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Chamber to Barrel Volume Ratio Explained

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Chamber to Barrel Volume Ratio Explained

Unread postAuthor: dbracer443 » Wed May 26, 2010 11:25 pm

I found a unit error in my calculation and as a result, reworked everything in metric. I converted back to english before graphing and then edited this post.

I'm new to this site so I don't know if this has ever been posted before but I figured it would be helpful to some people. There seems to be a lot of questions regarding chamber to barrel volume ratios. I wrote up a Matlab code that would allow me to input gun and projectile parameters so that the acceleration of a projectile could be calculated and graphed as a function of distance from the beginning of the barrel. I input projectile weight, barrel inside diameter, combustion chamber volume, combustion pressure generated (or tank pressure if pneumatic), frictional force of the barrel walls on the projectile, and barrel inclination angle. Then the program makes the proper calculations based on the code I've written and it spits out a graph of projectile acceleration on the vertical axis and distance the projectile has traveled from its starting point on the horizontal axis (this could be considered barrel length).

Now I'll get to the point. Looking at the second graph I posted, the curve corresponds to a particular set of gun and projectile parameters. As you can see, the curve begins to level out at a certain point. The acceleration starts out high and drops quickly before leveling out. A large portion of the projectile acceleration can be obtained without using a barrel that is a mile long. By making some simple calculations, it can be determined that the optimum chamber to barrel volume ratio is somewhere between .7:1 and 1:1. It is up to the builder. Each gun setup can be fine tuned for the best performance. There is no one "magic" ratio that will make your gun super fast or anything like that. Making an extra long barrel is not worth the inconvenience and cost for the small increase in gun performance. On the other hand, a barrel that is too short will not make the most out of the possible projectile acceleration.

It is important to note that these calculations and the resulting graphs are purely theoretically based. I tried to input every possible factor that I could but there are factors such as gas blow by and other losses that would affect the results and are not incorporated into the model. These are to be used as guidelines for building and they should get you in the ballpark for optimum performance.

The first graph is for the same gun as the first which is set-up with a 200 in^3 combustion chamber, a 2.5 in inner barrel diameter, combustion/tank pressure of 100 psi, .6 lb projectile, 6.75 pounds of frictional force, and 0 gun inclination angle. The three vertical lines are at chamber to barrel volume ratios of 1.02, .51, and .34 from left to right. You can see that you gain most of the possible velocity with a ratio of around .8.

I hope this helps some people. I can give extra clarification on any of this if needed.
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Attachments
velocity.jpg
Velocity Curve with C:B Ratios
velocity.jpg (21.83 KiB) Viewed 1530 times
Accel.jpg
Acceleration Curve
Accel.jpg (19.82 KiB) Viewed 1530 times
Last edited by dbracer443 on Thu May 27, 2010 11:07 pm, edited 3 times in total.

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Unread postAuthor: btrettel » Thu May 27, 2010 8:09 am

Welcome to Spudfiles!

Interesting plots. They seem to verify what was already known to a certain extent, though, the C:B ratios seem slightly higher than typical. Mind posting the code? I like looking at people's code.

It is important to note that these calculations and the resulting graphs are purely theoretically based. I tried to input every possible factor that I could but there are factors such as gas blow by and other losses that would affect the results and are not incorporated into the model. These are to be used as guidelines for building and they should get you in the ballpark for optimum performance.


Gas blowing by the projectile generally isn't too important because it doesn't seem to be much gas in all but the more egregious cases (which we want to avoid). I've found that the biggest differences come from how energy is tracked in the system. I think GGDT assumes the gas has no kinetic energy but has thermal energy; I make the same assumption in some of my simulations. This simplifies things a great deal and it does not seem to be much less accurate than giving the gas an average velocity.

You might want to non-dimensionalize what you wrote. I non-dimensionalized the adiabatic pneumatic gun problem and found some interesting things. Namely, I found a simple way to do some design tasks that would be nearly impossible otherwise (i.e. pick performance first and scale up the optimal configuration).
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Last edited by btrettel on Sun May 30, 2010 8:14 am, edited 2 times in total.
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Unread postAuthor: dbracer443 » Thu May 27, 2010 5:00 pm

Here is my EDITED code with all work done in metric. I used a force analysis to determine the projectile acceleration. If you want, I can justify it because the derivation isn't all that obvious. I used the ideal gas law to incorporate the change in pressure as the projectile moves along the barrel. I held n, R, and T as constants. I had never heard of the GGDT before, it seems much more thorough than my little 15 minute program. I considered incorporating flow losses and gas temperature variations but I haven't had the time yet. It would also be interesting to do an energy analysis instead of a force analysis and compare the results. I like the idea of non dimensionalizing the code as well, maybe I'll try it.

