Thermal analysis of functionally graded porous fins with spatially varying conductivity and porosity using a spectral collocation approach

Functionally graded materials have attracted considerable attention due to their enhanced thermal performance in advanced heat transfer applications. However, the coupled effects of spatially varying thermal conductivity and surface porosity on fin performance, particularly under transient conditions, remain insufficiently explored. This study investigates the thermal behaviour of longitudinal functionally graded material fins by analysing the combined influence of material grading, exponential porosity decay, radiation, and convective heat transfer under steady-state and transient conditions. A nonlinear heat transfer equation is formulated, nondimensionalised, and solved numerically using the spectral collocation method for steady-state analysis and the bivariate spectral collocation method for transient analysis, considering linear, quadratic, exponential, and power-law conductivity profiles. The obtained results are then validated using the finite element method, and a good agreement is achieved.The results reveal that increased surface porosity limits cooling by reducing the effective convective area without altering bulk conduction, leading to higher temperature distributions. Linear, quadratic, and exponential conductivity profiles increase the temperature, whereas a power-law profile enhances thermal performance by redistributing conductivity away from the base. Parametric analysis indicates that radiation and Biot numbers improve heat dissipation and lower fin temperatures, while higher Biot numbers reduce fin efficiency. In addition, internal heat generation intensifies thermal gradients. These findings provide a practical framework for designing high-efficiency, lightweight thermal systems.