Thermal Distribution in Fins of Different Geometries With Temperature and Wavelength Dependent Properties

In this research work, we developed a mathematical model of heat transfer for different profiles of fins. This paper investigates heat transfer dynamics in continuously moving fins (rectangular, trapezoidal and concave parabolic), focusing on temperature-dependent thermal conductivity, heat transfer coefficients, internal heat generation and emissivity that varies with temperature and wavelength. The different values of the heat transfer coefficient capture various types of convection, nucleate boiling, condensation, and radiation effects, while treating thermal conductivity as a linear function of temperature. This problem is converted into a dimensionless form, and we adopted the Legendre wavelet collocation method (LWCM) to get the solution of the fin problem for various profiles. An exact solution in specific cases shows congruence up to seven to eight decimal places with LWCM results. Moreover, our analysis explores the effect of numerous non-dimensional parameters such as thermal conductivity parameter $A$, Peclet number $\mathrm{Pe}$, surface emissivity parameter B, convention-conduction parameter $N_{\mathrm{cc}}$, radiation-conduction parameter $N_{\mathrm{rc}}$, internal heat generation $Q$, D fin taper ratio on the temperature profile and fin efficiency were studied in detail. As $N_{\mathrm{cc}}$, $N_{\mathrm{rc}}$, and B, D increases in magnitude, the temperature inside the fin decreases, while higher values of Peclet number ($P_e$), A, m, and Q cause lower heat transfer rate inside the fin. The results provide significant insights into the complex interplay of thermal processes across different fin geometries, emphasizing the importance of these dependencies for accurate modeling and optimization in thermal management systems.