Pc=100/.000145; 'gage combustion pressure (n/m^2)';
Vc=200*.00001639; 'Combustion Chamber Volume (m^3)';
D=2.5*.0254; 'inner barrel diameter (m)';
A=(pi*((D^2)/4)); 'projectile area (m^2)';
m=.3; 'projectile mass (kg)';
Ff=30; 'friction force (n)';
g=(9.81) 'gravitational constant (m/s^2)';
theta=0; 'gun inclination (degrees)';
x=(0:.5:200)*.0254; 'distance from beginning of barrel (m)';

a=(((Pc*Vc*A)./(m*((A*x) +Vc)))-(Ff/m)-(g*sin(theta)));
v=(sqrt(((2*Pc*Vc*A)*log(A*x+Vc)./(m*A))-(2*Ff*x./m)-(2*g*x*sin(theta))-(2*log(Vc)*Pc*Vc*A/(m*A))));

plot(x*39.37,a*3.281)
xlabel('Distance from 0 (in)')
ylabel('Acceleration (ft/s^2)')
title('Acceleration as a function of distance from beginning of barrel')

figure
plot(x*39.37,v*3.281)
xlabel('Distance from 0 (in)')
ylabel('Velocity (ft/s)')
title('Velocity as a function of distance from beginning of barrel')
y=(0:.5:500);
hold on
plot(40,y)
hold on
plot(80,y)
hold on
plot(120,y)
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Last edited by dbracer443 on Thu May 27, 2010 11:10 pm, edited 3 times in total.

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Unread postAuthor: jimmy101 » Thu May 27, 2010 5:07 pm

Without seeing how you actually did the calc's it is kind of hard to judge the results.

Did you model combustion (pressure rise from 0 PSIG) or just jump to 100 or 135 PSI or whatever?

Did you include both static and dynamic friction?

If I were to publish your results I would convert all the acceleration axes to either velocity or velocity squared (which is proportional to kinetic energy, KE, of the projectile). Acceleration isn't all that useful, at least compared to velocity or KE.

I really don't see how you can use the acceleration versus projectile position graphs to come to the conclusion that CB=0.8 is anything special. It is well established that a particular gun and ammo does indeed have special characteristics near CB=0.8, but I don't see that in your graphs.

BTW, CB=0.8 (or whatever it is for a particular gun + ammo) is really not all that important a parameter in spud gun design. Mostly it reflects the most efficient use of the energy in the fuel, which really isn't all that important. Most spudders are more than willing to spend the extra $0.002 per shot to get better muzzle velocity even though it comes at the expense of efficiency. Hence you see a fiar number of very good guns with CB well above 0.8.
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Unread postAuthor: dbracer443 » Thu May 27, 2010 5:28 pm

btrettel wrote:
You might want to non-dimensionalize what you wrote. I non-dimensionalized the adiabatic pneumatic gun project and found some interesting things. Namely, I found a simple way to do some design tasks that would be nearly impossible otherwise (i.e. pick performance first and scale up the optimal configuration).


I would like to do an energy analysis but I am unsure about what approach to take. What basic principles/equations did you use for the ''adiabatic pneumatic gun project''? I just need a jump start in the right direction and it will all come back to me.
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Unread postAuthor: dbracer443 » Thu May 27, 2010 5:59 pm

**Edited post
I generated a curve for velocity by doing a little math, The general shape and slope of the velocity curve reflect the acceleration results. I agree, Some interpretation has to be done with the acceleration graphs to learn anything useful. However, the same conclusions can be drawn from the velocity graph. The velocity increases sharply and the increase rate drops off as the pressure behind the projectile drops. That is what the graph suggests, and you can see that at certain ratios you make the most of these high acceleration values. My conclusion is that you end up with a real long inconvenient barrel to squeeze out a little more acceleration or velocity. Perhaps I was wrong to use the value of .8 but you can see what I was implying

I used the assumption that the pressure would be there instantaneously. I need to find a way to compensate for the fact that the pressure would have to build up before the projectile would move and it may continue to build if the valve isn't opened completely or if the chemical reaction isn't complete. Also, I assumed a constant dynamic friction force. Static friction has no effect on the results of my calculations. It would affect the magnitude of the pressure that the projectile begins to move at, but as I mentioned before, that isn't included in the calculation.

