Chapter 3, Problem 3.133P

The extent to which the tip condition affects the thermal performance of a tin depends on the fin geometry and thermal conductivity. as well as the convection coefficient. Consider an alloyed aluminum $\left(k=180\text{W/m}\cdot \text{K}\right)$ rectangular fin whose base temperature is $T_b=100°\text{C}$ .The fin is exposed to a fluid of temperature $T_{\infty }=25°\text{C,}$ and a uniform convection coefficient of $h=100{\text{W/m}}^{\text{2}}\cdot \text{K}$ may be assumed for the fin surface.

1. For a tin of length $L=10\text{mm,}$ thickness $t=1\text{mm,}$ and width $w\gg t,$ determine the tin heat transfer rateper unit width $q_f^{\text{'}},$ efficiency ${\eta }_f,$ effectiveness ${\epsilon }_f,$ thermal resistance per unit width $R_{t,f}^{\text{'}},$ and tip temperature $T\left(L\right)$ for Cases A and B of Table 3.4. Contrast your results with those based on an infinite fin approximation.
2. Explore the effect of variations in L on the heat rate for $3<L<50\text{mm}\text{.}$ Also consider the effect of such variations for a stainless steel fin $\left(k=15\text{W/m}\cdot \text{K}\right)$ .