What Makes Sense for PCPs?

[PakProtector]

rsterne-"The fact that the units worked out in the end was the validation"

It always feels good, doesn't it?  The first thing they try to beat into your head in physics class is to be sure to use dimensions throughout your calculations, and the last thing a student wants to do.

The courses I took used both English and metric units.  So, you could have a problem where 5 of the units were metric, and one English unit snuck in.  Always screwed you if you were not paying attention to the units.

LOL...slugs

[rsterne]

yes, the one unit I never understood.... *LOL*....

Bob

[PakProtector]

yes, the one unit I never understood.... *LOL*....

Bob

ha! slug-cubit/fortnight-squared...not going to cause a headache...back to metric...

[rsterne]

OK, I've been thinking about this stuff over the past couple of days, tried some stuff in my spreadsheet, some of which worked and some didn't.... I knew intuitively that there had to be an "optimum" bullet weight (or rather Sectional Density) and that it likely varied with pressure, where higher pressures would be able to handle heavier bullets.... The first chart I did above on Bullet Weight just gave what bullets of different SD's would weight in various calibers.... What I wanted was a way to decide what range of bullet weights made sense for different calibers and pressures.... It turned out to be a LOT simpler than the first few trials I made....

You remember that at 950 fps, the FPE is twice the bullet weight in grains?.... Well, I used that for the basis for the following graphs.... They are basically the FPE (for a 1 foot push) from above, divided by two.... As such, they represent what a bullet would weigh to achieve 950 fps while meeting the FPE possible from that graph, which I view as a "goal" for a high powered PCP....



You can use heavier bullets, but you will be less likely to achieve 950 fps, and if you use lighter bullets, they are likely to go faster in a really powerful PCP.... When I was doing this graph, I also calculated the Sectional Density for each combination, and it turned out that the SD was a constant for each pressure curve.... When I plotted that vs. the pressure I got the following:



Obviously, this is a linear trend.... The graph shows that as the pressure increases you can increase the Sectional Density.... and in fact you have to, to keep the same velocity.... Since I started this thread with an argument for using about 950 fps, it makes sense to bring the argument full circle.... One other thing I might mention.... For a given "shape" (ie length/diameter ratio) the SD increases in proportion to the caliber.... Here is an example, using roundball.... The BC is the SD divided by a "Form Factor", which for a sphere is 1.55, so the SD is also proportional to the caliber (ie linear)....



While we're on a roll here, this brings about one more relationship to ponder.... If the SD to achieve a given velocity (eg. 950 fps) is proportional to the pressure, and the SD for a given shape is also proportional to the caliber, then it follows that as the caliber increases, you need to increase the pressure to drive a given shape to that velocity.... 



The net result of this is that large calibers, because we currently can't use past about 4500 psi, can't drive bullets of high sectional density at our goal of 950 fps.... This makes them more practical for delivering lots of power at close range.... while the smaller calibers will tend to be better shooting bullets of high SD over long ranges, because the Ballistics Coefficient and the Sectional Density, for the most part, go hand in hand.... So that you can relate the above graph to actual bullet weights, here is an updated version of the graph offered in the first post....



I hope this information helps you to understand what is practical with the present state of the art for PCPs, and what makes sense to try and accomplish and what doesn't.... I've learned a lot developing these graphs, and hopefully they will prevent me from chasing unrealistic goals.... As an example, it would appear that bullets about 5x Roundball weight are only practical up to about .25 cal (eg. 117 gr. @ .257 cal).... those of 4 x Roundball to about .30 cal (eg. 170 gr. @ .308 cal).... and in .357 cal about the heaviest practical would be just over 3 x Roundball weight (roughly 220 gr.).... That is because we are currently limited to about 4500 psi.... If you drop the pressure to 3000 psi (much more common), that gives you the following maximum bullet weights to acheive roughly 950 fps.... This is the data from the blue line on the graph at the beginning of this post....

