In case of large scale microalgae cultivation in photobioreactors, reflection from the ground is low for the parallel placed panels and is not considered in the present study. The incident irradiance on the front and back side of the photobioreactors are given by the following equation:

Transmission of light through wall of flat panel photobioreactor

In order to model the transmission of light through the reactor to the culture volume, two interfaces, namely, the interface between air and reactor wall outer surface as well as the interface between reactor wall and culture volume need to be considered. The amounts of solar radiation reflected from each of the interface depend on the refractive indices of the material as well as the angle of incidence. The amount of light refracted is obtained by Snell’s Law from the following equation:

The angle of incidence for the diffuse light is assumed to be 60o as obtained in the literature 39. Reflection of solar radiation from flat panels is determined by using Fresnel equations and is provided in the following equations:

where, and are the reflection of the s-polarized and p-polarized light respectively, and and are the refractive indices of the medium before and after the interface respectively.

Since, normal sunlight is non-polarized in nature, the overall co-efficient of reflection R, is the average of and .

The overall reflection coefficient for the air and reactor wall interface ( as well as reactor wall and culture volume interface ( is computed by the above equations for the different incidence angle as refractive indices of the respective media. Thus, the total amount of irradiance transmitted to microalgae culture medium from the front and back side of the panel is calculated by the following equation:

where, is the factor incorporating the transparency of the medium.

Solar irradiance gradients within the culture volume

Two different light intensities need to be considered while modeling the light gradient inside the culture volume. The first light intensity is a function of height and penetration of diffuse radiation between the parallel plates of the photobioreactor and the second light intensity is the amount of radiation transmitted into the culture volume from the surface of the reactor wall. For microalgae growth, only the photosynthetic active radiation (PAR) of the spectrum is utilized, which is about 43% 18. The Beer-Lambart Law is used to assess the overall incident gradient in the culture volume.

To simplify the model, is integrated to obtain the mean value of irradiance for the entire culture volume ( inside the photobioreactor at time t. Thus,

Heat and Mass Transfer Model for Flat Panel Photobioreactor

Figure S2. Fluxes considered to develop the combined heat and mass transfer model to quantify reactor temperature 19

The temperature of the culture medium is determined by the formulating a heat transfer model for the photobioreactors. The governing equation is given by the following equation:

V = + + + + + + (35)

Where, is the dynamic temperature of the culture volume (K), is the density of water (kg m-3), is the specific heat capacity of water (J kg-1 K-1), V is the volume of photobioreactor (m3), is the radiation from the reactor surface (W),