12. Solved Problems

Example 1: Effect of Diameter and Thermal Conductivity on Heat Transfer through Fin 

A long, circular aluminium rod attached at one end to the heated wall and transfers heat through convection to a cold fluid.

  • If the diameter of the rod is triples, by how much would the rate of heat removal change?

  • If a copper rod of the same diameter is used in place of aluminium, by how much would the rate of heat removal change?
    aluminum: \(k=240\) W/(m.K); copper: \(k=400\) W/(m.K)

Solution: For long-fin, \[Q = \theta_o\sqrt{PhkA}\] For cyldrical fin, \[P = \pi D \qquad \text{ and } \qquad A = \frac{\pi}{4}D^2 \qquad \Longrightarrow \quad PA = \frac{\pi}{4}D^3\] Therefore, \[Q \propto \sqrt{kD^3}\] (a) \(D_2/D_1=3\). Therefore, \[\frac{Q_2}{Q_1} = \frac{\sqrt{D_2^3}}{\sqrt{D_1^3}} = \sqrt{\left(\frac{D_2}{D_1}\right)^3} = \sqrt{3^3}= 5.2\] i.e., there is a 520% increase in heat transfer.

(b) \(k_2/k_1=400/240=1.667\). Therefore, \[\frac{Q_2}{Q_1} = \sqrt{\frac{k_2}{k_1}} = \sqrt{1.667}=1.29\] i.e., there is a 29% increase in heat transfer.