$(9-12)$ An air$-$standard cycle is executed within a closed piston$-$cylinder system and consists of three processes as follows:

$1-2V=$ constant heat addition from $100$ kPa and $27C$ to $700$ kPa

$2-3$ Isothermal expansion until $V_3=7V_2$

$3-1,P=$ constant heat rejection to the initial state

Assume air has constant properties with $c_v=0\mathrm{.}718k\frac{J}{k}g*K,c_p=1\mathrm{.}005k\frac{J}{k}g*K,R=0\mathrm{.}287k\frac{J}{k}g*K,$ and $k=1\mathrm{.}4.$

$($a$)$ Sketch the $P-V$ and $T-$s diagrams for the cycle.

$($b$)$ Determine the ratio of the compression work to the expansion work $($the back work ratio$).$

$($c$)$ Determine the cycle thermal efficiency.

Answers: $($b$)0\mathrm{.}440,($c$)26\mathrm{.}6$ percent