Reduced Efficiency at Sea Level

[MattRob]

Liquid-fueled rocket engines almost always seem to have lower specific impulse at sea level than in a vacuum.

I know of one reason:
1. Nozzle expansion. Many engines are optimized for higher-altitude performance, so their nozzle is over-expanded for lower altitudes. I would say more, but I think most people familiar with the topic are rather familiar with this one:

Underexpansion - the nozzle doesn't expand far enough and the gas plume exerts force on the air instead of the nozzle, wasting energy from the combustion.

Good expansion - all of the exhaust is going directly to the right in the picture, so the change in momentum is maximized.

Overexpansion - the nozzle goes too far, and ends up re-compressing back to ambient atmospheric pressure. As a result, some momentum is going inward and rightward (right in the picture) motion isn't maximized.

But here's what I'm curious about:

2. Atmospheric Pressure

So, largely going off of physics guesswork here, but I imagine the fact that there's air exerting a pressure into the combustion chamber will reduce the efficiency of the engine. If the combustion chamber pressure is 1 atmosphere, for example, then the exhaust won't flow out and you'll have zero efficiency, heh. If it's only 2 atmospheres, then it will barely leak out. If it's 200, then it will blast out and create plenty of thrust.

So I was wondering, can you reclaim most, if not all, of the reduced sea-level efficiency by using a higher chamber pressure?

 

[boneh3ad]

Sure, increasing the chamber pressure will increase the resulting exhaust pressure and reduce the degree of overexpansion (or eliminate it entirely). The problem is that you will rapidly run into the material limits of the combustion chamber.

 

[MattRob]

Sure, increasing the chamber pressure will increase the resulting exhaust pressure and reduce the degree of overexpansion (or eliminate it entirely). The problem is that you will rapidly run into the material limits of the combustion chamber.

So it is theoretically possible to make an engine that gets 440 seconds of ISP in an arbitrarily dense atmosphere?

Really the idea is I'm poking around with a sci-fi setting, trying to imagine what kind of shuttle would fit the bill: something for an expedition to a distant Earth-like world, with a great deal of uncertainty as to the atmospheric density of the planet.

There's no infrastructure on the surface, obviously, so SSTO is a requirement. Superior materials and lower gravity help, but a thicker atmosphere would definitely hurt quite a bit. The question is a matter of making engines that won't have a horrific ISP at the surface, more than any of the other wide range and depth of technical challenges.

 

[boneh3ad]

I won't comment on specific numbers since I haven't done any actual math to back it up, but I am 100% sure that you could create an exhaust jet that is not overexpanded at all if you could raise the chamber pressure high enough.

 

[MattRob]

I won't comment on specific numbers since I haven't done any actual math to back it up, but I am 100% sure that you could create an exhaust jet that is not overexpanded at all if you could raise the chamber pressure high enough.

That seems straightforward enough - but it's not expansion of the nozzle that I'm really curious about, it's the effect of external pressure on trying to stop the exhaust gasses from leaving the combustion chamber at all.

The SSME's get about 450 seconds of ISP in a vacuum. But I don't think any nozzle could give them 450 seconds at sea level. So it's not only nozzle expansion, it's also something else.

Once again, if matching nozzle expansion with chamber pressure / ambient pressure is all there was to it, then you could have a combustion chamber work at 0.5 atmospheres, have a shrinking nozzle, and still get no overexpansion or underexpansion - but that clearly wouldn't actually create thrust. So I think something is missing... The same thing that could give the SSME's 450 seconds of ISP at sea level, theoretically.

 

[SCP]

The SSME's get about 450 seconds of ISP in a vacuum. But I don't think any nozzle could give them 450 seconds at sea level. So it's not only nozzle expansion, it's also something else.

No, the SSME's can't get 450 seconds at sea level, even theoretically.

The thrust of a rocket engine can be shown to be: $$T=\dot{m}V_{e}+\left ( P_{e}-P_{a} \right )A_{e}$$ where the $e$ subscript refers to the nozzle exit conditions, and the $a$ subscript refers to ambient/atmospheric conditions. It's more complicated than it looks because $V_{e}$ is a function of $P_{e}$. It can also be shown that your best case scenario for both thrust and Isp is when $P_{e} = P_{a}$, which is the good expansion (usually referred to as fully expanded) case you refer to above.

So at low altitudes, the best performance occurs when the exhaust is expanded to (but not below) atmospheric pressure. This is also true at higher altitudes, but, of course, atmospheric pressure is lower up there. So we can get better expansion in a vacuum than we can at sea level. In other words, the best-case thrust and Isp is worse at lower altitudes. But it's even worse than that. Conventional rockets use bell nozzles, which have a fixed expansion ratio. So they have a fixed design altitude. At all other altitudes, they operate below the best case because $P_{e}\neq P_{a}$.

I recently did a numerical study (project for grad school) on altitude compensating nozzles. For it, I looked at an RP-1/LOX powered rocket. The theoretical best case Isp was ~280 seconds at sea level and ~360 seconds at 60 km. Bell nozzles (with fixed expansion ratios) had Isp's ranging from ~75 seconds to ~270 seconds at sea level and from ~305 to ~335 seconds at 60 km. The variation depended on the expansion ratio, and the design altitudes ranged from ~6,000m to ~26,000m.

