A shell and tube heat exchanger $($single shell, two tube pass$)$ is to cool oil from a large marine engine. Lake water $($the shell$-$side fluid$)$ enters the heat exchanger at $2k\frac{g}{s}$ and $15C$ with a convection coefficient of $500\frac{W}{m^2}.$K$.$ Engine oil flows through $100$ copper tubes $(k=399\frac{W}{m}.K)$ at a rate of $1k\frac{g}{s}($the flow rate per tube is therefore $0\mathrm{.}01k\frac{g}{s}).$ The oil enters the tubes at $140C,$ where each tube is $1\mathrm{.}5$ m long with inner and outer diameters of $6$ and $8$ mm $,$ respectively. Assume the flow in the tubes is fully developed. $(40$ points$)$

Engine oil: $=854k\frac{g}{m^3},c_p=2118\frac{J}{k}g.K,=0\mathrm{.}0356N.\frac{s}{m^2},k=0\mathrm{.}318\frac{W}{m}.K$ Lake water: $c_p=4180\frac{J}{k}g.$ K

$($a$)$ Find the overall heat transfer coefficient UA$,$ in W$/$K$.$

$($b$)$ Calculate the oil outlet temperature.