$(15$ points$)$ Flow through a pipe $($internal flow$)$: a fluid $($with properties given below$)$ is heated from $25C(T_{m,\text{i}\text{n}\text{l}\text{e}\text{t}})$ to $75C(T_{m,\text{o}\text{u}\text{t}\text{l}\text{e}\text{t}})$ as it flows at $0\mathrm{.}2\frac{m}{s}$ through a pipe of $12\mathrm{.}7$ mm diameter. A uniform surface heat flux is maintained by a heater wrapped around the outer surface of the pipe. The pipe is $10$ m long.

a$.$ What is the required surface heat flux, $q_s^{\text{'}\text{'}}(\frac{W}{m^2})?$

b$.$ If the fluid enters the tube with a fully developed velocity profile and a uniform temperature profile, what is the surface temperature at the tube exit, $T_{s,o}(degC)?$

Properties for fluid:

$=2{10}^{-3}N\frac{s}{m^2}$

$=1000k\frac{g}{m^3}$

$C_P=4000\frac{J}{k}gK$

$k=0\mathrm{.}8\frac{W}{m}K$

$Pr=10$