Sonic Choke? Maybe not. 2162 FPS

[Scotchmo]

This pic is from an earlier post. Ignore the equation and 1640fps.  Im on my iPhone and cannot create a new pic right now:



At 50% efficiency, the 100's of lost FPE do not just vanish. It turns to heat. A portion of that heat continues to provide work. The heat accumulates in each "parabola" of air and expands it as progresses down the barrel. Providing some work, and some entropy gain (inefficiency).

Portions of the air are not as dense as you might think they are. And it may be even less dense when the temps are not allowed to stabilize before a shot.

We still need a baseline after letting the filled chamber rest several hours after fill disconnect, before taking the shot. There is a good chance that it will be much lower than 2162fps.

[lloyd-ss]

Yes, I agree with what you guys are saying.  It is fairly easy to visualize in general terms, but quantifying it is difficult.  I haven't done anything with the heat changes in the spreadsheet other than the exponent in the expansion ratios, but I realize that the thermal part is quite important.

Scott, the shot with the chamber first setting loaded and idle for several hours is upcoming.  And the nitrogen, too.  I have prep to do for the Funshoot in the meantime.

Lloyd

[match]

I think a slightly different point of view may be useful.

Think back to Bob's thought experiment with the packets flowing down the barrel:

 - these packets are really pre-compressed springs which expand while moving down the barrel.

The speed of the center of mass of the packets slows down as the packet expands and moves down the barrel but the speed of the forward "edge" increases.

These packets can also overlap, the molecules can mix with the adjacent packets, so a packet "behind" does not force a forward packet to move like a solid ball so we lose a bit of the combined forward velocity of the leading edge at the pellet.

[rsterne]

From Steve on the Yellow.... The new limit is 2515 fps....  http://www.network54.com/Forum/79537/message/1461247651/So+--+Just+how+fast+can+a+jet+of+air+REALLY+go-++1640fps+-+or+2515fps-

We shall see.... I think that is fast enough we likely can't test it anyway....

Bob

[rsterne]

Scott, I can't see how letting the chamber rest "for hours" can possibly make any significant difference.... The fact that the pressure did NOT drop more than 100 psi after nearly 8 hours, instead of well over 1000 psi, is proof positive that the air available for the shot after a couple of minutes is NOT at a high temperature.... Pressure and absolute temperature are highly correlated for a fixed volume and mass (ie density)....

John, I like your analysis, comparing the packets to decompressing springs moving down the barrel and intermixing, because of course there are no "finite" barriers between them.... that was just a way to visualize the process.... If the reservoir is infinite volume (relative to the barrel volume), the pressure should stay relatively constant, of course, not really allowing the "springs" to decompress, although they could still intermix as they travel down the barrel to some extent.... At some point, as the front packet and pellet reach a high enough velocity (driven by the ones behind), that further air released at the breech can have no further effect, and I think that point occurs at twice the molecular speed, ie about 3300 fps....

While I cannot rule out temperature effects, neither can I quantify them, because they are in two directions, cooling through Adiabatic expansion (with a gamma of about 1.7 ? ), and possible heating through friction, compressibility, or other unknown effects.... I still think we can have a successful model based just on F = mA.... Did anyone look at the equation at the Hyperphysics link I posted at the bottom of the previous page?.... This is for the KE of a fluid flowing in a tube....



At any given time increment, we know the Vm (maximum velocity at the center of the barrel = pellet velocity), the density (rho), and the barrel volume V.... It may even be possible to apply the equation at the moment the pellet exits.... For Lloyd's 2162 fps shot, where the pressure at the muzzle, from his L4 calculator (which uses the VanDerWaals corrections, and sonic Flow limiting), is 2213 psi, the air mass in the barrel is 0.0005746 slugs, the barrel volume is 2.792 CI, and the density is therefore 0.0002058 slugs/CI, and the MV is 2162 fps, so the equation would yield....

