Show me the steps to solveConsider a shell$-$and$-$tube heat exchanger where hot oil enters the shell side at a tem$-$ perature of Th$.150$C and leaves at The $90$C$.$ The mass flow rate of the hot oil is dot m$\_\{$h$\}=0\mathrm{.}5$kg $/$ s and its specific heat capacity is Cp $=2\mathrm{.}1$ kJ$/$kg$-$K$.$ Cold water enters the tube side at Tec $30$C and leaves at Too $70$C$.$ The mass flow rate of the cold water is dot m$\_\{$c$\}=0\mathrm{.}6$kg $/$ s and its specific heat capacity is Cp$.$c $=4\mathrm{.}18$kJ$/$kg$-$K$.$

$1.$ Derive the equations for the log mean temperature difference $($LMTD$)$ and effec$-$ tiveness $($e$)$ for the heat exchanger.

$2.$ Calculate the heat transfer rate $($Q$)$ and determine required heat transfer area $($A$)$ if the overall heat transfer coefficient is U $=500$W $/($m $^2)*$ K

$3.$ Compare the required area if the heat exchanger is operated in a counterflow con$-$ figuration versus a parallel flow configuration.