Calculations relating to a pneumatic launcher

[mikemo]

TL;DR Summary

After calculating (to the best of my ability) the acceleration of a slug in a pneumatic launcher then actually measuring the speed of the slug, I'm trying to understand why my results are drastically different than my calculations.

Hello, and thanks for reading. Any help with this will be greatly appreciated.

One of my hobbies is amateur radio. We often use wire for antennas, and getting those wires up in high places can be a challenge. I built a pneumatic "launcher" for this purpose, using ideas from others who have done the same. It is a fairly simple device consisting of an air tank, a valve and a barrel. Air is pumped into the tank and the slug (projectile) is loaded in the barrel. When the air is released from the tank it accelerates the slug down the barrel, then hopefully the slug will drag your line over the tree or branch that you want for your antenna.

The device works, and my backyard antenna is back up after several months of sagging on the ground. I was curious, however, as to the physics involved and if I could use my limited physics knowledge to roughly calculate the speed of the slug when it leaves the barrel. I made some assumptions to simplify the problem.
1) The tank volume was significantly larger than the barrel volume, so I assume the pressure on the back of the slug is constant as it is pushed down the barrel.
2) That there is no air escaping around the slug as it moves down the barrel.
3) That there is no friction between the slug and the barrel.

I know these are big assumptions, but the idea was to simplify the problem to a point where I could solve it, which would show me an absolute maximum value, which would be greater by some amount from the actual speed of the slug. I was happy with my work, but was still curious as to how different the calculated and measured values would be. I put together a simple electronic measurement device to attempt to measure the speed of the slug as it left the barrel. It is an infrared LED and a photo transistor, with an Arduino Uno microcontroller to measure the pulse width and do the math to display the calculated speed on a console. As the slug passed the LED it blocked the light and that change is seen by the computer. The width of the pulse, and the measured length of the slug gives the required info to display an approximation of the speed.

My calculated speed was 47 m/s, and I assumed that my measured speed would be significantly less than that. I tested 5 launches and measured an average of 80 m/s of the slug. I was surprised. Just as a sanity-check, I dropped the slug from 1m height and measured the speed ad 4.8 m/s, which is close to what I expected on earth. I also verified my pressure gauge with another gauge and they match.

Now I don't know why there is this discrepancy between calculated and measured values. I was hoping that someone could look at my work and point out any errors. Here is a summary of my calculations:

Tank pressure = 2.5 kg/cm^2
The slug is cylindrical.
Slug diameter = 3.5 cm
Slug mass = 0.075 kg
Barrel length = 36 cm

I started with the area of the back surface of the slug calculated to 9.6 cm^2. The air pressure at 2.5 kg/cm^2 should exert a force of 24 kg(f) on the slug.

I know that F=ma. I multiplied the 24 kg(f) by 9.81 to get the force in Newtons (235 Newtons). Then solving for a in Newton's 2nd law, 3145 m/s^2

Using d=1/2(a)(t^2) solving for time, I get 0.015 seconds to get down the barrel.
Using v=a(t) to find the velocity I get about 47 m/s (or around 155 f/s).

I know this was an awfully long post but I wanted to present as much info as I could. Any insight on this problem or my methods will be greatly appreciated. Mike M.

 

[Dullard]

I don't see anything wrong with your approach or calculations. I am suspicious about your slug parameters. Assuming that the length is twice the diameter, it is about the density of water; If it's the diameter is the same as the length, it could be aluminum - both seem unlikely choices for a slug. Slug material and length?

 

[mikemo]

Thanks for the reply. I think "slug" may be a poor choice of words. Projectile might be more appropriate. It is made from 3/4" schedule 80 PVC pipe and two pipe caps. Weight was added by inserting a 2 ounce lead fishing weight before gluing on the caps. The total weight is 75 grams and the length is 7.5 cm. I machined a groove around the outside of one of the caps for an o-ring, sized to just fit in the barrel. It slides down the barrel with just a little tap.

 

[Dullard]

Just in case / Please confirm:
Tank Pressure: 35 PSIG
Barrel Length: 14"

 

[mikemo]

Yes, that is correct. 35 PSI and a 14" barrel.

 

[Dullard]

Best Guess:
Something is wrong with your measurement. Possibilities:
Your phototransistor isn't turning 'ON' fast enough - the apparent pulse width is shortened. At low speeds (drop test) it is much less significant. Depending on how you're timing the pulse (interrupt, etc) some latency is also possible. A < 1mS error would explain the difference between your predicted/actual velocities, and would be almost undetectable in the drop test.

 

[mikemo]

Thank you for verifying my calculations. Yes, now it's time to have a closer look at my measurement setup. I have the gain of the phototransistor quite high, and that can cause slower responses (Miller capacitance multiplication). I'll measure pulse response on the bench and respond with my results.

Thanks!

