Question $(4)$: $(8$ marks$)$

A long cylindrical rod of diameter $200mm$ with thermal conductivity of $0\mathrm{.}5\frac{W}{m}.$ K experiences uniform volumetric heat generation of $24,000\frac{W}{m^3}.$ The rod is encapsulated by a circular sleeve having an outer diameter of $400mm$ and a thermal conductivity of $4\frac{W}{m}.$ K$.$ The outer surface of the sleeve is exposed to cross flow of air at $27C$ with a convection coefficient of $25\frac{W}{m^2}.K.$

$($a$)$ Find the temperature at the interface between the rod and sleeve and on the outer surface.

$($b$)$ What is the temperature at the center of the rod? $($Use the rate equation and Consider q conduction at the center $=0)$

Auxiliary data

$\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{x}}(k\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{x}})+\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{y}}(k\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{y}})+\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{z}}(k\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{z}})+q^{}=c_p\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{t}}$

$\frac{1}{r}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(kr\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{r}})+\frac{1}{r^2}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}}(k\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}})+\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{z}}(k\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{z}})+q^{}=c_p\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{t}}$

$\frac{1}{r^2}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(k{r}^2\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{r}})+\frac{1}{r^{2}si{n}^2}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}}(k\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}})+\frac{1}{r^{2}sin}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}}(ksin\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}})+q^{}=c_p\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{t}}$