Table 4.

Kinetic rate expressions (${\nu }_k$) of the biological processes within the biofilm

	Process	Kinetic rate expression
1	Growth of PH on ${\text{NO}}_3$	${\nu }_1={\mu }_{\text{max,PH}}\frac{S_{\text{IC}}}{K_{\text{PH,IC}}+S_{\text{IC}}}\frac{S_{{\text{NO}}_3}}{K_{{\text{PH,NO}}_3}+S_{{\text{NO}}_3}}\frac{K_{{\text{PH,NH}}_3}}{K_{{\text{PH,NH}}_3}+S_{{\text{NH}}_3}}\frac{K_{\text{PH},{\text{O}}_2}^{\mathit{in}}}{K_{\text{PH},{\text{O}}_2}^{\mathit{in}}+S_{{\text{O}}_2}}\frac{K_M^{\mathit{in}}}{K_M^{\mathit{in}}+M}\frac{I}{I_{\text{opt}}}{e}^{(1-(\frac{I}{I_{\text{opt}}})}{f}_{\text{PH}}$
2	Growth of PH on ${\text{NH}}_3$	${\nu }_2={\mu }_{\text{max,PH}}\frac{S_{\text{IC}}}{K_{\text{PH,IC}}+S_{\text{IC}}}\frac{S_{{\text{NH}}_3}}{K_{{\text{PH,NH}}_3}+S_{{\text{NH}}_3}}\frac{K_{\text{PH},{\text{O}}_2}^{\mathit{in}}}{K_{{\text{PH,O}}_2}^{\mathit{in}}+S_{{\text{O}}_2}}\frac{K_M^{\mathit{in}}}{K_M^{\mathit{in}}+M}\frac{I}{I_{\text{opt}}}{e}^{(1-(\frac{I}{I_{\text{opt}}})}{f}_{\text{PH}}$
3	Heterotrophic growth of PH	${\nu }_3={\mu }_{\text{max,PH}}^{\mathit{resp}}\frac{S_{\text{DOC}}}{K_{\text{PH,DOC}}+S_{\text{DOC}}}\frac{S_{{\text{O}}_2}}{K_{{\text{PH,O}}_2}+S_{{\text{O}}_2}}\frac{K_{\text{PH},I}^{\mathit{in}}}{K_{\text{PH},I}^{\mathit{in}}+I}\frac{K_M^{\mathit{in}}}{K_M^{\mathit{in}}+M}{f}_{\text{PH}}$
4	Aerobic growth of H	${\nu }_4={\mu }_{\text{max},H}\frac{S_{\text{DOC}}}{K_{H,\text{DOC}}+S_{\text{DOC}}}\frac{S_{{\text{NH}}_3}}{K_{H,{\text{NH}}_3}+S_{{\text{NH}}_3}}\frac{S_{{\text{O}}_2}}{K_{H,{\text{O}}_2}+S_{{\text{O}}_2}}\frac{K_M^{\mathit{in}}}{K_M^{\mathit{in}}+M}{f}_H$
5	Anoxic growth of H	${\nu }_5={\mu }_{\text{max},H}\frac{S_{\text{DOC}}}{K_{H,\text{DOC}}+S_{\text{DOC}}}\frac{S_{{\text{NO}}_3}}{K_{H,{\text{NO}}_3}+S_{{\text{NO}}_3}}\frac{S_{{\text{NH}}_3}}{K_{H,{\text{NH}}_3}+S_{{\text{NH}}_3}}\frac{K_{H,{\text{O}}_2}}{K_{H,{\text{O}}_2}+S_{{\text{O}}_2}}\frac{K_M^{\mathit{in}}}{K_M^{\mathit{in}}+M}{f}_H$
6	Growth of N	${\nu }_6={\mu }_{max,N}\frac{S_{\text{IC}}}{K_{N,\text{IC}}+S_{\text{IC}}}\frac{S_{{\text{NH}}_3}}{K_{H,{\text{NH}}_3}+S_{{\text{NH}}_3}}\frac{S_{{\text{O}}_2}}{K_{H,{\text{O}}_2}+S_{{\text{O}}_2}}\frac{K_M^{\mathit{in}}}{K_M^{\mathit{in}}+M}{f}_N$
7	Death of PH	${\nu }_7=k_{d,\text{PH}}{f}_{\text{PH}}$
8	Death of H	${\nu }_8=k_{d,H}{f}_H$
9	Death of N	${\nu }_9=k_{d,N}{f}_N$
where $K_{{\text{PH,O}}_2}^{\mathit{in}}=K_{{\text{O}}_2,\text{max}}^{\mathit{in}}\frac{\frac{S_{\text{IC}}}{S_{{\text{O}}_2}}}{\frac{S_{\text{IC}}}{S_{{\text{O}}_2}}+K_{R_{\text{IC}/{\text{O}}_2}}}$