Theoretical Maximum Velocity in a PCP

[rsterne]

I am not aware of hammer bounce, but it's possible, but would not matter, other than for pressure drop in the reservoir.... I am certain that the dwell is enough that the pellet is exiting the muzzle before the valve closes, because I have plotted velocity vs. hammer spring preload, and it is well up on the plateau at 1029 fps with the bullet.... The higher MV of the plastic BB (less time in the bore) would guarantee more than enough dwell to insure the same for it....

I'll delve into your paper when I have the time to properly devote, and a clear head.... thanks for that.... "PHEW".... just had a peek at your paper.... It will take a LOT more than a clear head for me to even BEGIN to understand it.... Perhaps I should just go back to "tinkering" in my shop....  …. Seriously that is a huge amount of work, Dr. Tavella…. thank you so much for sharing it.... 

One thing to consider about the pressure rise downstream of the poppet.... It takes a finite time (in this case about 0.00008 sec.) for the air molecules (travelling at about 1650 fps) to get from the valve seat to the base of the pellet, during which time the pellet is either stationary, or has moved only a fraction of an inch.... When you compare the gap between poppet and seat (at 0.00008 sec. after it cracks) to the size of the air molecules travelling through it, there is no reason for the pressure in the transfer port area to not reach "nearly" full reservoir pressure by the time the pellet has moved 0.1" or so.... I think using the full pressure (as expanded into the port volume) is valid at that point in time (say 0.0001 sec. after the poppet cracks off the seat).... Much later in the shot cycle, when the pellet, and the air column behind it, is moving at hundreds of fps, that would not apply, of course.... and there could be a significant drop in pressure across the poppet.... Once the valve starts to close, and the curtain area drops below the bore area, at some point the flow should choke, resulting in a drop in pressure downstream of the valve seat of ~ 47% (choked flow)…. Since the SOS at high pressure is over 1400 fps, however, for most shots with a PCP, that choking would only occur at the seat (assuming full bore-area porting), and only in the very last stages of the valve cycle.... The result is that the mass flow through the valve would drop off rapidly just before the valve slams shut.... squaring off the last portion of the flow cycle.... If the valve is open until after the pellet leaves the muzzle (and assuming full bore-area porting) that choking should not be an issue for shots with a MV of less than Mach 1 (outside the barrel)….

Bob

[DomingoT]

What you say makes sense. Things would be a lot simpler if the empty space between valve and TP didn't exist!

[rsterne]

I have read through your paper, without so much as looking at the equations.... You did an excellent job of describing what happens inside a PCP, from hammer strike to bullet exit.... The math is beyond me, unfortunately, despite my B.Sc. in Organic Chemistry.... I took Physical Chemistry 3 times, and the Calculus was my downfall.... Conceptualizing orbitals I got, but the math left me far behind....  …. My Physics is limited to 2nd year, and I was in over my head, math-wise, at that.... 

Your descriptions and assumptions are great for a typical PCP, but I wonder what would happen if you apply them to some of the more "far out" guns we are experimenting with.... When I build a PCP to shoot bullets (instead of pellets), I use full bore-area porting throughout.... The valve exhaust port and transfer port are caliber diameter, the annular valve throat area exceeds that by about 10%, the inlet area into the valve is at least as large as the throat diameter (sometimes much larger), and the barrel port is typically about 80% of the caliber wide x 120% long, so the area is again equal to the bore.... I use a flat-nosed, retracting bolt that after loading the bullet ahead of the barrel port, retracts to level with the back of it.... With this arrangement, all the passages are bore area, so effectively I am starting with the bullet part way along the barrel (about 1.5" from the valve would be typical)….

I have found that if you increase the hammer strike far enough, at whatever pressure, you will reach a velocity plateau where additional hammer strike only wastes air, because the pellet has left the muzzle....



Increased pressure, or to a lesser degree increased projectile weight, requires more hammer strike (preload) to get to the plateau, of course.... For a regulated PCP, tuning to a velocity 3-5% below the plateau velocity (which I call the "knee" of the curve) yields the best balance between power and efficiency.... Here is a plot of efficiency vs. hammer spring preload for a typical regulated PCP....



The other thing I would be curious about is how your formulae would work for predicting hammer bounce for the SSG (Stopping Spring Guide) I have championed over the last couple of years.... The concept is not mine, but I brought it into the mainstream in this thread.... https://www.gatewaytoairguns.org/GTA/index.php?topic=102095.0 …. The idea is that the hammer spring is installed on a spring guide which stops the spring travel just before the hammer strikes the valve stem.... Here is one of several versions possible....