Maximum Bullet Weight at 3000 psi to achieve ~950 fps

.177 cal - 35 gr. = 70 FPE
.224 cal - 55 gr. = 110 FPE
.257 cal - 75 gr. = 150 FPE
.308 cal - 115 gr. = 230 FPE
.357 cal - 150 gr. = 300 FPE
.408 cal - 200 gr. = 400 FPE
.457 cal - 250 gr. = 500 FPE
.510 cal - 300 gr. = 600 FPE
.580 cal - 400 gr. = 800 FPE
.720 cal - 600 gr. = 1200 FPE

I'm not saying that these are the absolute maximum bullet weights you can use to achieve 950 fps with those calibers at 3000 psi.... You can probably do a bit better with a long enough barrel and good efficiency.... It goes without saying that to achieve the above you will need bore-size porting or close to it.... as any ports less than 92% of the bore diameter will be subject to sonic throttling at that velocity....  However, it should be a pretty good guide to assess your goals by....
 

Bob

[SeanMP]

Is that So?

  Man I'm tellin ya. There is more useful information in this one post than NASA published in the 70's

Awesome stuff Bob.....rather amazed no one else has chimed in here.

How much would it take to convince you to publish this to our share drive?

[rsterne]

Oh, you noticed I added more to the spreadsheet I sent you.... *LOL*....

Consider it done, my friend....

Bob

[rsterne]

Here is just one way you can use the above information.... Let's say you want to find out the maximum bullet weight you can launch at 950 fps in a .457 cal PCP at two different pressures, to help you decide what to build.... On the top graph, find .457 cal on the bottom, and your pressures on the side.... Where those lines intersect is the bullet type.... In the case of 3500 psi, it's a bullet that weighs twice as much as a Roundball.... for 3000 psi,it would be about 1.7 times RB weight....



Then using the bottom graph, on the vertical line for .457 cal, find those corresponding spots on the graph, and read the bullet weight on the left.... For 3500 psi, about 280 gr., and for 3000 psi about 240 gr.... If you do a good job designing and building your PCP, you should be able to get about 950 fps with those combinations....

OK, how about you have a caliber and bullet weight in mind and you want to know how much pressure it will take to hit 950ish fps.... Using the graphs in the opposite order, this time for .300 cal and an 80 gr. bullet, on the top graph find .300 cal along the bottom, and 80 gr. on the left.... Where they intersect is the bullet type, which in this case is twice Roundball weight (the red line)....



Then on the bottom graph, you find where the red line (twice RB) intersects .300 cal, and read the pressure on the left, in this case 2300 psi.... If you are using a regulator, you would need to have it set for at least that pressure.... For an unregulated PCP, you will need that pressure at the BOTTOM of the shot string to get 950 fps at that point.... If you only want 950 at the peak, then you could use that for the mid-point of your fill range....

Remember, this information is only a guide.... but it does give you an idea about how to use just SOME of the above data....

Bob

[rsterne]

Sometimes I do my best work between 6-7 AM before I'm fully awake.... I know that sounds weird, maybe my mind is just relaxed and clear as I'm just waking up, but often that is when things suddenly become clear to me.... This morning I woke up thinking about the relationship between barrel length and pressure, and how increasing both allow you to push heavier bullets to a given velocity.... My brain then tied that in with the graph above showing how the Sectional Density of a bullet can increase with pressure, and I wondered if it might also increase with barrel length.... After breakfast I sat down and plunked some numbers into my spreadsheet and came up with the following graph....



As you can see, the trends are linear, going through zero as would be expected.... It shows an interchangeabilty between pressure and barrel length in their ability to push a given weight (actually SD) of bullet.... For example, look at a 36" barrel at 3000 psi and a 24" barrel at 4500 psi, and they should both be able to push a bullet with an SD of 0.252 to 950 fps.... Another example would be an 18" barrel at 2000 and a 12" barrel at 3000, pushing a bullet with an SD of 0.084.... Both combinations should in theory be able to produce about 950 fps.... Now my brain doesn't readily relate bullet weights in grains to the SD, so here are a couple of graphs to help with that....



