Problem $5$: $(20$ pts$)$

In your past fluid mechanics class, you have determined the thrust and$/$or specific impulse of a rocket engine, but have you ever designed one? This problem will actually teach you how to design a rocket engine!

The general shape of a rocket engine is given below. It consists of two main components:

$(1)$ Combustion chamber

$(2)$ Converging$-$diverging nozzle

The engine uses $H\frac{2}{O}2$ mixture that arrives inside the chamber at $T_{in}=300K.$ The subsequent combustion process generates $92,000\frac{J}{m}ol$ of energy $($please note: in chemical reactions we use moles instead of kilograms but they should be treated the same $-$ it will not matter in calculation as all moles will cancel out$).$

Find: $$

a$.$ Applying Reynolds Transport Theorem $($RTT$)$ over the combustion chamber, determine what is the combustion temperature assuming no heat transfer out of the chamber and no work.

$[$Use the following specific heats: $c{p}_{in}=29\frac{J}{m}ol-K$ and $c{p}_{\text{o}\text{u}\text{t}}=41\mathrm{.}9\frac{J}{m}ol-K]$

b$.$ Once the gases have combusted, they leave the combustion chamber $(1)$ and enter into the converging diverging nozzle $(2).$ Assuming an isentropic process through the nozzle and knowing that the nozzle exit gas temperature is $Te=600K,$ what is the exit mach number?

$[$assume $=1\mathrm{.}35]$

c$.$ Find the specific impulse of the rocket engine. $[$assume speed of sound is $c=360\frac{m}{s}]$

d$.$ What should the area ratio $\frac{A}{A_{\text{t}\text{h}\text{r}\text{o}\text{a}\text{t}}}$ be in order to achieve the desired specific impulse?

$[$Note: by solving point this last point you determine the actual geometry of the rocket nozzle$]$

Points b$-$d are "plug and chug" using the isentropic relations that were derived in your Fluid Mechanics class. They are repeated here for brevity:

$\frac{T_{ch}}{T_e}=(1+\frac{-1}{2}M{a}_e^2)$

$M{a}_e=\frac{V_e}{c}$

$I_{sp}=\frac{V_e}{9\mathrm{.}81}$

$\frac{A}{A_{\text{t}\text{h}\text{r}\text{o}\text{a}\text{t}}}=\frac{1}{M{a}_e}(\frac{\frac{+1}{2}}{1+\frac{-1}{2}M{a}_e^2}{)}^{\frac{+1}{2-2}}$