| Component | Description | Expressions |
|---|---|---|
| \(C\) | carbon content | \(C=\left[\begin{matrix}S\\D\\B\\E\end{matrix}\right]\) |
| \(I\) | input vector | \(I=\left[\begin{matrix}I_{S}\\I_{D}\\0\\0\end{matrix}\right]\) |
| \(T_{M}\) | transition operator | \(T_{M}=\left[\begin{matrix}-1 & 0 &\frac{a_{BS}\cdot r_{B}}{r_{B} + r_{E}} & 0\\1 & -1 &\frac{r_{B}\cdot\left(- a_{BS} + 1\right)}{r_{B} + r_{E}} & 1\\1 & E_{C} & -1 & 0\\0 & 0 &\frac{r_{E}}{r_{B} + r_{E}} & -1\end{matrix}\right]\) |
| \(N\) | decomposition operator | \(N=\left[\begin{matrix}\frac{E\cdot V}{K + S} & 0 & 0 & 0\\0 &\frac{B\cdot V_{U}}{D + K_{U}} & 0 & 0\\0 & 0 & r_{B} + r_{E} & 0\\0 & 0 & 0 & r_{L}\end{matrix}\right]\) |
| \(f_{s}\) | the right hand side of the ode | \(f_{s}=I+T_{M}\,N\,C\) |
\(\left[\begin{matrix}B\cdot a_{BS}\cdot r_{B} -\frac{E\cdot S\cdot V_{max}\cdot e^{-\frac{E_{a}}{0.008314\cdot T + 2.269722}}}{K_{0} + K_{s}\cdot T + S} + I_{S}\\-\frac{B\cdot D\cdot V_{Umax}\cdot e^{-\frac{E_{aU}}{0.008314\cdot T + 2.269722}}}{D + K_{U0} + K_{Us}\cdot T} + B\cdot r_{B}\cdot\left(- a_{BS} + 1\right) +\frac{E\cdot S\cdot V_{max}\cdot e^{-\frac{E_{a}}{0.008314\cdot T + 2.269722}}}{K_{0} + K_{s}\cdot T + S} + E\cdot r_{L} + I_{D}\\\frac{B\cdot D\cdot V_{Umax}\cdot e^{-\frac{E_{aU}}{0.008314\cdot T + 2.269722}}}{D + K_{U0} + K_{Us}\cdot T}\cdot\left(T\cdot\epsilon_{s} +\epsilon_{0}\right) + B\cdot\left(- r_{B} - r_{E}\right) +\frac{E\cdot S\cdot V_{max}\cdot e^{-\frac{E_{a}}{0.008314\cdot T + 2.269722}}}{K_{0} + K_{s}\cdot T + S}\\B\cdot r_{E} - E\cdot r_{L}\end{matrix}\right]\)
| Component | Description | Expressions |
|---|---|---|
| \(C\) | carbon content | \(C=\left[\begin{matrix}Y\\O\end{matrix}\right]\) |
| \(I\) | input vector | \(I=\left[\begin{matrix}i\\0\end{matrix}\right]\) |
| \(\xi\) | environmental effects multiplier | \(\xi=r\) |
| \(T\) | transition operator | \(T=\left[\begin{matrix}-1 & 0\\h & -1\end{matrix}\right]\) |
| \(N\) | decomposition operator | \(N=\left[\begin{matrix}k_{1} & 0\\0 & k_{2}\end{matrix}\right]\) |
| \(f_{s}\) | the right hand side of the ode | \(f_{s}=I+\xi\,T\,N\,C\) |
\(\left[\begin{matrix}- Y\cdot k_{1}\cdot r + i\\- O\cdot k_{2}\cdot r + Y\cdot h\cdot k_{1}\cdot r\end{matrix}\right]\)
| Component | Description | Expressions |
|---|---|---|
