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

[btrettel]

I wrote a dimensionless low-speed pneumatic gun simulation recently. Why dimensionless? 1) Results are independent of size and can be scaled and 2) the number of inputs decreases. In brief, I've found optimal C:B ratios as a function of pressure. Optimal was defined as when energy efficiency (percent of energy stored in the chamber that was transferred to projectile kinetic energy) hit a peak.

The table below and the right equations can be used to design efficient pneumatic guns and predict their performance accurately BY HAND. Still, it's always worthwhile to check against GGDT or another simulation and with a chronometer. The point of this table is to make some rules for design.

<table border="1"><tr><td>P_c* (atm.)</td><td>C:B ratio</td><td>% efficiency</td></tr><tr><td>1.5</td><td>4.03</td><td>11.0</td></tr><tr><td>1.75</td><td>2.54</td><td>17.2</td></tr><tr><td>2</td><td>1.88</td><td>22.2</td></tr><tr><td>2.5</td><td>1.25</td><td>29.8</td></tr><tr><td>3</td><td>0.956</td><td>35.4</td></tr><tr><td>4</td><td>0.662</td><td>43.3</td></tr><tr><td>5</td><td>0.514</td><td>48.7</td></tr><tr><td>6</td><td>0.425</td><td>52.6</td></tr><tr><td>7</td><td>0.364</td><td>55.6</td></tr><tr><td>8</td><td>0.321</td><td>58.1</td></tr><tr><td>9</td><td>0.287</td><td>60.1</td></tr><tr><td>10</td><td>0.261</td><td>61.8</td></tr><tr><td>12</td><td>0.222</td><td>64.6</td></tr><tr><td>14</td><td>0.195</td><td>66.7</td></tr><tr><td>16</td><td>0.174</td><td>68.4</td></tr><tr><td>18</td><td>0.158</td><td>69.8</td></tr><tr><td>20</td><td>0.145</td><td>71.0</td></tr></table>

P_c* is dimensionless initial chamber pressure. One way to think about this is that it's your pressure in atmospheres. P_c* = P_c / P_atm

C:B ratio is the C part of the C:B ratio. So 0.145 means 0.145:1. C:B ratio does not count dead space (I have another parameter for that).

The last column is energy efficiency at this configuration. This can be used in combination with the energy efficiency equation (which I'll edit in later) to find muzzle velocity at certain parameters.

Pressure has a strong effect on energy efficiency. Higher pressures mean higher efficiencies at optimal C:B ratios.

For optimal configurations, energy efficiency is largely independent of flow coefficient. This area is where the dimensionless projectile mass essentially is larger than a critical number. This critical dimensionless projectile mass is a function of the pressure, C:B ratio, and flow coefficient. In general for good flow coefficients, if this is greater than 200, you are in the range. However, sometimes a dimensionless projectile mass as low as 30 is okay. The only way to know is to check against my more general table, which isn't yet ready to be posted.

Dimensionless projectile mass is defined as m_p* = m_p / (V_c * rho_atm)

m_p is the projectile mass.

V_c is the chamber volume.

rho_atm is atmospheric air density.

Apologies if posting this seems a little premature as I'm not completely done, but I thought some might appreciate it.

In the coming months I'll post the code used to generate this table and a detailed explanation of its derivation. I made some fairly restrictive assumptions (minimal projectile friction, minimal dead space, air is the propellant, etc.), but they should be good for most high performance launchers. The simulation is approximately equivalent to GGDT without some valve dynamics. The valve is assumed to open instantaneously. The low effect flow coefficient had for "heavy" projectiles indicates that even if opening speed is suboptimal, performance isn't affected much.

[saefroch]

Can you post a table showing, independently, the effect of C:B ration and chamber pressure alongside the other?

[btrettel]

I'm not completely sure what you mean. Here's a spreadsheet full of more general tables: http://trettel.org/nerf/ndm/pneu-eng.csv

The two numbers above each individual table are Cd (valve efficiency with the barrel area as the reference area) and pressure ratio, respectively. The top row is dimensionless projectile mass. The left column is C:B ratio. The numbers in the table are efficiencies.

