Is there a limit on rocket engine exhaust velocity?

[tdp2010]

I've been trying to figure this out for days. I'm told that atmospheric pressure imposes a limit on maximum possible exhaust velocity in the Earth's atmosphere, and that under STP conditions that limit is approximately 15,000 feet per second. But that doesn't make any sense. Suppose you had an engine running at that exhaust velocity, and you halved the diameter of the throat of the combustion chamber through which the exhaust was flowing. That would reduce the area of the throat to a quarter of its previous size, and in accordance with Bernoulli's principle, the exhaust would have to accelerate to four times its previous velocity by the time it escaped the combustion chamber. At that point your engine would be producing four times its previous amount of thrust. Or, if you throttled back and reduced the fuel flow to a quarter of normal, it would produce the rated amount of thrust but burn for four times as long, dramatically improving its efficiency. Is there any reason, apart from the structural strength of the engine and materials, why this wouldn't work? Is there any reason why redesigning the chamber, throat and nozzle to to actively compress and accelerate the exhaust flow could not be done?

If anyone can explain this to me in simple terms without equations, I'd be extremely grateful. Thanks very much.

 

[Dr. Courtney]

Or you could have a CATO. At some point you realize that the weight of rockets is important and the non-burning components need to be a fraction of the fuel weight for these things to fly into space.

Fundamentally, you are probably right, given any physical configuration, raising the pressure in the combustion chamber will raise the exhaust velocity. It's not a hard upper limit, but matter diminishing returns.

 

[boneh3ad]

Actually it is a hard upper limit. First, Bernoulli's equation does not apply since this is a highly compressible flow. In a supersonic flow such as this, the velocity-area relationship is reverse and you actually have to make the nozzle exit larger to increase speed.

Second, as a gas expands, it gets colder. The maximum speed of gas coming from the nozzle is therefore limited by the gas's ability to cool further. Since it can't reach absolute zero, there is a hard maximum velocity.

Finally, this maximum is related to the reservoir conditions, not the outside pressure. Given that the flow is supersonic, information about atmospheric pressure can't propagate upstream through the nozzle throat anyway. I could provide more detail if you weren't anti-equations.

 

[Dr. Courtney]

Actually it is a hard upper limit. First, Bernoulli's equation does not apply since this is a highly compressible flow. In a supersonic flow such as this, the velocity-area relationship is reverse and you actually have to make the nozzle exit larger to increase speed.

Second, as a gas expands, it gets colder. The maximum speed of gas coming from the nozzle is therefore limited by the gas's ability to cool further. Since it can't reach absolute zero, there is a hard maximum velocity.

Finally, this maximum is related to the reservoir conditions, not the outside pressure. Given that the flow is supersonic, information about atmospheric pressure can't propagate upstream through the nozzle throat anyway. I could provide more detail if you weren't anti-equations.

Help me out here. The equations I've seen for the exhaust velocity depend on the square root of the temperature, which suggest diminishing returns rather than a hard upper limit. You can't lower the final temperature below zero, but you can keep increasing the initial temperature if you can contain the pressure, which is why it is impractical given weight constraints. I see how the techniques suggested by the OP will not work, but how does the hard upper limit apply if one instead increases the temperature?

 

[boneh3ad]

Help me out here. The equations I've seen for the exhaust velocity depend on the square root of the temperature, which suggest diminishing returns rather than a hard upper limit. You can't lower the final temperature below zero, but you can keep increasing the initial temperature if you can contain the pressure, which is why it is impractical given weight constraints. I see how the techniques suggested by the OP will not work, but how does the hard upper limit apply if one instead increases the temperature?

Maybe I just wasn't clear but that is what I was talking about.
$$u_{\mathrm{max}} = \sqrt{2h_0} = \sqrt{2c_p T_0}.$$
For a given reservoir condition, there is a hard upper limit. That limit can be moved by changing the reservoir conditions (specifically $T_0$). This also doesn't mean that the velocity will reach that speed for s given expansion, just that this is the theoretical maximum given the reservoir conditions in question.