KE = ( V / 2 ) x rho x (Vm^2 / 3) = ( 2.792 / 2 ) x 0.0002058 x ( 2162^2 / 3 ) = 1.396 x 0.0002058 x 1,558,081 = 447.4 FPE

The KE of the pellet is 80.0 FPE, so the total energy is 447.4 + 80.0 = 527.4 FPE, which is 49.3% of the Input energy (1070 FPE).... Seems entirely reasonable to me....

Regarding Lloyd's L4 spreadsheet, if I use 100% of the air mass, and an efficiency of 76.4%.... the flow chokes when the pellet is 12.9" from the breech.... If I use 50% of the air mass and 47.6% efficiency, the flow only chokes at 44.3", right near the muzzle.... but I get exactly the same results for the KE of the air and the total KE of the shot.... However, the "maximum fps" number increases from 1880 fps to 2580 fps, because less of the air mass is being accelerated.... It would appear that the latter choice makes more sense....

Bob

[rsterne]

Now for the next step....  can we turn that equation around, and predict the maximum velocity Vm from the volume (2.792 CI) , density (0.0002058 slugs/CI), and KE (1070 FPE) ?.... If we were just accelerating the air, and no pellet....

( Vm^2 / 3 ) = ( KE / V ) x (2 / rho)

Vm^2 = 2 x 3 x KE / V / rho

Vm = sqrt ( 6 x KE / V / rho ) = sqrt ( 6 x 1070 / 2.792 / 0.0002058 ) = sqrt ( 6420 / 0.0005745 ) = sqrt ( 11,174,934 ) = 3343 fps ....

hmmmmmmmmm that looks intriguing, doesn't it.... or maybe just a coincidence?.... If we look at the equation, it is really KE = 1/2 MV^2, but the mass M is being derived from the density (rho) times volume (V), which is then divided by 3 to correct for the losses caused by fluid flow in a tube, and the velocity used (Vm) is the maximum velocity at the center of the flow (in our case, the pellet velocity).... So let's see how we could include the pellet mass.... rewriting the equation yet again....

Vm^2 = 2 x KE / ( V x rho / 3).... where the denominator is the mass of the air used to calculate the KE, actually only 1/3 of the total air mass....  So let's add to that the pellet mass M (0.0000342 slugs)....

Vm^2 = 2 x KE / { ( V x rho / 3 ) + M }

Vm = sqrt [ 2 x KE / { ( V x rho / 3 ) + M } ] = sqrt [ 2 x 1070 / { ( 2.792 x 0.0002058 / 3 ) + 0.0000342 } ] = sqrt [ 2140 / {0.0001915 + 0.0000342} = sqrt [ 2140 / 0.0002257 ] = sqrt [ 9,481,613 ] = 3079 fps....

If I use only 1/3rd of the air mass in the barrel in Lloyd's L4 calculator, I have to reduce the overall efficiency to just 37.1% to make it balance.... however, when I do that, it spits out 3079 fps as the maximum theoretical velocity.... exactly the same answer.... Maybe I'm onto something here?.... or maybe not....

Please note, I had the "Input FPE" wrong at 1047, it is actually 1070, (I had used 4500 psi instead of 4600) and I went back and corrected this post using that....

Bob

[Scotchmo]

--------
Another way to look at it:

Start with the concept of a plug of air. The plug is bore area x a small fixed segment length.

FPE is energy that accumulates by applying power over a length of time. The power (not velocity or FPE) of the system is limited by pressure and bore diameter. At a Mach 1 inlet velocity, we have reached max power  (not max FPE, not max velocity).
Once Mach 1 at the inlet it achieved, momentum density becomes fixed. For every "plug" anywhere in the system, the mass x velocity is constant. To increase velocity, mass in that plug must decrease.

Mach 1 is 1128fps for air at 70F.

At 3307fps, the density of the air in the plug behind the pellet will be 1128/3307 or 34% of the density at the inlet. It will have expanded at about a 3:1 ratio. What air temperature will it take to accomplish that with an equal pressure throughout the system?