I intend to make a prettier version of that spreadsheet later.

This might be hard to believe, but generating that table took about 40 to 50 hours of computing time.

[boyntonstu]

C:B - the higher the pressure the shorter the chamber for a constant diameter.

A dynamic C:B ratio makes sense.

I have had little success in modeling GGDT to my actual chrono readings.

If GGDT fits a 100 gr projectile it won't fit a 15 gr one, etc.

i have heard this complaint from others.


Any ideas?

[btrettel]

If your measurements are above about Mach 0.5, I wouldn't put much faith in GGDT or my current simulation. That sounds like your problem. These simulations take what's called the "lumped parameter" approach, which breaks down for higher speeds. I'll eventually release a high speed simulation.

[boyntonstu]

If your measurements are above about Mach 0.5, I wouldn't put much faith in GGDT or my current simulation. That sounds like your problem. These simulations take what's called the "lumped parameter" approach, which breaks down for higher speeds. I'll eventually release a high speed simulation.

Correct!

All at 650 fps or higher.

I look forward to any improvement in air cannon modeling at the higher speeds.

Thanks for your hard work.

BoyntonStu

[Technician1002]

The original table is listing efficiency. In simple terms, as you go to a larger chamber for more power, efficiency drops. You do get more power, but in diminishing returns as the chamber becomes oversize. I have directly seen this with air cannons in using the 3 gallon chamber or the marshmallow cannon to launch tennis balls. Using the small compressor the amount of time to recover after each shot is very noticeable. Even though the marshmallow cannon with the large TB barrel is noticeable much less power, it make very efficient use of the power for some remarkable launches for the tiny chamber.

The Marshmallow cannon with the 3 foot TB barrel launches rolled T Shirts 200 feet at 90-100 PSI. The 3 gallon launcher can't quite reach 400 Feet on 80 PSI. The compressor takes quite a while to recover after the shot with the large cannon, but the small one is pretty much recovered in time for the next shot. It is load, fire, repeat with no wait.

[velocity3x]

If your measurements are above about Mach 0.5, I wouldn't put much faith in GGDT or my current simulation.

I found GGDT to be quite good. These are shots I did a couple months ago. Doppler radar was used to determine the velocities.

Shot 1. Golf Ball / nitrogen / 100psi / 635 fps.
Shot 2. Golf Ball / nitrogen / 100psi / 651.5 fps.
Shot 3. Golf Ball / nitrogen / 500psi / 1,036 fps.
Shot 4. Golf Ball / nitrogen / 700psi / 1,194.22 fps.

After each shot I checked GGDT against the test result. In each case GGDT predicted the velocity to be within a few < fps of the actual test shot. As for GGDT....I have no complaint.

[Technician1002]

When using GGDT at above 0.5 Mach, often other factors come into play as turbulence in bends, edges, valves, etc are not included in the calculations. I found this was an issue on test shots with the large t shirt cannon. With a high efficiency valve, and less plumbing than normal, I was getting higher than GGDT predicted results until I moved the valve efficiency up until I got a warning that the valve COF was optimistic. Then i started to match GGDT. I was not expecting to use above 60% on the cannon to match GGDT. It was a nice surprise.

[boyntonstu]

When using GGDT at above 0.5 Mach, often other factors come into play as turbulence in bends, edges, valves, etc are not included in the calculations. I found this was an issue on test shots with the large t shirt cannon. With a high efficiency valve, and less plumbing than normal, I was getting higher than GGDT predicted results until I moved the valve efficiency up until I got a warning that the valve COF was optimistic. Then i started to match GGDT. I was not expecting to use above 60% on the cannon to match GGDT. It was a nice surprise.

That is encouraging information.

Did GGDT predictions fit various weight projectiles with other parameters held constant?

