# CO2 propulsion

1. Jun 16, 2004

### salamander

Hi. I've been wondering about these disposable co2 capsules for airguns:

Please share opinions on my reasoning, any hints or ideas of how to calculate this, what to look for and so on will be appriciated.

What if I punched a hole in the seal of one of these capsules, how much thrust would I get from the ejection of the gas?
I know the pressure in the capsule must be way above critical, thus I can assume that the co2 is in liquid state inside the cylinder. The co2 will vaporize on a constant pressure around 70-80 bar at around 20-30 deg. C, if I haven't got the concept all messed up. However the rapid vaporization of the co2 will need energy which must be taken from the capsule shell, thus cooling the capsule dramaticly, and also, I believe from the co2, which will lower vapor pressure. A probable diameter of the hole is about 1 - 2.5 mm and a capsule contains 16 g of co2. I guess the volume of such a capsule is about 10 - 15 cm^3.

Now imagine I'd put an expansion chamber on this and let the gas accelerate through a de Lavall nozzle, what would happen? May it be possible to reach supersonic flow, or will the pressure drop to much due to temprature loss in flow acceleration? I guess such an engine could be designed in such manner to make it operate at roughly the same pressure as a [chemical] model rocket engine by using a proper chamber in/out flow ratio. However I guess the difference in temprature between model rockets and my cold co2 flow will limit the nozzle expansion ratio, and maybe even make a standard nozzle counter productive. What do you think?

What I'm really interested in is wheter you, making a rough estimation, think that a capsule of co2 will deliver enough impulse to make a useful rocket engine. I've heard of simple co2 powered cars and rockets on strings, but i have no idea how effective they are. And yes, I'm going for the non chemical rocket altitude world record

Last edited: Jun 16, 2004
2. Jun 16, 2004

### enigma

Staff Emeritus
What is a de Lavall nozzle?

3. Jun 17, 2004

### salamander

Isn't that what they call a rocket nozzle, you know converging-diverging...? with a throat.. I was quite sure they're called de Lavall nozzles or something, but I might as well be really really wrong. Sorry 'bout that. :uhh:

Anyway what I was wondering was wheter it would be possible to accelerate the co2 flow using a nozzle, the same type used for rockets, or wheter a different design is required.

4. Jun 17, 2004

### griffin

any type of flow can be accelerated using a nozzle, the problem will be how much you can accelerate it to. For that part I would suggest researching a little bit more about fluid mechanics. (bernulies equations are what govern acceleration of a fluid).

As for the rest of the problem, if there is 16 grams of co2 and there is 15 cm^3 area, then the density should be 1.0667 g/cm^3. pressure inside can be found by the perfect gas law which is Pressure exerted on your gas= Density of your gas *The gas Constant of your gas* Temperature of your gas. in the case of co2 the gas constant (R) would be 188.9 J/kg*K. once you have the pressure then use that as the initial pressure in the bernulies equations.

I hope this helps, but I very seriously doubt that you will be able to reach supersonic speeds with a paintball shooter (essentially), but then again I haven't crunched the numbers.

P.S. (The only place I ever heard De Laval nozzle is on October Sky, which was a very good movie, but I have not heard of it scince.)

5. Jun 17, 2004

### enigma

Staff Emeritus
I've never heard the term de Laval before. They may be "officially" called that, but I've only heard them called "converging-diverging" nozzles.

I don't think it will work too well.

First off, using Bernoulli's equation for this is a bad idea. Bernoulli makes the assumption of incompressible flow - fluid which will not increase its density as it speeds up. This assumption only holds for Mach numbers up to ~0.3. Higher than that, you'll need to use thermodynamics if you want any accuracy. As you accelerate a flow through a supersonic (converging-diverging) nozzle, the pressure, density, and temperature all drop. If you go much higher than Mach 1, which is the highest possible at the throat, the three variables drop off very rapidly.

Because your chamber pressure (the pressure of the CO2 gas in the canister) isn't very high relative to real engines, you're not going to be able to expand the flow very far before your exit pressure is lower than atmospheric pressure. If it does end up lower, it'll create a 'suction' at the nozzle exit which can (and my guess is will...) negate any positive impuse due to the mass flow of the gas.