I should add that this equation involves several assumptions that aren't strictly valid for a rocket nozzle such as adabaticity, but it's still illustrative.

 

[Dr. Courtney]

Thanks for the clarification.

 

[boneh3ad]

No there is no limit on rocket engine exhaust velocity. Amplified exhaust flow by change of lost motion in the exhaust flow environment.

Do you mind expanding on why you think the preceding discussion is incorrect?

 

[Johnny_Tsunami]

If exhaust velocity is the only question I'm guessing you want something useful, that can create a force. Otherwise just shine a flashlight! Photons instantly flying away with a velocity of c but not enough mass to create a useful force. If force is the actual goal you need mass and velocity but in a reasonable proportion. Take the mass of the propellant coming out of a Saturn V nozzle and squeeze it out of an 1/8" hole. Just for fun you could imagine that this would increase the velocity to something generating relativistic speeds. But that being an unreasonable set of equations you just get a massive overpressure and a big boom :-) Think Bugs Bunny with his finger in the end of the rifle, figuratively speaking.

 

[Gary Feierbach]

Jonny_Tsunami has the right idea. Once you are in space you can use light, accelerated ions, exploding nukes.
Ion propulsion engines are already in use.
A exploding nukes engine was proposed in the 1950s (Project Orion).
A successfully tested nuclear thermal engine (Superheated liquid hydrogen propellant)(Projects Rover and NERVA 1959-1972).
The NERVA engine, even thought it had twice the impulse power of a chemical rocket, was never flown. NASA lost its mojo (funding) after the moon shots.
(In government success often breeds failure because you then become a high profile target.)

 

[Baluncore]

The practical limit on exhaust velocity comes from the material properties of the combustion chamber.
The strength of the combustion chamber at the maximum temperature of operation determines the maximum pressure available to accelerate exhaust through the jet.
Cooling the outside of the combustion chamber with the fuels prior to injection into the chamber is one way of pushing that limit.

 

[berkeman]

Thread locked temporarily for Moderation...

Thread re-opened after some cleanup of misinformation posted by a PF newbie.

 

[betadave]

In a convergent, divergent nozzle, I think the limit might be imposed by the back pressure of the atmosphere causing the shock to form inside the nozzle. I am not an expert on the thermo involved in the calculations and so I don't feel qualified to take the argument much further than that. Any rocket nozzle is designed to a specific ambient pressure. When launched, the pressure is above ideal and it passes through ideal to where the pressure is below. If you have watched a launch at night, it is beautiful to watch the flame change as the rocket climbs and then the stage change occurs. It gives you a clear indication that the nozzle has limits.that depend on ambient pressure. It is a mistake to think you can change the throat diameter to get velocity changes, that changes thrust but not exit velocity for a given propellant.

 

[boneh3ad]

That's not really true though. Regardless of the ambient back pressure, there is always some level of pressure in the combustion chamber that would cause the nozzle to start (leave the nozzle). Of course that isn't exactly practical. Therefore, the theoretical limit on exhaust velocity is not related to back pressure, but the practical limit might be depending on the situation.

As previously discussed, the theoretical limit is due to the limited pool of thermal energy in the gas that can be converted to kinetic energy through the expansion. At some point the gas will theoretically approsch absolute zero and simply can't attain any more velocity. In practice, this manifests as material limits on the combustor, as the only way to raise that velocity limit is to raise the total temperature of the gas, but there are practical maxima on that.

 

[DaveC426913]

In a convergent, divergent nozzle, I think the limit might be imposed by the back pressure of the atmosphere causing the shock to form inside the nozzle. I am not an expert on the thermo involved in the calculations and so I don't feel qualified to take the argument much further than that. Any rocket nozzle is designed to a specific ambient pressure. When launched, the pressure is above ideal and it passes through ideal to where the pressure is below. If you have watched a launch at night, it is beautiful to watch the flame change as the rocket climbs and then the stage change occurs. It gives you a clear indication that the nozzle has limits.that depend on ambient pressure. It is a mistake to think you can change the throat diameter to get velocity changes, that changes thrust but not exit velocity for a given propellant.