A typical setup would use a longer than normal spring with a lower spring rate, set with about 3-5 lbs. of preload.... The gap between the hammer and the end of the spring guide is adjustable via the (red) hollow bolt, which basically changes the cocking distance, and hence the hammer velocity when it leaves contact with the spring and flies "free" before hitting the valve stem.... Properly designed and set up, this is easier to cock than a preloaded spring, and when the hammer is thrown back by the closing valve, it hits the end of the spring guide, which requires that 3-5 lbs. of force before moving and storing any energy in the hammer spring.... The result is an almost complete elimination of hammer bounce that is strong enough to open the valve a second (or third) time.... and hence a great savings in wasted air.... Basically, the hammer just rattles around between the end of the spring guide and the valve stem on rebound....

I have many more questions, but this is a start.... BTW, for simplicity, we model the hammer as travelling with the valve stem (instead of repeated collisions with it)…. With the new MDS (impregnated nylon) hammers I wonder if that may actually be the case?....

Lloyd Sikes, co-owner of this "Workshop Gate" has built some very unique PCPs and test guns, the most unusual is the gun which he used to first break 2000 fps, found in this thread.... https://www.gatewaytoairguns.org/GTA/index.php?topic=102604.0 …. The gun he used in that testing has no "transfer chamber", there is full pressure against the "pellet" before and at the instant of firing.... You may find it interesting.... When I built my 6mm over the winter, I just had to try it with an airsoft BB, and was pleasantly surprised that with a conventional layout I averaged 2092 fps for 3 shots.... https://www.gatewaytoairguns.org/GTA/index.php?topic=128036.320 …. Reply #324....

Lloyd is the author of the Internal Ballistics spreadsheet we use to model PCP performance.... He is just in the process of rewriting it, to take into account gas Density (VanDerWaals correction) and mass flow.... The goal is to try and get the "fudge factor" currently required as close to 100% as possible.... I have a feeling your paper will send him scurrying back to the drawing board.... 

Bob

[Scotchmo]

Scotchmo,
In fluid dynamics, "stationary" means steady-state, it doesn't mean quiescent - it is part of the jargon. It comes from the idea that if the flow is in steady state and you look at the streamlines, the streamlines don't move, even though the flow itself moves.  Like any jargon, the word "stationary" entails a bit of idiomatic abuse.
The flow in an airgun is not steady-state, and that is an essential difference in that total enthalpy behaves very differently in steady and unsteady flows. As I said earlier, I have seen experienced people trip on this one.
Fanno flow is steady-state, "one-dimensional" flow. You may be thinking, I suppose, that it has a radial component because Fanno flow refers to flow in a duct with wall friction, and the wall friction induces a radial component of movement.  But Fanno flow refers to a one-dimensional "equivalent" of duct flow with friction - there no radial component. The streamlines of Fanno flow and the streamlines of flow in a duct with friction do NOT coincide - even though both have the same cross-sectional mass, momentum and energy fluxes.
That said, some people may refer to Fanno flow "quasi" one-dimensional, but this is a misnomer; there are no "radial" dependencies in Fanno's relationships.

Understood, so the next points of concern with your velocity limit equations:

You said that the velocity limit is determined by:
Vmax = SOS x 2/(k - 1)

For 500 psi air at 70F, SOS = 1142fps, and k(gamma) = 1.461
Vmax = 4995fps

For 4500psi air at 70F, SOS = 1548fps, and k(gamma) = 1.744
Vmax = 4161fps

Those are very high numbers, even exceeding the highest velocity powder cartridges.  And it seems counterintuitive to say that the higher pressures will have a lower velocity limit.

For the Fanno derived equations, we get velocity limits that seem more reasonable:
500psi, 70F
Vmax = 2639fps

4500psi, 70F
Vmax = 2972fps

[DomingoT]

The theoretical limit is what would happen under ideal conditions, which cannot be implemented in real life. The fact that another formula yields a more realistic value doesn't make it right. If you divided your formula by the cubic root of 2, for example, you would get an even more realistic value.
Some people assume the max velocity is what you would get if you converted the stagnation enthalpy of the gas in the reservoir into kinetic energy of the gas. For air, this would give you about 2,500 ft/s as the limit. This value "looks" very reasonable, but is incorrect.

