# ABACO: A New Model of Microalgae-Bacteria Consortia for Biological Treatment of Wastewaters
## 3 Results
### ABACO Model
#### 3.2.1 Microalgae Biomass  
The microalgal cells are present in the photobioreactors, it not being feed to the system with the influent wastewater. Part of the microalgae biomass is removed every day with the effluent as a function of imposed dilution rate (inverse of hydraulic retention time). Microalgae biomass concentration increases due to autotrophic growth of microalgae, using light as energy source and of CO\({}_{2}\) as carbon source, whereas it reduces by endogenous respiration and decay of microalgae. These last phenomena represent the autoxidation of microalgae, where they metabolize their own cellular material. The global balance to estimate the microalgae biomass concentration is given by Equation (2). \[\text{V-X}_{\text{ALG}}\cdot\mu_{\text{ALG}}=\text{Q}_{\text{h}}\cdot\text{X} _{\text{ALG}}+\text{V}\cdot\frac{\text{dX}_{\text{ALG}}}{\text{dt}} \tag{2}\]  
where V [m\({}^{3}\)] is the volume in the reactor, X\({}_{\text{ALG}}\) [g m\({}^{-3}\)] is the microalgae biomass concentration, \(\mu_{\text{ALG}}\) [day\({}^{-1}\)] is the microalgae specific growth rate and Q\({}_{\text{h}}\) [m\({}^{3}\) s\({}^{-1}\)] represents the harvesting flow rate. The specific growth rate \(\mu_{\text{ALG}}\) is mainly a function of light availability inside the reactor, summarized by the average irradiance inside the culture I\({}_{\text{av}}\)[25], and modified by the influence of different variables such as temperature (\(\overline{\mu_{\text{ALG}}}(\text{T})\)), pH (\(\overline{\mu_{\text{ALG}}}(\text{pH})\)), dissolved oxygen (\(\overline{\mu_{\text{ALG}}}(\text{DO}_{2})\)) and CO\({}_{2}\) (\(\overline{\mu_{\text{ALG}}}(\text{CO}_{2})\)). In addition the influence of nutrients availability such as ammonium nitrogen (\(\overline{\mu_{\text{ALG}}}([\text{N}-\text{NH}_{4}])\)), phosphate phosphorus (\(\overline{\mu_{\text{ALG}}}([\text{P}-\text{PO}_{4}])\)) and nitrate nitrogen (\(\overline{\mu_{\text{ALG}}}([\text{N}-\text{NO}_{3}])\)), and the microalgae maintenance (m\({}_{\text{ALG}}\)), is considered as shown in Equation (3). \[\mu_{\text{ALG}}=(\mu_{\text{ALG}}(\text{I}_{\text{av}})\cdot\overline{\mu_{ \text{ALG}}}(\text{T})\cdot\overline{\mu_{\text{ALG}}}(\text{pH})\cdot \overline{\mu_{\text{ALG}}}(\text{DO}_{2})\cdot\overline{\mu_{\text{ALG}}}( \text{CO}_{2})\cdot\overline{\mu_{\text{ALG}}}(\text{N})\cdot\overline{\mu_{ \text{ALG}}}([\text{P}-\text{PO}_{4}]))-\text{m}_{\text{ALG}} \tag{3}\]  
Figure 3: Model input-output diagram. Microalgae can growth using both ammonium and nitrate as a nitrogen source. Then, there is a process rate for the growth of microalgae using ammonium and another one when using nitrate, thus Equation (3) becomes as Equations (4) and (5) for considering this phenomenon. Notice that Equation (5) considering the consumption of nitrate is only used when there is not ammonium in the system. \[\mu_{\text{ALG}}=(\mu_{\text{ALG}}(\text{I}_{\text{av}})\cdot\overline{\mu_{ \text{ALG}}}(\text{T})\cdot\overline{\mu_{\text{ALG}}}(\text{pH})\cdot\overline{ \mu_{\text{ALG}}}(\text{DO}_{2})\cdot\overline{\mu_{\text{ALG}}}(\text{CO}_{2}) \cdot\overline{\mu_{\text{ALG}}}([\text{N}-\text{NH}_{4}])\cdot\overline{\mu_{ \text{ALG}}}([\text{P}-\text{PO}_{4}]))-\text{m}_{\text{ALG}} \tag{4}\]  
\[\mu_{\text{ALG}}=(\mu_{\text{ALG}}(\text{I}_{\text{av}})\cdot\overline{\mu_{ \text{ALG}}}(\text{T})\cdot\overline{\mu_{\text{ALG}}}(\text{pH})\cdot\overline{ \mu_{\text{ALG}}}(\text{DO}_{2})\cdot\overline{\mu_{\text{ALG}}}(\text{CO}_{2}) \cdot\overline{\mu_{\text{ALG}}}([\text{N}-\text{NO}_{3}])\cdot\overline{\mu_{ \text{ALG}}}([\text{P}-\text{PO}_{4}]))-\text{m}_{\text{ALG}} \tag{5}\]  
As observed from Equation (3), during the microalgae growth, it is assumed that two main process occur: the microalgal growth and the microalgal maintenance.