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

Post questions and info about pneumatic (compressed gas) powered cannons here. This includes discussion about valves, pipe types, compressors, alternate gas setups, and anything else relevant.
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Unread postAuthor: velocity3x » Sun Apr 25, 2010 4:16 pm

@ btrettel,
What would be optimal C:B ratio at 2000psi?
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Unread postAuthor: btrettel » Sun Apr 25, 2010 4:45 pm

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.
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Unread postAuthor: btrettel » Sun May 02, 2010 11:50 am

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.
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