A double$-$pipe $($shell$-$and$-$tube$)$ heat exchanger is constructed of a copper $)=(380\frac{W}{m}*C$ inner tube of inner diameter $D_i=1\mathrm{.}2cm$ and outer diameter $D_0=1\mathrm{.}6$ cm and an outer shell of inner diameter $3\mathrm{.}0$ cm $.$ The convection heat transfer coefficient is given to be $h_i=700\frac{W}{m^2}*C$ on the inner surface of the tube and $h_o=1400\frac{W}{m^2}*C$ on the outer surface.

For a fouling factor of $R_{f,i}=0\mathrm{.}0005m^2*\frac{C}{W}$ on the tube side and $R_{f,o}=0\mathrm{.}0002m^2*\frac{C}{W}$ on the shell side, determine;

$($a$)$ The thermal resistance per unit length

$($b$)$ The overall heat transfer coefficients $(U_i)$ and $(U_o)$ based on the inner and outer surface areas of the tube.

$($c$)$ The rate of heat transfer per unit area if hot fluid enters at $150C$ and leaves at $100C.$