Calculating air gun energy vs PSI & bore diameter

[R_Rose]

So I'm trying to figure out what kind of energy that I can expect out of a few different setups.

Here are the barrel bore diameters.
.224
.308
.5
.77
The barrels are usually rifles and some projectiles are full size (size of bore) while others use sabot's where a .224 or .308 can be shot from the larger bores of .5 or .77 using a plastic adapter.
"bullet" or pellet weight can range from 20 grains up to 900 grains over the 4 bores.

There are also arrows or bolts that can be fired using something like a sabot to make a seal around the shaft and the arrows weigh about 150-300 grains.The pressure in these guns usually run from 2500-3000psi when fully charged but there are some that can reach 5000 with special equipment (and being built for such pressure). I'm trying to figure out what the calculations would be to determine the energy being put behind a projectile using these PSI ratings as well as how much energy might be in the projectile itself (how many joules or ft/lbs and maybe how fast it could be expected to travel).

Expect the barrels to run from 16-28 " and can be made for ideal power.

Can anyone help me in figuring out where to start for this?

 

[jrmichler]

Get a copy of The Airgun from Trigger to Target, by Cardew and Cardew. It's mostly about spring piston airguns, but they also cover precharged pneumatics. It's an excellent place to start.

 

[berkeman]

I think you may be able to ballpark this using some simplifying assumptions...

• Assume that the driving pressure doesn't drop during the shot (requires a high pressure, moderate volume reservior)
• Calculate the net force on the projectile using the driving pressure and atmospheric pressure (to give you the pressure difference) and the cross-sectional area of the barrel and projectile
• That net force will accelerate the projectile for the duration of the barrel length
  
• The projectile will exit the barrel with a velocity and energy defined by the force through the distance (length of the barrel)

Does that make sense? Can you post those calculations so we can check them? Also, can you post what the first order corrections should be to this simplified calculation to get a better match to real-world numbers?

EDIT / ADD -- and please be sure to include units in the equations you post -- that helps to provide a sanity check on calculations and unit conversions (like grains to grams to kilograms)...

 

[cjl]

berkeman: Unfortunately, that is likely to give a result that is substantially too high, since an airgun like this will be firing the projectile at a speed which is non-negligible relative to the mean molecular thermal speed in the gas itself. As a result, as the projectile accelerates, the driving pressure will decrease because a significant portion of the gas's energy is going into accelerating the driving gas, not just the projectile. You could maybe assume isentropic acceleration of the gas with the projectile and get a better estimate that way, though it'll be a significantly more complex problem.

 

[berkeman]

berkeman: Unfortunately, that is likely to give a result that is substantially too high, since an airgun like this will be firing the projectile at a speed which is non-negligible relative to the mean molecular thermal speed in the gas itself. As a result, as the projectile accelerates, the driving pressure will decrease because a significant portion of the gas's energy is going into accelerating the driving gas, not just the projectile. You could maybe assume isentropic acceleration of the gas with the projectile and get a better estimate that way, though it'll be a significantly more complex problem.

Sure, that's why I asked him/her to also post their thoughts on first order corrections.