Every gun has limitations, If you establish your own pressure limitations for a gun, then you would want to know how to make the most efficient use of the maximum available energy. It seems important to me.
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Unread postAuthor: jimmy101 » Fri May 28, 2010 4:49 pm

dbracer443 wrote:Every gun has limitations, If you establish your own pressure limitations for a gun, then you would want to know how to make the most efficient use of the maximum available energy. It seems important to me.

Fair enough point. But as a practical matter usually performance is the goal and not efficiency.

There are certainly exceptions to that basic statement though. For example, if you are pumping up a large pneumatic with a hand pump then efficiency becomes a much more important characteristic (unless you firing the gun for exercise). Or, if you are pressurizing from a limited source of compressed gas, like a CO2 cartridge.

Still, I think most spudders are much more concerned with power than getting everything possible out of the energy in the chamber. It is usually pretty easy, and often basically free, to increase the size of the chamber and increase the performance even though the efficiency drops a bit.

Besides, spud guns are pretty darn inefficient to start with. IIRC, 10% efficiency is about the norm for a combustion gun unless it is a fairly high pressure hybrid which might approach ~25% efficiency. Pneumatics may have higher efficiency (particularly if you ignore the inefficacies of the compressor system) but still, instead of tweaking the ratio to get that last 0.5% of efficiency you can just boost the chamber by 20% and get a couple percent increase in muzzle velocity, better "performance" increase than what you get by working for optimal efficiency.

BTW, in a combustion gun there is a very distinct thing that occurs at the optimal CB ratio. If you plot projectile velocity versus projectile position there is a maximum velocity at a unique projectile position. If the projectile is still in the barrel beyond that point then it is actually decelerating.

Pneumatics do the same thing. In both cases, maximum efficiency is obtained if the barrel ends at the exact distance where the following is true:
Pressure of the working gas - dynamic friction ammo-to-barrel (expressed in pressure units) = atmospheric pressure.

(ignoring compressive affects on the air in front of the round, friction between the working gas and the barrel, heat loss, adiabatic cooling of the working gas ...)

In a compressed air rifle, with a zero friction round and assuming zero blow buy (a Teflon round in a Teflon barrel?) then the efficiency of the gun is rising up until the point the chamber pressure equals atmospheric pressure. A shorter barrel wastes energy. A longer barrel starts to "suck the ammo back" towards the chamber and also lowers efficiency.

In a combustion gun things are a bit more complex. In a combustion gun heat loss is a significant factor in the operation of the gun. You don't have very long to take advantage of the temperature induced increased pressure. If you could build a combustion gun with zero temperature loss the optimal CB ratio would be in the vicinity of 0.1, as opposed to the measured optimal of about 0.8. In other words, the optimal barrel would have about 9x the volume of the chamber instead of a chamber and barrel of nearly the same volume.
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Unread postAuthor: dbracer443 » Sat May 29, 2010 11:11 pm

jimmy101 wrote:

BTW, in a combustion gun there is a very distinct thing that occurs at the optimal CB ratio. If you plot projectile velocity versus projectile position there is a maximum velocity at a unique projectile position. If the projectile is still in the barrel beyond that point then it is actually decelerating.


Good Point. I took your earlier advice and did generate this curve as you can see in the edited version of my first post of this thread. I didn't carry it out far enough to see the point where the velocity starts to drop, but it looks like this occurs very far along the barrel. It makes me wonder if my model is very accurate. I guess the temperature drop in guns must be very significant to cause such a large change in the optimum CB ratio, as you said, .1 to .8.
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Unread postAuthor: Technician1002 » Sat May 29, 2010 11:48 pm

This optimum is of course dependant on pressure. This curve applies to the pressure range modeled. If using a lower pressure, a larger chamber is required. The inverse is also true. Higher pressure can use a smaller chamber. With this in mind, I have removable barrels with a variety of lengths and diameters for various uses.

@ dbracer443, I don't know if you have seen my page on measuring projectiles in the barrel as they accelerate. I think the data may be of interest to you.