 

[mikemo]

Well, you nailed it. There is a major issue with my sensor. The capacitance in the phototransistor was high enough to cause a slow recovery of the signal when the sensor went dark. I lowered the gain by about a factor of 6 and the signal looks quite a bit better. Still, it seems that if I want an accurate measurement at faster speeds I may need to go to a photodiode and some additional amplifier circuitry.

With the original gain configuration, I changed the pulse width until I saw readings that were in the 80 m/s range, then worked backwards from the pulse width. This quick-and-dirty measurement puts me at an actual 33 m/s when the sensor was reading 80 m/s. Much more in line with what's expected. I'll use the modified sensor tonight and run some more tests (ambient light does affect the reading, so nighttime tests are necessary).

As is, I expect the modified sensor will provide fairly accurate readings below 40 m/s.

Thanks again for your assistance Dullard!

 

[Dullard]

Good Luck! File the serial numbers from any potatoes that you might have lying about.

 

[mikemo]

Will do :)

I measured 5 shots last night and averaged 33 m/s. Now I can sleep at night. Thanks again.

Any hints on the math needed to compensate for the drop in pressure as the air from the tank expands? My calculations assumed a constant pressure as the projectile traveled down the barrel.

 

[Dullard]

There will be a pressure reduction due to the increase in total volume (the barrel). Beyond that, I'm over my head. I suspect that the relative volumes of the tank and barrel make a more detailed thermo analysis unnecessary. Friction losses between the tank and barrel are probably the most significant factor in the difference between actual/theoretical instantaneous pressure, and that would require modelling (also over my head).

 

[jrmichler]

Any hints on the math needed to compensate for the drop in pressure as the air from the tank expands? My calculations assumed a constant pressure as the projectile traveled down the barrel.

Your initial calculations are good. I made the same assumptions and got the same result. The next step is a simulation program. Such a program can be done using a spreadsheet, with the equations for each time step across several columns, and successive time steps down the rows. A spreadsheet is a little clumsy, but gets the job done. I prefer Matlab, or its free clone, Octave, for this type of problem.

Without a good diagram / photo of your launcher, I have to guess a little. You have a reservoir, a valve, and a tube. I assume that the valve is directly connected to the tank, and the tube to the valve. The tank outlet has an entrance loss (Bernoulli equation), the valve has a Cv, the valve has a finite opening time, and the tube has friction loss. There is an initial gap between the valve and the projectile, calculate the volume of that gap. Ignore pressure drop in the tank, leakage past the projectile, and projectile friction for now.

If your valve is a ball valve with hand operator, valve opening time is significant. You can measure that by making a video and counting frames on playback. Now put together a series of equations:
1) Initial upstream pressure is the tank pressure, initial downstream pressure is atmospheric.
2) Initial projectile position, velocity, and acceleration are zero.
3) Valve percent open, calculate equivalent orifice area.
4) Calculate flow through the valve. The valve restriction will control flow when the valve starts to open, while some combination of valve Cv and entrance loss will control flow at full open. So include both in the calculation.
5) Calculate pressure between valve and projectile after one time step. You adjust the size of the time step to be small enough that the pressure does not change too much, and large enough so the the program runs in a realistic time. It will probably be between 0.001 and 0.0001 second.
6) Calculate the acceleration of the projectile from the pressure behind it. Ignore pressure in front of it for now.
7) Calculate the velocity from the acceleration.
8) Calculate the position from the velocity.
9) Loop to #3, repeat until the projectile leaves the tube.

Things to experiment with after you get it to work include length of tube, valve opening time, projectile friction, leakage past the projectile, tank pressure drop, adiabatic vs isothermal, etc. Keep it as simple as possible until it works correctly, then add in complexity as needed. I probably forgot a step or two, but the above should be enough to get started. I once wrote a simulation program for a high speed air cylinder that closely matched the experimental results, so this approach will work. I predict that valve opening time will be more significant than you think.

 

[mikemo]

jrmichler,
Thanks for that excellent information. I had to read it through a few times but I believe I understand what you are saying.

The construction is close to what you assumed. Parts are assembled in a straight line (no bends). The rear is the tank, the center is the valve, and the end is the barrel. I don't have my notes with me, but I do have all the measurements of volume and length.

The valve is a pressure actuated valve, consisting of a modified irrigation valve. There is a diaphragm that seals the inlet side. A small passage in the diaphragm allows the pressure on the inlet side to build to the same level on the back side. A spring adds additional force to hold the diaphragm against the inlet, sealing it. Also, the surface area on the back side of the diaphragm is larger than the inlet side, adding to the sealing force. There is a release valve that quickly vents the pressure on the back side of the valve, allowing the inlet pressure to force the diaphragm up and release the air into the barrel. When I pull the trigger to release the air, it fires fast enough for my human senses to say "instantaneous".

I've never used Matlab or Octave, but this looks like a good time to learn. I'm not sure how to estimate the speed at which the valve opens, but I suppose I could start by modeling the system with no loss, all the pressure arriving at the back of the projectile at t=0, no friction, etc. Then I could try adding other factors.

Thanks!