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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.

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    $\begingroup$ What do you mean by "ambient pressure"? From the way you wrote the question ("this method doesn't work anymore when considering an accelerating system"), it appears you want "stagnation pressure", which is $P + \frac12 \rho v^2$, where $P$ is the local static ("ambient") pressure, $\rho$ is the local density, and $v$ is the magnitude of the velocity with respect to the air. The latter quantity ($\frac12 \rho v^2$) is called "dynamic pressure". $\endgroup$ Oct 20, 2022 at 11:04
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    $\begingroup$ Hopefully you are really trying to optimize the exit plane pressure (i.e. the nozzle geometry) not the combustion chamber pressure. Because designing Pc based on average Pinfinity doesn't make a lot of sense. Pc >> Pinfinity. $\endgroup$ Oct 20, 2022 at 11:53
  • $\begingroup$ @DavidHammen This question makes sense for optimizing nozzle expansion ratio. Stanisverylow, as you mention, the spatial average won't be of much help. What you want is the burn-time average. Assume an expansion ratio and calculate height as a function of time. or estimate your height as a function of time any way you see fit. Then you can compute pressure as a function of time and the average of that across your burn time would make a nice design point. $\endgroup$
    – A McKelvy
    Oct 20, 2022 at 18:26
  • $\begingroup$ @AMcKelvy that is precisely what I want. I will try to do that thank you. Are there however any contents already covering this topic online? I am pretty sure that there should be but I haven't found any. $\endgroup$ Oct 21, 2022 at 7:13
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    $\begingroup$ @DavidHammen the ambient pressure is the pressure of the environment through which the exhaust gases are moving after exiting the nozzle. $\endgroup$ Oct 21, 2022 at 7:14

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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.

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