Acceleration and velocity of a pellet inside a PCP - internal ballistics

[lloyd-ss]

Here is the math geek stuff of what is going on inside a PCP. I did a similar post a while back, but this one I hope is better organized and explains things more clearly.

Here are a few graphs that tell a story and illustrate what is probably happening inside the barrel of a PCP. These graphs are from a model I have been developing for over 5 years, adding variables, tweaking, and refining.  Questions and suggestions for improvements have come from a handful of people, in particular Bob Sterne.  Thank you Bob. The spreadsheet model matches real-world data fairly well and can be used to predict how modifications to a PCP might impact its performance.

These 8 graphs use a .25 cal 20" barrel operating with a 3000 psi air supply, but the model will work with anything. They start with some raw inputs and each graph adds a level of refinement to the example.

The first graph uses a one half grain pellet, with absolutely no losses in the system for friction or flow or anything else. This is to demonstrate the theoretical max velocity if there were no constraints.  It is an impossible case, but we will work back to reality from here. I have graphed the velocity in FPS in green, labeled on the right vertical axis. The acceleration is the red line, values on the left axis. The horizontal axis is the length inside the barrel as the pellet travels through it.  Because the mass of the pellet and the force acting on it IN THIS EXAMPLE are constant, the acceleration is constant (slight drop because the reservoir is actually 2500ccs, not truly infinite). 3000psi acting on a .25 cal pellet is actually pushing the pellet with 147 pounds of force.
The numbers are pretty ridiculous. Read the values from the appropriate vertical scale. (the kink at the top of the velocity curve is just because of the increasing increments in the calculations because the velocity has gone through the roof). (Remember, an unlimited air supply and zero system losses.)



OK, so we know that the theoretical value is crazy, so let's add in the weight of the column of air that is behind the 0.5 gn pellet. That makes a tremendous difference as the weight of the column of air increases linearly with the pellet travel. Atmospheric air has little weight, but compress it to 3,000 psi and it is 200 times as dense! The analogy of the weighted sleds used in tractor pulls, where the weight of the sled is transferred from the wheels to the skid as it is pulled down the course, comes to mind. (Note- it looks like the accel drops close to zero, but that is only because of the expanded scale of the accel values.)



In this third graph, we have the same setup, except with a 25.4 gn pellet with an unlimited (2500cc) supply of 3000psi air, but again NOT adding in the weight of the column of air. Making the pellet 51 times heavier brings the numbers into something that is fathomable. Still way too high, but heading in the right direction.



In this 4th graph, I have again added the steadily increasing weight of the column of 3000psi air accelerating behind the pellet. I have calculated the column of air moving as a single entity with the pellet, so the air in the reservoir is accelerating itself and the pellet. Not exactly real world, but it is what I have modeled at this time. Looking at the velocity, this is actually starting to get believable.



In this fifth graph, I have closed the valve when the pellet is half way down the barrel. You can barely see the inflection point in the acceleration curve when the valve shuts at 10" of travel. But the air trapped between the valve and the pellet keeps expanding and accelerating the pellet. Also, at that time, the weight of the pellet and air column becomes a constant.
So it is no big surprise that the pellet lost very little velocity as a result of shutting the valve at half travel.  Think about that. Shut the valve half way down the barrel, use half as much air....
The velocity is still too high because I have not yet subtracted any deductions for the many losses within the system, such as: the pellet breakaway force, any barrel friction, any flow losses, or any general system inefficiencies, but that is for another time. I think this model is narrowing in on what the empirical (experimental) data shows.



This next graph gets even closer to the real world. It has additional variables modeled in: transfer port volume, pellet breakaway and drag, system efficiency, reservoir volume reduced back to 100cc, with the valve still closing at 10".  Getting believable?



This is same as the previous graph, except the valve now closes at 2.5" of travel down the barrel. Not a great loss of velocity, but a tremendous savings in air.   Look at this graph and see what you loose if you chop the barrel 3 inches, and then look at the previous graph and see what a 3" chop would do.  Do you understand why the difference?



