$(30$ points$)$ Water at $m^{}=0\mathrm{.}05k\frac{g}{s}$ and $T_{m,i}=30C$ enters an annular tube heat exchanger as shown in Figure $2.$ The annular tube heat exchanger is composed of an outer tube with radius $r_1=50mm$ and an inner tube with an outer radius $r_2=25mm$ and an inner radius of $r_3=10mm.$ The surface of the outer tube is well insulated. Saturated steam flows through the inner tube, maintaining a uniform temperature of $T_S=100C$ at the surface of the steam side. $.$ The thermal conductivity of the inner tube wall is $k_A=3\frac{W}{m}*K.$ Water has viscosity $=600{10}^{-6}\frac{m^2}{s},$ heat capacitance $c_p=4200\frac{J}{k}g*K,$ thermal conductivity $k=0\mathrm{.}6\frac{W}{m}*K,$ and Prandtl number $Pr=3\mathrm{.}8.$ The fully developed conditions may be assumed throughout the annulus.

$($a$)(10$ points$)$ Please obtain the value of heat convection coefficient of heat transfer at the outer tube wall.$(10$ points$)$ How long must the annular tube heat exchanger be to provide an outlet water temperature $T_{m,o}=80C?$ What is the overall heat flow rate that the water side receives?

$($b$)(10$ points$)$ After $5$ years of operation, the heat exchanger performance is degraded by fouling on the outer wall of the water side, and the water outlet temperature becomes $60C$ for the same fluid flow rates and inlet temperatures.

What is the fouling factor on the outer wall of the water side?

Figure $3$