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

`q=9.9w/m`

Step-by-step 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=(T_g-T_n)/(R_g+R_e)`

Where

`R_g=[kg((2\pi)/((in4d_g/d_x))]^(_1)`

`R_g=[(1.2)((2\pi)/((in4(2)/0.2))]^(-1)`

`R_g=0.489`

And

`R_e=(In((D_0)/(D_1)))/(2\pi ki)`

`R_e=(In(2))/(2*3.142 0.005)`

`R_e=22.1`

Therefore

`q=(223)/(0.489+22.064)`

`q=9.9w/m`