Problem $3(40$ Points$)$ An air$-$cooled heat sink is made of $100$ conical fins placed on a square metallic plate of side $W_b=3cm.$ The figure below illustrates all relevant variables so you can perform the next calculations.

$T_{}=20C$

$($a$)$ Calculate the efficiency of each conical fin. Consult Table $3\mathrm{.}5$ from the recommended textbook $($also in class notes$).$ Bessel functions can be calculated using Matlab or Mathematica.

$($b$)$ Represent the finned surface with an equivalent resistance network and calculate the total $($effective$)$ heat sink$\text{'}$s resistance $($fins and surface without fins$).$

$($c$)$ Plot the heat sink$\text{'}$s resistance as a function of the heat transfer coefficient from $20$ to $250\frac{W}{m^2}K.$

$($d$)$ Repeat $($a$)-($c$)$ considering that the fins are i$)$ square pin fins with diagonal $D_{\text{f}\text{i}\text{n}}$ and the same volume as conical fins; ii$)$ cylinder fins with diameter $D_{\text{f}\text{i}\text{n}}$ and the same volume as conical fins. If you could choose, what heat sink would you manufacture?