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I am going to calculate thrust for a theoretical solid propellant rocket engine I will be making. I have come across many equations for all sorts of aspects of the rocket engine.

The measurements I have available are the dimensions of the combustion chamber, thrust needed by the engine after everything is done, and combustion chamber temperature.

Thrust is defined as: $F={\dot {m}}*Ve+(Pe-Po)*Ae$

Where ${\dot {m}}$= mass flow rate Ve = exit velocity Pe = exit pressure Pa = ambient pressure Ae = exit area

Pe is very important because in order to get the highest thrust and avoid underexpansion it needs to be equal to ambient pressure. I need to find Pe and it is found by taking the throat pressure. Now I can't even being the exit pressure equation because I need throat pressure. Pressure at the throat is defined as:

$Pt=Pc (1 + (k-1)/2)^-k/(k-1)$

Is there an equation I could use to find the combustion chamber pressure so I can calculate throat pressure, exit pressure, and eventually nozzle dimensions?

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    $\begingroup$ Can you reassure us you will not hold us responsible if it all goes pear shaped. (There are genuine safety issues.) $\endgroup$
    – JMLCarter
    Commented Dec 22, 2016 at 21:01
  • $\begingroup$ @JMLCarter I assure you I will not hold anyone responsible except myself because I am the one doing calculations, and considering I have to calculate minimum thickness of the combustion chamber myself, that would all be my fault. $\endgroup$
    – user7310442
    Commented Dec 22, 2016 at 21:03

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The thrust is maximized when the momentum flux is maximized.The Mach number at the throat of the nozzle is 1.

The other constraint you have is that the exit pressure, $P_e$, is known and set to your ambient conditions.

The missing fact is that once the flow is choked, it is not possible for further changes in the ambient pressure to affect the mass flow rate and therefore the upstream pressure is unaffected. If you go through the equations and examples for an isentropic nozzle, you will find that for air (where $\gamma = 1.4$), the critical value for throat pressure and choked flow is:

$$P^\ast = 0.5283 P_0$$ where $P_0$ is the stagnation pressure in the chamber.

If you know the chamber pressure and need to find the throat pressure you reverse the equation:

$$ P_0 = \frac{P^\ast}{0.5283} $$

and that gives you the stagnation pressure required in your chamber. For your actual motor, with actual heat release and propellants, this will not be the same equation and you'll have to use the actual equation on the linked page.

But even that is only an estimate. It assumes a calorically perfect gas. And it assumes everything is isentropic. Your real motor won't be. It also won't have constant pressure due to the burning of the solid propellant changing the chamber volume.

So -- given an exit pressure and desired critical pressure (and associated temperature, density, and flow rate), you can figure out how much chamber pressure is needed, and then design a nozzle with the proper area ratios to ensure that the required exit pressure is met.

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  • $\begingroup$ The equation you give ($P_e=0.5283 P_0$) is for the critical pressure $p^\ast$, not for the exit pressure. You can make the exit pressure as low as you want. In order to find the chamber pressure the OP is looking for, he needs information on his propellant. More information on this can be found at nakka-rocketry.net/th_pres.html Oh, and the original question has nothing to do with mathematical physics. This tag should be removed. $\endgroup$
    – Pirx
    Commented Dec 22, 2016 at 19:52
  • $\begingroup$ I have information on my propellant, it is a mixture of 65% KNO3, 34% Surcose and 1% Powdered Graphite. I did notice it seemed to be the equation for Throat Pressure/Chamber Pressure just slightly different. Although the equation I had may be incorrect. Although I would like to say, that with this the ambient pressure, which I need my exit pressure to be, is 1 atm. $\endgroup$
    – user7310442
    Commented Dec 22, 2016 at 19:56
  • $\begingroup$ @Pirx The exit pressure is constrained by being ambient conditions (whatever those are in OP's case). And the critical pressure determines the minimum chamber pressure, which is what the OP was asking for. Going lower with exit pressure -- or higher with chamber pressure -- is not an optimal solution because it would require more work without changing the mass flow rate. Regarding the tag -- propose an edit to remove it if you disagree with it, no need to call it out or wait for somebody else to do it. $\endgroup$
    – tpg2114
    Commented Dec 22, 2016 at 20:01
  • $\begingroup$ @user7310442 Your question did say a theoretical rocket motor but now you're putting out some very specific fuels and conditions... I'm going to go ahead and say that if you are asking a question like how to find chamber pressures, you likely don't have the experience needed to actually design and build a solid rocket motor. I wont' give too much more help or advice regarding practical implementation details. $\endgroup$
    – tpg2114
    Commented Dec 22, 2016 at 20:03
  • $\begingroup$ @tpg2114: I repeat, the equation you give is incorrect, and should be changed by replacing your $P_e$ with the critical pressure. There is no definite relationship between chamber pressure and exit pressure. The ratio between those two depends on the nozzle geometry. $\endgroup$
    – Pirx
    Commented Dec 22, 2016 at 20:22

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