Scuba diving air enters a three$-$stage compressor at $1$ bar and $27C$ at a rate of $0\mathrm{.}02\frac{kg}{s},$ and is compressed to $250$ bar. The pressure ratio across each stage is the same, and the air is cooled back to the initial temperature after each stage of compression. Assume the compression processes to take place with an isentropic efficiency of $0\mathrm{.}8.$

$($a$)$ Determine the rate of work input for the three$-$stage compressor.

$($b$)$ Determine the rate of work input if$,$ instead, only a single$-$stage compressor is used.

$($c$)$ What are the maximum compression stage outlet temperatures in $($a$)$ and $($b$)?$

$($d$)$ Calculate the rate of entropy generation in the first compressor stage in $($a$).$