# What am I missing about rocket nozzle isentropic flow?

Playing a bit with Cpropep-Web, something looked wrong to me about how it models isentropic flow through a CD nozzle.

I'm taking the RS-25 characteristics as an example. I ask for a frozen equilibrium flow computation (to avoid molecular recombination and insure isentropy) and get this result :

First, we can see that entropy $$S$$ is conserved from CHAMBER to EXIT, so the whole process is by definition isentropic. (It is not a the throat though, which is weird...)

However, if I try to use isentropic flow relations on this output, things start to get weird.

If my understanding is correct, within an isentropic flow, total temperature $$T_{tot}$$ and pressure $$p_{tot}$$ should be conserved (and should be equal to chamber temperature and chamber pressure, as flow is assumed to be at velocity zero in the chamber).

Then,

at CHAMBER : $$p_{tot_c}=p_c=203.73atm ; T_{tot_c} = T_c = 3609.678K$$

At EXIT, $$M=\frac{I_{sp}}{V_{son}}=5.1245 ; \gamma = 1.28861 ; p_e = 0.153 atm ; T_e = 949.973K$$

Isentropic flow relations give $$\frac{T_e}{T_{tot_e}} = (1 + \frac{\gamma - 1}{2} M^2)^{-1}$$ and $$\frac{p_e}{p_{tot_e}} = \frac{T_e}{T_{tot_e}}^{\frac{\gamma}{\gamma - 1}}$$

Substituting all the values we get at EXIT : $$T_{tot_e} = 4550K ; p_{tot_e} = 166.77 atm$$ which means those quantities were not conserved. Some total pressure was lost and a lot of total temperature was gained.

Is Cpropep-Web wrong or am I missing something here ?

## 1 Answer

The isentropic relations you quote are derived for constant values of $$\gamma$$ (Gamma in the listing) and in real life $$\gamma$$ varies with temperature. I did a calculation with just the pressures and temperatures in your listing and found that the $$\gamma$$ that fits is 1.1855 which is between the chamber and exit Gamma in the listing. That is to be expected.

Secondly, the program did not do frozen flow since the molar fractions change from chamber to throat to exit. "Frozen equilibrium flow" that you state is not recognized by me.

• Sorry, I messed up when rerunning the computation for cryogenic propellants and used shifting equilibrium as I initially used gaseous ones, just to stick with the RS-25. I updated my question with the correct frozen equilibrium. Then I would ask : Are total pressure/temperature still constant throughout the nozzle ? If yes, why the computation doesn't work for non-constant gamma even if you know the local gamma ? Having to use an average gamma seems weird as I'm under the impression that the relation between state variables of the gas at one point shouldn't depend on its previous states. Commented Oct 19, 2022 at 17:30