Convecting–radiating fins: Explicit solutions, efficiency and optimization

We analyze the second order, non-linear, one dimensional differential equation describing the steady conduction of heat in convecting–radiating longitudinal fins. We introduce an auxiliary dependent variable, solving a first order differential equation and related to the thickness of the fin, giving the distribution of the heat along the fin. The purely convecting case, corresponding to a linear equation, and the convecting–radiating case, corresponding to a non-linear equation, are treated separately. For the linear case, different solutions, corresponding to different shapes of the fin, are analyzed. For the non-linear case, an explicit solution in terms of the auxiliary variable is obtained. The distribution of the temperature, the efficiency of the fin and the applicability of the results are discussed.