Thrust produced by a steam rocket

[Teslageddon]

I'm trying to calculate the thrust produced by a steam rocket, but I don't have that much information to work with (or at least information I don't know what to do with). It's 2m long and has a diameter of 1m, and it's constantly filled with water at a rate of 170.344L/sec, which is instantaneously superheated to 1000C and passed through a de Laval nozzle.

The convergent section of the nozzle is 7.62cm in length and 12.5cm in diameter at the opening. It converges to 5.08cm in diameter at the throat, which is 2.54cm in length. The divergent section is 30.48cm in length and expands to 50cm in diameter.

The only other information I have is that the mechanism that fills the chamber ejects the water with enough force to create a stream 6.096m high and .3048m wide when disconnected.

 

[Sunfire]

I read your post but am unsure what is the objective. Is the theory of thrust and propulsion unclear to you; or are you trying to plug in numbers into an equation but are unsure of the validity of the equation etc.

 

[Teslageddon]

Both, though more of the former I guess. I'm not sure how to calculate the needed variables based on the given information. For example, how would you calculate the exit velocity with the given information, or the exit pressure? Pretty much all I have now is the area(A), density(r) and specific heat ratio(gamma) for water, and the gas constant(R).

 

[Sunfire]

This thread may help. The links to other threads are very useful also, as well as other threads linked from those.

https://www.physicsforums.com/showthread.php?t=744967

 

[etudiant]

Well, you are claiming that you will put 170 liters of water as steam superheated to 1000 degrees C through a 5 cm diameter throat every second.
Excluding any changes of state, you have to push 170 liters through a 2.5x2.5 x 3.14 cm squared throat area every second. The throat area is about 20cm squared, so for 170,000 cubic centimeters per second the water flow must be 8,500 centimeters/second, or 85 meters/second, 310000 meters/hr, 310 km/hr.
Steam is about 800 times less dense than water at atmospheric pressure, so the steam will need to be under several hundred atmospheres just to get the volumes right, even before heating and even if the flow through the nozzle is supersonic, but doing the calculation is beyond me.

 

[russ_watters]

You can't have supersonic flow through the throat of the nozzle, so what is being asked here is not possible (not that the energy requirement would be easy either...).

What is the purpose of this question? Where does it come from?

 

[Teslageddon]

No purpose really, it was just a thought experiment proposed by my friend. He gave me the information about the fill rate and force, but I had to come up with the dimensions for the rocket. I couldn't get anywhere so I thought I'd find someone who might. Is there anyway to make this theoretically possible without changing the tank fill rate?

 

[cjl]

That's a nozzle expansion ratio of about 100:1, which gives a T_exit/T₀ of about 0.09. This means that at the exit plane of your nozzle, the temperature would be about 120 kelvin (based on your 1000C input temperature). This is clearly not going to happen, because this means the steam would tend to condense in your nozzle. To have any chance of having a viable steam rocket, you'll probably want to have a much, much lower expansion ratio on your nozzle so the exhaust still has a significant amount of heat.

 

[russ_watters]

No purpose really, it was just a thought experiment proposed by my friend.

...and your friend just pulled these numbers - to 2 or 3 decimal places - out of the air? Forgive me, but that seems unlikely.

He gave me the information about the fill rate and force, but I had to come up with the dimensions for the rocket. I couldn't get anywhere so I thought I'd find someone who might. Is there anyway to make this theoretically possible without changing the tank fill rate?

Make the nozzle large enough that your choked flow is exactly supersonic.

Also, have you considered how best to harness all of the output of a medium-sized power plant in order to power this thing? Is the goal to carry the power plant with you?

At the risk of that sounding a bit condescending to a new user, I want to make sure you understand just how bizarre what you are suggesting is and see if we can help you learn from that.

 

[mheslep]

You can't have supersonic flow through the throat of the nozzle, ...

Why is that? Particulars of this nozzle not sufficient?

 

[cjl]

Make the nozzle large enough that your choked flow is exactly supersonic.

That's actually not too hard - any nozzle with a pressure ratio greater than about 2 across it will have choked flow, which by definition means that the flow will be sonic at the throat. The mass flow rate will vary depending on the conditions (pressure, mainly) in the chamber though.

As for the large number of decimal places, that appears to be an artifact of the fact that whoever "designed" this initially did so in US standard units, then converted to metric - that flow rate is 45 US gallons/sec, for example, and that nozzle has a 2 inch diameter, 1 inch long throat, a 3 inch long convergent section, and a 1 foot long divergent section. These absolutely do sound like numbers that someone pulled out of thin air to me.

 

[russ_watters]

Why is that? Particulars of this nozzle not sufficient?

