Example $4\mathrm{.}11$ Entropy in a heat exchanger

A heat exchanger for cooling a hot hydrocarbon liquid uses $10k\frac{g}{m}in$ of cooling $H_{2}O$ which enters the exchanger at $25C.$ Five $k\frac{g}{m}in$ of hot oil enters at $300C$ and leaves at $150C$ and has an average specific heat of $2\mathrm{.}51k\frac{J}{k}g-K.$

$($a$)$Demonstrate that the process is irreversible as it operates now.

$($b$)$Assuming no heat losses from the unit, calculate the maximum work which could be obtained if we replaced the heat exchanger with a Carnot device which eliminates the water stream and transfers heat to the surroundings at $25C$