Q$3.($a$)$ Heat from an electronic device is to be dissipated by rectangular aluminium fins $(k=237\frac{W}{m}*C)$ that are $2$ mm thick and $20$ mm wide, with their bases attached at a constant temperature of $60C.$ These fins are dissipating heat by convection into ambient air at $20C$ with a heat transfer coefficient of $10\frac{W}{m^2}*C.$ Each fin should remove $2$ W of heat transfer from the base. Determine:

$($i$)$ the length $($in mm$)$ of each fin; and

$(8$ marks$)$

$($ii$)$ the temperature of fin tip.

$(2$ marks$)$

Given: $,Q_{\text{c}\text{o}\text{n}\text{v}\text{e}\text{c}\text{t}\text{i}\text{n}\text{f}\text{r}\text{o}\text{m}\text{f}\text{i}\text{n}\text{t}\text{i}\text{p}}^{}=\sqrt[\phantom{2}]{hpk{A}_c}(T_b-T_{})\frac{sinhmL+(\frac{h}{m}k)coshmL}{coshmL+(\frac{h}{m}k)sinhmL}$

$\frac{T(x)-T_{}}{T_b-T_{}}=\frac{coshm(L-x)+(\frac{h}{m}k)sinhm(L-x)}{coshmL+(\frac{h}{m}k)sinhmL},$ where $,m=\sqrt[\phantom{2}]{\frac{hp}{k{A}_c}}$