At some point, the power needed to progressively increase the temperature of every plug as they move down the barrel will consume all the power the inlet can provide. Acceleration ceases. Max velocity has been reached.

[rsterne]

In case anyone cares, Steve has revised 2515 fps downwards to 2259 fps because of a misplaced parenthesis.... I'm betting that can be broken, so we may be back to square one.... or would that be square two?.... I wonder what he would get using a gamma of ~ 1.7, which is applicable at 2000-6000 psi (within ~ 3%), instead of 1.4.... or would you need some intermediate value because the pressure at the muzzle is greatly reduced?....

Mach 1 is 1128fps for air at 70F.

not at high pressure, it's not.... 1547 fps @ 20*C @ 4500 psi....

Bob

[Scotchmo]

You are right Bob. Mach in air is a variable.

For a chamber pressure of 4500psi:
At 3307fps, the density of the air in the plug behind the pellet will be 1547/3307 or 47% of the density at the inlet. It will have expanded at about a 2:1 ratio. What air temperature will it take to accomplish that with an equal pressure throughout the system?

That's more like Lloyd's test system (with very large dump). I'll see if it looks doable when I get back home this weekend.

Since Mach is a variable, based on temp and pressure, I'm not sure if there is a set muzzle velocity limit for 70F air. The limit may change with pressure as well.

2259fps? Time for a 100cc dump and an 8 foot barrel.

[Scotchmo]

In case anyone cares, Steve has revised 2515 fps downwards to 2259 fps because of a misplaced parenthesis.... I'm betting that can be broken, so we may be back to square one.... or would that be square two?.... I wonder what he would get using a gamma of ~ 1.7, which is applicable at 2000-6000 psi...

I jumped in on the yellow, hopefully before a new myth springs up. I gave them the Wolfram Alpha link for high pressure specific heat ratios. Higher gammas soak up more energy, so it results in lower velocity. 1968fps. So it's already been broken.

[rsterne]

Scott, I still don't understand the temperature thing.... I have enough to do to warp my brain around the idea that only 1/3rd of the air mass should be used to calculate the KE of the air moving down the barrel, when using the velocity of the center of the stream.... I could grasp the idea of 1/2, because the volume of a parabola is half the volume of the same size cylinder of air.... I guess the rest of the reduction in mass has to do with using the velocity of the center of the stream, rather than the average velocity....

From what I have read, the idea of a parabolic shape to the velocity profile only applies for laminar flow, when the flow is turbulent the shape of the velocity profile is shortened.... That should make the mass used a larger percentage of the total mass, which would reduce the velocity, so maybe using the parabola, and the equation I showed above, is a good way to calculate the maximum velocity for a given combination of barrel volume and air density, based on the maximum KE value arrived at from bore area x pressure x barrel length, which does seem to be a valid maximum for the "input FPE".... However, a change in any of those parameters, and of course pellet weight, will change the maximum velocity.... Still, it may be a valuable way to create a maximum velocity for any given PCP setup....

For an absolute maximum velocity with room temperature air, I'll stick with my WAG of 3300 fps....

Bob

[Scotchmo]

3300fps could be right. I'm not ready to put number on it.

Laminar is easier to visualize with "parabola" shaped gradients of velocity or temperature. But at the pressures and velocities we have, it is a turbulent boundary layer. Probably very high Reynolds numbers, which I should look into more and actually try and come up with a reasonable friction coefficient.

So the other way to visualize it is as plugs of air. Each one homogeneous throughout it's short length. But after reaching Mach 1 at inlet, different temperature and density than the adjacent plugs. That's how I set up the numerical integration in my spreadsheet.

Whether you model it as moving only 1/2-1/3 of the air down the middle, or as progressively less dense segments, either method will get close to the same result. The momentum of each segment is the same. The difference between the two methods will be in how fast the dump chamber depletes during pellet acceleration. In reality, you can't really have full density farther along the barrel. If that were the case, pellet velocity would be limited to Mach 1 of chamber air. That's not the case, so the air farthest along the barrel has to "thin out" as the pellet outruns the inlet velocity.

