A ramjet engine is an air$-$breathing propulsion device with essentially no rotating machinery

$($no rotating compressor blades, turbine, etc.$).$ The basic generic parts of a conventional ramjet

are sketched below. The flow, moving from left to right, enters the inlet where it is compressed

and slowed down. The compressed air then enters the combustor at subsonic speed, where it is

mixed and burned with a fuel. The hot gas then expands through a nozzle. The net result is

the production of thrust toward the left. The ramjet shown is in a supersonic freestream with

a detached shock wave ahead of the inlet. For this problem, you may assume that the gas that

flows through the engine has a ratio of spocific heats $=1\mathrm{.}4,$ and $R=287\frac{J}{k}gK.$

$($a$)$ The portion of the shock between point $1$ and point $2$ is a normal shock. After passing

through the shock wave, the flow from point $2$ to point $3,$ located at the entrance to the

combustor, is isentropic$.$ The ramjet is flying at Mach $2$ at a standard altitude of $10km,$ where

the air pressure and temperature are $26500Pa$ and $223\mathrm{.}3K$ respectively. Calculate the air

temperature and pressure at point $3$ when the Mach number at that point is $0\mathrm{.}2.$

$($b$)$ Calculate the entropy change across the shock wave.

$($c$)$ Suppose heat is added from point $3$ to point $4$ in a constant pressure process, such that

$p_4=p_3,T_4=3990K,$ and $M_4=0\mathrm{.}633.$ The flow from point $4$ to $5$ is then accelerated in an

isentropic process isentropic$,$ where the flow becomes choked at point $5($i$.$e$.M_5=1).$ If the

area of the nozzle throat at point $5$ is $0\mathrm{.}1m^2,$ calculate $T_5,p_5,{}_5,c_5,$ and the mass flow rate of

air through the engine.