Problem $4$

a$)$ Use a control volume analysis along with conservation of momentum to derive an equation for the

thrust developed in a supersonic nozzle, from the throat where $M=1$ to the nozzle exit, ie$,$ the

control surfaces should be placed at the throat, and at the nozzle exit. Make a neat sketch, and

clearly show and label all terms in the momentum equation.

b$)$ The Raptor engine being developed by SpaceX is probably the highest$-$pressure rocket engine ever

made. Operating values for the Raptor according to Wikipedia were used to generate the CEA

outputs provided below. Flowrates are reported as as $m_{\text{L}\text{O}\text{x}}^{}=510k\frac{g}{s}$ and $m_{CH4}^{}=140k\frac{g}{s}.$

Consider the engine operates at sea level $),$ and use the equation from part a and data

from the CEA output to calculate the thrust generated in the nozzle.

THEORETICAL ROCKET PERFORMANCE ASSUMING EQUILIBRIUM

COMPOSITION DURING EXPANSION FROM INFINITE AREA COMBUSTOR

$\frac{O}{F}=,3\mathrm{.}60000,8$FUEL $=21\mathrm{.}739130,$ R$,$ EQ$.$RATIO $=1\mathrm{.}108129,$ PHI,EQ$.$RATIO $=1\mathrm{.}108129$

PERFORMANCE PARAMETERS

MASS FRACTIONS