Author Topic: FPE Predictions  (Read 1573 times)

Christopher

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FPE Predictions
« on: February 09, 2017, 03:14:20 AM »
Bob,

Always enjoy you predicting the energy outcome of airguns that people or yourself build and how close you get to what reality is.

Travis' new .357 Flex testing got me thinking and didn't want to clutter up his thread so I posted this question here, the question was for you anyway.

Let's take Travis .357 Flex for example. Bore(.2) (x) pressure (2600)=520 FPE (Theoretical max) I know that 50% of that is what you call lofty goals.

Now here's the question: How does the porting, in this case Cothran valve with .25 exhaust port, play into this equation? .25 is about 70% of .357, but a .357 caliber has roughly twice the bore of a .25 caliber so that is 50%.

Theoretical max (x) 70%=364fpe, Travis' gun is shooting close enough to say 150 fpe so that is eff. #'s of 41%, that's pretty good.
Theoretical max (x) 50%=260fpe, that comes out to about 58%, that's unheard of.

Both may be wrong, just need some direction here ::) Once again, it's probably not as simple as I think it is but looking forward to your answer.

Thanks again,
Chris




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rsterne

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Re: FPE Predictions
« Reply #1 on: February 09, 2017, 02:11:28 PM »
The formula for the maximum theoretical FPE from any PCP goes right back to Force (lbs.) times Distance (ft.) equals Energy (FPE).... The Force is the bore Area (sqin) times the Pressure (psi).... In the case of Travis' new .357, with a 24" barrel at 2600 psi, we get....

(0.357 x 0.357 x PI / 4) x 2600 x (24 /12) = 0.100 x 2600x 2 = 520 FPE.... However, to accomplish that, you would have to have an infinitely large reservoir (to get constant pressure throughout the shot), and 100% efficiency (no losses).... In addition, part of that 520 FPE goes into accelerating the mass of the air that is pushing the bullet, so there is a loss we can do nothing about.... This is also the reason that Helium can provide more usable FPE in the bullet than air, because it has lower mass, so more of the available energy can end up in the bullet....

I have a spreadsheet, based on the above, that uses half of the theoretical maximum to predict a "lofty goal" for PCPs on air, to take into account all the various losses.... I have never yet found a gun that will reach that 50% level unless it is shooting a very heavy bullet.... but to remove that from the variables, I now qualify my "prediction" by adding "when using a bullet where the weight in grains is half the predicted FPE".... That means 950 fps, so the 50% goal is at that velocity.... Typically, the very best of our custom guns need either a bit longer barrel, or a bit more pressure, to reach my "lofty goal" FPE at 950 fps.... If you get within 10% of the goal, you have done a pretty good job, IMO....

Using 50% of the 520 FPE absolute maximum, we get 260 FPE with a bullet weighing 130 gr.... To get there, however, would require using bore-size porting throughout (ie all ports, the valve throat etc. 0.100 sq.in in area).... While there is no hard and fast math to allow calculating the effect of smaller porting, I have come up with a modification to my spreadsheet that agrees quite well with empirical data.... Instead of using the (bore squared) to calculate the area, I use the (bore diameter times the smallest port diameter).... Can I offer a scientific / mathematical argument for this?.... No, but the units work out (area is still in sq.in), and it works very well in practice.... For me, that is good enough....

Applying this correction to Travis' gun (which has a 1/4" transfer port), instead of using 0.100 sq.in. we would use.... (0.357 x 0.250 x PI / 4) = 0.070 sq,in.... Using that, times 2600 psi, times 2 ft.... and derating it by 50%.... we get a "lofty goal" of 182 FPE using a 91 gr. bullet.... So, we're getting closer.... We have one other factor to consider.... The gun is regulated, and has only an 80 cc plenum.... I did the math, and with a 2600 psi setpoint, and the 1.3 FPE/CI efficiency Travis reported, the effective AVERAGE pressure available was only 2430 psi.... Use that instead of 2600 psi, and we're down to 170 FPE with an 85 gr. bullet for a "lofty goal".... Travis got 150 FPE, which is within spitting distance of 10% of the goal.... I would say you can't really expect a lot more.... Incidently, Travis asked what he could expect with a larger plenum, so I'll answer that here.... If you double the plenum to 160 cc, the AVERAGE pressure would be 2515 psi.... which works out to a lofty goal of 176 FPE.... The larger plenum should gain about 6 FPE at the same setpoint....

