The optimal dimensions of circular fins with variable profile and temperature-dependent thermal conductivity

The optimal dimensions of circular fins with variable profile, and temperature-dependent thermal conductivity are obtained. A profile of the form γ = (w/2)[1+(r₀/r)ⁿ] is studied, while the thermal conductivity considered is of the form k = k₀[1+ε(T-T_∞)/T₀]. The results have been expressed in terms of suitable dimensionless parameters. A correlation for the optimal dimensions of a constant and variable profile fins is presented in terms of reduced heat-transfer rate. It is found that a parabolic circular fin with n = 2 gives an optimum performance. For example, an increase in heat-transfer rate (as compared to constant thickness fin) by about 20% for the optimal fin profile is observed. The effect of thermal conductivity on the optimal dimensions is negligible for the variable profile fin. It is also observed that in general, the optimal fin length is greater for the optimal fin profile.