Once again, if matching nozzle expansion with chamber pressure / ambient pressure is all there was to it, then you could have a combustion chamber work at 0.5 atmospheres, have a shrinking nozzle, and still get no overexpansion or underexpansion - but that clearly wouldn't actually create thrust. So I think something is missing... The same thing that could give the SSME's 450 seconds of ISP at sea level, theoretically.

The combustion chamber has to have a pressure higher than ambient in order to drive the exhaust acceleration. Even at the nozzle exit, if the pressure is too low (compared to ambient), it can cause separation (that's the fourth picture in your graphic above). In the SSME, some serious design work was done around the lip of the nozzle in order to allow an exit pressure down to 4 psi (averaged across the exit plane, the pressure at the lip is actually higher to prevent separation).

 

[MattRob]

No, the SSME's can't get 450 seconds at sea level, even theoretically.

The thrust of a rocket engine can be shown to be: $$T=\dot{m}V_{e}+\left ( P_{e}-P_{a} \right )A_{e}$$ where the $e$ subscript refers to the nozzle exit conditions, and the $a$ subscript refers to ambient/atmospheric conditions. It's more complicated than it looks because $V_{e}$ is a function of $P_{e}$. It can also be shown that your best case scenario for both thrust and Isp is when $P_{e} = P_{a}$, which is the good expansion (usually referred to as fully expanded) case you refer to above.

So at low altitudes, the best performance occurs when the exhaust is expanded to (but not below) atmospheric pressure. This is also true at higher altitudes, but, of course, atmospheric pressure is lower up there. So we can get better expansion in a vacuum than we can at sea level. In other words, the best-case thrust and Isp is worse at lower altitudes. But it's even worse than that. Conventional rockets use bell nozzles, which have a fixed expansion ratio. So they have a fixed design altitude. At all other altitudes, they operate below the best case because $P_{e}\neq P_{a}$.

I recently did a numerical study (project for grad school) on altitude compensating nozzles. For it, I looked at an RP-1/LOX powered rocket. The theoretical best case Isp was ~280 seconds at sea level and ~360 seconds at 60 km. Bell nozzles (with fixed expansion ratios) had Isp's ranging from ~75 seconds to ~270 seconds at sea level and from ~305 to ~335 seconds at 60 km. The variation depended on the expansion ratio, and the design altitudes ranged from ~6,000m to ~26,000m.The combustion chamber has to have a pressure higher than ambient in order to drive the exhaust acceleration. Even at the nozzle exit, if the pressure is too low (compared to ambient), it can cause separation (that's the fourth picture in your graphic above). In the SSME, some serious design work was done around the lip of the nozzle in order to allow an exit pressure down to 4 psi (averaged across the exit plane, the pressure at the lip is actually higher to prevent separation).

Excellent. Fantastic reply. What are some thoughts on Linear aerospike engines?

Little bit of an "oh no" moment, though, is the realization that this means there's no real great way to get around a very thick atmosphere. It looks like somewhere with 4-5 atmospheres will ruin your ISP no matter how you cut it. What kind of ballpark ISP could I expect at 59-74 psi ambient? Is there any sort of (extremely) rough approximation equation I could use?
I just need something to look/sound believable, so just a very rough approximation for sci-fi reasons, but I do want it to be actually realistic.
Would something with an inverse-exponential form work well as a very rough approximation?

$I_{sp} = ~ ~ I_{sp max}e^{\frac{-P}{k}}$

And to fit to the two data points (0, 455), (14.7, 347),

$347 = Isp_{vacuum}e^{\frac{-14.7}{k}}$

$k = -\frac{14.7}{ln{\frac{347}{455}}}$

$I_{sp} = ~ ~ 455 e^{\frac{P ln( \frac{347}{455} ) }{14.7}}$

 

[SCP]

What are some thoughts on Linear aerospike engines?

I think they show a lot of promise. They look good on paper and the ground test data seems to support the theory. But we're still pretty light on flight test data.

Would something with an inverse-exponential form work well as a very rough approximation?

That might work well enough for writing science fiction, realistic and plausible, but not exactly accurate.

there's no real great way to get around a very thick atmosphere.

You'll certainly see a reduction in efficiency. But even at high pressures, you should still be able to produce thrust. The thrust and Isp are both driven by the ratio between the exit pressure and chamber pressure, but structural stresses in the chamber are driven by the difference between the atmospheric pressure and chamber pressure. So I'd be more concerned about structural limits being reached before efficiency problems. But if you're writing science fiction, it's easy enough to beat structural problems by invoking better/stronger/lighter/more heat resistant materials.

What kind of ballpark ISP could I expect at 59-74 psi ambient?

Not sure. I would say low, but usable. But in an atmosphere this heavy, drag will be a big problem -- 5 times the pressure means roughly 5 times the density, which means 5 times the drag.

It's worth noting that rockets need to travel horizontally to get into and remain in orbit. But when you watch launches, they start off vertical. This is done to get past the denser regions of the atmosphere -- primarily for drag avoidance, but to a lesser extent for engine efficiency.

Cheers,
Steve