$3$ Part $3$: Problems $(20$ Marks$)$

A factory uses a cylindrical steel tank to store a chemical liquid. The tank has an inner radius of $1\mathrm{.}5$ meters and a wall thickness of $5$ cm $.$ The outer surface of the tank is insulated with a layer of insulation material $10$ cm thick. The insulation has a thermal conductivity of $0\mathrm{.}04\frac{W}{m}K.$ The tank is exposed to an environment where the ambient air temperature is $25C.$ Inside the tank, the chemical is kept at a constant temperature of $80C.$ The thermal conductivity of the steel is $45\frac{W}{m}K.$

Find the following:

$($a$)$ The thermal resistance of the tank per unit length.

$($b$)$ The heat input required $(Q_{in}^{})$ per unit length to maintain the temperature inside the tank.

$($c$)$ The actual surface temperature of the assembly.

$($d$)$ BONUS: Find the total heat input required for a $3$ m long tank. hint: include the effect of the end caps

Notes: Estimate the surface temperature to be $30C$ for all but parts $C$ and $D.$ For efficiency, use a film temperature of air as $300$ K with properties:

$u=1\mathrm{.}589*{10}^{-7}\frac{m^2}{s},k=0\mathrm{.}0263\frac{W}{m}*K,Pr=0\mathrm{.}707,Cp=1007\frac{J}{k}g*k$