Fins and Extended Surfaces Questions and Answers

Answer: A

The key's answer is thin, closely spaced fins.

Fin effectiveness compares the heat transferred with the fin to the heat transferred from the bare base area it occupies. For a long fin:

\[ \varepsilon = \sqrt{\frac{kP}{hA_{c}}} \]

A thin fin has a large perimeter P for a small cross-section A_c, so each fin is more effective. Spacing them closely puts more fins, and so more surface, on a given base area.

Thick fins have a smaller P/A_c ratio and so lower effectiveness, and wide spacing wastes base area. Very close spacing can starve the gaps of air flow in practice, but the standard answer to this question takes thin and closely spaced as best.

Answer: A

The ratio is \(4/\pi\).

For a very long fin, \(q = \sqrt{hPkA_{c}}\,\theta_{b}\), so with the same material, coefficient and base temperature, \(q \propto \sqrt{PA_{c}}\). With side \(a\) and diameter \(a\):

\[ \begin{aligned} \text{square} &: P A_{c} = 4a \cdot a^{2} = 4a^{3} \\ \text{circle} &: P A_{c} = \pi a\cdot\frac{\pi a^{2}}{4} = \frac{\pi^{2}a^{3}}{4} \\ \frac{q_{sq}}{q_{circ}} &= \sqrt{\frac{16}{\pi^{2}}} = \frac{4}{\pi} \end{aligned} \]

\(16/\pi^{2}\) forgets the square root. \(1/\pi\) and \(1/\pi^{2}\) are below 1, but the square fin has more perimeter and more area than the inscribed circle, so it must transfer more heat.