| \(I\) | input vector | \(I=\left[\begin{matrix}0\\\Phi_{l}\\0\\0\\\Phi_{i} -\Phi_{o} -\Phi_{up}\end{matrix}\right]\) |
| \(C\) | carbon content | \(C=\left[\begin{matrix}C_{s}\\C_{f}\\C_{ds}\\C_{df}\\N\end{matrix}\right]\) |
| \(A_{GM}\) | decomposition operator | \(A_{GM}=\left[\begin{matrix}-\frac{A}{C_{s}}\cdot C_{ds} & 0 & s & s & 0\\0 & - y & 0 & -\frac{\alpha\cdot r}{\alpha -\beta} & -\frac{i}{\alpha -\beta}\\\frac{A}{C_{s}}\cdot C_{ds} & y & - r - s & 0 & 0\\0 & 0 & 0 &\frac{\alpha\cdot r}{\alpha -\beta} - r - s &\frac{i}{\alpha -\beta}\\0 & y\cdot\left(-\alpha +\beta\right) &\alpha\cdot r & 0 & - i\end{matrix}\right]\) |
| \(f_{s}\) | the right hand side of the ode | \(f_{s}=I+A_{GM}\,C\) |
\(\left[\begin{matrix}- A\cdot C_{ds} + C_{df}\cdot s + C_{ds}\cdot s\\-\frac{C_{df}\cdot\alpha\cdot r}{\alpha -\beta} - C_{f}\cdot y -\frac{N\cdot i}{\alpha -\beta} +\Phi_{l}\\A\cdot C_{ds} + C_{ds}\cdot\left(- r - s\right) + C_{f}\cdot y\\C_{df}\cdot\left(\frac{\alpha\cdot r}{\alpha -\beta} - r - s\right) +\frac{N\cdot i}{\alpha -\beta}\\C_{ds}\cdot\alpha\cdot r + C_{f}\cdot y\cdot\left(-\alpha +\beta\right) - N\cdot i +\Phi_{i} -\Phi_{o} -\Phi_{up}\end{matrix}\right]\)
| Component | Description | Expressions |
|---|---|---|
| \(C\) | carbon content | \(C=\left[\begin{matrix}A\\B\end{matrix}\right]\) |
| \(I\) | input vector | \(I=\left[\begin{matrix}m\\0\end{matrix}\right]\) |
| \(A_{GeM}\) | decomposition operator | \(A_{GeM}=\left[\begin{matrix}-\alpha & 0\\K\cdot\alpha & -\beta\end{matrix}\right]\) |
| \(f_{s}\) | the right hand side of the ode | \(f_{s}=I+A_{GeM}\,C\) |
\(\left[\begin{matrix}- A\cdot\alpha + m\\A\cdot K\cdot\alpha - B\cdot\beta\end{matrix}\right]\)
| Component | Description | Expressions |
|---|---|---|
| \(C\) | carbon content | \(C=\left[\begin{matrix}C_{1}\\C_{2}\\C_{3}\\C_{4}\\C_{5}\end{matrix}\right]\) |
| \(I\) | input vector | \(I=\left[\begin{matrix}J\cdot\gamma\\J\cdot\left(-\gamma + 1\right)\\0\\0\\0\end{matrix}\right]\) |
| \(\xi\) | environmental effects multiplier | \(\xi=f_{T}\,f_{W}\) |
| \(A\) | decomposition operator | \(A=\left[\begin{matrix}- k_{1} & 0 & 0 & 0 & 0\\0 & - k_{2} & 0 & 0 & 0\\a\cdot k_{1} & a\cdot k_{2} & a\cdot k_{3} - k_{3} & a\cdot k_{4} & 0\\b\cdot k_{1} & b\cdot k_{2} & b\cdot k_{3} & b\cdot k_{4} - k_{4} & 0\\0 & 0 & 0 & 0 & 0\end{matrix}\right]\) |
