Chapter 4, Problem 4.32P

In Chapter 3 we that, whenever fins are attached to a base material, the base temperature is unchanged. What in fact happens is that, if the temperature of the base material exceeds the fluid temperature, attachment of a fin depresses the junction temperature $T_j$ below the original temperature of the base, and heat flow from the base material to the fin is two-dimensional.

Consider conditions for which a long aluminum pin fin of diameter $D=5\text{mm}$ is attached to a base material whose temperature far from the junction is maintained at $T_b=100°\text{C}\text{.}$ Fin convection conditions correspond to $h=50{\text{W/m}}^{\text{2}}\cdot \text{K}$ and $T_{\infty }=25°\text{C}\text{.}$

1. What are the fin heat rate and junction temperature when the base material is (i) aluminum $\left(k=240\text{W/m}\cdot \text{K}\right)$ and (ii) stainless steel $\left(k=15\text{W/m}\cdot \text{K}\right)?$
2. Repeat the foregoing calculations if a thermal contact resistance of $R_{t,j}^n=3\times {10}^{-5}{\text{m}}^{\text{2}}\cdot \text{K/W}$ is associated with the method of joining the pin fin to the base material.
3. Considering the thermal contact resistance, plot the heat rate as a function of the convection coefficient over the range $10\le h\le 100{\text{W/m}}^{\text{2}}\cdot \text{K}$ for each of the two materials.