Another major problem will be pressure regulation. As the canister depletes, the chamber pressure will drop significantly. Because you'd be expanding the flow through the nozzle, any drop in chamber pressure drops the exit pressure by a much larger amount. That will not only reduce the mass flow of the engine, but you'll still have to contend with the overexpansion 'suction' effect. Real engines, even cold gas thrusters which are sometimes used for satellite stationkeeping have the propellant tank seperated from the chamber by pressure regulators to counteract that problem.

All that being said, it will be possible to put a nozzle on, you'd just have to design it correctly. My guess is that the added mass and aerodynamics of the nozzle will end up not being worth it. That is only an educated guess though. Why don't you run a few numbers to find out for sure.

$$\frac{T_0}{T}=1+\frac{\gamma-1}{2}M^2$$
$$\frac{p_0}{p}=(1+\frac{\gamma-1}{2}M^2)^{\frac{\gamma}{\gamma-1}}$$
$$\frac{\rho_0}{\rho}=(1+\frac{\gamma-1}{2}M^2)^{\frac{1}{\gamma-1}}$$
$$\frac{A}{A*}=(\frac{\gamma+1}{2})^{\frac{-\gamma+1}{2(\gamma-1)}}*(1+\frac{\gamma-1}{2}M^2)^{\frac{gamma+1}{2(\gamma-1)}}/M$$
$$a=\sqrt{\gamma*R*T}$$
$$\frac{dm}{dt}=\rho*V*A$$
$$Thrust=\frac{dm}{dt}V_e + A_e*(p_e-p_{amb})$$

Those are the 3 isentropic flow relations, the expansion ratio/Mach number relation EDIT: Latex doesn't like this one... you can find it here, equation #9, the formula for the speed of sound, the formula for mass flow, and the thrust equation.

The way you find the thrust follows:

Like griffin stated, from the ideal gas laws, determine your chamber conditions. You then apply the isentropic flow equations at M#=1 to determine your conditions at the throat. At the throat, you can take the density, with the speed of sound detmine the flow velocity. With those, you can find the mass flow. Once you've got the mass flow, you can play around with the nozzle expansion ratio to get the exit M#, and from that run through the isentropic equations again to get exit temp & pressure. With those you can get the exit velocity and determine the rocket's thrust. I'd recommend using a spreadsheet.

Remember: use consistant units, X0 is chamber conditions, Xe is exit conditions, R is the specific gas constant (universal gas constant/molecular mass), gamma for CO2 is 1.3.

Give it a go, and post what you find out. I'm curious what comes of it.
If you've got any questions, just ask.

Last edited: Jun 17, 2004
6. Jun 17, 2004

### Staff: Mentor

Even if your idea isn't practical, your intuition is correct - supersonic wind tunnels work exactly the way you describe, just on a larger scale. The one I used at USNA had nitrogen tanks that discharged into a converging/diverging nozzle. IIRC, the pressure was on the order of 1200psi.

7. Jun 17, 2004

### Parker

hello,

I have used the small co2 canisters for rockets....the set-up was very simple...I was truly amazed at what kind of speed was attained by the rocket.It flew too fast for me to follow it.The only bad thing about flying them is that they don't have a parachute, so when they land, they sink into the ground a few inches...so take cover and launch them away from any homes.....They can be easily made with just; electrical, or duct tape, a Bic pen, a nail, some cardboard, and a few feet of pvc pipe....I would like to calculate the height it may reach...but we need to some testing first....

8. Jun 21, 2004

### salamander

Ok. Hey enigma thanks for the advice. I've seen most of these equations before and yet I think I know how to use them now, I still lack the knowledge to derive or get any deeper meening in most of them, which is quite annoying... but well, it'll pass. I got all summer...

It would be nice though if you could help me with the 6th of your equations:

dm/dt = rho * V * A

I'm not quite sure how to use this. I guess it is the mass flow... or? What V and A should i use? Exit conditions?

EDIT: GAH, Just pay no attention to the above.
Mass flow is ofcourse constant through the entire nozzle so i can use any V and A as long as they are from the same spot in the nozzle, right? ... I hate it when my brain gets stuck.

Large gas cylinders containing liquid gas under high pressure gives a fairly steady pressure since the gas vaporize at a given pressure (critical pressure). However those cylinders (like welding tubes) work with signifficaly slower gas flows than I wish to achieve. Could I use this principe to stabilize the "chamber pressure"? Can I find the drop of chamber pressure when the engine is running by using the ideal gas law (pressure is simply lost due to escape of molecules)? Or will I need a new set of fancy thermodynamical equations? (Challenge me! I'm not afraid of maths )

Yes the weight and aerodynamics of such a nozzle probably do make it questionable. However... there are canisters for shotguns that contain roughly 100 g of co2. On such a capsule the nozzle weight fraction is lower and so is probably also its contrbution to the drag coefficient. Btw. the nozzle will be lathed from a block of pvc plastic.

