# Very nice, analysis, Frank! I only wish I had as good an answer to offer...

June 1 2012 at 1:31 PM

Response to Analysis HDD improvements on a MRod 25

...to your question about an analytical solution to the equations of airgun interior ballistics. Unfortunately, I don't.

However, if "(un)reasonable approximations," interest you, here're a couple.

The "average" driving pressure at the breech required to achieve a given muzzle velocity is determined by...

1. Caliber.
2. Barrel length.
3. Pellet weight.
4. Propellant density.

Here's the semi-empirical formula I've had some predictive success with...

Feet_per_second = 172 x sqrt(Q / (grains + Z x Q))

Where: Q = Pavg x barrel_inches x caliber^2
Pavg = average driving pressure in psi
Z = 0.011 for air, 0.02 for CO2, 0.002 for helium.

In a PCP, the generation of the driving pressure occurs in two phases...

1. "Isobaric" phase = the interval while the firing valve is open and connecting the breech to the reservoir, maintaining (approximate) constant pressure.

2. "Adiabatic" phase = after the valve closes and the dispensed charge expands (and cools) behind the pellet, causing an exponential drop in pressure proportional (for a - mostly - diatomic gas like air) to volume1.4.

If there's a close-form solution to this mish-mash, I have no idea what it would look like. But simple iterative solutions converge rapidly and seem to lead to fairly realistic results.

Have fun.

Steve

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