What part of the temperature thing that is not understood? It all balances out fairly well. The wasted FPE is from air friction. Friction causes a temperature rise. The temperature rise causes the decrease in air density.

[phoebeisis]

 In does an increase in gamma mean
    More work energy available?
or Less work energy available?
Thanks
Charlie
PS at 300k via eyeball looks like 2-3% of air molecules are 1000m/s or more-if the pellet is light enough-low friction enough-no reason you can't hit 1000m/s-

[Bryan Heimann]

I have to ask, how close are ya'll to launching a "real" projectile at over 2,000 fps, with accuracy?  It is cool ya'll have taken it this far, but that is what I want to see!

[rsterne]

I also haven't checked the Reynold's numbers, Scott, but I suspect Turbulent flow would dominate as well.... The shapes I have seen for the velocity profile would suggest that the mass compared to a cylinder is larger than the 1/2 you get with a parabola, which would logically mean that the mass for the KE formula could easily be about 1/2 instead of 1/3.... a number that just seems more "comfortable" to me.... (see, I told you I'm not an Engineer, I seek the simple explanations).... You are 100% correct, the "plugs" of air must be less dense (lower pressure) towards the muzzle, after all, you need a pressure gradient to cause flow to happen.... I suspect that the mass inside each plug is relatively constant (and once the velocity at the breech reaches Mach 1, starts decreasing slowly as the reservoir pressure drops), but that it expands somewhat as it moves along the barrel, lowering pressure and increasing velocity, while conserving momentum....

Charles, when you are talking about extracting energy from compressed air, the higher the gamma the more cooling you will see as the air expands, which results in lower pressure because of the lower temperature.... This calculator will quickly show you....  http://www.tribology-abc.com/abc/thermodynamics.htm .... If you change Cp or Cv you will see the effect on the energy released and the temperature.... gamma is Cp/Cv, of course.... If you are talking Adiabatic compression, like occurs in a Springer, the situation is reversed, the higher the gamma the more energy, because the temperature increases more....

Bryan, if by "real projectile" you mean something heavy enough to have reasonable FPE, I don't think that will ever happen with air.... To achieve such high velocities requires extremely light projectiles, which means that the mass of the gas driving it becomes dominant in drawing energy from the expanding air.... In Lloyd's 2162 fps shot, the pellet weighed 7.7 gn. but the air in the barrel when the pellet exited weighed 127.4 gn.... Changing to Helium, which is only 1/7th as dense, changes things in a hurry, and modern PCPs can be pushed into the 1500 fps range relatively easily with "real projectiles".... so 2000 fps would not be out of the realm of possibility.... However, you are then faced with the situation of the drastic increase in drag in the Transonic and Supersonic zones slowing the bullet down again once outside the barrel..... The Supersonic drag is 3-5 times as high as the Subsonic drag.... You have to drive bullets over 2500 fps to get the wind drift down to what can be achieved at 900 fps with the same bullet.... While this is exciting, and interesting, from a Scientific point of view, it realistically isn't practical....

Bob

[lloyd-ss]

..................................................
On another note, I've been working on trying to find an inexpensive ADC (analog to digital converter) to instrument a barrel.  I still think it would be instructive, to know what the air pressure is doing at various points in the barrel.  The problem I'm having is finding a fast enough ADC with sufficient channels without a whole lot of extra stuff, and is affordable. 

On the analog side of the system with strain gauges and signal conditioning, I have a good handle on those design principles.  It is with the ADC and type of input to the computer (serial, or parallel)  that I'm having trouble.  Software will depend on how data is delivered to the computer, which is yet another hurdle. 

