Mathematical framework for predicting solar thermal build-up of spectrally selective coatings at the Earth’s surface

A transient finite element thermal model is formulated valid for surface coatings on any substrate material and based on the continuum conduction equations with solar loading as a heat source. The model allows cooling to be applied at outer surfaces of the body, by natural convection and accounts for ambient radiative heat loss. Hemispherical spectral reflectivities are obtained for various polymer-based coatings on a steel substrate using spectrophotometers in the 0.1 μm to 25 μm wavelengths. A time-dependent solar irradiation energy source (blackbody equivalent) is applied to an object with spectrally diffuse outer surfaces, and the incoming heat flux is split by a band approximation into reflected and absorbed energy and finally integrated over the complete spectrum to provide thermal source terms for the finite element model.Results show that cyclic diurnal thermal build-up of temperature can be predicted for a body with different spectrally selective coatings. While the model exhibits the classic relationship for thermal build-up with colour, i.e., dark colours absorb more heat and lighter colours remain cooler, it also shows that colours, which appear similar, can have very different thermal build-ups, depending on the infrared reflectivity of the coating.The general suitability of the finite element method to describe geometrically complex bodies coupled with additional parameters such as latitude, longitude and a variable ambient temperature can be used to simulate a variety of scenarios for a diverse number of applications.