Transmittance of transparent horizontal and tilted windows supporting large non-absorbing pendant droplets

Abstract

This study establishes that the deviation of large pendant droplets from an ideal cap-shape due to gravity can have significant and complex impacts on the normal-hemispherical and directional-hemispherical transmittances of light through horizontal and tilted transparent windows. First, the shape of pendant droplets larger than the capillary length was predicted numerically by balancing gravitational and surface tension forces for various droplet volumes, contact angles, and window tilt angles. Then, light transfer through windows supporting such numerically generated droplets was simulated using the Monte Carlo ray-tracing method. The window transmittance for large droplets was found to be nearly independent of droplet spatial arrangement and size distribution for relatively narrow size distributions. Furthermore, the droplets could be assumed to be cap-shaped in predicting the normal-hemispherical transmittance for droplet volumes V< 10 µL and contact angles θc<θcr where θcr is the critical angle for total internal reflection at the droplet/air interface. However, for larger droplets and/or contact angles, assuming droplets to be cap-shaped caused the transmittance to be overestimated by as much as 37% for horizontal windows. This was due to gravity-induced deformation of the droplet shape resulting in increased reflection at the droplet/air interface. For tilted windows, the droplet deformation caused the normal-hemispherical transmittance to increase with increasing droplet volume and window tilt angle. For both horizontal and tilted windows, transmittance decreased linearly with increasing droplet surface area coverage. These results and numerical tools can be used to design energy efficient solar stills, greenhouses, and covered photobioreactors, for example.