Steam $(1$Mpa, $250$C$)$ flowing at the rate of $1$kg$/$s expands through a well$-$ C insulated nozzle to $10$ kPa. Neglecting the velocity of the steam at the inlet of the nozzle. The steam exiting the nozzle is discharged to a condenser wherefrom it exits as saturated water. The condenser is a counter$-$flow heat exchanger cooled by cooling water whose inlet and exit temperatures are $25$C and $35$C$.$ Show the expansion process within the nozzle on a T$-$s plot and determine the following:

$($a$)$ velocity of the steam at the nozzle exit

$($b$)$ exit area of the nozzle

$($c$)$ mass flow rate of the cooling water