# ABACO: A New Model of Microalgae-Bacteria Consortia for Biological Treatment of Wastewaters
## 3 Results
### ABACO Model
#### 3.2.1 Microalgae Biomass
The microalgae growth rate is modeled as the product of a maximum growth rate (\(\mu_{\text{ALG,max}}\)) as expressed in Equation (6), algae biomass concentration (\(X_{\text{ALG}}\)) as shown in Equation (7), and switching functions for environmental parameters (irradiance, temperature, pH and dissolved oxygen), carbon dioxide, nitrogen and phosphorous (Equation (9)-(15)). The rate of the microalgae maintenance (m\({}_{\text{ALG}}\)) considers the endogenous respiration of the microalgae and the microalgae decay (Equation (8)). Taking the model described by [PERSON] et al. in [25], the light limitation growth model can be expressed as follows:  
\[\mu_{\text{ALG}}(\text{I}_{\text{av}})=\frac{\mu_{\text{ALG,max}}\cdot\text{I} _{\text{av}}\text{n}}{\text{I}_{\text{k}}\text{n}+\text{I}_{\text{av}}\text{ n}} \tag{6}\]  
where \(\mu_{\text{ALG,max}}\) [day\({}^{-1}\)] is the maximum microalgae growth rate, \(\text{I}_{\text{av}}\) [\(\mu\)E m\({}^{-2}\) s\({}^{-1}\)] is the average irradiance inside de culture it summarizing the light availability inside the reactor, \(\text{I}_{\text{k}}\) [\(\mu\)E m\({}^{-2}\) s\({}^{-1}\)] is the irradiance constant (equivalent to irradiance required to achieve half of the maximal growth rate) and n is a form parameter. The average irradiance is expressed as follows:  
\[\text{I}_{\text{av}}=\frac{\text{I}_{0}}{\text{K}_{\text{a}}\cdot\text{X}_{ \text{ALG}}\cdot\text{h}}\left(1-\text{e}^{-\text{K}_{\text{a}}\cdot\text{X}_{ \text{ALG}}\cdot\text{h}}\right) \tag{7}\]  
where \(\text{I}_{0}\) [\(\mu\)E m\({}^{-2}\) s\({}^{-1}\)] is the irradiance on the reactor surface, \(\text{K}_{\text{a}}\) [m\({}^{2}\) g\({}^{-1}\)] is the biomass extinction coefficient and h [m] is the culture depth in the reactor. The endogenous respiration term can be expressed as follows:  
\[\text{m}_{\text{ALG}}=\text{m}_{\text{min,alg}}+\frac{\text{m}_{\text{max,alg }}\cdot\text{I}_{\text{av}}\text{n}_{\text{resp}}}{\text{I}_{\text{k,resp}} \text{n}_{\text{resp}}+\text{I}_{\text{av}}\text{n}_{\text{resp}}} \tag{8}\]  
where \(\text{m}_{\text{min,alg}}\) and \(\text{m}_{\text{max,alg}}\) [day\({}^{-1}\)] represent the minimum and maximum respiration rates, \(\text{I}_{\text{k,resp}}\) [\(\mu\)E m\({}^{-2}\) s\({}^{-1}\)] is the irradiance required to stop photosynthesis and start respiration process, and \(\text{n}_{\text{resp}}\) is the form parameter for respiration. The influence of temperature, pH, dissolved oxygen and nutrients concentration into the microalgae growth rate are included as normalized values, then it varying between 0 and 1. Therefore, when the culture conditions are optimal these terms are equal to 1 and the specific growth rate is only a function of light availability, achieving the maximal value at irradiances upper than saturation irradiance. However, if culture conditions are not optimal the respective normalized values are lower than 1, directly reducing the microalgae growth rate whatever the irradiance. The temperature index \(\overline{\mu_{\text{ALG}}}(\text{T})\), expressed by [PERSON] et al. in [26], represents the influence of temperature on microalgae growth. The temperature index can be expressed as follows:  
\[\overline{\mu_{\text{ALG}}}(\text{T})=\frac{\left(\text{T}-\text{T}_{\text{ min,ALG}}\right)\left(\text{T}-\text{T}_{\text{min,ALG}}\right)^{2}}{\left(\text{T}_{ \text{opt,ALG}}-\text{T}_{\text{min,ALG}}\right)\left(\left(\text{T}_{\text{ opt,ALG}}-\text{T}_{\text{min,ALG}}\right)\left(\text{T}_{\text{opt,ALG}}-\text{T}_{ \text{min,ALG}}\right)\right)-\left(\left(\text{T}_{\text{opt,ALG}}-\text{T}_{ \text{max,ALG}}\right)\left(\text{T}_{\text{opt,ALG}}+\text{T}_{\text{min,ALG}}- \text{2}\text{T}\right)\right)\right)} \tag{9}\]  
where T [\({}^{\circ}\)C] is the culture temperature, whereas \(\text{T}_{\text{max}}\) [\({}^{\circ}\)C], \(\text{T}_{\text{min}}\) [\({}^{\circ}\)C] and \(\text{T}_{\text{opt}}\) [\({}^{\circ}\)C] are the respective maximal, minimal and optimal temperature for the microalgae strain.