but if the nozzle is adpated the pressure at the exit of the nozzle is the same of the atmosphere around it...
so there is no pressure gradient
Note to Papa Legba: Don't read past this point, as it is way above your pay grade!
One tiny correction, it is only for the "optimal nozzle" that "the pressure at the exit of the nozzle is the same of the atmosphere around it..."
This is a problem for rockets that are launched from ground and have to reach high altitude (100 km for Space-X stage 1 at separation).
The nozzle is then designed for an intermediate altitude. At launch the nozzle is over-expanded and at maximum altitude it is under-expanded, leading in both cases to a little lost efficiency.
A nozzle that is too much over-expanded can lead to instability of the exit stream. This is more serious for high-altitude jet planes (like the SR-71A) which must spend most of their flight time at extreme altitudes, yet be able to take off from sea level. In this case, some form of variable nozzle becomes necessary.
Notwithstanding all this, the rocket nozzle works perfectly well even when not optimised. And, of course, under vacuum conditions it is completely impractical to "optimise" the nozzle - it would need an "infinite" area! So, an
optimised nozzle very often is not the
best nozzle for a particular application.
An important point to note is that, even for the non-optimal nozzle, the thrust of a rocket always increases as the external pressure falls, right down to a vacuum, as in:
Here: T=static thrust, m.dot=rate of change of momentum (it's the rocket's mass that is changing), A
e=exhaust area, p
e=exhaust pressure and p
o=outside or ambient pressure.
The first term is the major one and is simply the time derivative momentum, the second smaller term shows
Just watch this trigger a memory dump from the AI we all know as Papa Legba - wait for it.........................!
(I hope you will forgive me for hijacking your post a bit to get up
you know who's nose a bit.)