Symmetry Classification and Exact Solutions of the Convective-Radiative Fin Equation with Variable Thermal Conductivity

Fins at large temperature differences lose heat by convection and radiation. With temperature-dependent conductivity, the nonlinear problem is usually treated numerically. Using λ-symmetries, the extended Prelle–Singer method and Lie symmetries, this study identifies the conductivity laws that make the problem linearizable. In this class, one relation gives the temperature for every heat-loss function, and the problem is well posed for heat loss with positive derivative, vanishing at ambient temperature. For power-law convection with Stefan–Boltzmann radiation, explicit solutions exist for purely convective or radiative loss and for constant heat transfer coefficients. They also exist for mixtures at negligible ambient temperature reducing to polynomials of degree at most four, including turbulent natural convection and nucleate boiling. Other mixtures have implicit exact solutions. The fin efficiency equals tanhμ/μ throughout the class, so the classical design relations hold. Tip losses and internal heat generation admit closed forms. For temperature-dependent heat loss, the class consists of the fin equations with maximal point symmetry sl(3,R). Within the Rosseland and thin-fin approximations, its constant-coefficient member is realized by semitransparent media of high refractive index satisfying a matching condition. A perturbation theory gives the efficiency correction outside the class. The exact solutions provide benchmarks for numerical methods.