[DomingoT]

Bob, if I understand your sketch correctly, you want the guide+spring to dissipate the hammer energy on the rebound.  For this to happen you need friction or material that dissipates, rather than store, energy on impact.  I am not familiar with the impregnated nylon material you mention. Aluminum and copper have low restitution coefficient.

[MJP]

More of a theoretical type of guy?
The spring slows the hammer down on rebound and not exerting enough force to open the valve second time.
I'm a little confused on your maximum velocity theory, how can less pressure have greater theoretical maximum?
Doesn't that go against everything that is teached at school?

Marko

[Scotchmo]

...
I'm a little confused on your maximum velocity theory, how can less pressure have greater theoretical maximum?
Doesn't that go against everything that is teached at school?

Marko

The max velocity theory that shows higher pressures giving lower max velocities is not mine and I don't currently accept that theory, but - I see how variation in k(gamma) with respect to the starting/ending temperature and pressure might give what seems like a counterintuitive result. When k is not constant (and maybe not even linear), strange things happen.

[MJP]

The comment was more directed to Domingo.  You just pointed it out Scott.

Marko

[rsterne]

Domingo, the SSG works just fine with a steel hammer and steel spring guide.... although I do think that the small 70D O-ring between the back of the spring guide and the gap adjusting nut (red) helps absorb the hammer rebound.... That wasn't it's purpose, it was to quiet the stopping of the guide, and reduce the "shock load" on the parts, but may have a beneficial effect on reducing the hammer bounce as well.... I know it isn't essential, however, because lots of these have been made, and work fine, without it....

The MDS hammer is a separate device, and can be used with or without an SSG.... Some have a metal (steel or brass) nose that strikes the valve stem, others are all MDS.... still others have a metal core to add weight, but the nose of the striker is MDS.... If the point where the sear latches the hammer is MDS, you must be using a trigger with a "drop sear" where the sear falls away on release.... If the sear gradually creeps down the face of the hammer, and drags on the side of the hammer at release, it will quickly destroy the reliable latching of the sear, and the gun may eventually misfire.... Here is a photo of an MDS hammer with steel core, in this case the core extends through to strike the valve, but the sear latches only on the MDS....

Rear view showing steel core, and the recess in the back where the Delrin spring guide pushes... and the clearance hole for the stop rod....



Top/front view.... The sear catches on the slightly tapered front of the MDS filled Nylon sleeve.... Here are the properties of MDS....
 http://kmac-plastics.net/data/technical/mds.htm#.W1Y9CeSWyUk



SSG from the same PCP (my 6mm)…. Small threads adjust spring preload, large threads adjust the gap from guide to hammer.... both 28 TPI.... The white Delrin guide slides back to compress the spring, and drives the hammer.... The steel rod is stationary when firing or cocking.... The threaded gap adjuster is drilled to allow room for the 3" long hammer spring inside it.... In this style of SSG, the small O-ring bumper is at the front, between the white Delrin spring guide and the head on the front of the stop rod....



The hammer has its own cocking handle, it is not cocked by withdrawing the bolt.... It weighs about 1/3 what a steel hammer would.... The grooves in the sides of the hammer are to allow air to move past it, to prevent a positive pressure in front, or a partial vacuum behind, on firing.... That can cause you fits (inconsistent velocity and increased preload required) if you don't realize it's happening.... 

Bob

[lloyd-ss]

Stimulating conversation!  Domingo, your use of the terms stationary and non-stationary flow, and your additional associated descriptions and comments have shed new light on the discussion of internal ballistics of PCPs. The fact that a steady state is never reached, that is, the air is always in a dynamic "entry length" and never becomes "fully developed," is what seems to complicate the math of the PCP.  I have always (well, almost always) know that the the pressure, temperature, and velocity, at all points within the PCP pressure system are always changing..... a continuum of changes. There are never any steady state, fully developed regions within the system. The changes occur both axially and radially within the system.
My spreadsheet, due to my lack of expertise in several areas, treats the calculations as a series of snapshots of "stationary flow", along the axial length of the flow path. Despite the spreadsheet's lack of sophistication, it does provides an approximation of what might be occurring inside the system.  I appreciate your sharing your research paper and hopefully will use your presentation to see what areas of my spreadsheet might be improved upon.

