HW#$7-1(15$ pt$)$ Consider a Carnot cycle executed in a closed system with air as the working fluid. The minimum pressure $(P)$ in the cycle is $100$kPa while the minimum temperature $(T)$ is $300K.$ The entropy increase during the isothermal heat addition process is $0\mathrm{.}287k\frac{J}{k}g*K$ and the net work output is $143\mathrm{.}5k\frac{J}{k}g.$

$($a$)$ Determine the maximum $T(T_H)$ and find the constant$-$pressure and constant volume specific heats and their ratio at the average $T_{\text{a}\text{v}\text{g}}$ of $T_H$ and $T_{?}$

$u$ i$.$e$.,c_v{c}_p$ and $k$ at $\frac{T_L+T_H}{2}($using Table A$-2)$

Using the constant air properties at $T_{\text{a}\text{v}\text{g}},$ determine

$($b$)$ the maximum $P$ in this cycle $(P_1),$

$($c$)$ the heat rejection from the cycle, and

$($d$)$ the thermal efficiency of the cycle.

$($e$)$ If an actual heat engine cycle operates between the same temperature limits and produces $7$ MW of power for an airflow rate of $100k\frac{g}{s},$ determine the second$-$law efficiency of this cycle $(=\frac{W_{\text{a}\text{c}\text{t}\text{u}\text{a}\text{l}}}{W_{\text{C}\text{a}\text{r}\text{n}\text{o}\text{t}}}).$