This final graph is the same as the one above except it shows the pressure inside the barrel, the velocity, and the energy. Make sure you are reading off of the correct scales. 
Not perfect, but I feel it reflects the real world empirical data fairly accurately. Comments and questions are welcome.  Lloyd-ss


Lloyd-ss
www.airgunlab.com

[Motorhead]

Suffering temporary Brain freeze ..... Must look away   

Dang Lloyd some serious thought going on upstairs in your head !!

Nice data indeed.

[rsterne]

Lloyd has been working on the spreadsheet used to develop these graphs for much longer than I have know him.... It is incredibly complex, but we are getting some insight as to how it grew right here in this post.... You start with the (relatvely) simple physics of how an object accelerates over distance when a constant force is applied (F=ma).... Then you add an input to change the weight of the object (pellet), stir in the increasing mass of the air while the valve is open, add in what happens when you close that valve, and then once it starts becoming a reasonable model, you start adding refinements which degrade the theoretical potential, like the transfer port volume, the pellet breakaway and drag, and an efficiency factor.... I have only had a very few suggestions on refinements, but I'm certainly honoured that Lloyd trusted me with his "baby", and gave me that opportunity....

I would like to comment about the efficiency factor, and the valve dwell.... Most of the inputs can be measured directly (pressure, reservoir volume, barrel length, pellet weight, etc.) or closely estimated.... Once you put those into the spreadsheet, you are faced with what values to use for the dwell and efficiency.... It turns out that if you have some empirical data (ie a muzzle velocity in fps for a given pellet and the amount of air used in psi, volume, or FPE/CI).... that there is only ONE combination of dwell and efficiency that produces a pressure drop in the tank (and hence a value for FPE/CI) that matches reality.... By adjusting those two "unknowns" until the velocity and air use match what you measured, I feel you have a very accurate representation of what is going on inside your PCP (or pumper, or CO2 gun)....

Once you have an efficiency factor that works for "your" gun, then changing the dwell (only) gives a very accurate representation of how the velocity and air use track with each other.... Likewise, changing the pellet weight gives a very accurate prediction of the velocity and energy with the new pellet.... You can also change the barrel length and see the effect on velocity and energy.... Changing the pressure provides good results as well, and if you leave everything else the same and change the pressure and dwell, you can show how things change during the shot string of an unregulated PCP as the pressure drops and the dwell (through self-regulation) increases to keep the velocity relatively constant, producing our beloved "bell-curve".... The FPE/CI output in this case very much tracks with real world results....

I have also found that you can even predict how a caliber change will change the FPE output, although sometimes the efficiency needs to be tweeked to agree with reality once you actually do the conversion.... but it gives you a goal you can likely realize.... More obscure things like how the volume of the air reservoir (or in the case of a regulated gun, the plenum between regulator and valve seat) affect the results can also be explored with relative ease and the results are very accurate.... It even works well (as I recently found out with my .25 cal pumper) for quite small valve volumes.... Lastly, it allows you to see if that latest change you made increases the efficiency of your gun.... as making a gun "breathe" better generally increases the efficiency factor required to make the spreadsheet "balance"....

Overall, I think Lloyd's spreadsheet, which was used to produce these graphs, it a wonderful model of what is happening inside our PCPs (also pumpers and CO2).... It is an honour and a privilege to have been able to use it, and I have enjoyed watching it mature....

Bob

[Cal]

LLoyd

What factors are used in consideration of transfer port volume?

Also,  to  Bob's explanation of efficiency, I assume volumetric efficiency.  Do you feel the starting pressure  at initial Pellet accel. is nearly equal to reservoir pressure?  The graph depicts as much.  i.e. 200Bar

And Greetings,  first post here, though I have been dropping in at times for some while

Cheers

Cal

[Plekto]

Heh.  If you want your head to really break, start doing this for CO2 or firearms.  Adding expanding gasses to the equation is a whiole other mess to deal with. 