| \(f_{s}\) | the right hand side of the ode | \(f_{s}=I+\xi\,A\,C\) |
\(\left[\begin{matrix}- C_{1}\cdot f_{T}\cdot f_{W}\cdot k_{1} +\frac{DR\cdot J}{DR + 1}\\- C_{2}\cdot f_{T}\cdot f_{W}\cdot k_{2} + J\cdot\left(-\frac{DR}{DR + 1} + 1\right)\\\frac{0.46\cdot C_{1}\cdot f_{T}\cdot f_{W}\cdot k_{1}}{2.672\cdot e^{- 0.0786\cdot pClay} + 4.0895} +\frac{0.46\cdot C_{2}\cdot f_{T}\cdot f_{W}\cdot k_{2}}{2.672\cdot e^{- 0.0786\cdot pClay} + 4.0895} + C_{3}\cdot f_{T}\cdot f_{W}\cdot\left(- k_{3} +\frac{0.46\cdot k_{3}}{2.672\cdot e^{- 0.0786\cdot pClay} + 4.0895}\right) +\frac{0.46\cdot C_{4}\cdot f_{T}\cdot f_{W}\cdot k_{4}}{2.672\cdot e^{- 0.0786\cdot pClay} + 4.0895}\\\frac{0.54\cdot C_{1}\cdot f_{T}\cdot f_{W}\cdot k_{1}}{2.672\cdot e^{- 0.0786\cdot pClay} + 4.0895} +\frac{0.54\cdot C_{2}\cdot f_{T}\cdot f_{W}\cdot k_{2}}{2.672\cdot e^{- 0.0786\cdot pClay} + 4.0895} +\frac{0.54\cdot C_{3}\cdot f_{T}\cdot f_{W}\cdot k_{3}}{2.672\cdot e^{- 0.0786\cdot pClay} + 4.0895} + C_{4}\cdot f_{T}\cdot f_{W}\cdot\left(- k_{4} +\frac{0.54\cdot k_{4}}{2.672\cdot e^{- 0.0786\cdot pClay} + 4.0895}\right)\\0\end{matrix}\right]\)
| Component | Description | Expressions |
|---|---|---|
| \(C\) | carbon content | \(C=\left[\begin{matrix}C_{1}\\C_{2}\\C_{3}\\C_{4}\\C_{5}\\C_{6}\\C_{7}\end{matrix}\right]\) |
| \(I\) | input vector | \(I=\left[\begin{matrix}F_{s}\cdot J_{1}\\F_{m}\cdot J_{1}\\F_{s}\cdot J_{2}\\F_{m}\cdot J_{2}\\0\\0\\0\end{matrix}\right]\) |
| \(\xi\) | environmental effects multiplier (DEFAG) | \(\xi=f_{T}\,f_{W}\) |
| \(A\) | decomposition operator | \(A=\left[\begin{matrix}- k_{1} & 0 & 0 & 0 & 0 & 0 & 0\\0 & - K_{2} & 0 & 0 & 0 & 0 & 0\\0 & 0 & - k_{3} & 0 & 0 & 0 & 0\\0 & 0 & 0 & - K_{4} & 0 & 0 & 0\\\alpha_{51}\cdot k_{1} & 0.45\cdot K_{2} &\alpha_{53}\cdot k_{3} & 0.45\cdot K_{4} & - k_{5} & 0.42\cdot K_{6} & 0.45\cdot K_{7}\\\alpha_{61}\cdot k_{1} & 0 &\alpha_{63}\cdot k_{3} & 0 &\alpha_{65}\cdot k_{5} & - K_{6} & 0\\0 & 0 & 0 & 0 & 0.004\cdot k_{5} & 0.03\cdot K_{6} & - K_{7}\end{matrix}\right]\) |
| \(f_{s}\) | the right hand side of the ode | \(f_{s}=I+\xi\,A\,C\) |
