How to find the average ambient pressure during rocket ascent?

Question

I am trying to find the average ambient pressure that my rocket will flight through during its ascent to then determine the most optimum conditions in my combustion chamber and nozzle.

I have initially used the barometric formula and took the average integral of the barometric formula from launch site's altitude to the desired altitude (in my case 30km). I have realized however that this gives me the "spatial" average, and that this method doesn't work anymore when considering an accelerating system.

Does anyone know a formula I could use to find the average ambient pressure my rocket would fly through?

I presume that besides a few constants, the important parameters would be thrust, burn time, weight and final altitude.

---

¹For the purposes of this question "ambient pressure" is the pressure of the environment through which the exhaust gases are moving after exiting the nozzle for the purposes of modeling the performance of a given combustion chamber and nozzle.

Answer 1

I assume you want to optimize the area of the nozzle at the exit so the total impulse that the rocket provides is maximum, taking into account that the "ambient pressure" changes throughout the flight. So you want to optimize $I_t=f(P_e)$, where $P_e$ is the pressure of the gases at the exit of the nozzle and is controlled through your nozzle and rocket motor design. You can't really do this analytically. Simplifying the equation to 1D and having thrust and aerodynamic drag forces only:

$I_t = \int F dt = \int (\dot m c + (P_e - P_a)A_e - C_D \frac{1}{2} \rho v^2 A_f) dt$

This integral is not generalizable for all cases. Each motor will have a different thrust curve $F(t)$, and the rest of the variables are coupled together. For example, the drag coefficient depends on the velocity, $C_D(v)$. The only way to do this is by numerically integrating the trajectory, coupled with a model for $C_D(v)$ of your rocket and a model for the ambient pressure in function of the altitude, $P_a(x)$. You need to solve numerically:

$F = \dot m c + (P_e - P_a)A_e - C_D \frac{1}{2} \rho v^2 A_f= m \frac{d^2x}{dt^2}$,

for $x(t)$. Once you have that, you can find the total impulse. You could do a grid search of different $P_e$ to see which one yields best results.

As a side note, you'll find that it will have little impact on the overall total impulse of the rocket motor. I wouldn't stress about it and pick for now an area that expands at about $P_e = 0.6-0.7$ bar which is in between 0-30 km ambient pressure and ensures that you don't overexpand too much at low altitudes (it could lead to vibrations and instabilities). Good luck on your sounding rocket.