A designer finds that because a new material has been discovered, she can increase the allowable turbine inlet temperature $($i$.$e$.,$ that ${}_{}$ can be increased$).$ She already has a turbojet design $-$ say engine $1-$ and decides to redesign the engine to engine $2,$ so that engine $2$ has the same air specific impulse $I_{sp}$ as engine $1($at the same flight Mach number$).$

$1$

a$.)$ Show that the compressor stagnation temperature ratio of the new engine, ${}_{c_2},$ is related to ${}_{c_1},{}_{{}_1},{}_r,$ and ${}_{{}_2}$ by the following:

$(\frac{{}_{c_2}}{{}_{c_1}})=\frac{1}{2}\{\begin{array}{c}[\frac{({}_{{}_2}-{}_{{}_1})}{{}_r{}_{c_1}}+1+\frac{{}_{{}_1}}{({}_r{}_{c_1}{)}^2}]+\end{array}$

$\{$:$[(1-\frac{{}_{{}_1}}{({}_r{}_{c_1}{)}^2}{)}^2+\frac{{}_{{}_2}-{}_{{}_1}}{({}_r{}_{c_1}{)}^2}\{({}_{{}_2}-{}_{{}_1})-4+2({}_r{}_{c_1}+\frac{{}_{{}_1}}{{}_r{}_{c_1}})\}{]}^{\frac{1}{2}}\}$

HINT: your solution will require use of the quadratic formula. Reason why you have chosen one particular sign to obtain the above solution.

b$.)$ Given the conditions for engine $1$ of $=1\mathrm{.}4,T_0=222K,M_0=2,Q_R=4\mathrm{.}42{10}^7\frac{J}{k}g,c_p=1005\frac{J}{k}gK,{}_{{}_1}=7,$ and ${}_{c_1}=20,$ calculate the air specific impulse and specific fuel consumption for engine $1.$

c$.)$ Given that ${}_{{}_2}=8,$ calculate ${}_{c_2}$ and the SFC for engine $2.$

d$.)$ Compare the thermal and propulsive efficiencies, ${}_{th}$ and ${}_p,$ respectively, of the two engines.