Final velocity of an air rifle pellet from compressed gas

[szopaw]

As in the title, I'm trying to establish the approximate velocity (sans friction and other losses) of a pellet propelled by compressed gas from a tank. Below is what I have came up with myself, I would appreciate if someone could review this as say whether the end values are reasonable.

I have looked for a formula that would give me the energy of the pellet (for which I know the mass and diameter), and I've found the formula for work done in volume on the wikipedia page for the adiabatic process, which goes like this:

W=P1 * V1^γ * ( V2^(1-γ) - V1^(1-γ) ) / (1-γ)

Where P1 is the pressure (in Pascals) in the tank of volume V1 (cubic meters), V2 is the total volume after work has been done (so tank+barrel), γ is the heat capacity ratio (~1.4 for air for this purpose).

Assuming that all of the work is put towards the kinetic energy of the pellet (please tell me if the assumption is unreasonable), that would make the result be

v = sqrt(2*W/m)

Where W is the work from the previous equation and m is the mass (kg) of the pellet

After that, I've run a calculation for what reasonably could be a real air gun with the following values, for a simple scenario:

The starting pressure is 10 bar = 1000000 Pa

Volume V1 of the tank is half volume of the barrel. The barrel has the diameter of the pellet and length of 0.5m.

The pellet has a diameter of 6mm and mass of 0.25 g (0.00025 kg)

So that's V1 ~ 0.00000706858 m3 (~7068.58 mm3)

V2 is the barrel+tank, which is 2 * V1 + V1 = 3 * V1

For γ = 1.4, that gives us
V1^γ ~ 6.15 *10^-8
V2^(1-γ) - V1^(1-γ) ~ 74.032 - 114.886 = -40.866

Therefore

W ~ 10^6 * 6.15*10^-8 * -40.866 / -0.4 ~ 6.3 J

From that

v = sqrt(2 * 6.3 / 0.00025) = sqrt(50400) ~ 224.5 m/s

Does that seem like a number that's anywhere close to what should be true for a 6mm 0.25 g plastic BB pellet shot with 10 bar? Are there any considerable errors in either the assumptions or calculations?

 

[scottdave]

I assume that your 10 bar is gauge pressure. There is 1 atm acting against motion on the barrel side of the bb as it moves through the barrel. I think that air resistance will play a role. You may find the following sites (below) interesting.
The one from NASA has some cool animations showing how air flows around a sphere. But inside the barrel, air does not get to flow around the BB, it is getting shoved out the barrel, rather than flowing around the sphere.
If you look at the chart that i found on Crosman, there are several air rifle velocities much higher than the 225 m/s that you calculated, but these are with lead pellets. So I think with a plastic pellet, it won't take as much energy to accelerate it, but I think you should expect the actual muzzle velocity to be less than your non-friction calculations. How much less? I'm not sure.

https://www.grc.nasa.gov/www/k-12/airplane/dragsphere.html
http://hyperphysics.phy-astr.gsu.edu/hbase/airfri.html
http://www.crosman.com/pdf/CrosmanAirgunPelletCapabilitiesChart20120713.pdf

 

[szopaw]

I assume that your 10 bar is gauge pressure. There is 1 atm acting against motion on the barrel side of the bb as it moves through the barrel. I think that air resistance will play a role. You may find the following sites (below) interesting.
The one from NASA has some cool animations showing how air flows around a sphere. But inside the barrel, air does not get to flow around the BB, it is getting shoved out the barrel, rather than flowing around the sphere.
If you look at the chart that i found on Crosman, there are several air rifle velocities much higher than the 225 m/s that you calculated, but these are with lead pellets. So I think with a plastic pellet, it won't take as much energy to accelerate it, but I think you should expect the actual muzzle velocity to be less than your non-friction calculations. How much less? I'm not sure.

https://www.grc.nasa.gov/www/k-12/airplane/dragsphere.html
http://hyperphysics.phy-astr.gsu.edu/hbase/airfri.html
http://www.crosman.com/pdf/CrosmanAirgunPelletCapabilitiesChart20120713.pdf

Hey, thank you for those.

Yes, I'm considering gauge pressure and yes, I agree, surface area compared to it's mass dosn't do a plastic BB any favours. Certainly not compared to the drag/mass of a lead pellet.

However my main question is still rather about the adiabatic work and whether 10 bar gauge pressire could reasonably bestow 6 joules of energy on a plastic bb (or appropriately less with drag considered), or are my assumptions off.

 

[scottdave]

The lead pellets (at same velocity) have more kinetic energy, because of higher mass. The amount of energy output looks reasonable. Just that some of it is going to overcome friction. My guess is not a very high percentage, but noticeable.

 

[szopaw]

The lead pellets (at same velocity) have more kinetic energy, because of higher mass. The amount of energy output looks reasonable. Just that some of it is going to overcome friction. My guess is not a very high percentage, but noticeable.

All right, that's brilliant to hear. I'm not trying to aim for a specific velocity, that's why I'm not concerned about drag or exact values, but I was worried whether the pressure/energy could have been off by an order of magnitude.