Theoretical Maximum Velocity in a PCP

[lloyd-ss]

Tom, Yes, that is basically how it works.  The cross shaft and the barrel and the bullet all have matching half moon notches and the  centerlines are offset by the radius of the notch.  That way, when the cross shaft is in the fired position, the barrel can be screwed in and out f the breech for loading.  For firing, the cross shaft is rotated as quickly as possible Clockwise to achieve (usually) instantaneous release of the bullet. The watch spring shown in the pic of the trigger wasn't quite strong enough to twist the shaft once the bullet was pressurized, so that needs to be improved.

The key to this set-up is that the the bullet is fully pressurized (without leaking !), and then its holding mechanism releases it like a cork from a champagne bottle. That way the bullet feels full force from the very beginning without having to wait for the air pressure to build up behind it.
Lloyd

[rsterne]

Lloyd.... Now that I see how the mechanism works, I put the following into your spreadsheet....

.25 cal, 54 gr., 2800psi, 0.003 sec. dwell (dump shot), 20cc reservoir, 0 TP volume, 8" barrel, 0 drag and breakaway, air mass included....

To get it to balance out at 707 fps, the efficiency factor needed is 0.88.... If I then change only the weight to 17 gr., guess what your spreadsheet predicts for a velocity?....

1119 fps....     

I think your model works pretty well, even with a very unusual design.... 

Bob

[lloyd-ss]

More testing.  Originally, I had hoped to achieve some really high velocities, but now it seems that the identification of trends could be of more immediate value.  Actually, I am validating some of the ideas that Bob has proposed along the way.

The data in the following table has shots from the past few weeks, and also some of my original hi vel testing of the electronic valve back in 2009.  All of the testing correlates surprisingly well.  Consider all of the shots to be dump shots from unlimited reservoirs.
The columns with Maximum Theoretical Velocity and Energy are exactly that. Theoretical, with absolutely zero losses accounted for. These Theoretical values are impossible to achieve.

The Actual Values shown in the columns are broken down by barrel length, and then by sectional density of the bullet. 
As Bob suggested, it appears easier to achieve a higher percentage of the theoretical values with bullets of higher sectional density. Another way to look at it is, for a given barrel length, the lower the theoretical max velocity, the closer you will be able to get to it.
The max velocity I was able to achieve (so far) was 1517 fps, that was .22 cal, 10.3 grain, 23.3" barrel, 2800 psi.
Lloyd
 
If you click on the table, it will enlarge so that you can read it.

[rsterne]

Yes, that certainly shows the trend toward reaching a higher percentage of the possible FPE with bullets of higher Sectional Density (heavier for the caliber).... You are likely seeing some degree of either partial sonic choking (mass flow limiting), or molecular velocity effects at the higher velocities required of the lighter bullets, or rather a reduction in those effect with the heavier ones....

I notice one other trend, and that is that the longer barrel had more difficulty reaching the same percentages as the shorter barrel, with the same bullet and pressure.... Since the tests were done with a virtually unlimited reservoir (ie constant pressure), I think most of that decrease is due to the larger mass of gas required in the longer barrel.... A 3 times longer barrel will contain 3 times the mass of air when the bullet exits.... air that has to be accelerated as well, reducing the theoretical velocity possible to achieve for the bullet.... I wonder if this experimental method might help quantify how much of the mass of the gas needs to be incorporated in the "bullet weight" in your spreadsheet model?.... The two obvious choices are half the mass, or all of it.... You may already have the data to determine that....

There is definitely more friction to be overcome in the longer barrel as well.... both between the bullet and bore, and resistance to the gas flowing down the bore.... but my gut feel is that the mass of the gas is the biggest factor.... Doing the same tests with Helium to see if the percentage drop with a long barrel is similar might tell the tale....