To make an exact calculation of the weight if you know the SD and the caliber, the formula is:  Weight (gr.) = Sec.Dens. x Cal.^2 x 7000

Let's do an example, for my .257 Hayabusa, which runs on 3000 psi and has a 28" barrel.... That says I should be able to push a bullet with an SD of 0.196 at 950 fps.... That works out to a weight of 0.196 x 0.257 x 0.257 x 7000 = 90.6 gr (181 FPE) .... The best I've ever done is an 88 gr. bullet at 908 fps (161 FPE) not too far off (89% of target).... Another example would be the Haley Scandalous .257 with a 36" barrel running 4500 psi.... The SD is 0.379, which works out to a bullet weight of 175 gr. at 950 fps (350 FPE).... and Haley has reported that he has managed over 300 FPE (~86% of target).... but I don't actually know what bullet weight and pressure was used to acheive that.... If anything, it would appear that driving the weights produced by this method to 950 fps is a very lofty goal, and very difficult to achieve.... but that's a good thing, because the whole point of this thread was "what makes sense for a PCP".... I think using the weights predicted by this system as an absolute maximum if you want to try and achieve 950 fps (2 FPE/gr.) is a pretty good guide.... For a 24" barrel and 3000 psi, that works out to a maximum SD of 0.168, which gives the following bullet weights....

.177 cal:  37 gr.
.217 cal:  55 gr.
.250 cal:  74 gr.
.308 cal:  112 gr.
.357 cal:  150 gr.
.408 cal:  196 gr.
.457 cal:  246 gr.
.508 cal:  304 gr.
.580 cal:  396 gr.
.720 cal:  611 gr.

Using heavier bullets than the above, IMO you are pretty much conceding that you won't acheive 950 fps unless you use more pressure or a longer barrel.... In fact, achieving 900 fps in a 24" barrel with the above at 3000 psi would be excellent results.... At 900 fps, the FPE is 1.8 times the bullet weight in gr....

Bob

[wwonka]

Great insights and info....

Using your graphs and formula, would the way to calculate the maximum .357 yield for an 18" barrel at 3000 PSI be like the following?

0.120 x 0.357 x 0.357 x 7000 = 107 grain bullet at 950 fps, or 214 FPE?

[rsterne]

Close, the maximum SD is actually 0.126 (I have the benefit of the numbers, not reading the SD from a graph, where it's just over 0.12), so it's actually:

0.126 x 0.357 x 0.357 x 7000 = 112 gr. @ 950 = 224 FPE.... Getting over 200 FPE, in reality, will be quite a challenge, however.... The most useful result, IMO, is that if you want to approach 950 fps, I wouldn't use a bullet heavier than 112 gr....

Bob

[Sfttailrdr46]

 Every time I read your posts I develope a massive headache then reread and begin to understand. When I was young all I did was shoot and hit what I aimed at no science . Now I am learning what should work best before I pull the trigger. Sometimes trial and error is ok but knowledge is a better way to achieve good results   

[bnowlin]

I love your charts Bob, If I just knew what I was looking at LOL.  Na I am beginning to understand some of them.  Not really much difference that the standard deviation charts back when I was in the concrete testing business.
Love your work and explanations.
Bob

[cambriajohn]

It seems that supersonic velocities were eliminated at the beginning of this discussion because of the low efficiency and turbulence above mach .9, but... Those efficiencies and turbulence significantly improve once you get above mach 1.2. It seems it might be possible to build a PCP gun that shoots above mach 1.2 for the duration of the flight?

