Question

Consider two infinitely long fins made of the same material and exposed to the same convective environment. One fin has a square cross-section of side \(a\), and the other has a circular cross-section of diameter \(d\). Assume \(a = d\). The ratio of the steady-state heat transfer rate from the square fin to that from the circular fin, \(\frac{\dot{Q}_{\text{square}}}{\dot{Q}_{\text{circular}}}\), is:

• A. \(\frac{\pi}{4}\)
• B. \(\frac{4}{\pi}\)
• C. \(\frac{2}{\sqrt{\pi}}\)
• D. \(\frac{\sqrt{\pi}}{2}\)

Hint:

For fin comparison problems, identify the constant parameters first. If only the shape changes, \(\dot{Q} \propto \sqrt{P A_c}\) for infinite fins, and \(\dot{Q} \propto \sqrt{P A_c} \tanh(mL)\) for finite fins with insulated tips.

Answer 1

The Correct Option is B

Step 1: Identify the Governing Equation. For an infinitely long fin, the rate of heat transfer is: \[ \dot{Q} = \sqrt{hPkA_c} \cdot \theta_0 \] Since both fins are of the same material (\(k\)), same environment (\(h\)), and same base temperature (\(\theta_0\)), the ratio depends on the geometry: \(\dot{Q} \propto \sqrt{P \cdot A_c}\).
Step 2: Calculate Geometric Parameters for Square Fin (Side \(a\)). Perimeter \(P_s = 4a\); Area \(A_{cs} = a^2\). Geometric factor \(\sqrt{P_s A_{cs}} = \sqrt{4a \cdot a^2} = \sqrt{4a^3}\).
Step 3: Calculate Geometric Parameters for Circular Fin (Diameter \(d = a\)). Perimeter \(P_c = \pi a\); Area \(A_{cc} = \frac{\pi}{4} a^2\). Geometric factor \(\sqrt{P_c A_{cc}} = \sqrt{\pi a \cdot \frac{\pi a^2}{4}} = \sqrt{\frac{\pi^2 a^3}{4}} = \frac{\pi}{2} \sqrt{a^3}\).
Step 4: Find the Ratio. \[ \frac{\dot{Q}_{sq}}{\dot{Q}_{cir}} = \frac{\sqrt{4a^3}}{\frac{\pi}{2}\sqrt{a^3}} = \frac{2}{\pi/2} = \frac{4}{\pi} \]