Characterization of directional and anisotropic scattering dependency of emissivity for fibrous heat shields under non-isothermal conditions

The spectral and directional dependence of emissivity along with radiative emissions of scattering media has been characterized as functions of the optical thickness, scattering albedo, and anisotropic scattering phase function. Emissivity is found to be a strong function of the scattering albedo and the anisotropic scattering phase function of the medium, and a generalized relation between spectral emissivity and scattering behavior for thick optical media is developed through the use of Bézier curves. A comparison of the newly developed relation with the spectral emissivities of silica- and carbon-based fibrous media demonstrated excellent agreement, with a root mean square error of 0.19%. The characterization of directional emissivity through scattering media indicates that directional emissivity is higher than hemispherical emissivity in directions normal to the surface, but drops at polar angles greater than 50° because of increased reflection. A novel model is also developed to capture emissions from a medium with a non-uniform temperature distribution using the same emissivities derived under isothermal conditions, but with an effective blackbody temperature, which differs from the surface temperature of the medium. The new emission model exhibited excellent agreement compared to the true emissions under various conditions, with a root mean square error of 1.27%. This new approach offers a practical computationally tractable solution for the radiation energy balance at surfaces that can be used in current state-of-the-art computational fluid dynamics and thermal response simulations.