HTHs....

Bob
« Last Edit: February 09, 2017, 02:15:34 PM by rsterne »
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Christopher

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Re: FPE Predictions
« Reply #2 on: February 10, 2017, 02:12:50 PM »
Yes sir, that helps out tremendously. I appreciate the response.

There was something I didn't understand though about calculating the avg. air in the plenum. I understand that air pressure will drop during the shot cycle due to the cycling of the valve will "outrun" the response of the regulator. Just wondering how you got to numbers. I'll lead you through my archaic, and possibly wrong, way of thinking.

Again I'll use Travis' .357 Flex as an example.
80cc plenum set at 2600 psi...that's 4.88ci of plenum capacity at a pressure of 179.3 bar which gives us a total of 875 ci of air. The avg. of the shot string was 903 fps, which is 146.7 fpe, at an impressive eff. of 1.3. So it is taking almost 113 ci of air per shot. So, 113/4.88=23.15 bars of pressure drop multiplies by 14.5=335 psi....2600-335=2265, giving us an average pressure in the plenum of 2430, just like you said. I guess the real question isn't how you calculate it but rather why do you use the avg. pressure? Not implying you're wrong, just wondering.

Continuing on...could this info be use to figure out the precise time, probably more accurately, the place that the valve closes?

That means (I think anyways) that when the valve closes the pressure from the base of the pellet back all the way to the valve stem the pressure is 2265 psi....that's a total of 156.2 bar of pressure, with 113 ci of air....which would indicate that all that air in 1.38 ci of space, in this case the barrel, TP, and what little room that is in the valve.

.357 has a bore area of .1 in squared, so the pellet is 13" (1.3ci of capacity), approximately, down the barrel (leaving the other .08ci for TP and valve, just a guess) when the valve closes.

I find all of this just fascinating and understand probably just enough to sound like an educated idiot ::). But I do know a short dwell time has much to do with efficiency and report of the airgun as well.

I thought that I would meander just a little, but is my thinking anywhere close on the valve timing?

Thanks again,
Interrogating Chris
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rsterne

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Re: FPE Predictions
« Reply #3 on: February 10, 2017, 03:06:11 PM »
Your method of predicting the pellet position at the moment of valve closure is, to say the least, unique, but very intuitive.... I think, however, you made a miscalculation somewhere.... Using your numbers, 113 CI of air at 156 bar occupies (113 / 156) = 0.723 CI of space, not 1.3 CI.... I think you simply inverted the calculation....  ??? .... Using that 0.723 CI, and carrying on with your logic, and using your 0.08 CI estimate for the wasted volume of valve,transfer port, and barrel chamber (pretty good guess),  that leaves us 0.643 CI of air in the barrel, which at 0.1 CI per inch works out to 6.43" down the barrel for the pellet when the valve closes....

I use Lloyd's proprietary PCP Internal Ballistics Spreadsheet to give a very close estimate of the valve dwell.... There are inputs for just about everything, but two are left for you to play with by "trial and error" to bring the spreadsheet into agreement with the experimental results you are trying to model.... Those are the valve dwell, and a "fudge factor" for the overall efficiency, which usually runs from 55-75%.... There is only one combination of dwell and efficiency that will duplicate what the gun did for velocity and FPE/CI....

Some of the inputs you have to estimate, but fortunately those don't make much difference to the results.... These include the transfer port (wasted) volume, pellet drag down the bore, and starting friction.... Most of the other inputs are known quantities, such as reservoir (or plenum) volume, caliber, bullet weight, pressure, barrel length, etc.etc.... You are absolutely correct that in general terms, the earlier the valve closes, relative to the barrel length, the higher the FPE/CI.... If the valve closes after the pellet has passed halfway down the bore, the efficiency starts to drop like a stone, and of course after the pellet exits the muzzle, any further dwell just wastes air completely....

