## 2 Materials and Methods
### Microalgae Growth Model  
In this study, a high-rate algae pond (reactor) model was just simplified to focus on microalgae growth as a function of air temperature (T), photosynthetically active radiation (PAR), and a reduced number of impurities present in the cultivation medium. Phosphorus (\(P\)) and nitrogen (N) are two of the impurities in wastewater (cultivation medium) that microalgae can remove for their growth. Wastewater contains a number of nitrogen compounds, such as ammonium, organic nitrogen, nitrate, and nitrite. In this study, total nitrogen is considered equal to the sum of total Kjeldahl nitrogen, nitrite, and nitrate. Similarly, wastewater is relatively rich in phosphorus compounds, including phosphate ions, inorganic forms (ortho and polyphosphates), and organic forms (organically bound phosphates). Therefore, total phosphorus is given here as a combination of these different phosphorus compounds. These parameters were taken as limiting substrates for the growth of these photosynthetic microorganisms. In addition, the initial concentrations of nitrogen and phosphorus in the culture medium are favorable to the growth of microorganisms. These substrates are consumed exclusively by microalgae. Only PAR and temperature are beyond our control and can sometimes act as inhibitors. This simplification has made it possible to reduce the complexity of the model and study biomass productivity including climatic variability. The remaining micropollutants, heavy metals, other nutrients, and organic pollutants are considered non-limiting nutrients. The model does not include the population of bacteria or the maintenance of microalgae. The energy balance [25] and the gas exchange [8; 26] between the system and its environment were also neglected. Finally, the depth of the pond was the only geometric parameter of the system included in the model and was equal to 10 cm. This was used to calculate the mean PAR value in the culture medium. The cell mass concentration of the microalgae in the culture medium was determined using the material balance approach. For a batch culture model, the influent and effluent of the materials in the medium are null, that is, it does not add nutrients to the culture medium, and the volume is considered constant. Furthermore, assuming that all cells have the same mass, the growth kinetics of microalgae is proportional to its specific growth rate (\(\mu\)) which depends on the availability of a limiting substrate. A proportion of microalgae that die during the cultivation period was added to the model formulation and represented by the specific mortality rate (\(\mu_{D}\)). \[\frac{dX}{dt}=\mu(P,N,PAR,T)\cdot X-\mu_{D}\cdot X \tag{4}\]  
\[\mu(PAR,T,P,N)=\mu_{max}\cdot\mu(P)\cdot\mu(N)\cdot\mu(PAR)\cdot\mu(T)\]where \(X(\text{mg}/\text{L})\) is microalgae concentration, \(\mu_{max}\), in day\({}^{-1}\), is the maximum specific growth rate of microalgae. To calculate the specific growth rate, these factors \(\mu(P)\) and \(\mu(N)\) were estimated from the Monod model [27], \(\mu(PAR)\) from modified Monod model [28; 29], and \(\mu(T)\) from the so-called cardinal temperature model [14; 30]. The empirical Monod model, which is easy to calibrate and links growth with a culture parameter, is widely used to simulate wastewater treatment using microorganisms [31]. For PAR, [PERSON]'s modified model involved adding a term to the denominator to account for inhibition when the substrate (PAR) is highly above the optimal limit. The same consideration was included in the cardinal temperature model. Equation (5) gives the expression of the factors \(\mu(P)\) and \(\mu(N)\), where S represents the concentration of phosphorus (P) or nitrogen (N). \(K_{S}\) is a constant, expressed in mg/L, which represents the half-saturation concentration of the limiting nutrients (P and N). \[\mu(S)=\frac{S}{K_{S}+S} \tag{5}\]  
For Equations (6) and (7), an intermediate calculation was performed to find the mean values for the available PAR and the medium temperature of cultivation, respectively.