Optimization of circular fins with variable thermal parameters

The optimization of rectangular profile circular fins with variable thermal conductivity and convective heat transfer coefficients is discussed. The linear variation of the thermal conductivity is considered to be of the form k = ka(1+β(T−Ta)), and the heat transfer coefficient is assumed to vary according to an exponential function with the distance from the bore of the form h = hb exp (γ(r−rb)\(re−rb)). The nonlinear conducting–convecting–radiating heat transfer equation is solved by the differential transformation method. The effective of convective–radiative heat transfer at the fin tip is considered. It is shown that, considering the thermal conductivity and heat transfer coefficient are both constant, for a given fin volume, the optimum fin length is almost independent of the fin base temperature for pure convection. However, for both convection–radiation and pure radiation, the length of the optimum fins for higher temperatures is shorter than the length of the fins with lower temperatures.