If I understand it correctly, the only problem with an overexpanded nozzle is (aside from technical difficulties) that the surrounding atmosphere puts pressure on the exhaust gases and reduces engine thrust and efficiency. So isn't it rather that the nozzle needs to be expanded enough to make the exhaust velocity as high as possible and still pay off due to the pressure? But this point of balance doesn't have to happen just when the pressures are equal, right?
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1$\begingroup$ Sure, that's why booster engines like the SSME are overexpanded for much of the ascent. The trick is to design the nozzle so that the flow doesn't separate. $\endgroup$– Organic MarbleCommented May 20 at 21:53
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$\begingroup$ An overexpanded nozzle may cause less thrust and increased dry mass, so less payload mass. $\endgroup$– UweCommented May 21 at 12:30
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$\begingroup$ Rocket nozzles are ALWAYS overexpanded for sea level $\endgroup$– Lawn Hollander LawnCommented May 22 at 13:53
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$\begingroup$ Clarification request: By "point of balance", do you mean "point at which thrust is maximized" or "best practice when designing an engine"? I answered assuming the former, but the answer to the latter is quite different. $\endgroup$– Charles StaatsCommented May 23 at 16:43
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An engine's thrust is transmitted to the rocket via the pressure that the exhaust exerts on the interior surface of the nozzle. Note that this pressure is not uniform: it is very high where the nozzle is narrow, and lower on the wider parts of the nozzle. (IIUC, the interior pressure at a given "height" within the nozzle is inversely proportional to the cross-sectional area at that height.) If a nozzle is overexpanded, then on the lowest portion of the nozzle, the interior pressure is less than the ambient pressure of the atmosphere on the nozzle's exterior surface; so that portion of the nozzle is actually providing net negative thrust.
Note that flow separation actually alleviates this problem, but causes a bunch of other problems (like destroying the nozzle).
TL;DR Yes, the "point of balance" is necessarily exactly when the pressures are equal; this is the point where further expansion will actually reduce thrust (for the given ambient pressure).
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1$\begingroup$ I understand that any increase in nozzle expansion causes the resulting thrust to decrease due to pressure, but can't that still increase thrust as a result because the exhaust gases are accelerated more? $\endgroup$– Saturn VCommented May 21 at 6:07
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$\begingroup$ Note that the converging part of the nozzle also produces negative thrust. Also only the axial component of the pressure force matters for thrust, and this is quite small towards the nozzle exit. $\endgroup$ Commented May 21 at 11:04
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$\begingroup$ @SaturnV Overexpanded exhaust compresses when it leaves the engine, slowing it down. $\endgroup$ Commented May 21 at 16:32
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$\begingroup$ @SaturnV The acceleration of exhaust gas is not separate from the exertion of pressure. Just as the rocket is accelerated by the pressure of the gas on the nozzle, so the gas is accelerated by the equal and opposite pressure of the nozzle on the gas. Expressing everything in terms of pressure just makes some calculations easier (and others harder). $\endgroup$ Commented May 21 at 17:38
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$\begingroup$ @SaturnV An extension of the nozzle does increase the velocity of the exhaust gases. But it also increases the amount of ambient air that the nozzle has to push out of the way. And for an overexpanded nozzle, the latter effect dominates. $\endgroup$ Commented May 21 at 17:43