Fins are extended surfaces designed to enhance heat transfer by increasing the surface area. The steady 1D heat balance along a fin of constant cross-section \(A_c\) and perimeter \(P\) is:

\[\frac{d^2 \theta}{dx^2} - m^2 \theta = 0\]

Where \(\theta(x) = T(x) - T_\infty\) is the temperature excess, and \(m^2 = \frac{h P}{k A_c}\).

Fin Tip Boundary Conditions

  • Adiabatic Tip (\(d\theta/dx|_{x=L} = 0\)): The temperature profile is \(\theta(x) = \theta_b \frac{\cosh(m(L-x))}{\cosh(mL)}\). The total fin heat transfer rate is:
    \[q_f = M \tanh(m L) \quad \text{where} \quad M = \sqrt{h P k A_c} \theta_b\]
  • Fin Efficiency (\(\eta_f\)): Ratio of actual fin heat transfer to the maximum possible rate if the entire fin were at base temperature \(T_b\):
    \[\eta_f = \frac{\tanh(m L)}{m L}\]

References

  • Kern, D. Q., & Kraus, A. D. (1972). Extended Surface Heat Transfer. McGraw-Hill.