[rsterne]

The transfer port volume used should be the total volume from the valve seat to the base of the pellet.... It produces an intantaneous drop in pressure when the valve opens and before the pellet starts to move....

The "efficiency factor" is a collective one, not aimed at any one specific thing, from what I understand.... Lloyd, please correct me if I'm wrong....

This will work for CO2 providing you know the temperature (and therefore the pressure).... and providing there is no liquid CO2 present in the valve at the moment of firing.... If the valve is full of liquid, all bets are off.... Since firearms PRODUCE their own, very hot, gasses, this calculator won't work.... nor will it work for a springer in which any form of combustion is taking place....

Bob

[Bent]

Wow, that is some serious @$%^.... Respect .. wouldn't mind having a copy of that... 

Is the flow of air also included in the chart, the disturbanse in flow through the port and barrel?

It also explains what I have found when I was tweaking, the best result in speed vs. airconsumpsion was having the hammer move shorter with a harder spring, and a higher reg. pressure.
Especially with light pellets... heavier bullets preffered a little less reg. pressure, and a harder hammerblow.

Could it be that this opened the valve a bit longer, giving the heavy bullet time (doesn't accellerate as fast) to move long enough to have "a good shot"?

Sorry for my English.... 

[lloyd-ss]

Thanks for the interest, everyone,

Bob,
Thanks for the kind words and explanation on the usage of the spreadsheet.  You are pretty much spot-on.  Your high level of interest in this has keep it fresh and evolving.

Cal,
Welcome aboard.  Bob's explanation of the transfer port volume is correct.  In application, the instantaneous pressure drop causes less force to applied than the gauge pressure would imply.  The push on the back of the pellet is reduced.  Pellet breakaway force immediately comes into play at its full value and then decreases linearly to zero over a distance equal to half the caliber.  The distance of 1/2 caliber is just something I chose that seemed appropriate.

The efficiency factor shows up in 2 places with 2 totally different meanings.  The output from the spreadsheet has a whole listing of output data including the traditional efficiency in FPE/cuin, (and also barcc/FPE).  The other efficiency factor is for the mechanical system as a whole.  Most mechanical systems have efficiencies in the neighborhood of 40% to 80% (choose what ever value you like) for a collection of a multitude of small losses.  The airgun is the same.  All sorts of flow losses, heating and cooling, valve transitions, etc, etc.  Plus many we don't have a clue about.  So call it a fudge factor or whatever you want.  What I have found over time as this spreadsheet has evolved, is that as I have added new quantifiable force reductions, say pellet drag for instance, the magnitude of the adjustment applied  by the efficiency factor has become less and less. So over the years, the results have gotten better, but there is still a ways to go.

Bentxbetjent,
No, I have not modeled in the flow dynamics, but I am sure they are significant.  That is not within my skill set, LOL, or at least being able to develop it it accurately into excel.

Lloyd-ss

[Cal]

LLoyd.

Thanks for the welcome,  and the response.

[Cal]

I am sure that the following extraction from another form has it's complications,  but wondering how we might explore the topic (and if it could be introduced to the spread sheet of course)

My "gut feeling" is that the actual expansion is neither purely adiabatic or isothermal.... In the first case, there would have to be NO interchange of heat between the expanding air and the surroundings.... In the second, the temperature of the expanding air would be constant.... While I don't believe the latter for a second, I can think of several things that could instantaneously add heat to the expanding air column....

1. The compression of the air in the "transfer port", ie the air lying between the valve seat and the pellet, which goes from 15 psi to (say) 2980 in a fraction of a second, heating rapidly....   [end quote]

TIA

Cal

[lloyd-ss]

Yes, I recognize the text from my math compadre, LOL.  I am all for trying things that will make this model more accurate. 
The heart of the calculator (well....actually it has multiple hearts) is F=ma, or a=F/m.  Different phenomenon either add to the mass or deduct from the force.  These variables are applied as appropriate during the travel of the pellet through the barrel.  So how would the heating in the transfer port manifest itself in the form of an anomaly in the force???  Flesh this out as some sort of function and I will see if I have a way to quantify it in the model.  I may not, but you never know.

