Problem $2$: Heat Conduction in Composite Materials $(25$ Points$)$

A cylindrical tank $($figure to the right$)$ of inner radius $5$ m

and height $15$ m is used to store LNG at $0\mathrm{.}1$ bar and $110$ K $.$

The surface temperature along the inner surface of the tank

is also at $110$ K $.$ Assume the air temperature $($on the

outside$)$ is $300$ K $.$ The convective heat transfer coefficient

on the outside is $25$W$//$m$^(2)-$K $($i$.$e$.,$ the tank is gaining heat

via convection$).$

LNG storage tanks are composed of three layers $($assume

top and bottom caps are insulated. So$,$ heat transfer occurs

only along the circumferential face$)$:

$10$ mm thick steel shell $($Thermal conductivity: $15$W$//$m$-$K $)($in contact with LNG$)$

$100$ mm thick insulation material $($Thermal conductivity: $0\mathrm{.}02$W$//$m$-$K$,)$

$10$ mm thick steel shell $($Thermal conductivity: $15$W$//$m$-$K $)($in contact with

surroundings$)$

Use the $0\mathrm{.}02$W$//$m$-$K for the thermal conductivity of the insulation

a$.$ Calculate the heat transfer $($in W$)$ through the circumferential walls of the tank when

there is no insulation $(5$ points$)$

b$.$ Calculate the heat transfer $($in W$)$ through the circumferential walls of the tank with

$100$ mm thick insulation $(10$ points$)$

c$.$ Assuming that the thicknesses of the inner and outer steel shells are fixed. Determine

the thickness of the insulation needed to reduce your heat transfer $($determined in Part

$[$b$])$ by $50\%(10$ points$)$