Parameters of differential equations in four-flux and two-flux models approximated for scattering and absorption results on solar thermal collector black paints

The intrinsic and extrinsic scattering and absorption coefficients of a thin black paint substrate used as a solar thermal collector, deposited on a thick glass substrate, were obtained in an approximated single-layer method, using the new solved equations recently determined from the collimated and diffuse differential equations of the four-flux model, for the intrinsic coefficients, and considering the differential equations of the two-flux model with total light intensities, for the extrinsic coefficients. In a first step, the optical constants were determined from the measurements of regular transmittance and specular reflectance, fitting them to the collimated transmittance and reflectance solutions of the four-flux model. In a second step, the intrinsic coefficients were obtained from the diffuse transmittance and reflectance measurements, considering four forward scattering ratios (for collimated and diffuse light intensities and for forward and backward light directions) and two average crossing parameters (for forward and backward light directions). In a third step, the extrinsic coefficients were obtained, using the new solved equations recently determined considering the total differential equations as the sum of the collimated and diffuse differential equations. Due to the approximated single-layer method, a simple rule of three was applied to the intrinsic and extrinsic results, relating the extinction coefficients calculated from the optical constants to those calculated from the collimated differential equations. The same external and internal diffuse interface reflectance was observed by integrating the collimated interface reflectance up to the critical angle of total internal reflection.