Your company has received complaints about condensation on the outside surface of their new model of refrigerator. The walls have fiberglass insulation with thickness L f in between two layers of $1$ mm thick sheet metal. For the insulation, kf $=0\mathrm{.}035$ W$/($m$·$K$)$; for the sheet metal, km

$=15$ W$/($m$·$K$).$ The operating manual suggests that the refrigerator$’$s internal temperature should be kept at $3º$C$.$ The heat transfer coefficients at the inner and outer surfaces of the wall are h in $=5$ W$/($m $2·$K$)$ and h out $=10$ W$/($m $2·$K$),$ respectively. The kitchen temperature averages $23º$C$.$

In initial testing, you find that condensation occurs when the temperature of the outer surface drops below $18º$C$.$

A$)$ Calculate the maximum heat flux through the refrigerator wall that will prevent condensation.

B$)$ Calculate the minimum thickness L f of fiberglass insulation needed to prevent condensation.

C$)$ If this thickness is used, determine the temperatures on both sides of the fiberglass layer.

$($Use L f $=0\mathrm{.}5$ cm if you could not solve part B$.)$