Table 7.

Kinetic rate expressions (${\nu }_k^{\ast }$) of the biological processes within the bulk liquid

	Process	Kinetic rate expression
1	Growth of PH on ${\text{NO}}_3$	${\nu }_1^{\ast }={\mu }_{\text{max,PH}}\frac{S_{\text{IC}}^{\ast }}{K_{\text{PH,IC}}+S_{\text{IC}}^{\ast }}\frac{S_{{\text{NO}}_3}^{\ast }}{K_{{\text{PH,NO}}_3}+S_{{\text{NO}}_3}^{\ast }}\frac{K_{{\text{PH,NH}}_3}}{K_{{\text{PH,NH}}_3}+S_{{\text{NH}}_3}^{\ast }}\frac{K_{{\text{PH,O}}_2}^{in,\ast }}{K_{{\text{PH,O}}_2}^{in,\ast }+S_{{\text{O}}_2}^{\ast }}\frac{K_{M^{\ast }}^{\mathit{in}}}{K_M^{\mathit{in}}+M^{\ast }}\frac{I_0}{I_{\text{opt}}}{e}^{(1-\frac{I_0}{I_{\text{opt}}})}{\psi }_{\text{PH}}^{\ast }$
2	Growth of PH on ${\text{NH}}_3$	${\nu }_2^{\ast }={\mu }_{\text{max,PH}}\frac{S_{\text{IC}}^{\ast }}{K_{\text{PH,IC}}+S_{\text{IC}}^{\ast }}\frac{S_{{\text{NH}}_3}^{\ast }}{K_{{\text{PH,NH}}_3}+S_{{\text{NH}}_3}^{\ast }}\frac{K_{{\text{PH,O}}_2}^{in,\ast }}{K_{{\text{PH,O}}_2}^{in,\ast }+S_{{\text{O}}_2}^{\ast }}\frac{K_{M^{\ast }}^{\mathit{in}}}{K_M^{\mathit{in}}+M^{\ast }}\frac{I_0}{I_{\text{opt,PH}}}{e}^{(1-\frac{I_0}{I_{\text{opt}}})}{\psi }_{\text{PH}}^{\ast }$
3	Heterotrophic growth of PH	${\nu }_3^{\ast }={\mu }_{\text{max,PH}}^{\mathit{resp}}\frac{S_{\text{DOC}}^{\ast }}{K_{\text{PH,DOC}}+S_{\text{DOC}}^{\ast }}\frac{S_{{\text{O}}_2}^{\ast }}{K_{\text{PH},{\text{O}}_2}+S_{{\text{O}}_2}^{\ast }}\frac{K_{\text{PH},I}^{\mathit{in}}}{K_{\text{PH},I}^{\mathit{in}}+I_0}\frac{K_{M^{\ast }}^{\mathit{in}}}{K_M^{\mathit{in}}+M^{\ast }}{\psi }_{\text{PH}}^{\ast }$
4	Aerobic growth of H	${\nu }_4^{\ast }={\mu }_{max,H}\frac{S_{\text{DOC}}^{\ast }}{K_{H,\text{DOC}}+S_{\text{DOC}}^{\ast }}\frac{S_{{\text{NH}}_3}^{\ast }}{K_{H,{\text{NH}}_3}+S_{{\text{NH}}_3}^{\ast }}\frac{S_{{\text{O}}_2}^{\ast }}{K_{H,{\text{O}}_2}+S_{{\text{O}}_2}^{\ast }}\frac{K_{M^{\ast }}^{\mathit{in}}}{K_M^{\mathit{in}}+M^{\ast }}{\psi }_H^{\ast }$
