A model for light distribution and average solar irradiance inside outdoor tubular photobioreactors for the microalgal mass culture

A mathematical model to estimate the solar irradiance profile and average light intensity inside a tubular photobioreactor under outdoor conditions is proposed, requiring only geographic, geometric, and solar position parameters. First, the length of the path into the culture traveled by any direct or disperse ray of light was calculated as the function of three variables: day of year, solar hour, and geographic latitude. Then, the phenomenon of light attenuation by biomass was studied considering Lambert- Beer's law (only considering absorption) and the monodimensional model of Cornet et al. (1900) (considering absorption and scattering phenomena). Due to the existence of differential wavelength absorption, none of the literature models are useful for explaining light attenuation by the biomass. Therefore, an empirical hyperbolic expression is proposed. The equations to calculate light path length were substituted in the proposed hyperbolic expression, reproducing light intensity data obtained in the center of the loop tubes. The proposed model was also likely to estimate the irradiance accurately at any point inside the culture. Calculation of the local intensity was thus extended to the full culture volume in order to obtain the average irradiance, showing how the higher biomass productivities in a Phaeodactylum tricornutum UTEX 640 outdoor chemostat culture could be maintained by delaying light limitation.