# ABACO: A New Model of Microalgae-Bacteria Consortia for Biological Treatment of Wastewaters
## 3 Results
### ABACO Model
#### 3.2.3 Nitrifying Bacteria  
Nitrifying bacteria can be also supplied to the reactor with the wastewater supplied to the reactor, it being also removed during harvesting. Nitrifying bacteria perform the nitrification process, thus oxidizing ammonium to nitrate. These microorganisms are aerobic, then requiring oxygen, also using CO\({}_{2}\) as a carbon source. The concentration of nitrifying bacteria increases due to growth but also decrease by endogenous respiration and decay. The global balance to estimate the concentration of nitrifying bacteria is given by Equation (24). \[\text{Q}_{\text{d}}\cdot\text{X}_{\text{NIT,in}}+\text{V}\cdot\text{X}_{\text {NIT,out}}\cdot\mu_{\text{NIT}}=\text{Q}_{\text{h}}\cdot\text{X}_{\text{NIT, out}}+\text{V}.\frac{\text{dX}_{\text{NIT,out}}}{\text{dt}} \tag{24}\]  
where \(\text{X}_{\text{NIT,in}}\) [g m\({}^{-3}\)] is the nitrifying bacteria inlet concentration, \(\mu_{\text{NIT}}\) [day\({}^{-1}\)] is the nitrifying bacteria specific growth rate and \(\text{X}_{\text{NIT,out}}\) [g m\({}^{-3}\)] is the nitrifying bacteria concentration in the reactor. The processes related with nitrifying bacteria include both autotrophic growth and maintenance. The rate of the autotrophic growth is modeled as the product of maximum growth rate (\(\mu_{\text{NIT,max}}\)) and switching functions for environmental parameters, such as temperature (\(\overline{\mu_{\text{HET}}}(\text{T})\)), pH (\(\overline{\mu_{\text{HET}}}(\text{pH})\)) and dissolved oxygen (\(\overline{\mu_{\text{HET}}}(\text{DO}_{2})\)); in addition to ammonium nitrogen (\(\overline{\mu_{\text{HET}}}([\text{N}-\text{NH}_{\text{4}}])\)) and phosphate phosphorous (\(\overline{\mu_{\text{HET}}}([\text{P}-\text{PO}_{\text{4}}])\)) (Equation (25)). The rate of maintenance (\(\text{m}_{\text{NIT}}\)) considers the endogenous respiration of the nitrifying bacteria and nitrifying decay. The following equation represents the nitrifying bacteria specific growth rate:  
\[\mu_{\text{NIT}}=\mu_{\text{NIT,max}}\cdot(\overline{\mu_{\text{NIT}}}(\text {T})\cdot\overline{\mu_{\text{NIT}}}(\text{pH})\cdot\overline{\mu_{\text{NIT} }}(\text{DO}_{2})\cdot\overline{\mu_{\text{NIT}}}(\text{CO}_{2})\cdot \overline{\mu_{\text{NIT}}}([\text{N}-\text{NH}_{\text{4}}])\cdot\overline{ \mu_{\text{NIT}}}([\text{P}-\text{PO}_{\text{4}}]))-\text{m}_{\text{NIT}} \tag{25}\]  
where \(\mu_{\text{NIT,max}}\) [day\({}^{-1}\)] is the maximum specific growth rate for nitrifying bacteria and \(\text{m}_{\text{NIT}}\) [day\({}^{-1}\)] is the endogenous respiration of the nitrifying bacteria and the nitrifying maintenance. As for the heterotrophic bacteria, the temperature and pH terms (\(\overline{\mu_{\text{NIT}}}(\text{T})\) and \(\overline{\mu_{\text{NIT}}}(\text{pH})\)) are expressed the same form. These terms depend on the maximum (\(\text{T}_{\text{max, NIT}}\) and \(\text{pH}_{\text{max, NIT}}\)), minimum (\(\text{T}_{\text{min, NIT}}\) and \(\text{pH}_{\text{min, NIT}}\)) and optimal (\(\text{T}_{\text{opt, NIT}}\) and \(\text{pH}_{\text{opt, NIT}}\)) values of temperature and pH for nitrifying bacteria, such as expressed in Equations (26) and (27), respectively.