Problem $2$: $[10$ points$]$

For this problem The environment is at $101\mathrm{.}30$ kPa and $288\mathrm{.}15$ K $.$

$($a$)$ Model air as a perfect gas with specific heat capacity at constant pressure C$\_$p$=1,030$ J$/$kg$/$K$,$ and gas

specific constant $R=286\mathrm{.}96\frac{J}{k}\frac{g}{K}.$ The outlet of a small industrial compressor is at stagnation

conditions $485\mathrm{.}00$ kPa and $492\mathrm{.}45$ K $.$ What is the isentropic stagnation$-$to$-$stagnation efficiency?

What is the polytropic stagnation$-$to$-$stagnation efficiency? Which of these two efficiencies is

bigger?

$($b$)$ Model the working fluid through the $($free$)$ power turbine of a CBE cycle as a perfect gas with

specific heat capacity at constant pressure C$\_$p $=1,085\frac{J}{k}\frac{g}{K},$ and gas specific constant $R=286\mathrm{.}96$

$\frac{J}{k}\frac{g}{K}.$ At turbine inlet stagnation conditions are $422\mathrm{.}67$ kPa and $850\mathrm{.}32$ K $.$ At turbine outlet

stagnation conditions are $115\mathrm{.}25$ kPa and $635\mathrm{.}16$ K $.$ What is the isentropic stagnation$-$to$-$stagnation

efficiency? What is the polytropic stagnation$-$to$-$stagnation efficiency? Which of these two

efficiencies is bigger?

$($c$)$ What is the specific power of the turbine in $($b$)?$