Table 2 Autotrophic process model in batch operation

Differential equations

\(\frac{dX}{dt}=(\mu -{\mu }_{d})\cdot X\)

(Eq. 23)

\(\frac{d{c}_{{H}_{2},g}}{dt}=-{k}_{L}a\cdot ({c}_{{H}_{2}}^{*}-{c}_{{H}_{2},l})\cdot \frac{{V}_{L}\cdot R\cdot T}{{V}_{G}\cdot P}\)

(Eq. 28)

\(\frac{d{c}_{P,A}}{dt}={q}_{P,A}\cdot X\)

(Eq. 24)

\(\frac{d{c}_{{O}_{2},l}}{dt}={k}_{L}a\cdot \left({c}_{{O}_{2}}^{*}-{c}_{{O}_{2},l}\right)-X\frac{\mu }{{Y}_{X,{O}_{2}}}\)

(Eq. 29)

\(\frac{d{c}_{P,B}}{dt}={q}_{P,B}\cdot X\)

(Eq. 25)

\(\frac{d{c}_{{CO}_{2},l}}{dt}={k}_{L}a\cdot \left({c}_{C{O}_{2}}^{*}-{c}_{C{O}_{2},l}\right)-X\frac{\mu }{{Y}_{X,{CO}_{2}}}\)

(Eq. 30)

\(\frac{d{c}_{{O}_{2},g}}{dt}=-{k}_{L}a\cdot ({c}_{{O}_{2}}^{*}-{c}_{{O}_{2},l})\cdot \frac{{V}_{L}\cdot R\cdot T}{{V}_{G}\cdot P}\)

(Eq. 26)

\(\frac{d{c}_{{H}_{2},l}}{dt}={k}_{L}a\cdot \left({c}_{{H}_{2}}^{*}-{c}_{{H}_{2},l}\right)-X\frac{\mu }{{Y}_{X,{H}_{2}}}\)

(Eq. 31)

\(\frac{d{c}_{{CO}_{2},g}}{dt}=-{k}_{L}a\cdot ({c}_{{CO}_{2}}^{*}-{c}_{C{O}_{2},l})\cdot \frac{{V}_{L}\cdot R\cdot T}{{V}_{G}\cdot P}\)

(Eq. 27)

Kinetic rates

\(\mu = {\mu }_{max}\cdot \frac{{c}_{{O}_{2},l}}{{c}_{{O}_{2},l}+{K}_{{O}_{2}}}\cdot \frac{{c}_{C{O}_{2},l}}{{c}_{{CO}_{2},l}+{K}_{C{O}_{2}}}\cdot \frac{{c}_{{H}_{2},l}}{{c}_{{H}_{2},l}+{K}_{{H}_{2}}}\)

(Eq. 32)

\({\mu }_{switch}= {\mu }_{max}\cdot \frac{{c}_{{O}_{2},l}}{{c}_{{O}_{2},l}+{K}_{{O}_{2}}}\cdot \frac{{c}_{{H}_{2},l}}{{c}_{{H}_{2},l}+{K}_{{H}_{2}}}\)

(Eq. 40)

\({\mu }_{d}= {\mu }_{d,min}+{\mu }_{d,max}\cdot \frac{{c}_{{O}_{2},l}}{{c}_{{O}_{2},l}+{K}_{{O}_{2}}}\cdot \frac{{c}_{C{O}_{2},l}}{{c}_{{CO}_{2},l}+{K}_{C{O}_{2}}}\cdot \frac{{c}_{{H}_{2},l}}{{c}_{{H}_{2},l}+{K}_{{H}_{2}}}\)

(Eq. 33)

\({\mu }_{d,switch}= {\mu }_{d,min}+{\mu }_{d,max}\cdot \frac{{c}_{{O}_{2},l}}{{c}_{{O}_{2},l}+{K}_{{O}_{2}}}\cdot \frac{{c}_{{H}_{2},l}}{{c}_{{H}_{2},l}+{K}_{{H}_{2}}}\)

(Eq. 41)

\({q}_{P,A}= \frac{\mu }{{Y}_{X,A}}\)

(Eq. 34)

\({c}_{{O}_{2}}^{*}={c}_{{O}_{2},g}\cdot P\bullet {H}_{{O}_{2}}\)

(Eq. 42)

\({q}_{P,B}= \frac{\mu }{{Y}_{X,B}}\)

(Eq. 35)

\({c}_{C{O}_{2}}^{*}={c}_{C{O}_{2},g}\cdot P\cdot {H}_{C{O}_{2}}\)

(Eq. 43)

\({q}_{f1,A}={q}_{f1,A,max}\cdot \frac{{c}_{P,A}}{{c}_{P,A}+{K}_{f1,A}}\)

(Eq. 36)

\({c}_{{H}_{2}}^{*}={c}_{{H}_{2},g}\cdot P\cdot {H}_{{H}_{2}}\)

(Eq. 44)

\({q}_{r,B}={q}_{r,B,max}\cdot \frac{{c}_{P,B}}{{c}_{P,B}+{K}_{r,B}}\)

(Eq. 37)

\({q}_{f2,A}={q}_{f2,A,max}\cdot \frac{{c}_{P,A}}{{c}_{P,A}+{K}_{f2,A}}\)

(Eq. 45)

\({q}_{f1,BA}={Y}_{f1,B,A}\cdot {q}_{f1,A}\)

(Eq. 38)

\({q}_{f2,BA}={Y}_{f2,B,A}\cdot {q}_{f2,A}\)

(Eq. 46)

\({q}_{r,AB}={Y}_{r,A,B}\cdot {q}_{r,B}\)

(Eq. 39)