Problem $1$

A microchip with a bottom surface temperature of $344$ K uses an array of fins to enhance heat transfer.

The fins are square in shape $(5$ mm on each side$),$ are insulated on their lower end, and have the

dimensions specified in the figure below. Assume a constant convection coefficient for all surfaces of

$h=40\frac{W}{m^2}$ and a thermal conductivity $k=30\frac{W}{m^{*}}K.$ The system is at steady state.

a$.$ What is the temperature $($in K $)$ at the end of a fin $($at the insulation fin$/$insulation interface$)?$

b$.$ How much heat does one fin dissipate?

c$.$ What is the effectiveness of a single fin?

d$.$ What is the efficiency of a single fin?

e$.$ How much heat does the array dissipate $($in W$)$ under the conditions shown?

f$.$ What is the effectiveness of the array shown?

Approximate solutions: $($a$)300$ K $,($b$)1$ W $,($c$)17,($d$)80\%,($e$)60$ W $,($f$)6$