5	Anoxic growth of H	${\nu }_5^{\ast }={\mu }_{max,H}\frac{S_{\text{DOC}}^{\ast }}{K_{H,\text{DOC}}+S_{\text{DOC}}^{\ast }}\frac{S_{{\text{NO}}_3}^{\ast }}{K_{H,{\text{NO}}_3}+S_{{\text{NO}}_3}^{\ast }}\frac{S_{{\text{NH}}_3}^{\ast }}{K_{H,{\text{NH}}_3}+S_{{\text{NH}}_3}^{\ast }}\frac{K_{H,{\text{O}}_2}}{K_{H,{\text{O}}_2}+S_{{\text{O}}_2}^{\ast }}\frac{K_{M^{\ast }}^{\mathit{in}}}{K_M^{\mathit{in}}+M^{\ast }}{\psi }_H^{\ast }$
6	Growth of N	${\nu }_6={\mu }_{max,N}^{\ast }\frac{S_{\text{IC}}^{\ast }}{K_{N,\text{IC}}+S_{\text{IC}}^{\ast }}\frac{S_{{\text{NH}}_3}^{\ast }}{K_{H,{\text{NH}}_3}+S_{{\text{NH}}_3}^{\ast }}\frac{S_{{\text{O}}_2}^{\ast }}{K_{H,{\text{O}}_2}+S_{{\text{O}}_2}^{\ast }}\frac{K_{M^{\ast }}^{\mathit{in}}}{K_M^{\mathit{in}}+M^{\ast }}{\psi }_N^{\ast }$
7	Growth of ${\text{PH}}_d$ on ${\text{NO}}_3$	${\nu }_7^{\ast }={\mu }_{\text{max,PH}}\frac{S_{\text{IC}}^{\ast }}{K_{\text{PH},\text{IC}}+S_{\text{IC}}^{\ast }}\frac{S_{{\text{NO}}_3}^{\ast }}{K_{\text{PH},{\text{NO}}_3}+S_{{\text{NO}}_3}^{\ast }}\frac{K_{\text{PH},{\text{NH}}_3}}{K_{\text{PH},{\text{NH}}_3}+S_{{\text{NH}}_3}^{\ast }}\frac{K_{\text{PH},{\text{O}}_2}^{in,\ast }}{K_{\text{PH},{\text{O}}_2}^{in,\ast }+S_{{\text{O}}_2}^{\ast }}\frac{K_{M^{\ast }}^{\mathit{in}}}{K_M^{\mathit{in}}+M^{\ast }}\frac{I_0}{I_{\text{opt}}}{e}^{(1-\frac{I_0}{I_{\text{opt}}})}{\psi }_{d_{\text{PH}}}^{\ast }$
8	Growth of ${\text{PH}}_d$ on ${\text{NH}}_3$	${\nu }_8^{\ast }={\mu }_{\text{max,PH}}\frac{S_{\text{IC}}^{\ast }}{K_{\text{PH},\text{IC}}+S_{\text{IC}}^{\ast }}\frac{S_{{\text{NH}}_3}^{\ast }}{K_{\text{PH},{\text{NH}}_3}+S_{{\text{NH}}_3}^{\ast }}\frac{K_{\text{PH},{\text{O}}_2}^{in,\ast }}{K_{\text{PH},{\text{O}}_2}^{in,\ast }+S_{{\text{O}}_2}^{\ast }}\frac{K_{M^{\ast }}^{\mathit{in}}}{K_M^{\mathit{in}}+M^{\ast }}\frac{I_0}{I_{\text{opt,PH}}}{e}^{(1-\frac{I_0}{I_{\text{opt}}})}{\psi }_{d_{\text{PH}}}^{\ast }$
9	Heterotrophic growth of ${\text{PH}}_d$	${\nu }_9^{\ast }={\mu }_{\text{max,PH}}^{\mathit{resp}}\frac{S_{\text{DOC}}^{\ast }}{K_{\text{PH},\text{DOC}}+S_{\text{DOC}}^{\ast }}\frac{S_{{\text{O}}_2}^{\ast }}{K_{\text{PH},{\text{O}}_2}+S_{{\text{O}}_2}^{\ast }}\frac{K_{\text{PH},I}^{\mathit{in}}}{K_{\text{PH},I}^{\mathit{in}}+I_0}\frac{K_{M^{\ast }}^{\mathit{in}}}{K_M^{\mathit{in}}+M^{\ast }}{\psi }_{d_{\text{PH}}}^{\ast }$
