Fin Optimization $(100$ Points$)$

You are planning to use copper fins with a circular arc quadrangle in a heat exchanger

application. The quadrangle cross$-$section is shown below $(P=2r,A=(4-)r^2).$ You are

asked to optimize the radius r so that heat transfer is maximized from the fin. The volume

of the material you are using in creating the fin is known. The fin has a length $L,$ which also

needs to be optimized. Assume the fin to be insulated at the tip $($use the fin equation given

below$).$ The base of the fin has a temperature of $T_b.$ Set up the problem and generate the

characteristic equation that will be used to solve for the optimal radius, $r_{\text{o}\text{p}\text{t}}$ and Lopt. Then

solve for the maximum amount of heat removal, $Q_{\text{m}\text{a}\text{x}}^{},$ from this optimized fin. Finally,

compare this amount of removed heat to that of an optimized cylindrical fin $($see class notes$)$

using the same properties listed below. Which fin is better, the one with a quadrangle cross$-$

section or the one with a circular cross$-$section?

$Q_b^{}$~$=(T_b-T_{})(K{A}_{c}hP{)}^{\frac{1}{2}}$TANH$\{(\frac{hP}{K{A}_c}{)}^{\frac{1}{2}}(L+\frac{A_c}{P})\}$

Define:

$M=(T_b-T_{})(K{A}_{c}hP{)}^{\frac{1}{2}}$

$m=(\frac{hP}{K{A}_c}{)}^{\frac{1}{2}}(L+\frac{A_c}{P})$

Data:

$h=25\frac{W}{m^2}*K,V=25c{m}^3,K=401\frac{W}{m}*K,T_b=100C,T_{}=20C$