Parker:

Sweet. Do you think this sort of propulsion have any potential, I mean, do you think you could actually lift some payload with a canister? If you put the capsule in a launch tube, did you estimate or otherwise elliminate the ammount of impulse due to overpressure behind the canister in the tube?

Cheers fellows.

Last edited: Jun 21, 2004
9. Jun 22, 2004

### Parker

I'm sorry that I really don't understand what you mean, about the "overpressurization in the launch tube."...is what you mean, that the rocket is shot out of the launch tube, like a bullet out of a gun? If, so this is not the case.....the co2 canister releasing the gas, produces the thrust, not the build-up of pressure behind the rocket forcing it out (of which really didn't exist in my set-up). the diameter of my rockets were around 5-6 inches at the most, while the pvc tube I used was around 8 inches in diameter. The pvc was just used as a launching guide, just like the steel bars that are used for grain engine model rockets.....I have been thinking about setting up a system to measure the thrust these co2 canisters produce....I build all kinds of model rockets before I went to college (I don't have a lot of time now)....If I were to compare the weight and liftoff acceleration of the co2 rockets I build vs. the store bought models, I would have to say that co2 rockets weighed roughly the same as the medium sized "D" powered rockets. The acceleration off the launch pad of the co2 rocket was by FAR faster....almost too fast to see!!! so, it probably has a relatively large impulse....SO, if the two types of rockets weigh about the same, then I would guess that the co2 canisters produce a relatively larger amount of thrust.....just a very rough guess. I just found out on the estes site that their "D" engines have an impulse of about 10-20 Newton seconds, a max lift of of 283 grams, and a max thrust measurement of 30 Newtons....so not bad, if the co2 cansisters are in the same ballpark of values.....

10. Jun 23, 2004

### salamander

Sorry if it was unclear, but you got it right, I was wondering wether the rocket had been blown ut of the tube like a bullet from a gun. D engine doesn't sound to bad. How large canisters did you use? Please tell me more about your experiments.

Last edited: Jun 23, 2004
11. Jun 23, 2004

### Parker

I used the very small canisters....the 12 gram canisters used in bb and pellet guns....what size were you looking to use?

parker

12. Jun 24, 2004

### salamander

I'm planning to use 12 g.
There are 90 g and 46 g canisters as well that are quite interesting, but the 46 g 's have to small lenght/diameter ratio (they are short and chubby, almost spherical) so if initial tests give positive results, the next step would probably be 90 g shotgun canisters.

I'd suppose a bigger cylinder has better thrust/weight ratio as well as a longer burntime. Might there be some other factors?

13. Jun 25, 2004

### dingpud

Just a thought. Could you somehow get the CO2 canister to drop off after it is empty?

Also, safety issue, I know that this was a dumb idea, and I know that it does not pertain to the applications in which you are researching, however I thought I would mention it. Regarding the small CO2 canisters, do not fill them with any explosive powders. I knew of a friend who was doing that with the powder from a firework sparkler and one time, kaa-boom. Shrapnel everywhere. Blew out 3 or 4 windows of a barn and left shards in the walls. Previous attempts, the CO2 canister would launch into the air, some the idea was persued. After the explosion, all further attempts were halted.

Best of luck with your rocket though.

14. Jun 26, 2004

### salamander

Hi dingpud!
I could probably make the canister drop off with some kind of mechanic ejection, but that would probably be both heavy and complicated (think springs or parachutes... detection systems...). A simpler approach would be giving the engine section a larger diameter or othervise significantly magnify the drag on the lower part of the rocket to make it fall off after engine burnout. However dropping the lower section would most likley make my rocket aerodynamicly unstable and make it start tumble through the air, which is not desireable.

As I want to reach as high as possible, I find it hard to see what benefit I would have from a sudden reduction in rocket mass at top speed. Think of it like this: after engine burnout the rocket is slowed down by aerodynamic drag that is proportional to the square of it's velocity, as well as it is slowed down by gravity. How high the rocket will reach is dependent of the energy it has at burnout, the more energy, the higher it gets. Kinetic energy is 1/2 * m * v^2, hence, a greater mass is desireable. Sure, during acceleration the rocket is slowed down by it's mass, but when top speed is reached ejection of mass is a waste of energy.