\(\left[\begin{matrix}- C_{1}\cdot K_{1}\cdot f_{T}\cdot f_{W}\cdot e^{- 3\cdot Ls} + J_{1}\cdot\left(0.018\cdot LN + 0.15\right)\\- C_{2}\cdot K_{2}\cdot f_{T}\cdot f_{W} + J_{1}\cdot\left(- 0.018\cdot LN + 0.85\right)\\- C_{3}\cdot K_{3}\cdot f_{T}\cdot f_{W}\cdot e^{- 3\cdot Ls} + J_{2}\cdot\left(0.018\cdot LN + 0.15\right)\\- C_{4}\cdot K_{4}\cdot f_{T}\cdot f_{W} + J_{2}\cdot\left(- 0.018\cdot LN + 0.85\right)\\C_{1}\cdot K_{1}\cdot f_{T}\cdot f_{W}\cdot\left(- 0.55\cdot A_{l} + 0.55\right)\cdot e^{- 3\cdot Ls} + 0.45\cdot C_{2}\cdot K_{2}\cdot f_{T}\cdot f_{W} + C_{3}\cdot K_{3}\cdot f_{T}\cdot f_{W}\cdot\left(- 0.45\cdot A_{l} + 0.45\right)\cdot e^{- 3\cdot Ls} + 0.45\cdot C_{4}\cdot K_{4}\cdot f_{T}\cdot f_{W} - C_{5}\cdot K_{5}\cdot f_{T}\cdot f_{W}\cdot\left(- 0.75\cdot Tx + 1\right) + 0.42\cdot C_{6}\cdot K_{6}\cdot f_{T}\cdot f_{W} + 0.45\cdot C_{7}\cdot K_{7}\cdot f_{T}\cdot f_{W}\\0.7\cdot A_{l}\cdot C_{1}\cdot K_{1}\cdot f_{T}\cdot f_{W}\cdot e^{- 3\cdot Ls} + 0.7\cdot A_{l}\cdot C_{3}\cdot K_{3}\cdot f_{T}\cdot f_{W}\cdot e^{- 3\cdot Ls} + C_{5}\cdot K_{5}\cdot f_{T}\cdot f_{W}\cdot\left(- 0.75\cdot Tx + 1\right)\cdot\left(0.68\cdot Tx + 0.146\right) - C_{6}\cdot K_{6}\cdot f_{T}\cdot f_{W}\\0.004\cdot C_{5}\cdot K_{5}\cdot f_{T}\cdot f_{W}\cdot\left(- 0.75\cdot Tx + 1\right) + 0.03\cdot C_{6}\cdot K_{6}\cdot f_{T}\cdot f_{W} - C_{7}\cdot K_{7}\cdot f_{T}\cdot f_{W}\end{matrix}\right]\)
| Component | Description | Expressions |
|---|---|---|
| \(C\) | carbon content | \(C=\left[\begin{matrix}S\\D\\M\\E\end{matrix}\right]\) |
| \(T\) | transition operator | \(T=\left[\begin{matrix}-1 & 0 & 0 & 0\\1 & -1 &\frac{K_{r}\cdot k_{t}}{SUE\cdot k_{m} + k_{t}} & 0\\0 & - K_{e} + SUE & -1 & 0\\0 & K_{e} & 0 & -1\end{matrix}\right]\) |
| \(N\) | decomposition operator | \(N=\left[\begin{matrix}\frac{E}{S}\cdot k_{d} & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & SUE\cdot k_{m} + k_{t} & 0\\0 & 0 & 0 & k_{l}\end{matrix}\right]\) |
| \(f_{s}\) | the right hand side of the ode | \(f_{s}=T\,N\,C\) |
\(\left[\begin{matrix}- E\cdot k_{d}\\- D + E\cdot k_{d} + K_{r}\cdot M\cdot k_{t}\\D\cdot\left(- K_{e} + SUE\right) + M\cdot\left(- SUE\cdot k_{m} - k_{t}\right)\\D\cdot K_{e} - E\cdot k_{l}\end{matrix}\right]\)
| Component | Description | Expressions |
|---|---|---|
| \(C\) | carbon content | \(C=\left[\begin{matrix}S\\D\\M\\E\end{matrix}\right]\) |
| \(T\) | transition operator | \(T=\left[\begin{matrix}-1 & 0 & 0 & 0\\1 & -1 &\frac{K_{r}\cdot k_{t}}{SUE\cdot k_{m} + k_{t}} & 0\\0 & - K_{e} + SUE & -1 & 0\\0 & K_{e} & 0 & -1\end{matrix}\right]\) |
