$1(50$ points$)$

An oil preheater consists of a round tube of $10$ mm diameter and $30$ m length. The first half of the tube is heated by a heat flux of $q_s^{\text{'}\text{'}}=5x^2[\frac{W}{m^2}],$ where x is the distance from the entrance, and the second half is well insulated. Before entering the tube, the oil has a uniform temperature of $25C$ and a flow rate of $0\mathrm{.}0125k\frac{g}{s}.$ The hydrodynamic boundary layer has been fully developed.

$($a$)$ Determine the location where the thermal boundary layer is fully developed.

$($b$)$ Calculate the mean fluid temperature at the tube exit, $T_{m,o}.$

$($c$)$ What is the tube surface temperature at the tube exit, $T_{s,o}?$

Properties of the oil: $=865\mathrm{.}8k\frac{g}{m^3},c_p=2035\frac{J}{k}g*K,=8\mathrm{.}36{10}^{-2}N*\frac{s}{m^2},k=0\mathrm{.}141\frac{W}{m}*K,=0\mathrm{.}8{10}^{-7}\frac{m^2}{s},$ and $Pr=1206\mathrm{.}6.$