Transmittance of semitransparent windows with absorbing cap-shaped droplets condensed on their backside

Abstract

This study aims to investigate systematically light transfer through semitransparent windows with absorbing cap-shaped droplets condensed on their backside as encountered in greenhouses, solar desalination plants, photobioreactors and covered raceway ponds. The Monte Carlo ray-tracing method was used to predict the normal-hemispherical transmittance, reflectance, and normal absorptance accounting for reflection and refraction at the air/droplet, droplet/window, and window/air interfaces and absorption in both the droplets and the window. The droplets were monodisperse or polydisperse and arranged either in an ordered hexagonal pattern or randomly distributed on the backside with droplet contact angle θc ranging between 0 and 180° The normal-hemispherical transmittance was found to be independent of the spatial distribution of droplets. However, it decreased with increasing droplet diameter and polydispersity. The normal-hemispherical transmittance featured four distinct optical regimes for semitransparent window supporting nonabsorbing droplets. These optical regimes were defined based on contact angle and critical angle for internal reflection at the droplet/air interface. However, for strongly absorbing droplets, the normal-hemispherical transmittance (i) decreased monotonously with increasing contact angle for θc< 90° and (ii) remained constant and independent of droplet absorption index kd, droplet mean diameter dm, and contact angle θc for θc≥ 90° Analytical expressions for the normal-hemispherical transmittance were provided in the asymptotic cases when (1) the window was absorbing but the droplets were nonabsorbing with any contact angles θc, and (2) the droplets were strongly absorbing with contact angle θc> 90° Finally, the spectral normal-hemispherical transmittance of a 3mm-thick glass window supporting condensed water droplets for wavelength between 0.4 and 5 μm was predicted and discussed in light of the earlier parametric study and asymptotic behavior.