I have a few questions about your early statement of the pellet riding the front of the expansion wave  in the barrel. For there to be air expansion, there must be a pressure differential in the length of the barrel, with the lowest pressure at the back of the pellet, and air flowing toward that low pressure. Is the air velocity the greatest at the back of the pellet? Since this is non-stationary flow, i.e., always in transition, is it true to say that the average air velocity immediately at the back of the pellet is equal to the velocity of the pellet?

I will digress a little here, but I have often envisioned the pellet as analogous to a piston being driven in a cylinder by air pressure.... a piston air motor.  Similar to a single cylinder piston steam engine, without the additional thermodynamics (and power) of the heat energy in the steam. Is that piston air motor analogy reasonable? Thank you.

Lloyd

[Rocker1]

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

[WhatUPSbox?]

stationary and non-stationary flow

Interesting how terminology varies. My old schooling used "steady flow" for time invariant conditions.

is it true to say that the average air velocity immediately at the back of the pellet is equal to the velocity of the pellet?

The back of the pellet forms a boundary condition. The axial velocity of the air contacting the back of the pellet has the velocity of the pellet.

[lloyd-ss]

is it true to say that the average air velocity immediately at the back of the pellet is equal to the velocity of the pellet?

The back of the pellet forms a boundary condition. The axial velocity of the air contacting the back of the pellet has the velocity of the pellet.

Then might we also say that inside the barrel, the pressure VELOCITY (edit) in the barrel the instant before the pellet exits the muzzle is lowest at the breech end and highest at the muzzle, and the pressure is highest at the breech end and lowest at the muzzle, with no pressure or velocity reversal anomalies along the length of travel?

Can this also be taken a step farther and stated that throughout the entire pressure system, the air velocity increases toward the muzzle, and the pressure decreases toward the muzzle, except for localized regions that have physically constricted air passages?

Just trying to nail down some basics.
Thanks,
Lloyd

[WhatUPSbox?]

Lloyd,
The back of the pellet boundary condition is a constraint on the gas behavior at that location and does not in itself imply anything about the rest of the flow. The two statements you make are conclusions to be evaluated. While they both sound reasonable, there may be conditions where there are pulses in the flow that may affect them. The breech pressure measurements George made with the CP-2 CO₂ gun show a generally smooth gross pressure profile.

[Scotchmo]

...
Just trying to nail down some basics.
Thanks,
Lloyd

"...is it true to say that the average center air velocity immediately at the back of the pellet is equal to the velocity of the pellet?"

"Then might we also say that inside the barrel, the pressure velocity in the barrel the instant before the pellet exits the muzzle is lowest at the breech end and highest at the muzzle, and the pressure is highest at the breech end and lowest at the muzzle, with no pressure or velocity reversal anomalies along the length of travel?"

That is what I would say is more correct.

[QVTom]

I think Lloyd was testing us to see if we can actually read and comprehend.  LOL

...
Just trying to nail down some basics.
Thanks,
Lloyd

"...is it true to say that the average center air velocity immediately at the back of the pellet is equal to the velocity of the pellet?"

"Then might we also say that inside the barrel, the pressure velocity in the barrel the instant before the pellet exits the muzzle is lowest at the breech end and highest at the muzzle, and the pressure is highest at the breech end and lowest at the muzzle, with no pressure or velocity reversal anomalies along the length of travel?"

That is what I would say is more correct.

. 

[rsterne]

^X2....

Bob

[lloyd-ss]

Scott, Thank you catching my using pressure where I should have used velocity. My mistake, and I did go back and edit that in reply 313 to correct it.  I am still leaning toward average velocity at the back of the pellet as opposed to center of the pellet.  It is arguable either way, I guess.

Stan, yes, the 2 statements might be conclusions, but if you were a gambling man, what kind of odds would you put on them if we assumed PCPs that did not have any unusually odd or restrictive passageways or characteristics.

Tom and Bob, I was trying to see if I could get down on paper (ok, computer screen) what was in my head. At work I would always print out any of my important emails/documents and give them a final read on paper to catch those last few silly mistakes.  Not getting any younger, so I guess I still need to do that, LOL.

Lloyd

[Scotchmo]

... I am still leaning toward average velocity at the back of the pellet as opposed to center of the pellet.  It is arguable either way, I guess....

Lloyd

For what purpose are you using that assumption?

"average" implies that there is greater and lesser. None of the air that is behind the pellet will be at a greater velocity than the pellet. The average velocity of the air behind the pellet will be less than the velocity of the pellet.