIMUM is as I stated.... you can't get more FPE out than you put in.... HOWEVER, various factors, such as a smaller than infinite reservoir, the fact that the gas actually has mass, friction between the bullet and/or gas and the barrel, and a host of others can REDUCE that.... Your equations, which while elegant theory are unproven (in this context), are just part of that group of things that mean we can never get to the maximums Lloyd predicted in his original post in this thread.... and he freely admits that, and so do I.... If they are correct (and I'm not smart enough to understand them), then would not the velocity (eg. 2970 fps on 4500 psi air@ 20*C) only be achievable with a bullet of zero mass?.... In that case the FPE achieved by the projectile would also be.... zero.... all the energy would have to go into accelerating the gas.... As to your answer to my question....

"Now, how do I apply that to determine what a PARTICULAR PCP, running a given gas, at a given pressure, with a given barrel caliber and length, can theoretically achieve?."

Answer: In your spreadsheet, choke the flow at the throat when it reaches Mach 1. And get rid of the efficiency fudge factor.

I haven't a clue how to do that.... Perhaps you can show us?.... In the meantime, I am getting perfectly satisfactory results by using a "fudge factor" of 50% with air and 70% with Helium, of the theoretical maximum predicted by FPE = barrel volume (in^3) x pressure (psi) / 12 (in/ft).... Exceeding that is difficult, and doing it with a bullet of half the FPE (in gr.) is something I haven't yet seen.... Is it a "limit", no (the limit is the FPE predicted by the equation, with no fudge factor).... but it's a darn good indication of a "lofty goal" that if you accomplish it, you can pat yourself on the back....

Please don't misunderstand me, I'm not saying your method is incorrect, or even flawed.... I'm just saying I don't understand it, and I therefore can't use it....

Bob

Showing is a tall order. We both did spreadsheets that can model Lloyd's tests fairly closely. Yours needs a substantial fudge factor to get close. Mine does not.

I can't tell you exactly what to do as our spreadsheets are different. I use a distance based increment. I think you and Lloyd are using a time based increment. Each approach has it's advantages.

Here is a screen shot of a late version of my spreadsheet modeling one of Lloyd's test shots.



Rows 1 thru 4 are user inputs and data needed to run. I should probably move that to another sheet. The main spreadsheet uses row 6 as the column labels. I use a "True" flag to mark if/when the projectile transitions from subsonic to supersonic. See where I drew a green oval around cell N17 to mark that point? Column G is the speed of sound in the driving gas. Column R is the projectile velocity. When column R exceed column G, we have transitioned to the supersonic regime. At that point, the mach-1 cylinder/slug of gas starts to expand. It is not obvious in this model because of the relatively small 55cc dump chamber. In the next screen shot, I increased the dump chamber to 10000cc in order to keep pressure more constant, so you can see what is happening.



In that second picture, each cm of barrel in the subsonic regime contains approximately the same mass of air (circled in red). The air flowing in can no longer keep up with the projectile. Subsequent segments of barrel in the supersonic regime contain less and less dense air.

Since you are using a time based increment in your spreadsheet, the air mass per increment will increase until you transition to supersonic. At which point, the air mass, entering the barrel, will be constant for each time increment.

If the fudge factor works for you, use it. But the nice thing about my model is this:

I decrease the pellet mass to 0.1gr
I increase the dump chamber to 100,000cc
remove the fluid friction factor

With a 3000 ft barrel, velocity is over 2900fps and rising very,very slowly with each increase in length.

If I extended out the barrel for miles, I would likely approach the 2972 velocity predicted for 4500psi air.

Vmax = SOS x sqrt((2/(k-1))+1)
That is the absolute limit that you will never reach.

Edit: If you are trying to figure out the rest of the spreadsheet from the pictures, don't bother. Some of the columns in the pictures are from older what/ifs and no longer apply to the supersonic regime. Some of the additional temp K columns are monitors that I used earlier to see if things would adhere to the RMS maximum velocity theory.

[rsterne]

Actually, the only spreadsheet that models Lloyd's tests quite well is Lloyd's, not mine.... and yes, it requires a significant fudge factor.... I won't even pretend to understand your spreadsheet, because I don't.... That doesn't mean it isn't 100% correct, however....

My spreadsheet is far simpler, and is intended to only do what was started in this thread.... predict with some degree of repeatability what can be achieved by real world PCPs.... starting from very few, simple variables.... It deals with FPE, not velocity, and the further the projectile varies from the norm, the poorer the results.... That is where Lloyd's is far superior, because it takes into account the mass of the propelling gas, which is a larger and larger percentage of the total FPE as the projectile gets lighter.... Think of mine as a boiled down, practical, meat and potatoes kind of spreadsheet.... not so good with wine or dessert....   

I don't think I can really offer any more at this time....

Bob

[jhm757]

Wow! Reading this is giving me brain pain!

I don't have a clue! Wish I did but I don't!

Jim - jhm757

[Sfttailrdr46]

Wow! Reading this is giving me brain pain!

I don't have a clue! Wish I did but I don't!

Jim - jhm757

I enjoy the information and occasionally after reading several times it begins to make some sense but the physics has always made my brain hurt and my eyes cross 

I am also married to an RN so I am frequently in way over my head when I sit and listen to her conversations with other nurses and doctors so I am getting used to having a puzzled expression on my face and a dull ache behind my eyes 

[Scotchmo]

"There is a maximum FPE that any PCP can reach in theory, and that is Force times Distance.... The force (in lbs.) is bore area times pressure, and the distance is the barrel length (in ft.).... Multiply them together, and you get the maximum FPE that can theoretically be achieved"

Assuming we ignore air mass, that actually works perfectly. But only until flow becomes choked.

Example:
imaginary gas = 2000 psi mass-less gas with a SOS of 1000fps
caliber: .357
projectile mass: 45gr
barrel length: 24"

What is the FPE? velocity?

force = 2000psi x pi x (.357/2)^2 = 200 lb.

FPE = 200 lb. x 24"/12in/ft = 400 ft-lb

In the first foot, the projectile reached 200fpe. In the second foot, it gained another 200fpe.

velocity = sqrt(400/(45gr/450436)) = 2000fps

That would work perfectly, except that the flow becomes choked at SOS of 1000fps.

After 1000fps, the FPE gain is no longer 200 lb. x distance.

The FPE gain after 1000fps is power x time.

power(choked) = ft-lb/s = 200 lb. x 1000fps = 200,000 ft-lb/s

The problem transitions from constant force to constant power: given a constant power, how much time does the projectile take to cover the last 12" of barrel length?

In the first foot, the projectile reached 200fpe. The projectile needs another .001 second in order to gain 200 more ft-lb. At super sonic velocity, it will take less than a .001 second to cover the second foot of barrel. So FPE will be less than 400 ft-lb.

Some calculus or numerical integration is needed to solve that. Numerical integration allows us to incrementally add in the air mass and have a more realistic model.

The whole gist of what I've been trying to get across (here and over on the yellow forum), is that we can look at it as two distinct segments. The subsonic regime (constant force), and the supersonic regime (constant power). And then add them together to get the final solution.

[rsterne]

NT - see next post....

Bob