| \(N\) | decomposition operator | \(N=\left[\begin{matrix}\frac{D_{C}}{S} & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & SUE\cdot k_{m} + k_{t} & 0\\0 & 0 & 0 & k_{l}\end{matrix}\right]\) |
| \(f_{s}\) | the right hand side of the ode | \(f_{s}=T\,N\,C\) |
\(\left[\begin{matrix}-\frac{E\cdot k_{d}}{E + K_{es}}\\- D +\frac{E\cdot k_{d}}{E + K_{es}} + K_{r}\cdot M\cdot k_{t}\\D\cdot\left(- K_{e} + SUE\right) + M\cdot\left(- SUE\cdot k_{m} - k_{t}\right)\\D\cdot K_{e} - E\cdot k_{l}\end{matrix}\right]\)
| Component | Description | Expressions |
|---|---|---|
| \(C\) | carbon content | \(C=\left[\begin{matrix}S\\D\\M\\E\end{matrix}\right]\) |
| \(T\) | transition operator | \(T=\left[\begin{matrix}-1 & 0 & 0 & 0\\1 & -1 &\frac{K_{r}\cdot k_{t}}{SUE\cdot k_{m} + k_{t}} & 0\\0 & - K_{e} + SUE & -1 & 0\\0 & K_{e} & 0 & -1\end{matrix}\right]\) |
| \(N\) | decomposition operator | \(N=\left[\begin{matrix}D_{C} & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & SUE\cdot k_{m} + k_{t} & 0\\0 & 0 & 0 & k_{l}\end{matrix}\right]\) |
| \(f_{s}\) | the right hand side of the ode | \(f_{s}=T\,N\,C\) |
\(\left[\begin{matrix}-\frac{E\cdot S\cdot k_{d}}{E + K_{es}}\\- D +\frac{E\cdot S\cdot k_{d}}{E + K_{es}} + K_{r}\cdot M\cdot k_{t}\\D\cdot\left(- K_{e} + SUE\right) + M\cdot\left(- SUE\cdot k_{m} - k_{t}\right)\\D\cdot K_{e} - E\cdot k_{l}\end{matrix}\right]\)
| Component | Description | Expressions |
|---|---|---|
| \(C\) | carbon content | \(C=\left[\begin{matrix}P\\M\\Q\\B\\D\\EP\\EM\end{matrix}\right]\) |
| \(I\) | input vector | \(I=\left[\begin{matrix}I_{P}\\0\\0\\0\\I_{D}\\0\\0\end{matrix}\right]\) |
| \(T\) | transition operator | \(T=\left[\begin{matrix}-1 & 0 & 0 &\frac{F_{E}}{F_{E} + F_{R}}\cdot\left(- g_{D} + 1\right)\cdot\left(- p_{EM} - p_{EP} + 1\right) & 0 & 0 & 0\\- f_{D} + 1 & -1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & -1 & 0 &\frac{F_{A}}{F_{A} + F_{U}} & 0 & 0\\0 & 0 & 0 & -1 &\frac{F_{U}}{F_{A} + F_{U}} & 0 & 0\\f_{D} & 1 & 1 &\frac{F_{E}\cdot g_{D}}{F_{E} + F_{R}}\cdot\left(- p_{EM} - p_{EP} + 1\right) & -1 & 1 & 1\\0 & 0 & 0 &\frac{F_{E}\cdot p_{EP}}{F_{E} + F_{R}} & 0 & -1 & 0\\0 & 0 & 0 &\frac{F_{E}\cdot p_{EM}}{F_{E} + F_{R}} & 0 & 0 & -1\end{matrix}\right]\) |