Such the aerospike engine for the Venturestar back in the 90's.

Ah VentureStar. We miss ye. You were going to take us to orbit in style.

 

[Jeff Rosenbury]

That's not really true though. Regardless of the ambient back pressure, there is always some level of pressure in the combustion chamber that would cause the nozzle to start (leave the nozzle). Of course that isn't exactly practical. Therefore, the theoretical limit on exhaust velocity is not related to back pressure, but the practical limit might be depending on the situation.

As previously discussed, the theoretical limit is due to the limited pool of thermal energy in the gas that can be converted to kinetic energy through the expansion. At some point the gas will theoretically approsch absolute zero and simply can't attain any more velocity. In practice, this manifests as material limits on the combustor, as the only way to raise that velocity limit is to raise the total temperature of the gas, but there are practical maxima on that.

I seem to recall from my days of math porn that the hard maximum is dependent on the molecular weight of the propellent with lower weight causing higher impulse. That was from the thought of using a gas gun as a space launch system.

I'm older now and my desires are fading ... My days of solving differential equations are behind me. No more evil math porn for me.

 

[256bits]

That's not really true though. Regardless of the ambient back pressure, there is always some level of pressure in the combustion chamber that would cause the nozzle to start (leave the nozzle). Of course that isn't exactly practical. Therefore, the theoretical limit on exhaust velocity is not related to back pressure, but the practical limit might be depending on the situation.

As previously discussed, the theoretical limit is due to the limited pool of thermal energy in the gas that can be converted to kinetic energy through the expansion. At some point the gas will theoretically approsch absolute zero and simply can't attain any more velocity. In practice, this manifests as material limits on the combustor, as the only way to raise that velocity limit is to raise the total temperature of the gas, but there are practical maxima on that.

And also practical limits from the divergent section. The increase in area of the divergent section would have to be how large to obtain the temperature to approach absolute zero. After a certain exit area, diminishing returns on the exit velocity and thrust might not be worth it, just due to the extra mass of the nozzle.

 

[cjl]

I seem to recall from my days of math porn that the hard maximum is dependent on the molecular weight of the propellent with lower weight causing higher impulse. That was from the thought of using a gas gun as a space launch system.

I'm older now and my desires are fading ... My days of solving differential equations are behind me. No more evil math porn for me.

This is correct - the limit will basically be dependent on the mean molecular weight of the exhaust products and the temperature in the combustion chamber. Ideally, you'd want extremely low molecular weight and extremely high temperature. This isn't related to being in the atmosphere either. The presence of an atmosphere just limits how low you can make the exit pressure before you'll have flow separation, but you can still run your pressure ratio arbitrarily high (in theory) just by increasing chamber pressure. This will be limited by the structure of the chamber and pressure capability of your turbopump, but that's fundamentally a design/engineering problem, not a limit of the fundamental physics.

As for 15kft/s being the limit? Certainly not. The Space Shuttle main engine had an exhaust velocity around 4.3-4.4 km/s, which is around 14,400 feet per second, but higher is definitely possible. Experiments performed by Rocketdyne in the 1960s with certain highly volatile tripropellant rockets (liquid fluorine, liquid lithium, and hydrogen, if memory serves...) achieved specific impulses upwards of 540 seconds, which corresponds to an exhaust velocity of 5.3 km/s (17,500 feet per second). NERVA, a nuclear thermal rocket using hydrogen as its reaction mass achieved exhaust velocities of over 8 km/s (26k ft/s). Admittedly, in that last case, it didn't run a high enough pressure ratio to really be useful in the atmosphere, but with a higher chamber pressure, there's no physical reason it couldn't.