Transient heat transfer in a functionally graded convecting longitudinal fin

The study investigates the transient thermal performance of a constant area longitudinal fin made of a functionally graded material. Such a fin offers advantages that are not attainable with a traditional fin made of a homogeneous material. A numerical approach has been used to study the transient response of the fin with a step change in its base temperature. The fin is assumed to have an adiabatic tip. Three types of variations in the thermal conductivity with the longitudinal distance along the fin are considered: (a) linear, (b) quadratic, and (c) exponential. New analytical solutions for the steady state performance of the fin are derived in terms of the Bessel functions for cases (a) and (c) and in terms of the Legendre functions for case (b). These solutions provide a check on the accuracy of the transient numerical predictions for large times. The thermal performance of the fin is governed by the classical fin parameter, N _c, and the fin thermal conductivity grading parameter, a. Results are presented for the transient temperature distribution, base heat flow, convective heat loss, the energy stored in the fin and the fin efficiency for representative values of N _c and a. It is found that the transient, as well the steady state performance of the fin, is significantly affected by the functional grading of the fin material. The results presented are not only of fundamental interest but can also be used to design a functionally graded fin with the desirable steady and transient thermal characteristics.