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.

Answer: A

The statements accepted are (a), (c) and (d), so the answer is (a), (c) and (d).

(a) Fins are worth adding on the side with the smaller heat transfer coefficient, because that side controls the overall resistance. This is why a car radiator carries its fins on the air side, not the coolant side.

(d) A high thermal conductivity keeps the fin's temperature close to the base temperature along its length, so more of it works effectively. (c) Fins should be thin, so that many fit on a surface, and not long: the temperature falls along a fin towards the surrounding fluid, so the far end of a long fin adds very little heat transfer for the material it uses.

Statement (b), long and thick fins, is therefore the one that is wrong, and option A is the only choice that leaves it out.