A $0\mathrm{.}5$ kg$/$s oil stream with a heat capacity of $2050$ J$/$kg$/$K is cooled from $380$ K to $350$ K via heat exchange with a $0\mathrm{.}2$ kg$/$s water stream with an inlet temperature of $280$ K$.$ The heat capacity of water is $4186$ J$/$kg$/$K$.$ The heat transfer coefficient is $200$ W$/$m$2/$K$.$

$($a$)$ What is the outlet temperature of the water stream? $(2$ points$)$

$($b$)$ What are the LMTD and required heat transfer area for a co$-$current double$-$pipe exchanger? $(1$ point$)$

$($c$)$ What are the LMTD and required heat transfer area for a counter$-$current double$-$pipe exchanger? By what percentage is the counter$-$current heat exchanger area lower than that of the co$-$current heat exchanger? $(1$ point$)$

$($d$)$ What are the correction factor, Ft$,$ and required heat transfer area for a $2-4$ shell$-$and$-$tube exchanger?