Fin Efficiency

Temperature of a fin gradually drops along the length. Typical variation is given in Fig.(7). In the limiting case of zero thermal resistance (\(k\rightarrow \infty\)), the temperature of the fin will be uniform at the base value of \(T_0\). The heat transfer from the fin will be maximized in this case: \[Q_{\text{fin,max}} = hA_{\text{fin}}(T_0-T_\infty)\] Fin efficiency (\(\eta_{\text{fin}}\)) can be defined as: \[\eta_{\text{fin}} = \frac{Q_{\text{fin}}}{Q_{\text{fin,max}}} = \frac{\text{actual heat transfer rate from the fin}}{\begin{array}{c}\text{ideal heat transfer rate from the fin}\\ \text{(if the entire fin were at base temperature)}\end{array}}\] Fin efficiency decreases with increasing fin length because of decrease in fin temperature with length. For the ‘long-fin’ \[\eta_{\text{fin}} = \frac{Q_{\text{fin}}}{Q_{\text{fin,max}}} = \frac{\sqrt{PhkA}(T_0-T_\infty)}{hA_{\text{fin}}(T_0-T_\infty)} = \frac{\sqrt{PhkA}}{hPL} = \frac{1}{L}\sqrt{\frac{kA}{hP}} = \frac{1}{mL}\]


Fin Effectiveness

The performance of fins is judged on the basis of the enhancement in heat transfer relative to the no-fin case, and expressed in terms of the fin effectiveness: \[\varepsilon_{\text{fin}} = \frac{Q_{\text{fin}}}{Q_{\text{no fin}}} = \frac{\text{heat transfer rate from the fin}}{\text{heat transfer rate without fin}}\]  
  \[\varepsilon_{\text{fin}} = \left\{ \begin{array}{ll}<1 & \text{fin act as insulation} \\ = 1 & \text{fin does not affect heat transfer} \\ >1 & \text{fin enhances heat transfer} \end{array} \right.\] For the long-fin \[\varepsilon_{\text{fin}} = \frac{Q_{\text{fin}}}{Q_{\text{no fin}}} = \frac{\sqrt{PhkA}(T_o-T_\infty)}{hA(T_o-T_\infty)} = \frac{\sqrt{PhkA}}{hA} = \sqrt{\frac{kP}{hA}}\] and, \[\frac{\varepsilon_{\text{fin}}}{\eta_{\text{fin}}} = \frac{PL}{A}\] Effectiveness of a fin must be greater than 2; otherwise it is not recommended to use that fin.

Fins - Requirements

  • Fins are generally used where convective heat transfer coefficient (\(h\)) values are relatively low. i.e., when air or gas is the medium and heat transfer is by natural convection.

  • Fin material should be of highly conductive materials.

  • Lateral surface area of the fin should be as high as possible.

  • The efficiency of most fins used in practice is above 90%.

  • Note: \(\eta_{\text{fin}} < 1 \qquad \text{ but } \qquad \varepsilon_{\text{fin}} > 1\)