So, I ran the numbers for Travis' regulated .357, and here is what I got using Lloyd's spreadsheet.... I get a dwell of 1.54 mSec, with the pellet 6.55" down the bore at valve closure.... Makes your simple method look pretty good, no....  :o 8) ;D .... Incidently, it predicts a residual muzzle pressure of 401 psi when the pellet exits.... ie once that 113 CI of air expands to fill the barrel.... I would say that at least based on this example, your method of calculating where the pellet is when the valve closes is pretty darn good.... CONGRATULATIONS  8) 8) 8)

I use the average pressure because I think it makes the most sense, and the results by doing so, match real world results the best.... Travis just did some tests without regulator, but still at 2600 psi, and got an increase of 33 fps.... I ran the numbers, and getting rid of the regulator, and using a total volume of 510 cc for the reservoir (80 + 430) and no other changes, should increase the velocity by 34 fps.... so the math definitely works.... Going back the other way, and trying a reduced pressure with the 510 cc reservoir, to get back to 903 fps needs 2440 psi.... so the 2430 psi average I use looks pretty good....

Bob
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Christopher

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Re: FPE Predictions
« Reply #4 on: February 10, 2017, 04:29:09 PM »
Thanks sir, yeah...got my numbers backwards. :-[ Good catch.

Something else I've been thinking about too along those same lines, although far from the original post, is those efficiency losses you speak of.

Continuing with Travis' gun and not taking into account the TP and valve space just for sake of discussion:

Bore volume in the 24" barrel is 2.4ci and we already know that there is 113ci of atmospheric air working for us behind the pellet. So by the time the pellet leaves the barrel that 113ci of air has expanded to fill the 2.4ci of bore volume, obviously resulting in a pressure drop.

Converting 2.4ci to feet=.2.....and we have a known output energy of about 147fpe....So, 147 divided by .2 =735 psi. Double checking my math ;D, Force(735psi) times Distance (.2) = 147FPE.....Theoretically speaking, but how does that line up with what's really going on inside the barrel?

113ci inside of 2.4ci of volume that is 47 bar, or 682psi.  Then you said Lloyd's spreadsheet says it's 401 psi at pellet exit.

What's going on??? I had a point about efficiency losses that happen inside the barrel but now I'm just confused...

Chris



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rsterne

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Re: FPE Predictions
« Reply #5 on: February 10, 2017, 05:48:25 PM »
Not sure about the 2.4 CI = 0.2 ft.... how can you convert cubic measure into linear?

Barrel length is 2 feet, so 147 FPE result requires continuous force of 147 / 2 = 73.5 lbs. over that distance (or come other combination, see below).... Yes, the pressure you got is correct, however, as (73.5 / 0.1) = 735 psi.... but that pressure would have to be applied constantly, throughout the pellet's travel, to be the right answer.... The pressure accelerating the pellet is anything BUT constant.... When the valve opens, the plenum pressure drops to fill the wasted volume between valve seat and pellet.... The second stage is while the valve is open, which is a linear drop in pressure relative to pellet position in the barrel (note distance/volume, not time).... The third stage is after the valve closes, and then the pressure decays as that air expands to fill the barrel as the pellet travels (still accelerating) to the muzzle.... That is why Lloyd's spreadsheet uses integration (actually row by row calculations) to determine the pressure, force, acceleration, and velocity, every few microseconds.... The pellet gets more of it's acceleration at the beginning, when the pressures are much higher, so you are getting 735 psi of "average effect" from air that would only fill the barrel to 682 psi, and be at 401 psi at pellet exit.... The 401 psi comes from using Adiabatic expansion of the air after the valve closes (ie the air is cooling), rather than Isothermal (which gives 658 psi when you take into account the 0.08 CI of wasted volume)....

Lloyd's spreadsheet can use Adiabatic or Isothermal (or in between) for either valve open, valve closed, or both.... My default is Isothermal when the valve is open, and Adiabatic once it closes....

HTHs....

Bob
« Last Edit: February 10, 2017, 06:23:08 PM by rsterne »
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Christopher

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Re: FPE Predictions
« Reply #6 on: February 10, 2017, 07:07:56 PM »
Yes sir that helps.  As usual it's much more complex than I realized.