| \(N\) | decomposition operator | \(N=\left[\begin{matrix}\frac{EP\cdot V_{P}}{K_{P} + P} & 0 & 0 & 0 & 0 & 0 & 0\\0 &\frac{EM\cdot V_{M}}{K_{M} + M} & 0 & 0 & 0 & 0 & 0\\0 & 0 &\frac{K_{des}}{Q_{max}} & 0 & 0 & 0 & 0\\0 & 0 & 0 & F_{E} + F_{R} & 0 & 0 & 0\\0 & 0 & 0 & 0 & F_{A} + F_{U} & 0 & 0\\0 & 0 & 0 & 0 & 0 & r_{EP} & 0\\0 & 0 & 0 & 0 & 0 & 0 & r_{EM}\end{matrix}\right]\) |
| \(f_{s}\) | the right hand side of the ode | \(f_{s}=I+T\,N\,C\) |
\(\left[\begin{matrix}B\cdot m_{R}\cdot\left(- g_{D} + 1\right)\cdot\left(- p_{EM} - p_{EP} + 1\right) -\frac{EP\cdot P\cdot V_{P}}{K_{P} + P} + I_{P}\\-\frac{EM\cdot M\cdot V_{M}}{K_{M} + M} +\frac{EP\cdot P\cdot V_{P}}{K_{P} + P}\cdot\left(- f_{D} + 1\right)\\D\cdot K_{ads}\cdot\left(-\frac{Q}{Q_{max}} + 1\right) -\frac{K_{des}}{Q_{max}}\cdot Q\\\frac{B\cdot D\cdot\left(V_{D} + m_{R}\right)}{E_{C}\cdot\left(D + K_{D}\right)} + B\cdot\left(-\frac{D}{D + K_{D}}\cdot\left(-1 +\frac{1}{E_{C}}\right)\cdot\left(V_{D} + m_{R}\right) - m_{R}\right)\\B\cdot g_{D}\cdot m_{R}\cdot\left(- p_{EM} - p_{EP} + 1\right) + D\cdot\left(-\frac{B\cdot\left(V_{D} + m_{R}\right)}{E_{C}\cdot\left(D + K_{D}\right)} - K_{ads}\cdot\left(-\frac{Q}{Q_{max}} + 1\right)\right) +\frac{EM\cdot M\cdot V_{M}}{K_{M} + M} + EM\cdot r_{EM} +\frac{EP\cdot P\cdot V_{P}}{K_{P} + P}\cdot f_{D} + EP\cdot r_{EP} + I_{D} +\frac{K_{des}}{Q_{max}}\cdot Q\\B\cdot m_{R}\cdot p_{EP} - EP\cdot r_{EP}\\B\cdot m_{R}\cdot p_{EM} - EM\cdot r_{EM}\end{matrix}\right]\)
| Component | Description | Expressions |
|---|---|---|
| \(C\) | carbon content | \(C=\left[\begin{matrix}X\\S\end{matrix}\right]\) |
| \(I\) | input vector | \(I=\left[\begin{matrix}0\\BGF + Exu\end{matrix}\right]\) |
| \(T\) | transition operator | \(T=\left[\begin{matrix}-1 & Y\\K_{r} & -1\end{matrix}\right]\) |
| \(N\) | decomposition operator | \(N=\left[\begin{matrix}\frac{D_{max}\cdot K_{d}}{K_{d} +\frac{S}{\theta}} & 0\\0 &\frac{X\cdot\mu_{max}}{Y\cdot\left(K_{s}\cdot\theta + S\right)}\end{matrix}\right]\) |
| \(f_{s}\) | the right hand side of the ode | \(f_{s}=I+T\,N\,C\) |
\(\left[\begin{matrix}-\frac{D_{max}\cdot K_{d}\cdot X}{K_{d} +\frac{S}{\theta}} +\frac{S\cdot X\cdot\mu_{max}}{K_{s}\cdot\theta + S}\\BGF +\frac{D_{max}\cdot K_{d}\cdot K_{r}}{K_{d} +\frac{S}{\theta}}\cdot X + ExuM\cdot e^{- ExuT\cdot t} -\frac{S\cdot X\cdot\mu_{max}}{Y\cdot\left(K_{s}\cdot\theta + S\right)}\end{matrix}\right]\)