[velocity3x]

With GGDT valve efficiency set at 65%, its predictions were almost a perfect match of the actual shots.

[btrettel]

GGDT does do something for better accuracy at high speeds, but it still uses the lumped parameter approach. Pressure, density, temperature, etc. are independent of location in the barrel or gas chamber.

The problem is that the velocity of the projectile is comparable to the velocity of the pressure waves. A rarefaction wave will travel from the back end of the projectile, reducing pressure. Compression waves could turn into shock waves on the other side of the projectile. If the velocity of the projectile is far slower than the velocity of the pressure waves, the pressure, density, etc., equalizes very quickly, so the lumped parameter assumption is okay.

Imagine a tsunami. Is the height of the tsunami at the shore the same as that of the entire ocean? Of course not. Assuming that it is won't be a good way to model that phenomena. That's why the lumped parameter approach can be bad at high speeds. It doesn't capture the wave phenomena.

Now, the lumped parameter approach isn't necessarily bad at high speeds, but it's not always good, and I hope my analogy shows why.

Enough of high speed stuff. Let's get back to C:B ratios for low Mach numbers.

Anyone think the results of the simulation are reasonable? Checking against real world data is always good. I don't have access to any pneumatics at the moment, so I can't check.

When I have the time, I'll post some instructions on how to use this table to design spud guns. Basically, you can choose what performance you want and then find the dimensions and pressure that get that performance at high efficiency. Working backwards is part of the idea.

[Technician1002]

That is encouraging information.

Did GGDT predictions fit various weight projectiles with other parameters held constant?

Predictions held well once I figured out the valve COF was above 60. This was tested with apples in a 2 inch barrel, oranges and shirts in the 3 inch barrel and a Poof brand foam ball in the 4 inch barrel. I was surprised when the 4 inch 2 oz foam ball busted out the bottom of a HDPE bucket at 10 feet. I didn't think that could be done with a foam basketball.

A rolled up t shirt punched holes through 2 layers of wall to wall carpet draped over a sawhorse at about 10 feet.



All testing was done under 100 PSI. This was not due to the use of insane pressure.

The poly bucket broken with a foam ball.


Disclaimer, bucket bottom cracked with the foam ball. The rest of the damage was done with the t shirt.

Marshmallows make lousy projectiles. They don't fly straight and don't go more than about 180 feet. Moving to the Marshmallow cannon, I didn't expect to put dents in a car door at 15 feet. Oops. Somewhere in the old posts, I have photos of the car door dented with marshmallows. Marshmallows can bust full cans of soda.

Edit; Found a photo of the launcher with the various barrels and projectiles used for testing. The closest one is the foam ball and 4 inch barrel. If you look close, you can see the chrono wire for the magnet pickup spaced every foot on the 3 inch and 2.5 inch barrels. The 2 inch is not shown. A lot of data was collected to tune the launcher for the competition. The 3 inch barrel gave the best performance on the 2 inch valve. Shorter fatter shirts fly further.

[btrettel]

I've run the simulation for a wider range of pressures now and have found that energy efficiency peaks at about 150 atmospheres with an efficiency about 82%. Optimal C:B ratio seems to stop decreasing around there. This makes sense and I'll explain why.

Efficiency increases with pressure because higher pressures allow for lower C:B ratios (a longer barrel), which in turn allow more energy to be extracted from the gas because the projectile travels down the barrel for a longer time. However, as the chamber gets smaller relative to the barrel, the gas cools down more. So the C:B ratio can't continue decreasing at a certain point, and this means that a smaller percent of the total energy can be extracted. And as the total potential energy increases with pressure, efficiency will start to decrease at a certain point.

So about 2200 psi is the way to go!

Edit: This peak was an artifact from not using a high enough dimensionless projectile mass. There is no actual peak.

[Gun Freak]

Yes, I can see everybody on SpudFiles switching to high pressure tanks and nitrogen filling systems to achieve 2200 psi W00T!!