Bob

[lloyd-ss]

I did some more calculations with my spreadsheet and added the increasing weight of the column of air behind the bullet.  Adding this additional weight, significantly reduced the "Theoretical" max vel.  Dividing the actual velocity by the NEW max theoretical significantly improved the percentage of theoretical velocity achieved.
Now, in the first green column for the 8" barrels, the percentages instead of being 67% to 84%, were 88% to 92%.  For the 24" barrels, the percentages instead of being  50% to 73%, were now 84% to 93%.  So just adding the column of air behind the bullet greatly reduced the discrepancy between theoretical and actual. The trend of achieving higher percentages with the heavier S.D. bullets continues. 
But what I find rather baffling is the fact that the percentages achieved in both the short 8" barrels and the long  24" barrels were very similar.  I would have expected the percentages in the shorter barrels to be much better than in the long barrels, but such was not the case. 

I believe that the remaining losses (the 16% to 7%) can be accounted for mostly by the expansion of the air behind the bullet not being able to keep completely up with the accelerating bullet, and the force pushing on the bullet therefor slowly dropping off. 

It must be remembered that these are ideal circumstances, with no U-turns in the air passages, no valve heads to squeeze past and no T-ports to be accelerated through.

So now, on to designing the next part of the experiment.  Suggestions?  First, making these shots takes a lot of time, so taking lots of shots with lots of variables is not an option.  I do have another 24" .22 barrel that I could cut into into anther length, if that would be useful.  Or I could install orifice plates at the breech of the barrel to possibly find that illusive sonic choking velocity.   There should be a cusp or break point somewhere where the velocity changes/drops/increases much less/more than it should and I think that will yield some valuable information regarding the modeling of the output from a PCP.

Lloyd

[rsterne]

I thought that accounting for the air in the barrel would bring the longer barrel more in line with the short one, but I am surprised it made them nearly equal.... It shows again just how good your spreadsheet model really is.... BTW, is the progressive addition of the air mass behind the bullet something you have been using all along?.... That does seem like the best way to handle it until the valve closes (which in this case it doesn't)....

As far as additional experiments are concerned, the idea of inserting a series of smaller orifices to see when Sonic Choking occurs would be an excellent idea.... At 2800 psi the speed of sound in air at 70*F is about 1340 fps.... so if you plot efficiency vs. velocity for a given bullet you should see something happening as you approach that number at the orifice.... Perhaps you may need to plot efficiency vs. calculated orifice velocity instead?.... Using multiple weights to do multiple tracks should give you some useful data.... The bullet velocity may continue to increase after that velocity is reached in the orifice, but the mass flow rate should reach a maximum there.... so I would think the efficiency should show a marked decrease....

Using the fourmula I was given for maximum mass flow through an orifice, I get the following for your test conditions:

Mach 1 in air at 70* and 2800 psi = 408 m/s
Air density at 2800 psi = 224 kg/m^3
Orifice diameter of 1 cm; area = 0.785 cm^2 = 0.0000785 m^2

Maximum mass flow = 408 m/s x 224 kg/m^3 x 0.0000785 m^2 = 7.174 kg/s through a 1 cm orifice.... which is also 7.174 g/mS.... This apparently needs to be derated 5-10% for frictional losses, so the maximum should be 6.5-6.8 g/mS through a 1 cm orifice.... and proportional to orifice area....

Bob

[lloyd-ss]

Bob,
OK, given what you have said, it seems like the .22 cal 23.3" lg barrel can serve as a test bed.  Max vel with the 10.3 gn projectile was 1517fps and I don't think I can make the projectile any lighter so we are stuck with that as the max Muzzle Velocity. Because this arrangement is strictly a dump valve, the reservoir volume needs to be sized properly. What do you think makes sense? 20 or 30 ccs? Also, I don't know if any efficiency calculations would be valid because there will be no way to tell how much air is being "wasted".

Maybe this is a way to start:  We know max vel with unlimited air and max flow passage is 1517fps.  Install an orifice plate of full bore dia ( .22 dia orifice) and verify the max vel again. Reduce the reservoir volume until the velocity drops off to about 1500.  Permanently fix the reservoir to that volume. Then start reducing the orifice plate and graph the velocities.  My guess is that the velocity will drop off linearly (with the area of the orifice) but when the max mass flow is reached, the velocity will fall off much more rapidly.  Or as you said, the efficiency will get very bad.

What do you think?

Oh, and the question about the air mass variable in the spreadsheet.  The way I do the calculation is to calculate the air mass at each increment along the travel within the barrel, in proportion to the pressure in the barrel at that time. So the air mass continually increases until the valve closes. Once the valve closes, the air mass remains constant. 

