Problem $3$: $(25$ pts$)$

Air at $100$ kPa and $280$ K is compressed steadily to $800$ kPa and $500$ K $.$ The mass flow rate of the air is $0\mathrm{.}05k\frac{g}{s},$ and a heat loss of $20k\frac{J}{k}g$ occurs during the process. Assuming the changes in kinetic and potential energies are negligible, determine the necessary power input to the compressor. The specific heat for air is $.$ K$).$

Tasks:

$($a$)$ Compare the magnitudes of $h,ke,$ and $pe(10$ pts$)$

$($b$)$ Determine the work done per unit mass of the steam flowing through the turbine $(7$ pts $)$

$($c$)$ calculate the mass flow rate of the steam $(8$ pts $)$