$($b$)1\mathrm{.}5kg{s}^{-1}$ of steam is to be condensed to saturated liquid at constant pressure by passing it over the tubes of a shell$-$and$-$tube heat exchanger. The steam enters the shell side of the counter$-$current heat exchanger with dryness fraction $=0\mathrm{.}9$ at $80C.$

$($i$)$ Find the total rate of heat transfer from the steam.

Liquid water with an inlet temperature of $T_{c,in}=20C$ is passed through the heat exchanger tubes. The condenser contains $124,10$ mm diameter, thin$-$walled tubes $($of negligible thermal resistance$).$ The total water flow rate through the tubes is $15kg{s}^{-1}.$ The average heat transfer coefficient associated with condensation on the outer surface of the tubes may be taken as $5000W{m}^{-2}{K}^{-1}.$ For the water in the tubes take ${?}_{\text{I}}{C}_p=4180Jk{g}^{-1}{K}^{-1},=0\mathrm{.}001kg{m}^{-1}{s}^{-1},k=0\mathrm{.}61W{m}^{-1}{K}^{-1}$ and $Pr=6\mathrm{.}85.($Check you know where to find these values in HLT$).$

$($ii$)$ Find the effectiveness of the heat exchanger and use an appropriate graph from part $($a$)$ to find the number of transfer units, NTU. $[$Hint: think carefully about the appropriate value of capacity rate ratio for this question$].$

$($iii$)$ Find the heat transfer coefficient inside the tubes, and hence both the overall heat transfer coefficient, $U,$ and the length of each heat exchanger tube.