## 2 Mathematical Modelling  
In microalgal research, several light models have been proposed that are able to describe the photosynthetic behavior of microalgae growing under different conditions. [PERSON] et al. [11] proposed classification into three types. Type I models are able to predict the rate of photosynthesis as a function of the incident or average light intensity, while type II models compute productivity as the sum of local productivities. Type III models consider, instead, the impacts of light gradients and of short light cycles. Furthermore, the distribution of light may be described in different ways, for example, through the employment of the Lambert Beer model, the two flux model or Hyperbolic Beer-Lambert law [13].  
In this work a [PERSON]'s type II light model was used to describe microalgal growth in photobioreactors with different geometries enlightened from one-side and predict the growth in reactors with variable depths. The growth is related to the light distribution inside the reactor, which takes into account the light attenuation according to the Lambert-Beer model, widely used to describe the light attenuation pattern in algal culture broth, especially in the presence of low-cell densities [13]. When using this equation, the hypothesis is that the medium inside the reactor is isotropic and that there are no scattering phenomena, which is a good approximation of reality. Furthermore, if scattering is taken into account, even smaller deviations from uniformity are predicted, with respect to the limiting \"zero reflectance\" case we propose [12]. Considering that the cell concentrations experimentally obtained are below 2 g L\({}^{-1}\), the use of the Lambert-Beer model was considered to be acceptable [13].  
The Lambert-Beer law describes an exponential decay of light intensity from the external surface of the cultivation system, as described in the following equation [11]:  
\[I=I_{0}\;e^{-k_{a}Xz} \tag{1}\]  
where \(I\) is the local light intensity of light, \(z\) is the distance of the external surface to the considered position in a flat geometry, \(I_{0}\) is the intensity of the incident light, \(k_{a}\) is the attenuation constant and \(X\) the cellular concentration.  
The attenuation constant \(k_{a}\) can be estimated as the ratio between the area of the cell capturing light (A\({}_{capt}\)) and the mass of the cell (m\({}_{cell}\)) and can be simply estimated as:  
\[k_{a}=\frac{3}{2}\frac{d_{capt}^{2}}{\rho_{cell}d_{cell}^{3}} \tag{2}\]  
where \(\rho_{cell}\) and \(d_{cell}\) are respectively the density and the diameter of a microalgal cell and \(d_{capt}\) is the diameter of the transversal section of light capture. According to the proposed model, the medium is ideally mixed, the biomass concentration is uniform inside the reactor and all the nutrients are in excess.  
The growth rate \(\mu\) is then expressed according to a Monod-like model [11] considering light as a substrate for algal photosynthesis, as expressed by the following expression:  
\[\mu=\frac{\mu_{max}\;I}{I_{K}+I} \tag{3}\]  
where \(\mu\) is the specific growth rate, \(\mu_{max}\) is the maximum specific growth rate and \(I_{K}\) is the light saturation constant. It is worth noting that the range of light intensities used in this work is far away from photo-inhibition phenomena [14], so that [PERSON]'s law can be applied without any limitation.