BTW,
I have been trying make some chips in the shop all day, but this is awfully distracting!
Lloyd-ss

[Cal]

Lloyd

I know the feeling.  I too had some goals set "in the shop"  But....

I keep a good work shop,  Three good lathes, H/V mill. surface and T&C grinder, press etc.

Plus supporting "stuff", then the wood working assortment.  Hard to get anything done for the options ;-)

[Bill G]

The modeling tool that I have been developing produces very similar results.  Lloyd and I have compared notes and it seems that we get results that are extremely close, but have different starting places.  My model utilizes relationship of pressure and volume applied to know variables to render velocity, as a heart of the calculator.  As Lloyd said, it is made of many hearts.  Also, Bob is very accurate in the observation on knowns.  Using empirical shot strings along with start and end psi, when the calculated and actual are the same, you are obviously on to something. I think that we both use the same approach in the fact that we start with the absurd and figure in reality until it matches begins to accurately depicts results.  Then we change guns, cal, ports, ect... to see if the calculated will accurately depict the calculated.  Most of all, this thing never seems to end.  These calculators are really a sort of comparing/reasoning engine.  A method of using math to prove or disprove the previous series of math.  For me, The biggest problem is to keep yourself in check and not falling victim to confirmation bias.  Lloyd and Bob have been the greatest of help in leading me into the right direction.  They have both enlightened and refreshed me on many approaches but with out teaching me to just regurgitate information (real teachers).   

[rsterne]

to paraphrase.... "the best teachers teach you how to think".... All of this leads me back to what my Dad used to say....

"The man who knows HOW will always be found working for the man who knows WHY".... I always strive to know WHY things work the way they do....

Bob

[PakProtector]

hey-Hey!!!,
I'd like to see the early silly velocity dealt with by applying the average molecular velocity speed limit.
cheers,
Douglas

[Bent]

hey-Hey!!!,
I'd like to see the early silly velocity dealt with by applying the average molecular velocity speed limit.
cheers,
Douglas

Next step helium... 

[PakProtector]

hey-Hey!!!,
I'd like to see the early silly velocity dealt with by applying the average molecular velocity speed limit.
cheers,
Douglas

Next step helium... 

ummmm, no. No need to go for half measures. Hydrogen will do just fine...and with a well purged tank poses no great risk.
cheers,
Douglas

[Bill G]

couldn't have stated it better Bob.

[rkr]

That's extremely interesting. Could you try some more graphs with pellet weights matching .22 and .177 calibers. Also pressure around 2000 psi would be more like the middle of the power curve and closer to operating pressure of regulated guns. If you ever get that spreadsheet finalized it would be a wonderful tool for those who like to tinker with regulator pressures and barrel lengths.

[PakProtector]

From Bob's Hydrogen thread:
"i agree, i tried helium in my mrod once and saw about 23% increase in velocity but, the helium gas molecules being much smaller then oxygen/nitrogen mix (air) leaked out of my pressure cylinder in the rife. "

With the acceleration of the pellet in the bore measured, say a high-speed x-ray camera( and I happen to know where one resides) you can determine the pressure behind it. If that 23% increase is valid( assuming it is for the moment ), then it looks like two things are going on. One, Helium flows through the valve better, and two at the higher velocity it delivers more pressure behind the pellet.

If the air pressure drop at high velocity isn't accounted for, the other numbers will be less-than-accurate. Judging by how well it works, that last bit will generate better numbers...I'll bet heavily that the model will do quite adequately on low velocity running heavy pellets( say less than 700 fps ). I suspect there is some curvature to the valve behaviour as MV is increased that should not actually be there( or maybe its response is linear when it should be quadratic for example ).
cheers,
Douglas