10	Aerobic growth of $H_d$	${\nu }_{10}^{\ast }={\mu }_{max,H}\frac{S_{\text{DOC}}^{\ast }}{K_{H,\text{DOC}}+S_{\text{DOC}}^{\ast }}\frac{S_{{\text{NH}}_3}^{\ast }}{K_{H,{\text{NH}}_3}+S_{{\text{NH}}_3}^{\ast }}\frac{S_{{\text{O}}_2}^{\ast }}{K_{H,{\text{O}}_2}+S_{{\text{O}}_2}^{\ast }}\frac{K_{M^{\ast }}^{\mathit{in}}}{K_M^{\mathit{in}}+M^{\ast }}{\psi }_{d_H}^{\ast }$
11	Anoxic growth of $H_d$	${\nu }_{11}^{\ast }={\mu }_{max,H}\frac{S_{\text{DOC}}^{\ast }}{K_{H,\text{DOC}}+S_{\text{DOC}}^{\ast }}\frac{S_{{\text{NO}}_3}^{\ast }}{K_{H,{\text{NO}}_3}+S_{{\text{NO}}_3}^{\ast }}\frac{S_{{\text{NH}}_3}^{\ast }}{K_{H,{\text{NH}}_3}+S_{{\text{NH}}_3}^{\ast }}\frac{K_{H,{\text{O}}_2}}{K_{H,{\text{O}}_2}+S_{{\text{O}}_2}^{\ast }}\frac{K_{M^{\ast }}^{\mathit{in}}}{K_M^{\mathit{in}}+M^{\ast }}{\psi }_{d_H}^{\ast }$
12	Growth of $N_d$	${\nu }_{12}^{\ast }={\mu }_{max,N}^{\ast }\frac{S_{\text{IC}}^{\ast }}{K_{N,\text{IC}}+S_{\text{IC}}^{\ast }}\frac{S_{{\text{NH}}_3}^{\ast }}{K_{H,{\text{NH}}_3}+S_{{\text{NH}}_3}^{\ast }}\frac{S_{{\text{O}}_2}^{\ast }}{K_{H,{\text{O}}_2}+S_{{\text{O}}_2}^{\ast }}\frac{K_{M^{\ast }}^{\mathit{in}}}{K_M^{\mathit{in}}+M^{\ast }}{\psi }_{d_N}^{\ast }$
13	Death of PH	${\nu }_{13}^{\ast }=k_{d,\text{PH}}{\psi }_{\text{PH}}^{\ast }$
14	Death of H	${\nu }_{14}^{\ast }=k_{d,H}{\psi }_H^{\ast }$
15	Death of N	${\nu }_{15}^{\ast }=k_{d,N}{\psi }_N^{\ast }$
16	Death of ${\text{PH}}_d$	${\nu }_{16}^{\ast }=k_{d,\text{PH}}{\psi }_{d_{\text{PH}}}^{\ast }$
17	Death of $H_d$	${\nu }_{17}^{\ast }=k_{d,H}{\psi }_{d_H}^{\ast }$
18	Death of $N_d$	${\nu }_{18}^{\ast }=k_{d,N}{\psi }_{d_N}^{\ast }$
where $K_{\text{PH},{\text{O}}_2}^{in,\ast }=K_{{\text{O}}_2,max}^{\mathit{in}}\frac{\frac{S_{\text{IC}}^{\ast }}{S_{{\text{O}}_2}^{\ast }}}{\frac{S_{\text{IC}}^{\ast }}{S_{{\text{O}}_2}^{\ast }}+K_{R_{\mathrm{IC}/{\text{O}}_2}}}$