Problem $1(5$ points$)$: A $15$ mm radius current carrying solid cylindrical copper wire $(k=200\frac{W}{m}\frac{?}{K})$ is tightly covered with $5$ mm thick plastic

insulation whose thermal conductivity is $k=0\mathrm{.}15\frac{W}{m}\frac{?}{K},$ and is exposed to convection with ambient air at $20$ C with $h=30\frac{W}{m^2}\frac{?}{K}.$ The heat

generated within the copper wire is $1M\frac{W}{m^3}.$

a$.$ In the absence of insulation $($without the $5$ mm thick insulation$),$ determine outer surface temperature of the copper wire if its outer surface

is exposed to ambient conditions directly. Use HDE and energy balance for this part.

b$.$ In the presence of insulation of thickness $5$ mm and using ideas from thermal resistance networks $($as well as results from HDE$),$ determine

the temperature at the outer surface of the copper wire.

c$.$ Sketch temperature profile as well as the heat flux as a function of radius, respectively. No need to find explicit values.

d$.$ If a contact resistance were present between the insulation and the copper wire, sketch the approximate temperature profile for this case

$($no need to solve for anything$).$

e$.$ What is the critical radius of insulation for the conditions of this problem?

f$.$ What should be the minimum thickness of the insulation such that the outer surface temperature of the copper wire does not fall below

its value calculated from part $($a$)?$