Evaluation of direct beam energy received by convex solar collectors and their optimal orientations

We use analytical and numerical methods to evaluate the daily beam energy received by some convex surfaces. Spherical, hemispherical, cylindrical, and semi-cylindrical convex surfaces with arbitrary tilt angles have been investigated and compared with a flat surface of unit area. Diffusive irradiations (sky and ground) were not considered. For hemispherical and semi-cylindrical surfaces, the optimal orientations at which the received beam energy is maximal were obtained for each day of the year. The dependence of the optimal tilt angle on the day of the year is qualitatively the same as for the flat surface. Clear sky condition has been assumed to hold in this work. It is shown that a flat surface per unit of its area receives highest beam energy among other convex surfaces whereas a sphere receives the least amount. Furthermore, the received daily beam energy per unit of the ground-occupied area has been calculated. In this case, a cylindrical surface with a range of values of radius to height ratio receives the highest amount of energy whereas a flat surface receives the least. This aspect becomes noticeable in places where there are some limits, e.g., land price or any other limitation on the available surface area.