Question

A long power transmission cable is buried at a depth (ground-to-cable-centerline distance) of 2 m. The cable is encased in a thin-walled pipe of 0.1-m diameter, and, to render the cable superconducting (with essentially zero power dissipation), the space between the cable and pipe is filled with liquid nitrogen at 77 K. If the pipe is covered with a super insulator (ki = 0.005 W/mK) of 0.05-m thickness and the surface of the earth (kg = 1.2 W/mK) is at 300 K, what is the cooling load (W/m) that must be maintained by a cryogenic refrigerator per unit pipe length?

Answer 1

Answer:

$q=9.9w/m$

Explanation:

From the question we are told that:

Ground-to-cable-center line Distance $d_g=2m$

Diameter of Cable case $d=0.1m$

Temperature of Nitrogen $T_n=77K$

Insulator $ki = 0.005 W/mK$

Thickness $t=0.05$

Mass $kg = 1.2 W/mK$

Temperature of earth surface $T_e=300K$

Generally the equation for Heat rate per unit length is mathematically given by

 $q=\frac{T_g-T_n}{R_g+R_e}$

Where

 $R_g=[kg(\frac{2\pi}{(in4d_g/d_x)}]^{_1}$

 $R_g=[(1.2)(\frac{2\pi}{(in4(2)/0.2)}]^{-1}$

 $R_g=0.489$

And

 $R_e=\frac{In(\frac{D_0}{D_1})}{2\pi ki}$

 $R_e=\frac{In(2)}{2*3.142 0.005}$

 $R_e=22.1$

Therefore

 $q=\frac{223}{0.489+22.064}$

 $q=9.9w/m$