Lloyd

[rsterne]

I would think you would want the largest possible reservoir volume, so that the pressure was a constant, or nearly so, until after the pellet leaves the barrel.... Your valve is never closing, so the air velocity at the orifice is a function of the pellet velocity at any given instant, by the inverse of the areas (orifice to bore).... If the orifice is half the bore diameter, the velocity should be 4 times the pellet velocity at any given instant....

I'm not sure why you would want to decrease the reservoir volume until you hit any given velocity.... You are obviously thinking of something different than I am.... If you reduce the volume, the pressure will drop as the pellet moves down the bore.... At 1500 fps, you should already be experiencing Sonic Choking by the bore itself, as Mach 1 at 2800 psi is about 1340 fps.... I would think that should mean that the mass flow should be directly proportional to orifice area.... What that means to the velocity I don't know, however.... You may not see any "step" in the efficiency, as you are already choked....

In order to see the test cross the threshold from unchoked to choked flow, I would think that you need to start wayyyyyyyyyy under Mach 1, in fact I see no reason why you couldn't start at, say, 800 fps unchoked, by using a heavier pellet.... Then you should see little reduction in velocity until the orifice chokes, and then a drastic reduction in efficiency.... We need to separate "efficiency" (what we normally think of, in FPE/CI) from the percentage of maximum achieved (per this thread).... as they are quite different....

Bob

[lloyd-ss]

Bob, Yes, we are indeed looking at it a bit differently.  First, regarding the reservoir volume and the 1517fps velocity.  That velocity was achieved with a reservoir of about 66ccs.  For practical reasons (having to refill air tanks) there is no need to have the reservoir any larger than what is needed by everything downstream.  so for that pellet 10.3gn pellet and barrel I first want to see if going smaller on the reservoir makes any difference.  If it makes no difference why use the extra air?  I planned on choking the orifice down even farther, which would use even less air.  But after your explanation, i see a couple of experiments.  With the 10.3 gn pellet, start with the wide open port, and start reducing it to verify that the vel does indeed drop proportionally to the drop in orifice area. Since we are assuming that sonic choking has already occurred, it will be interesting to see how linear the change is.

For the second experiment, I agree with your suggestion of starting with a heavier pellet, with a full bore size orifice, and muzz vel below, say 1000fps, and then start reducing the orifice to see when choking occurs, and if it indeed agrees with calculations. This continues to be interesting.

Lloyd

[rsterne]

HI Lloyd....

Using your own spreadsheet, and inputting 2800 psi, infinite reservoir, 10.3 gr. in a 23.3" barrel, no drag or transfer port losses, no air mass, 100% efficiency, you get a theoretical maximum of 2896 fps, 204 FPE (not quite sure why it isn't 207?).... Dropping down to a 66 cc reservoir drops it to 2844 fps (185 FPE).... Add in the air mass drops it to 1782 fps (72.6 FPE).... Dropping the efficiency down to 72.5% put you where you are at 1517 fps (52.6 FPE).... with an residual muzzle pressure when the pellet leaves the barrel of 2296 psi.... The lock time is 0.00193 sec (pellet in the barrel this long).... If you could close the valve at a dwell of 0.000193 sec. your efficiency would be 0.38 FPE/CI....

If you cut the reservoir volume in half to 33 cc, you should get the following.... 1500 fps, 51.5 FPE, residual pressure (at pellet exit) of 1945 psi, lock time of 0.00194 sec, efficiency of 0.43 FPE/CI if you closed the valve then.... If you cut it down to 20 cc, you should get.... 1479 fps, 50.0 FPE, residual pressure at pellet exit of 1623 psi, lock time of 0.00196 sec., efficiency of 0.51 FPE/CI if you closed the valve then.... As you can see, the smaller you make the reservoir, the less the maximum theoretical velocity and FPE because you no longer have 2800 psi for the entire shot.... For the last example, the average pressure over the shot is less than 2200 psi.... I think this complicates things, it certainly means that your prediction of 207 FPE is no longer relevant....