Choked flow is a fluid dynamic condition associated with the Venturi effect. When a flowing fluid at a given pressure and temperature passes through a restriction (such as the throat of a convergent-divergent nozzle or a valve in a pipe) into a lower pressure environment the fluid velocity increases. At initially subsonic upstream conditions, the conservation of mass principle requires the fluid velocity to increase as it flows through the smaller cross-sectional area of the restriction. At the same time, the Venturi effect causes the static pressure, and therefore the density, to decrease downstream past the restriction. Choked flow is a limiting condition which occurs when the mass flow rate will not increase with a further decrease in the downstream pressure environment while upstream pressure is fixed.

http://en.wikipedia.org/wiki/Choked_flow

 

[russ_watters]

That's actually not too hard - any nozzle with a pressure ratio greater than about 2 across it will have choked flow, which by definition means that the flow will be sonic at the throat. The mass flow rate will vary depending on the conditions (pressure, mainly) in the chamber though.

Right, but having choked flow and the desired mass flow rate is a contradictory set of conditions.

 

[cjl]

Right, but having choked flow and the desired mass flow rate is a contradictory set of conditions.

I wouldn't say it's contradictory. Starting with choked flow + a desired mass flow rate as assumptions, you can calculate a set of chamber conditions (pressure + temperature) that will deliver the desired flow rate. There's not a unique solution either, since many possible combinations of pressure and temperature will give the desired mass flow rate. An additional constraint with water as the working fluid is of course that the chamber conditions must be such that the water is all in vapor form (which would probably be true at 1000C, but I haven't worked out the necessary chamber pressure and looked at the phase diagram to verify it).

 

[boneh3ad]

That's not contradictory at all. Given, this is a steam rocket and therefore a two-phase flow and therefore more difficult than just using air, but you could get an effective ratio of specific heats for the steam/water mix and calculate the necessary nozzle dimensions almost exactly for a nozzle for a given reservoir pressure, temperature, and throat diameter. It really is quite easy unless you want a more exact answer treating it like a truly two-phase flow.

 

[cjl]

That's not contradictory at all. Given, this is a steam rocket and therefore a two-phase flow and therefore more difficult than just using air, but you could get an effective ratio of specific heats for the steam/water mix and calculate the necessary nozzle dimensions almost exactly for a nozzle for a given reservoir pressure, temperature, and throat diameter. It really is quite easy unless you want a more exact answer treating it like a truly two-phase flow.

The biggest issue I remember having when I tried to work out the math for a steam rocket a few years ago is that it is pretty difficult to not have the majority of your mass flow condense out of the gas phase in the nozzle due to the temperature drop in the expanding region. Admittedly, the rocket will still work with the majority of the exiting mass as liquid rather than gas, but it does pretty severely hurt the efficiency compared to a pure gaseous flow.

 

[russ_watters]

I wouldn't say it's contradictory. Starting with choked flow + a desired mass flow rate as assumptions, you can calculate a set of chamber conditions (pressure + temperature) that will deliver the desired flow rate.

But we've already been given the chamber conditions. All of the conditions are already fully constrained.

 

[AlephZero]

But we've already been given the chamber conditions. All of the conditions are already fully constrained.

Yes, but the real rocket won't care if you over-specify its design conditions. It will just ignore some of your assumptions and obey the real laws of physics, not the imaginary laws you were hoping for.

 

[russ_watters]

Yes, but the real rocket won't care if you over-specify its design conditions. It will just ignore some of your assumptions and obey the real laws of physics, not the imaginary laws you were hoping for.

That's fine as long as everyone is clear that going down that path leads away from what the OP asked: we'll be making-up the problem as we go along by choosing some of our own design constraints and ignoring some of those of the OP.

It doesn't make the OP's question answerable, it makes a somewhat different, somewhat related question that we make up answerable.

 

[russ_watters]

But we've already been given the chamber conditions. All of the conditions are already fully constrained.

Er, wait - we only got the chamber temperature, not the pressure. So I guess we would have to see if you can compress steam enough to get it through the nozzle at that rate.

Here's a superheated steam table that gets us close to the conditions we are talking about. If we max-out the program at 999C and 216bar, we get:
http://www.spiraxsarco.com/us/resources/steam-tables/superheated-steam.asp

density: 37.2 kg/cubic meter
speed of sound: 854 m/sec

That's 64 kg/sec or a water equivalent of 64 l/sec (assuming that pressure will provide sonic flow). Way below the requested flow rate.

Also, a quick pass at the energy requirement: something on the order of 600 megawatts is required to vaporize the water and raise its temperature to 1000C. That's most of a nuclear reactor or about the same as a space shuttle main engine.