          Sorry folks, sidetracking the thread a tiny bit. Lloyd

David,
I have fiddled a little with strain gages in the past and used a Parallax Stamp microprocessor to process the data.  I don't know if that microprocessor has the speed to do what you want, but the basic set-up ought to be similar.

Here is the basic circuit to convert the low level analog output from the strain gage to a digital signal.  It has an instrument amp to boost the output voltage, then a A/D convertor (I think mine was a 12 bit, but you'd probably need a 16 bit to get the speed), then the digital output was processed by the Stamp and its programming, and finally, output to a laptop via RS232 serial.  The connections are usually USB now.  The instrument amp and A/D convertor are a few bucks each.  The microprocessor (Stamp, Propeller, Pic, Arduino, etc), maybe $75+, which includes the builtin I/O and power supply, etc.  I liked the Stamp because the programming was so easy.  Just another version of Basic.  In fact, in the end, I didn't need the 2 adjustment pots on pins 1 and 3 of the A/D convertor because I could make those compensations in the programming.



Here is the strain gage, which is actually a pressure transducer that I made.  You'd need a strain gage with a pretty high frequency to accurately detect the strain in the barrel. (Sorry for the fuzzy picture.  I forgot to have the camera in macro mode when I took the pic years ago.)



And here is the circuit board with some other I/O, a multi-channel solenoid driver, RS232 connection, additional regulated power supplies, and output to a 3 line LCD display.  It looks more complicated than it is.  I know very little about electronics, but these hobbyist microprocessors have a pretty good support community to lend help with projects.

[match]

Bob -

I am not sure momentum is strictly conserved in the macro "packet" sense.

The air packet absorbs energy cooling as it moves forward and expands - That is the molecules slow down in translational modes but speed up in vibrational/rotational modes.

In Helium you get the strange effect of while expanding the molecules speed up in translational modes (and temp increases) because there are NO rotational/vibrational modes available to absorb the energy due to He being a single atom.

This may also be another reason He works so well to increase fpe at the muzzle.

[rsterne]

John, there you go, making it complicated again.... *LOL*.... I need it simple....

Scott, you remember the equation Steve had that we discussed way back about post #310 and page 16, and discussed the constant being 194, not 172, because of the omission of a PI/4 term in the denominator?.... With that corrected, the maximum velocity (with zero pellet mass) worked out to 1854 fps, using 1/2 the mass of air in the barrel (Steve got 1644).... If we use only 1/3rd of the air mass, as per the equation above, that increases to 2257 fps, assuming air is an ideal gas.... Once we apply the VanDerWaals corrections, which effectively mean that the pressure is higher for a given density, once the pressures are above about 2000 psi, the equation is no longer independent of pressure.... Here is a chart comparing Steve's original equation (with the 172 constant, and as an Ideal Gas), my version, using VanDerWaals, (with the 194 constant) using 1/2 the mass and 1/3 the mass of air in the barrel.... compared to the output of Lloyd's L4 spreadsheet (in red).... for the 2025 fps shot @ 4000 psi with a 47.5" barrel....



You can see that only the version using 1/3 of the mass of air in the barrel predicts maximum velocities higher than Lloyd achieved in that shot.... So if there is any validity to that equation, something on the order of that ratio of mass is necessary for it to predict high enough numbers.... Note I am not suggesting this is the answer, just what will not work, based on existing results.... I wonder if there is any relationship between Steve's latest number of 2259 fps and the 2257 fps I obtained using his previous formula, corrected to include the PI/4 omission, and using only 1/3 of the mass of air in the barrel.... or if that is just another coincidence.... With air not acting as an Ideal Gas at over 2000 psi, it doesn't matter anyway, I guess.... because the maximum is higher than that at high pressures....

Bob

[match]

aren't diatomic molecules fun!

[phoebeisis]

Bob
Got it-
- so in PCP   lower gamma gases have relatively  more  energy stored in not translation motion- 
so the  the temperatures-speeds- don't drop as much when it starts expanding  and doing work-
Thanks-I had been puzzling over that-
Charlie