Chapter 2, Problem 2.50P

An electric cable of radius $r_1$ and thermal conductivity $k_e$ is enclosed by an insulating sleeve whose outer surface is of radius $r_2$ and experiences convection heat transfer and radiation exchange with the adjoining air and large surroundings. respectively. When electric current passes through the cable, thermal energy is generated within the cable at a volumetric rate $\stackrel{.}{q}.$

1. Write the steady-state forms of the heat diffusion equation for the insulation and the cable. Verify that these equations are satisfied by the following temperature distributions:

$\begin{array}{l}\text{Insulation}:T\left(r\right)=T_{s,2}+\left(T_{s,1}-T_{s,2}\right)\frac{\ln \left(r/r_2\right)}{\ln \left(r_1/r_2\right)}\\ \text{Cable:}T\left(r\right)=T_{s,1}+\frac{\stackrel{.}{q}{r}_1^2}{4k_c}\left(1-\frac{r^2}{r_1^2}\right)\end{array}$
Sketch the temperature distribution, $T\left(r\right),$ in the cable and the sleeve, labeling key features.

Applying Fourier's law, show that the rate of conduction heat transfer per unit length through the sleeve may be expressed as
$\stackrel{.}{q}=\frac{2\pi k_s\left(T_{s,1}-T_{s,2}\right)}{\ln \left(r_2/r_1\right)}$
Applying an energy balance to a control surface placed around the cable, obtain an alternative expression for $q_r^{\text{'}},$ expressing your result in terms of $\stackrel{.}{q}$ and $r_1.$

Applying an energy balance to a control surface placed around the outer surface of the sleeve. obtain an expression from which $T_{s,2}$ may be determined as a function of $\stackrel{.}{q},r_{1,}h,T_{\infty },\epsilon ,$ and $T_{\text{sur}}.$

Consider conditions for which 250A are passing through a cable having an electric resistance per unit length of $R_e^{\text{'}}=0.005\Omega \text{/m},$ a radius of $r_1=15\text{mm,}$ and a thermal conductivity of $k_e=200\text{W/m}\cdot \text{K}\text{.}$ For $k_s=15\text{W/m}\cdot \text{K,}{r}_2=15.5\text{mm},h=25{\text{W/m}}^{\text{2}}\cdot \text{K,}\epsilon =0.9,T_{\infty }=25°\text{C,}$ and $T_{\text{sur}}=35°\text{C,}$ evaluate the surface temperatures, $T_{s,1},$ and $T_{s,2},$ as well as the temperature To at the centerline of the cable.

With all other conditions remaining the same. compute and plot $T_o,T_{s,1,}$ and $T_{s,2}$ as a function of $r_2$ for $15.5\le r_2\le 20\text{mm}\text{.}$