[rsterne]

The turbulence and stability problems lessen about Mach 1.2 or so, as you say, but the drag is still some 5-6 times as great as it is below Mach 0.8-0.9, even with bullets.... so they will still lose their velocity quite quickly.... In addition, once the air velocity inside the porting and barrel exceed Mach 1, you start getting a sonic choking effect which drastically affects the efficiency.... While it is possible to drive pellets (or bullets) supersonic with a PCP, unless you start using Helium, you run into another "limit" at about 1650 fps due to the RMS velocity of the air molecules at 70*F.... If you actually experiment with achieving high velocities in a PCP (using air) you will find a huge drop in efficiency (FPE/CI) at around Mach 1 (or below if the ports are smaller than the boresize) and what amounts to a brick wall at around 1500 fps, with the efficiency going essentially to zero at 1650 fps (about Mach 1.46).... I have seen one recorded shot at 1706 fps on air, using a Condor in .177 drivng a 1.4 gr. plastic sabot, but the efficiency was about 3%.... Since by definition, the RMS velocity is part of a curve, I guess there are a few molecules moving fast enough to break the theoretical limit of 1650 fps.... The nitrogen in the air actually has an RMS molecular velocity of 1680, so it's not that surprising.... but who wants a PCP that dumps 500 psi to produce a very loud shot of only 9 FPE?....

Therefore, even if you were willing to accept a very low effiiciency (ie extremely low shot count) you have a very narrow window of velocities that are fast enough to get the bullet to still be travelling at Mach 1.2 (which is 1350 fps) at, say, 100 yards.... Since the title of this thread is "What Makes Sense for PCPs" I feel that eliminating velocities over about Mach 0.9 is logical....

Bob

[TimmyMac1]

Good job teacher. Very articulate way of saying "you don't need to go fast to get-r done". Quite the opposite actually. We have a relatively limited FPE window in each caliber. Going from 500-900 .177 can deliver from 4-20 FPE, .20 can deliver 6-40 FPE, .22 can deliver from 7-50FPE or .25 cal can deliver 10-60+ FPE and stay both accurate and efficient.

TimmyMac1

[rsterne]

I figure the maximum FPE to be 2X the pellet weight in grains, which occurs at 950 fps.... so yes, those numbers are about it, unless you go to bullets.... You can now add to the list, with pellets, .30 cal 25-100 FPE, and .35 cal 48-156 FPE, using 500-950 fps....

Bob

[Sfttailrdr46]

They promised me there would not be any algebra on this forum.......

Every time I re-read this thread I see another reason I need to go back to school for Mech Engineering classes . Fortunately there is a SUNY 5 miles away that has an excellent engineering program and I have a Geek buddy to help me hone up my math skills  they also have an old fart program

[HYspd]

so I was thinking about a post I read where the problem was an entirely down sloping velocity plot, a couple of fixes were discussed in the post but I wondered if increasing the projectile weight would raise the far end of the curve...

then I reread this thread and in the second post ( correct fourth post from the bottom of the first page) there is a graph showing that happening as the projectile weight gets heavy enough to drop the velocity below 1100fps...

hmmmmmm...more feud for thought....

[Sfttailrdr46]

  Thank you Bob this thread has been a huge PIA (learning experience  ) but it has enlightened this relative newbie  to the potential and limitations of AG's in general and the  very valid reason why for longer range repeatable accuracy bigger is better. So again thanks for all your hard work this info needs to find its way int the library . PS: now I need a .357 or .45 possibly a Recluse and molds and lead bars and a melting pot ladle and more STUFF as my wife is fond of calling my new obsession

[cambriajohn]

Do you think that pellets in general are over stabilized when shot from a barrel that has enough spin to stabilize bullets? (and thus loosing some accuracy) After looking over some calculations for model rocket stability, It seems to me that the diabolo shaped pellets are far closer to being stable than bullets before you add spin. Model rockets don't spin so they rely on their shape and tail fins for stability. So it seems logical that pellets must be over stabilized when you add the same amount of spin, of course this assumes "geometric stability" and "spin stability" are additive or multiplicative and not just different?