Whether or not this makes any difference to trying to figure out what happens with Sonic Choking I don't know.... but because the pressure is less, the speed of sound will also drop some as the pellet moves down the barrel, and the smaller the reservoir, the more it will drop.... I think you will find that reducing the reservoir size has the same effect as using too small a plenum in a regulated gun.... the power drops off.... Because you are wasting so much of the shot to start with, however, it won't be as pronounced.... and I can certainly see the advantage to saving air....

Bob

[rsterne]

HI Lloyd.... Using the formula above, I calculated the Mass Flow Limit for a .22 cal at 2800 psi.... for various port sizes of interest (1/64" increments)....



The units are grams per millisecond.... The flow has been derated 10% to allow for frictional losses.... Hope you find it useful....

Bob

[lloyd-ss]

Bob,
Changing units of measure on me. That is not nice, LOL.  My calculator uses slugs and there are 14,593 grams per slug, so let's do some back figuring.  Using the highest velocity sample that I shot, 1517 fps, the lock time is approx .0019 seconds. Mass of the air in the 23.3" barrel at 2550 psi (average for 66 cc reservoir) is .00021 slugs, or 3.06 grams.  Lock time was about 1.9 millisec, therefore average mass flow was 1.61 grams/millisec. That is less flow than the limit of 2.0 grams/millisec on your chart.  To reach 2.0 grams per millisec, lock time would need to be about 1.5 milli sec.  Changing only the grain weight in the spreadsheet, projectile weight would need to be reduced to about 5 grains to get locktime down to 1.5 millisec, and the MV would be about 1781fps. So to reach the mass flow limit, the MV would greatly exceed Mach 1.  What do you think? Is the mass flow limit the "hard stop" for velocity?
Lloyd

[Bill G]

Mass flow is how I have based my calculations..... Actually I have kinda gone about it ignorantly, because I calculated the flow in ci's per sec or ms.  So, let me organize this line of thought.

 establish cal
establish projectile weight
establish desired velocity.  This gives a calculated duration that the projectile will be exposed to pressure for accelerating.  I do the calculation 1000 times (divide the barrel length by 1000).  That leaves me with a final velocity at the muzzle. 
establish porting size and pressure
So if I use a established porting, I should then know how much volume I can move in calculated duration as established above.  If the velocity isn't as desired, I must increase or decrease port size and/or pressure.   There are a lot of back ground calculations that affect the above mentioned such as friction, porting length (dead space) ect.  Ultimately the mass flow is the calculation that I use, it is expressed as Volume which in retrospect is confusing.  As the psi drops obviously the amount of volume that can flow through the porting will change.  For the sake of experiment, a dump valve would seem to me as the best platform to base it all from.  as to efficiency, that's a hard one to base line for a dump valve. I suppose using a volume of dump compared to volume of barrel might be a place to start.  Bob has done such work already. If we were to move from the test platform to a more typical valve design, we could assume constant psi ( regulated) and a fixed volume plenum.  This plenum is essentially your dump valve and the reservoir volume feeding it allows us to use Bob's established efficiency calculations.  The hard part there would be getting a set-up that is inline and has no dead space.  I suppose it could be done it the rig were designed like one of those quick exhausting potato gun valves.  Not sure if you guys have ever seen one but the set up would be pretty sweet for single shot testing.

Lloyd, your test rig is bore diameter porting isn't it?  I don't guess I'm seeing how you are going to change the port size.  Maybe I should look at the photos a bit closer. Is the barrel you are using a smooth bore? 

Bill G   

 

[rsterne]

Lloyd, I'm not sure you can average the pressure and use that to calculate the mass flow, as when the pressure drops, so does the maximum flow rate.... and if you do average the pressure, then the maximum flow rate would not be 2.01 grams/mSec, but about 1.83, and you are at 1.61.... which is 88% of the predicted maximum mass flow rate.... Perhaps the 5-10% derating that I was given was not enough for a "long" orifice like a barrel (more friction)?.... I freely admit to having a poor understanding of sonic choking.... but it appears it is not a sudden event but builds up as the velocity approaches the speed of sound in the system.... Perhaps you can get something useful from post # 132 (and others) in this thread.... http://www.gatewaytoairguns.org/GTA/index.php?topic=66737.120

The fact remains that your muzzle velocity of 1517 fps is exceeding the speed of sound at 2800 psi of 1340 fps, so there must be a choke forming in part of all of the barrel, shouldn't there?.... According to the definition of choked flow, the pressure downstream of the choke point cannot be more than 52% of the source pressure.... So much to learn and understand.... 

