Yep you are pretty much on the right track there. The latter part of the nozzle is mainly to match the gas pressure to the ambient pressure at the exit. Because as you point out the two should match up to provide the most thrust. The terms you want to look into are over/underexpanded plumes.

Thank you!

If I may ask an out-of-context beginner question, I've always wondered why the nozzles aren't longer. Sometimes the pressure drives the exhaust up the sides of the rocket.

Compared to the Saturn V engine exhaust thrust, I'm surprised the sea-level air pressure even needs to be in the equation.

In space the theoretic best performance would with a nozzle of an infinite size/length, so the real size comes from a tradeoff between performance and mass.

In atmosphere the size is heavily limited by the atmosphere itself as pressure differences between inside and outside of the nozzle would destroy the nozzle structure.

For example the planned Starship upper stage by SpaceX will have two kinds of engines and nozzles.

Needing to match exit pressure to ambient pressure is also why increasing the chamber pressure gets you a higher efficiency in atmosphere, because it lets you use a greater expansion ratio for the same exit pressure. For example, the Rocketdyne F-1 and the SpaceX Merlin are both kerosene-oxygen rocket engines that use a gas generator cycle, and both have relatively similar specific impulse in a vacuum (F-1 304 s, Merlin 311 s). But where the F-1 has 7 MPa of chamber pressure, the Merlin has 9.7 MPa, which translates into a significantly higher specific impulse at sea level (F-1 263 s, Merlin 282 s).

Your assumption that the exhaust is high pressure is wrong

Generally, you want to design the nozzle so that the exhaust pressure matches the ambient pressure. So the exhaust is very fast, but low pressure. At least, that's the case by the time it leaves the nozzle (at the throat, it's high pressure, low speed, before undergoing controlled dilatation in the nozzle)

Ultimately, it's a trade off. Nozzles have a weight associated with them, after all, which is extremely important when launching something to orbit. For the first stage of a rocket, the nozzles are typically optimized for lower altitudes, as they will spend the most time at lower altitudes gaining speed. However, once they are higher up, their plumes become absolutely huge because of this. Once you get to high altitude or vacuum, the second+ stage nozzles can get pretty big and it becomes a tradeoff between how much weight the extra nozzle ends up adding.

>Ideally, you want the pressure at exit to be equal to ambient pressure. (This is mainly about efficiency?)

Yes, this is ideal. I'll expand on this a bit. Take a look at this image of possible nozzle flows. The first one is underexpanded (as in the nozzle didn't expand enough, so the flow is higher pressure than ambient), the second is perfectly expanded, and the third is overexpanded.

Let's go back to Newtonian physics. Remember how every action has an equal and opposite reaction? This is how rockets work - gas is pushed out the back, and that pushes the rocket forward. However, if the nozzle is not perfectly expanded, then that means the gasses are moving out at an angle - either radially outwards (underexpanded) or inwards (overexpanded). So if that gas is moving away from the rocket, there must be an equal and opposite reaction in the same direction - if some gas is moving up from the rocket, then the gas must have pushed it down. Since it's a circle, it cancels out, but this is the primary issue with exit pressure not at ambient - some of the energy coming out of your rocket is being futilely used to push on the walls of the nozzle rather than to push the rocket forward. With a perfectly expanded nozzle you don't have that problem, as all of the momentum is pushing straight up the rocket's velocity vector.

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> Am I right in understanding that this is the tricky balance you need to strike? And shooting for Pe=Pa is where it happens to be optimized for thrust?

The tradeoff is complicated.

First stage nozzles are always compromises. A smaller nozzle is a better match at takeoff and it lets you fit more nozzles underneath the base of the rocket. A larger nozzle is better later in the first stage burn as it's a better match to the lower ambient pressure. However, if it's too big there will be significant flow separation at sea level and that can break the nozzle.

And it gets more complicated with design choices. Rockets like the Falcon 9 stage fairly early, so they have nozzles that are decent compromises between sea level and vacuum. Rockets like the Atlas V stage much higher and therefore the first stage spends more time in vacuum. And rockets like the shuttle run their engines all the way to orbit, so vacuum performance is far more important.

Second stage engines are generally easier; you generally just put on the largest nozzle that will comfortably fit.

I’m going to actually assume you’re an out of context beginner. The earlier responses are right but they aren’t actually helping you understand the concept.

The perfect nozzle has the ambient pressure exactly match the exhaust pressure. This because any excess pressure in the exhaust gases is wasted energy that could have been done. And if the pressure drops below ambient, that means the atmosphere will be pushing in on your exhaust gases, reducing velocity and thus performance.

But ambient pressure changes with altitude. So the perfect exit size changes as you ascend with it. In theory, an infinitely adaptable nozzle would be ideal, but to accomplish that objective requires weight. And the rocket equation tells us that the pounds of mass we have to take to orbit isn’t worth what you would have to spend to get it there .

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