Optimal dimensions of convective-radiating circular fins

The optimal dimensions of convective-radiating circular fins with variable profile, heat-transfer coefficient and thermal conductivity, as well as internal heat generation are obtained. A profile of the form y = (w/2) [1+(r_o/r)ⁿ] is studied, while variation of thermal conductivity is of the form k = k_o[1+ε((T-T_∞)/(T_b-T_∞))m]. The heat-transfer coefficient is assumed to vary according to a power law with distance from the bore, expressed as h = K[(r-r_o)/(r_e-r_o)]^λ. The results for λ = 0 to λ = 1.9, and -0.4≤ε≤0.4, have been expressed by 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 (quadratic) hyperbolic circular fin with n = 2 gives an optimum performance. The effect of radiation on the fin performance is found to be considerable for fins operating at higher base temperatures, whereas the effect of variable thermal conductivity on the optimal dimensions is negligible for the variable profile fin. It is also observed, in general, that the optimal fin length and the optimal fin base thickness are greater when compared to constant fin thickness.