The Effect of Temperature Dependency of Surface Emissivity in 1-D and 2-D Nonlinear Heat Radiation Equations

Knowledge of the temperature dependency of the physical properties such as surface emissivity, which controls the radiative problem, is fundamental for determining the thermal balance of many scientific and industrial processes. The surface emissivity generally depends on surface temperature, wavelength, surface material geometry (curvature, roughness), and direction of observation, and often changes with oxidation, melting, coating, and even surface pollution. Current work studies the influences of temperature dependency of surface emissivity on heat transfer in a lumped system. The problem was investigated for both one-dimensional (1-D) and two-dimensional (2-D) systems. For 1-D equations, two recent analytical methods called the homotopy perturbation method (HPM) and parameterized perturbation method (PPM) are presented. Unlike classic perturbation methods, these techniques do not need small parameters for nonlinear heat transfer equations. Thus, we can apply them for large values of surface emissivity. For the 2-D domain, a finite-element code is used to obtain the unsteady distribution of temperature. Three different functions were chosen to describe the thermal behavior of surface emissivity. The solutions of 1-D nonlinear equations are compared with the accurate numerical fourth-order Runge–Kutta method, and excellent agreement (maximum error of 0.0021%) was observed. The capabilities of employed analytical methods are discussed and it is shown that HPM needs fewer series terms in comparison with PPM. For both 1-D and 2-D equations, it is illustrated that the temperature gradient increased by adding to the order of emissivity variation versus temperature.