Optimal C:B ratio as a function of pressure in pneumatics

[velocity3x]

@ btrettel,
What would be optimal C:B ratio at 2000psi?

[btrettel]

Below is an updated (slightly more accurate and including higher C:B ratios) table. Remember that the assumptions involved make this most accurate for low-speed guns, though, it should be a decent starting point for high-speed guns.

<table border="1"><tr><td>P_c*</td><td>C:B</td><td>efficiency</td></tr><tr><td>1.75</td><td>2.54352</td><td>0.17151</td></tr><tr><td>2</td><td>1.87803</td><td>0.22151</td></tr><tr><td>2.5</td><td>1.25462</td><td>0.29807</td></tr><tr><td>3</td><td>0.95535</td><td>0.35443</td></tr><tr><td>3.5</td><td>0.77841</td><td>0.39807</td></tr><tr><td>4</td><td>0.66092</td><td>0.43314</td></tr><tr><td>5</td><td>0.51371</td><td>0.48655</td></tr><tr><td>6</td><td>0.42449</td><td>0.52582</td></tr><tr><td>7</td><td>0.36424</td><td>0.55625</td></tr><tr><td>8</td><td>0.32058</td><td>0.58073</td></tr><tr><td>9</td><td>0.28740</td><td>0.60098</td></tr><tr><td>10</td><td>0.26121</td><td>0.61808</td></tr><tr><td>12</td><td>0.22239</td><td>0.64558</td></tr><tr><td>14</td><td>0.19482</td><td>0.66690</td></tr><tr><td>16</td><td>0.17417</td><td>0.68404</td></tr><tr><td>18</td><td>0.15805</td><td>0.69820</td></tr><tr><td>20</td><td>0.14503</td><td>0.71014</td></tr><tr><td>25</td><td>0.12137</td><td>0.73329</td></tr><tr><td>30</td><td>0.10527</td><td>0.75017</td></tr><tr><td>40</td><td>0.08451</td><td>0.77328</td></tr><tr><td>50</td><td>0.07163</td><td>0.78836</td></tr><tr><td>75</td><td>0.05358</td><td>0.80935</td></tr><tr><td>100</td><td>0.04408</td><td>0.81895</td></tr><tr><td>150</td><td>0.03415</td><td>0.82387</td></tr><tr><td>200</td><td>0.02909</td><td>0.82010</td></tr><tr><td>250</td><td>0.02615</td><td>0.81221</td></tr><tr><td>300</td><td>0.02433</td><td>0.80208</td></tr><tr><td>350</td><td>0.02320</td><td>0.79067</td></tr><tr><td>400</td><td>0.02254</td><td>0.77853</td></tr><tr><td>450</td><td>0.02221</td><td>0.76601</td></tr><tr><td>500</td><td>0.02215</td><td>0.75335</td></tr></table>

You can use linear interpolation to get points between the ones in the table.

P_c* is absolute pressure in atmospheres. So P_c* = 1 refers to atmospheric pressure and P_c* = 2 is twice atmospheric pressure.

So 2000 psig is about 137 atma. g refers to gauge pressure (referenced against atmospheric) and a refers to absolute pressure (referenced to zero pressure). So the optimal C:B ratio is about 0.0367:1 and the efficiency at that pressure is 82.3%.

Remember that for this to be true, m_p* must be large (> 1000 should be adequate in the majority of situations--this simulation was run at m_p* = 10000). I define m_p* as m_p / (rho_atm * V_c) where m_p is the projectile mass, rho_atm is atmospheric air density, and V_c is the gas chamber volume.*

For those who note that I don't have P_c* = 1.5 in the table above, note that I'm rerunning the simulation at some lower pressures and I needed to use a slightly different configuration to do that. Those numbers aren't done yet.

* I just realized that an higher m_p* might result in a different plot. I'm going to investigate that after the current simulation is done running.

[btrettel]

I've found that the trend I noted earlier (that efficiency asymptotically approaches the maximum for high projectile masses) isn't true. The projectile can be too heavy.

Right now I'm rerunning the simulation over a wider range of projectile masses and pressures. We'll have some results in a few days.