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I am currently designing a serial staged rocket with 3 stages. The first stage brings me up to 10 km. When I do my calculations for the required expansion ratio and nozzle length (Rao Nozzle) for the second nozzle I get a length of nearly 350 mm. This will shrink the amount of fuel I can carry in my second stage. What is the drawback if I design my nozzle for 3 km (only 250 mm long) and use this nozzle in 10 km altitude? So I will use a underexpanded nozzle .

If I calculate the Isp of a nozzle designed for 10 km it is: Isp= 276 s If I calculate the Isp of a nozzle designed for 3 km and use it in 10 km: Isp=274 s

So there should be not a big performance difference?

The easiest thing to shorten the nozzle is to use several nozzles. But I want to avoid this.

Thanks for your feedback Lucas

Lucas
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    It seems like you already understand the trade-offs involved. Maybe just go a little further in your performance calculations and see where your design takes you? I imagine you'll find that nozzles tuned for 3km aren't very different from 10km nozzles, and that real gains start happening closer to ambient vacuum, but a proper answer would go into more detail than I have time to right now. – Erin Anne Feb 10 '23 at 03:05
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    Not an answer to question as asked, but some spacecraft solve your original space problem by either putting lower stage structure inside the upperstage nozzle extension (sea drago) or use sliding extensions to pack a wide nozzle into a shorter height. – GremlinWranger Feb 10 '23 at 05:54
  • What exactly is your question? You have 3 sentences ending in question marks in your post, and they all are quite different. – Organic Marble Feb 10 '23 at 13:13

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