Application of Collocation Spectral Method to Solve a Convective-Radiative Longitudinal Fin with Temperature Dependent Internal Heat Generation, Thermal Conductivity and Heat Transfer Coefficient

摘要

In this paper, spectral collocation method (SCM) is used to predict the temperature distribution in the fin with temperature dependent thermal conductivity, heat transfer coefficient and internal heat generation. These temperature dependent properties or parameters cause multiple nonlinearities of energy equation. In order to reduce these multiple nonlinearities, a local linearization approach is adopted. The spatial distribution of temperature in the fin is discretized by Chebyshev-Gauss-Lobatto collocation points and Chebyshev polynomials. Accordingly, the differential form of energy equation is transformed to the matrix form of algebraic equation. The accuracy of the SCM model is verified by comparing SCM results with those from analytical solutions for a special case and those available data in references. Meanwhile, compared with analytical solutions, it can be found that the convergence rate of SCM approximately follows exponential law. In addition, effects of various significant properties and parameters, such as thermal conductivity, heat transfer coefficient, convective-conduction parameter, radiation-conduction parameter, convection sink temperature and radiation sink temperature on the temperature distribution and the fin efficiency are comprehensively analyzed.