Chapter 3, Problem 3.131P

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 tin of length $L=10\text{mm,}$ thickness $t=1\text{mm,}$ and width $w\gg t.$ base temperature of the fin is $T_b=100°\text{C,}$ and the tin is exposed to a fluid of temperature $T_{\infty }=25°\text{C}\text{.}$

1. Assuming a uniform convection coefficient of $h=100{\text{W/m}}^{\text{2}}\cdot \text{K}$ over the entire tin surface,determine the tin heat transfer rate per unit width $q_f^{\text{'}},$ efficiency ${\eta }_f,$ effectiveness ${\epsilon }_f,$ thermal resistance per unit width $R_{t,f}^{\text{'}},$ and the tip temperature T(L) 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 the convection coefficient on the heat rate for $10<h<1000{\text{W/m}}^{\text{2}}\cdot \text{K}\text{.}$ Also consider the effect of such variations for a stainless steel fin $\left(k=15\text{W/m}\cdot \text{K}\right).$