Under steady state, a compressor is used to increase the pressure of an ideal gas air from $100$ kPa to $1$ MPa $.$ Meanwhile the temperature is also raised from $300$ K to $500$ K $.$ To be safe, a separate stream of water is used to cool the air down to $320$ K while the pressure of air remains unchanged during the cooling process. The temperature of the cooling water increases from $15C$ to $30C,$ while the pressure of the cooling water remains the same at $1$ bar approximately. Under steady state, the compressor consumes a power of $102$ kW $,$ and the compressor is warm and transfers heat to the surroundings. The mass flow rate of air is $0\mathrm{.}5k\frac{g}{s}.$ Assume the kinetic and potential energy change is negligible, there is no stray heat transfer from the heat exchanger to the surroundings. Gas constant of air is $R=0\mathrm{.}287k\frac{J}{k}g*K$

Draw two $p-$v diagrams, one for air and one for water, identify state $1$ to $3$ on air $p-$v diagram and identify state $4$ and $5$ on water $p-$v diagram.

What is the heat transfer rate from the compressor to the surroundings in kW $?$

To maintain a safe operation of air supply, what is the amount of cooling water needed in $k\frac{g}{s}?$

If the heat exchanger also loses heat to the environment in a rate of $2$ kW $,$ what is the amount of cooling water needed to maintain a safe operation of air supply, in $k\frac{g}{s}?$