Exact and approximate solutions of convective-radiative fins with temperature-dependent thermal conductivity using integral equation method

Abstract The nonlinear thermal performance of convective and radiative longitudinal cooling fins with temperature-dependent thermal conductivity is studied. For linearly temperature-dependent thermal conductivity, the exact temperature distribution is obtained analytically in an implicit integral form. By converting the resulting integral equation to an algebraic equation, a simpler explicit expression of quadratic polynomials for the temperature excess in the whole convective-radiative fins is derived and the numerical results are compared with the exact one and the previous ones. Additionally, analytical expressions for the temperature change at the fin tip and for the fin efficiency are respectively given in terms of the thermal and geometric parameters of extended surfaces. The accuracy of approximate solution is examined. The influences of nonlinearity and hybrid Biot number on the temperature distribution, the fin-tip temperature change, and the fin efficiency are analyzed.