Well, I'm fresh out of ideas.  I guess I'll leave you alone...............for now! :)

Thanks for the conversation,
Chris
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rsterne

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Re: FPE Predictions
« Reply #7 on: February 10, 2017, 08:26:01 PM »
Chris, my compliments for coming up with a simple and elegant solution for where the pellet is when the valve closes, with very little information to work from.... To me it shows a clear understanding of what is going on, and the principles and math behind them.... WELL DONE !!!  8) 8) 8)

Bob
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Alan

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Re: FPE Predictions
« Reply #8 on: February 11, 2017, 03:56:31 AM »
Reading all of the palaver makes me wonder if anyone has used a strain gauge to graph the barrel's internal pressure???
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lloyd-ss

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Re: FPE Predictions
« Reply #9 on: February 11, 2017, 06:38:04 AM »
Christopher,
I am impressed with your intuitive reasoning on the internal ballistics.  And obviously, Bob's contributions and explanations are enabling your behavior.
I am sorry to say to that I think you have the sickness and will now start waking up in the middle of the night wondering what will happen if change the port size or the dwell or larger bore.  My sympathies to you.  :'(
But the good news is that are among friends who attend the same meetings to keep us functional in normal life outside of the airgun world.  ;)  But seeing we are all here together we should enjoy the geekiness.  This is one of the parts of this hobby that I really love.  And Bob is right there as we prod each other along, and believe me, Bob is full of ideas, LOL.

=============
Edit- Sorry if I got a bit side tracked and off topic, but I do find this very fascinating. Lloyd
=============
So, seeing that this is hot in your mind, please indulge me as I throw out some progressively growing info about internal ballistics.....at least, my take on it.

This first Chart 1 is what I think is going on inside Travis 357 Flex as you and Bob have described it in this post.  I might have gotten a few numbers off slightly, but it is close enough for now.
The cells in blue at the top are the input variables and the green cells show some of the calculated values. The chart represents the calculations that continue on down the page as the pellet progresses through the barrel.
The blue line represents the air pressure in the barrel behind the pellet.  The red is the FPE, and the Green is the FPS.
Please note that this chart INCLUDES the mass of the air as it fills the barrel behind the pellet.  That mass (mass is used in the calc, not weight) of air must also be accelerated behind the pellet.  Think of this like the sleds they use in tractor pulls where the load is transfered forward as the tractor moves down track, making it harder and harder to accelerate.  That mass of air continues to increase until the valve closes.  Then the mass being accelerated remains constant as the pressure continues to drop off.  But the residual pressure in the barrel is still expanding and doing work.


Chart 2 shows the acceleration of the pellet in the barrel and it closely tracks the pressure that is behind the pellet.
Force = mass x acceleration, or, acceleration = Force divided by mass.  The mass is increasing; the force is dropping off.  It all is simple physics. No magic.  Except that the complicated airflow losses are unfortunately just a fudge factor.  Lok at cell M3 in chart 1.  That is the real bugaboo.


Now, for Chart 3, let's say all parameters are kept the same, but the air plenum has infinite volume.  Not nearly as much of an increase in FPS as you might have thought.


For Chart 4, let's leave the valve open and use as much air as we can.  A lot more air is used for not much gain.  Remember, this is all based on, or predicted from, the internal characteristics and performance of the Flex as established from Chart 1.

 
Chart 5 shows the acceleration from Chart 4.  The force is constant throughout the barrel length, but the mass that is being accelerated continues to increase and therefor the acceleration drops off.



Chart 6 is a real eye opener and shows just how much work must be done to move the air in the barrel. The expanding air moves itself and it comes at a high price.  This chart shows what would happen if the air had no mass.  Major power increase.  The FPE increases in a straight line.  This is part of the secret to helium guns.


And here is the last Chart 7, which shows the pellet acceleration from Chart 6.  The mass is constant and the force is constant and therefore the acceleration is constant at a ridiculous 13,353 g's. Unfortunately, as soon as the pellet leaves the muzzle, the acceleration goes negative.  Oh well, that's physics.  Cool stuff and I hope you enjoy it.


Lloyd-ss

« Last Edit: February 11, 2017, 08:08:27 AM by lloyd-ss »
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Christopher

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Re: FPE Predictions
« Reply #10 on: February 11, 2017, 11:32:04 AM »
Thanks to Bob and Lloyd for words of encouragement.....like the old saying goes,"even a blind hog finds an acorn every once in a while" (at least they say that in here in Ky)

Lloyd, those charts are awesome. That'll give me something to chew on for a while. Just looking at the different scenarios of valve dwell and available plenum pressure and it's interesting to see the levels of air consumption based on the dwell.

And didn't think you sidetracked the thread any, glad you jumped in  :D

Chris
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