Problem $5$

The Sizing procedure for a fin generally requires the selection of more than one physical dimension

$($Length$).$ In a plate fin, for example, there are two such dimensions, which are the length of the fin, $L,$ and

the thickness, $t.$ One of these dimensions can be selected optimally when the total volume of fin material is

fixed. Such a constraint is often justified by high cost of the high$-$conductive metals that are employed in

the manufacturing of finned surfaces $($e$.$g$.,$ copper, aluminum$)$ and by the cost associated with the weight

of the fin $($e$.$g$.,$ the finned cylinders of air$-$cooled motors for model airplanes and motorcycles$).$ Assuming

a rectangular fin shown below,

what would be the optimal dimensions of the fin?

Hint: To select one of the two dimensions of a rectangular fin optimally, we should maximize the total heat

transfer rate $q_f$ while keeping the known volume of fin, $V,$ constant. For simplicity assume that the fin

profile is sufficiently slender