$4)$

A counterflow double$-$pipe heat exchanger is designed to cool lubricating oil for a large industrial gas turbine engine. The flow rate of cooling water through the inner tube is $\backslash (0\mathrm{.}2\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$s$\}\backslash ),$ and the water enters the tubes at $\backslash (20^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ),$ while the flow rate of oil through the outlet annulus is $\backslash (0\mathrm{.}4\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$s$\}\backslash ).$ The oil and water enter the heat exchanger at temperatures of $\backslash (60^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ and $\backslash (30^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ),$ respectively. The heat transfer coefficients in the annulus and in the inner tube have been estimated as $8$ and $\backslash (2445\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}^\{2\}\backslash$cdot $\backslash$mathrm$\{$~K$\}\backslash ),$ respectively. The outer diameter of the inner tube is $25$ mm $,$ and the inner diameter of the outer tube is $45$ mm $.$ The total length of the double$-$pipe heat exchanger $($total length per hairpin$)$ is $15$ m $,$ and total heat exchanger area is $\backslash (325\backslash$mathrm$\{$~m$\}^\{2\}\backslash ).$ In the analysis, the tube wall resistance and the curvature of the wall are neglected. What is the total value of the fouling resistance used in this design?