Modeling Air Compression Cylinder

[erobz]

So as a simple physics model applying conservation of energy treating the gas trapped between piston and bb as incompressible and without significant mass:

$$ \frac{1}{2}kx_o^2 = \frac{1}{2}M_{p} v^2 + \frac{1}{2} \frac{1}{3}M_s v^2 + \frac{1}{2}M_{bb} \left( \frac{D_p}{D_{bb}}\right)^4 v^2 $$

Is this in the ballpark upper bound for the velocity of piston as the BB leaves the barrel?

Giving for the BB and upper bound of:

$$ v_{bb} = \left( \frac{D_{p}}{D_b} \right)^4 v \approx 1500~\text{m/s} $$

If its even half that in actuality, I've done some other ballpark calculations that say the back pressure from accelerating the air ahead of the bb out of the barrel could be significant.

 

[Tom.G]

WOW! At mach 4.5 you could save a lot on gun powder.

 

[erobz]

WOW! At mach 4.5 you could save a lot on gun powder.

 

[erobz]

WOW! At mach 4.5 you could save a lot on gun powder.

It could be significant at mach 1 (or less) too. As far as I know the OP is looking for deviations in simulation vs. measurement.

So I just imagined a tube of length $L$ and diameter $D$ and constant friction factor $f$. For unsteady- incompressible flow of the gas in the barrel initially at rest I get this for the pressure in front of the projectile as it clears the barrel:

$$ P = P_{atm} + \rho l \dot v + \rho f \frac{L}{D}\frac{v^2}{2} $$

So for a constant a pressure differential across the tube the flow would begin to accelerate from rest. I ignore the decrease in mass, so a solution is able to be obtained.

I'm trying to answer "what pressure would I need to reach a certain velocity in the flow by the time an element would travel the length of the barrel":

with $u = \frac{v^2}{2}$ the ODE becomes:

$$ P = P_{atm} + \rho L \frac{du}{dz} + \rho f \frac{L}{D}u$$

After I solve for everything in terms of $v$ ( the desired projectile velocity at the end of the barrel) I get this:

It quadratic and here is a plot with some reasonable parameters I believe.

Sorry to mix English and metric but I just have more sense of scale with psi over pascals. Is a 10 psi back pressure ignorable, what is the pressure behind the projectile pushing it?