And thanks for the warning. :)

Hey, Parker!
How big a hole should i make in the canister? I'm making a puncture device now (basicly a spring and a needle) and don't want to waste any canisters on finding it out. Just a hint would be great... 0,5 mm? 1 mm? 1,5 mm? ...2 mm?? (I will of course try different needles later on but right now I could use some guidelines.) (By the way, using Enigmas equations i found that a hole of something like .75 - .85 mm would accelerate the flow to mach 1. However I don't think they're fully apliceable in this case so I'd rather take advise from someone who have tried this than from my own delerious calculations...)

Also, what kind of force is required and do I need to incorporate a mechanism to drag the needle out of the canister or will it just blow off by it self?

By the way, I'm building a launch mechanism that will serve as a prototype to find out wheter multi staging is practical.

Cheers fellas.

Last edited: Jun 27, 2004
15. Jun 27, 2004

### Parker

I really never actually measured the diameter of the hole in the canister...I would guess that your calculations of about .75 - .85 mm sounds about right. I am also going to do some experiments and calculations in order to tweak the canister to produce the most thrust theoretically possible. The object that you decide to make the puncture, should be expelled from the escaping co2, I never had a problem with the nail getting stuck in it....As for the force required to make the hole isn't too much. The spring mechanism should deliver plenty of force. I will draw a picture of my set-up in paint or something to show the basics of my design....but its not that good...

parker

16. Jun 28, 2004

### dingpud

just thoughts

I was thinking that you could use an e-match, gunpowder charge all off of an electric timer. I wouldn't think that the weight of it would matter that much, and dependent on the mass of your empty tank of CO2, you might be able to get just a little bit higher.

Just suggestions, take them as you wish.

17. Jun 28, 2004

### salamander

Parker

Thanks! Man, what would I do without you?

Dingpud

Oh, you mean that I could get some extra impulse by ejecting the engine after burnout...? it is an interesting thought and using an electrical timer and gunpowder the setup won't be that hard nor heavy. However, I'm doubt that such a rocket could be concidered non-pyrotechnical... and so gunpowder is not an option. Left is having a spring, and that is probably also possible. The impulse would be less, though. I'm not sure if I'm going to try it. Perhaps in a later project. I was planning to get the rocket down as one piece. Great idea though.

Ejecting the canister could also be a good way of recovery. Parachute deploy and canister ejection could easily be integrated. If the main rocket contains valuable equipment, a lower rocket mass att descent would require a smaller parachute and hence make the entire vehicle lighter at takeoff, or allow a gentler landing. This might assume that the canister is allowed to fall free. Grab your helmets boys and take cover!

The current vehicle is just a canister with trailing fins, very simple, and dangerous, made from straws and cardboard. It is not designed to take payload. Having center of pressure approximentally 7-8 rocket diameters behind center of gravity, it has proven stable even at low speeds. Which is nice, scince i do not want it to go astray if Parker is right about the acceleration of these things... I am concidering a simple recovery system, perhaps a helicopter recovery (you know like paper-and pin choppers) or a chute. Most likley however is that I will never see it again.
I have manifactured a ridicolously simple accelerometer from a couple of ballpoint pens. The principe is to have a wheight pulling a spring and a small piece of pencil led record the peak load on a paper. This equipment will require a working recovery system, most likley time-controlled. I will use the following one or two rockets to develop such technology, before I launch the accelerometer, or some other payload.

Oh lord how I love rocket science...
Cheers fellows.

18. Jun 28, 2004

### enigma

Staff Emeritus
If you have a way to contain the CO2 after puncturing it (or performing the puncture mid ascent), you could just do a 'fire-in-the-hole' staging method. Take a look at the interstage "fairing" on the Soviet N-1 Since yours is made from straws, you could have the first stage with vertical straws and the 2nd with toothpicks (or similar). Basically, have the upper stage merely sitting on the lower stage with no hard point connections.

Fire off the upper stage when the lower stage is nearly depleted (or out), and basically blow off the lower stage.

19. Jun 28, 2004

### Parker

My paint file is too large to file on the forum, would you like me to email it to you? It isn't anything special....

It's just the very basic construction of my first rockets....

Thanks

parker

20. Jun 29, 2004

### dingpud

If you could post a link to any video that you have when you fly it, I think that all of us would love to see it.