I see you have a low post count and are fairly new. This page may be of interest to you. If you need help with the page or wish to reproduce the experiment, feel free to contact me for details on using a sound card instead of a scope and printer. I still have a limited collection of the test results.
https://inteltrailblazerschallenge.wikispaces.com/Barrel+length+trim+method
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Unread postAuthor: btrettel » Sun May 30, 2010 8:02 am

dbracer443, I don't exactly follow what you did in my quick run though, but thanks for posting that. I always like looking at people's code. Here's a somewhat old version of my non-dimensional simulation: http://trettel.org/nerf/ndm/pneu-sim.py

If you run that Python code it'll generate a giant table. I'm not a particularly efficient programmer, so that could take 40 hours or more. :lol:

My simulations are not based solely on energy. I start with the conservation of energy, mass, and momentum and use the Reynolds transport theorem to get some integral equations. Okay, actually, the fluid mechanics book I use did this for me and I just use the integral equations with appropriate substitutions. I then use the lumped parameter approach (i.e. barrel pressure/density/temperature is independent of location in the barrel and is different from the chamber pressure) on some of the variables to simplify things, and ignore some other things that have turned out to be okay to ignore (gas kinetic energy and momentum often are fine to ignore). After that there's a lot of simplifying and rearranging of the differential equations.

I've been meaning to post up a derivation of my most recent simulation. I'll see what I can do over the next few days. Feel free to email me if I forget. btrettel at umd dot edu

Edit: In my first post I meant problem, not project. I'm prone to substituting similar words when I make a typo.
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Unread postAuthor: jimmy101 » Sun May 30, 2010 3:57 pm

An estimate of the optimal CB for efficiency is generally pretty trivial to calculate and you should be able to do it in your head. Yes, actually you should be able to do it in your head.

The simple optimal ratio for efficiency is just the point at which the pressure pushing the projectile out of the barrel equals the atmospheric pressure in front of the ammo trying to push the ammo back into the barrel.

For a pneumatic it is trivially easy to calculate. For a combustion it is much more complicated to get even a remotely accurate estimate of the optimal CB.
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Unread postAuthor: c11man » Sun May 30, 2010 4:52 pm

jimmy101 wrote:The simple optimal ratio for efficiency is just the point at which the pressure pushing the projectile out of the barrel equals the atmospheric pressure in front of the ammo trying to push the ammo back into the barrel.
.


what about barrel friction? you would have to add the force of friction to the atmopheric pressure side of the equation
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Unread postAuthor: D_Hall » Sun May 30, 2010 5:06 pm

c11man wrote:what about barrel friction? you would have to add the force of friction to the atmopheric pressure side of the equation

Not so much.

Static friction can be significant for combustion launchers because it can affect burn dynamics, but that little tidbit aside, friction is largely irrelevant in a well designed launcher.
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Unread postAuthor: Technician1002 » Sun May 30, 2010 5:24 pm

jimmy101 wrote:
For a pneumatic it is trivially easy to calculate.


This is a common error. Calculating the valve performance at higher pressure is far from trivial. As near SOS air speeds in the valve have an impact on the maximum flow through a valve, many builders find performance in the real world is much less than trivially calculated.

This often results in barrels that reach peak velocity well before the projectile reaches the muzzle as peak flow through the valve is less than assumed.

On this page, some of the factors are mentioned.
http://www.crazybuilders.com/item.php?id=000002&type=project_section
In his videos, he shows calculated performance is quite a bit higher than measured performance.
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Unread postAuthor: btrettel » Sun May 30, 2010 5:41 pm

jimmy, based on your idea, I used adiabatic relationships to come up with the following relationship for ideal C:B ratio in pneumatics:

CB = 1 / (Pc*^(1 / k) - 1)

(The ratio is CB:1. I'm using the notation I used in my optimal C:B ratio thread.)

This is actually very reasonably accurate for medium pressures and up. I had never considered trying this to compare, and I'm now glad that I had. This is not a bad estimate at all for pressures above 30 psig. It's about 10% low (i.e. the barrel is too big) at 30 psig. Note that this only applies for subsonic projectiles.

I'm going to see if it's more accurate for a frictionless system.

Edit: Note that the equation above assumes the efficiency is 100%.

Edit 2: I've found what I believe is the exact solution to optimal C:B ratio for pneumatics. This fits my numerical results nearly perfectly.

CB = (1 + Vd*) / ((Pc* / (1 + Pf*)) ^ (1 / k) - 1)

Pf* = Pf / Patm (Pf is the "pressure of friction" much like how GGDT does friction)
Vd* = Vd / Vb (Vd is the "dead" volume)

I assumed Pf* = 0.1 and Vd* = 0.001 for my simulations.

Neat. I didn't know it would be this simple.

Note that for these equations to be valid, mc* (dimensionless projectile mass) needs to be over a certain amount.
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Last edited by btrettel on Mon May 31, 2010 4:58 pm, edited 1 time in total.
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