Thrust from a compressed gas cylinder

[Techno97]

How do I calculate the thrust from a CO2 gas bottle?
2,000 PSI pressure.
Looking for maximum thrust available from this power source.
Does nozzle size matter?

 

[astroclone]

Nozzle size matters, no matter what people tell you. And it's been a while for me, but I think it's all about mass ejection over time. John Anderson's book Fundamentals of Aerodynamics has some exellent nozzle information and even pressure discharge stuff.

Anecdotal info on nozzle size:
I was delivering pizzas in 1999 and I dropped the 2-Liter of Coke that someone wanted (only a dollar or 2 over grocery store price) and it hit on the top. The cap cracked and broke off and the 2-Liter shot across the street. It was pretty cool.

 

[princy]

I work in the compressed gas industry, some years ago a stuntman approached our company to build a co2 rocket engine that would spin his car in circles upon activation. With a full cylinder 60Lb of liquid CO2 & an exit port on the electric valve of about 3/8" we were surprised that the thrust lasted about 3 seconds before the balance of the liquid (about 40Lb) just froze into dry ice in the cylinder. We tried the same setup with N2 at 2000psi (CO2 only good for 1000psi) and it worked beautifully without freezing.
Cheers.

 

[borjame]

Size does matter, but that's another forum.

If you really want to know the max thrust you can get, here is the easist way: one sqin. hole = (2000 - 14.7) lb thrust (thrust=deltaP*area). Pretty simple right? But like Princy pointed out, you'll want to know how long. That goes well beyond the scope of my typing as it is a very long differential problem involving the temperature and pressure drop of a rapid decompression. It's also a matter of choked flow; the nozzle could get choked so that differential eqn becomes a computer program that takes into acct choked flow.
You can alos blackbox the problem to get energy instead of thrust and then you can convert that into anything you want (i.e. can I move an underwater scooter with a CO2 tank, the answer in "no" by the way unless you have A LOT of tanks or you don't want to go too far). Calculate the energy required to compress a tankful of CO2 to 2000 psi and that is the max amount of energy you can get back out.

Assume tank vol1 = 1 ft^3
p1 = 2000 psi
p2 = 14.7 psi
T1 = 70 degreesF or 530 degreeR

m(co2) = pV/RT R(CO2)=35.11 ft-lbf/lbmol-R
m(C02) = 15.5 lbm

v2 = 136.3 ft^3, now reverse it so we are compressing the gas

If the process is polytropic and steady flow (not to sure this assumption):
delta h = n (p2V2 - p1V1)/n-1 n~1.25 for a good compressor
delta h = -2599.2 ft-lbf or 3.3 BTUs

3.3 BTU ain't much

 

[gsmollin]

AS princy noted, CO2 has a complicated phase diagram. At 68 degrees F, the material in a cylinder is a liquid with a vapor pressure of about 800 psi. As it vaporizes, it will cool rapidly and freeze. If you heat it, at about 85 degrees F it will come to a critical point, at about 1800 psi. Above that temperature CO2 becomes a supercritical fluid, with density properties varying between liquid and gas with temperature and pressure. Still, with the rapid expansion required in a rocket engine, it will cool to its freezing point of about -100 F pretty fast.

Since any fluid will have the same temperature drop, you need a working fluid that remains gas or liquid down to the lowest temperature you will expand it too. Nitrogen may work, but you may want a heavier gas to get the specific impulse. As the fluid gets heavier, its freezing point rises, of course.

I imagine there is cold-gas-jet literature out there. Most rockets use helium for cold gas jets, but those tanks are also used to pressurize liquid propellants, so helium is preferred since it is not reactive.

 

[gsmollin]

Arrgh! I don't see a way to edit posts here. I wanted to add that the CO2, or other working fluid only gets cold where it expands. In princy's case, they allowed an entire cylinder of gas to expand. It got cold and froze. To avoid this problem, the liquid CO2 would be piped into an expansion chamber (aka rocket engine) where it would expand and exit through a nozzle. The rocket engine would become very cold, but the CO2 in the cylinder would remain warm, and liquid, until the tank was down to about 10% fill, when all the CO2 would become gas.

This method is used in a simple fire extinguisher. If you have ever used a CO2 extinguisher, you know the CO2 comes out frozen as "snow". It falls around the fire, then sublimes to a gas and smothers it. You also do gets some reactive "kick" from the extinguisher, but its not really too exciting. Not rocket engine stuff. I think you'll want a better working fluid than CO2.

 

[space1space]

Highest specific impulse is with the lowest molecular weight gas. As was pointed out, CO2 will not produce 2000 psi as the critical pressure is about 1100 psi at 95 F. Using a separate chamber just gives you another location to produce the ice. That chamber would quickly go to the freezing point of the CO2.

Helium's specific impulse at 2000 psi is right up there with a fair rocket propellant. Trouble is it is so low in density you can't carry much mass. There is certainly a tradoff between density and Isp.

Expelling liquid instead of gas in the case of CO2 improves the results. The Cricket sounding rocket used a low freezing point liquid with the CO2 with excellent results.

 

[LarryC]

The simplest representation of thrust is the Zucrow rocket equation, F = [ M / 32.1 ] x Ue where F is the thrust in pounds, M is the ejected massflow rate in lb/sec and Ue is the ejection velocity. This is not adequate for design, but it helps in understanding a couple of important points.

The velocity Ce will vary with the pressure difference between the internal cylinder pressure and the ambient pressure you're working in. With a gas as your working fluid, this will vary from a very high to a very low value, approaching zero velocity as the pressure becomes fully equalized. This is a very wide range of velocities to have to tolerate during operation.

With a liquid as the working fluid, the action is improved in several ways. Suppose you start with half the cylinder filled with liquid, and half compressed gas. The range of pressure variation will only be 2:1, since the gas will only double in volume by the time the liquid is fully ejected. The jet velocity will be much lower [for a given nozzle area] due to the viscosity of the liquid, but this is more than compensated for by the MUCH larger mass flow as long as liquid is available. Also, the gas pressure is available for a much longer time, again because of the reduced exit velocity. All this explains why a kid's water rocket works so well when half full of water, and so pitifully when you try it with air alone [even though there is twice as much air available, and you can pump it up to the same starting pressure].

Now, there is an additional advantage if you really MUST use CO2: If you size the nozzle so ejection is slow enough, the expansion of the CO2 won't cause freezing of the gas. Getting the nozzle size just right would be a matter of experimentation. As suggested above, water probably isn't the best choice for the working fluid, since it will freeze before the CO2 does; however, it is probably the only 'non-polluting' liquid you can use under most circumstances. I don't think an old-style Seltzer bottle ever freezes up, no matter how constantly it gets used [but the nozzle area is so small, you probably don't get much thrust, either].

The trick would be, rather than go for absolute maximum thrust, try to size the nozzle so you get good usable thrust as long as the liquid lasts, without freezing either the liquid or the CO2 above it. Naturally, when using a liquid working fluid in this way, I'm assuming that you can always ensure that the liquid stays in contact with the nozzle until it's exhausted. In the water rocket, both gravity and forward acceleration take care of this for you. If you need horizontal ejection, or you're expecting inertial effects [e.g. sloshing of liquid during cornering], you'll have to do a lot more complex design to keep it working. [This would not be the case with gas as the working fluid.]

There is no point in trying to use a deLaval [convergent/divergent] nozzle for a liquid rocket stream, since the velocity will be relatively low; a simple, very smooth-surfaced convergent nozzle with a straight exit should be ideal.

LarryC