Air enters a water$-$jacketed air compressor operating at steady state with a volumetric flow rate of $37\frac{m^3}{m}in$ at $136$kPa,$305K$ and exits with a pressure of $680$ kPa and a temperature of $400$ K $.$ The power input to the compressor is $155$ kW $.$ Energy transfer by heat from the compressed air to the cooling water circulating in the water jacket results in an increase in the temperature of the cooling water from inlet to exit with no change in pressure. Heat transfer from the outside of the jacket as well as all kinetic and potential energy effects can be neglected.

$($a$)$ Determine the temperature increase of the cooling water, in K $,$ if the cooling water mass flow rate is $82k\frac{g}{m}in$

$($b$)$ Plot the temperature increase of the cooling water, in K $,$ versus the cooling water mass flow rate ranging from $75$ to $90k\frac{g}{m}in.$