$8\mathrm{.}20$ Engine oil flows through a $25-mm-$diameter tube at a rate of $0\mathrm{.}5k\frac{g}{s}.$ The oil enters the tube at a temperature of $25C,$ while the tube surface temperature is maintained at $100C.$

$($a$)$ Determine the oil outlet temperature for a $5-$m and for a $100-$m long tube. For each case, compare the log mean temperature difference to the arithmetic mean temperature difference. $($b$)$ For $5L100m,$ compute and plot the average Nusselt number ${}^{-}D$ and the oil outlet temperature as a function of $L.$