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

A cylindrical tank $($figure above$)$ 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 $25\frac{W}{m^2}-K($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: $15\frac{W}{m}-K)($in contact with LNG$)$

$100$ mm thick insulation material $($Thermal conductivity: $0\mathrm{.}02-0\mathrm{.}03\frac{W}{m}-K,$ Please see table below$)$

$10$ mm thick steel shell $($Thermal conductivity: $15\frac{W}{m}-K)($in contact with surroundings$)$

Use the following values for the thermal conductivity of the insulation

$\backslash$table$[[$If your last two$],[$numbers of your$],[$student ID are...$]]$

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$)$