Prob. $3.($Liquid nitrogen tank$)$

Liquid nitrogen is stored in a container made of two concentric spheres with the space between them evacuated. The inner sphere has an outside diameter of $1$ m and the space between the two spheres is $0\mathrm{.}1$ m $.$ The outer surface of the inner sphere and the inner surface of the outer sphere are gray with an emissivity of $0\mathrm{.}2.$ If the saturation temperature for nitrogen at atmospheric pressure is $78$ K and its latent heat of vaporization is $\backslash (2\backslash$times $10^\{5\}\backslash$mathrm$\{$~J$\}/\backslash$mathrm$\{$kg$\}\backslash ),$ estimate its boiloff rate under each of the following conditions:

$($a$)$ The outer sphere is at $300$ K $.$

$($b$)$ The outer surface of the outer sphere is black and transfers heat by radiation to surroundings at $300$ K $.$ Assume convection is negligible.

$($c$)$ The outer sphere is at $300$ K and a heat shield in form of a concentric sphere made of a thin metallic sheet with emissivity of $0\mathrm{.}2$ is placed half$-$way between the two spheres.