Bob

[lloyd-ss]

Bill G, A lot f info in your post. Thank you.  It sounds like you have a pretty good data base of ports/calibers/velocities, etc. 
My set up does use full bore size porting and a .22 cal Discovery barrel.  I planned on modifying the cone-shaped entry into the barrel so that I could use interchangeable ports. A quick exhaust valve (similar to the one described by Ron in Reply# in this topic, or something like the potato guns or paint ball markers could work well and eliminate the need for special projectiles.  I have fiddled with some of the QEVs on my prototypes and the difference in solenoid actuation speed is noticeable.

Bob, even though the sonic choking is a progressive phenomenon, if the pressure downstream of the choking can be no more than 52% of the source pressure, I would expect to see some loss of linearity, or change of slope, in the velocity.  If indeed my 1517fps shot is already being hampered by sonic choking, maybe just taking a series of shots with progressively lower pressures would produce results that quantified the effect of the choking.  I might give that a try.  It will be easy to do without changing my set-up.
Lloyd

[rsterne]

Lloyd, I would agree, when choking occurs there should be a change in slope of the velocity vs. pressure curve.... Your suggestion should work.... As the pressure drops so does the speed of sound, but not drastically....

Bob

[lloyd-ss]

I am reviving this post for at least 3 reasons:
1) I m recovering from some surgery    and  I can't work outside, so I can stay in my shop all day and play with airguns, he, he, he.
2) Bill G was taking me to task in another thread  http://www.gatewaytoairguns.org/GTA/index.php?topic=74919.60  (see Replys # 76 and 77)
about trying to get high velocities out of itty bitty air reservoirs. Bill, I know you can do this yourself, LOL, but I am going to help you out anyway.
3) I can't stop myself from doing this sort of experimentation.

So, today,  I modified the test rig by replacing the way too big reservoir, with an adjustable one: max vol 9.17 cc, but plugged down to 5.08ccs.

I set this series of tests up with the same 23.3" long .22 cal Disco barrel and the same funny bullets.
Look at the pictures starting around Reply # 69 to see what everything looks like.

I only took 3 shots (so far) and here are the results:

.22 cal, 23.3" barrel, zero effective T-port volume,
5.08cc dump reservoir, 2150 psi.
10.3 gn 1180 fps  31.9  fpe
13.6 gn 1065 fps  34.3  fpe
42.7 gn 708  fps   47.5  fpe

Bill, this doesn't answer your original question of whether you can get a .25 cal 25 gn pellet to go 950 fps from a 20" barrel with a 7cc reservoir at 2000 psi, but I am headed in that direction.  So just to make you happy, give me a gn weight and volume and velocity for this 23.3" 22 cal barrel and I'll see what I can do.  Sorry, I don't have a .25 cal barrel that I want to chop up at this time.

I think these numbers show that if you can get rid of all the air constipation associated with valves and T-ports and making U-turns, it doesn't take a lot of air to make a pellet fly pretty fast.

I'm going to try some other configurations.

Lloyd




 

[Rescue912]

Wow, surprising numbers. Supersonic. With 5cc reservoir in .22 is more than I would have expected. Gonna watch this closely...

[Motorhead]

Just need a configuration where exit port from valves throat is right at the breech/pellet.
Only losses to volume/pressure within plenum would be that of @valves throat.
You would get the Amplitude of pressure pulse instantly with minimal loss and be quite effective too one would think !

[lloyd-ss]

Just need a configuration where exit port from valves throat is right at the breech/pellet.
Only losses to volume/pressure within plenum would be that of @valves throat.
You would get the Amplitude of pressure pulse instantly with minimal loss and be quite effective too one would think !

Yes, exactly. If the valve were 90 degrees to the breech and maybe the hammer traveled up and down. Maybe use some sort of tap loader for compactness.  I think part of the speed to my test setup is that the bullet is more like a cork in a champagne bottle, holding back all the pressure until the bullet is mechanically released.
Lloyd