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I'm designing a solid rocket motor with a fixed pressure in the combustion chamber $p_{cc} = 70~bar$. When it comes to the choice of the right nozzle, I have to consider the proper expansion ratio, in order to maximize the average thrust coefficient $C_f$ over the burning time $t_b$: $$\bar{C_f} =\frac{1}{t_b} \int_{0}^{t_b}{C_f(t)dt}.$$ The design is based on isentropic relations and so, if the expansion ratio of the nozzle is too small - say 5 - it follows that the nozzle will be underexpanded already at sea level.

Are there any cons of designing a nozzle that will never be adapted? Why should the nozzle be adapted at a particular altitude if the ascension trajectory is exactly vertical?

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    $\begingroup$ Using an under-expanded nozzle means a loss of efficiency, using an over-expanded nozzle means excessive additional weight of the nozzle. A compromise between weight and efficency valid for all ambient pressures should be found. $\endgroup$
    – Uwe
    Commented May 31, 2019 at 10:57

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For a vertical ascent from sea level, the usual situation is that the nozzle is over-expanded at lift-off, under-expanded at the end of the burn, and achieves correct expansion and optimal performance somewhere in the middle. It's only "optimal" at one instant in time, but it's fair-to-good over most of the flight.

Starting off under-expanded at sea level is a bad place to be; performance is suboptimal from the start and only gets worse as you ascend.

The only reason you'd want to start under-expanded is if you're not capable of providing a lightweight nozzle (because you're using stone knives and bearskins), and the weight of a larger nozzle is unacceptable.

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