Anaerobic digester
Introduction and Model¶
The anaerobic digester is designed to break down organic matter in sewage sludge to produce methane gas through four key stages: hydrolysis, acidogenesis, acetogenesis and methanogenesis. Most of this processes rely on microorganisms that metabolize substrates in the absence of oxygen, typically at mesophilic temperature range
(20-45 °C).
The implementation is based on the Anaerobic Digestion Model No. 1 (ADM1) by Batstone et al. (2002), with some deviations from the original model due to computational issues and to ensure consistency with BSM2. ADM1 represents the anaerobic digester as two separate volumes: one for the liquid phase and one for the gas phase. Both phases are assumed to be completely mixed (CSTR).
The model incorporates 24 relevant components, which are categorized into insoluble components (X) and soluble components (S). In the liquid phase these components are transformed due to 19 biochemical processes that can be described by four key processes:
- Disintegration (more complex hydrolysis steps)
- Hydrolysis
- Substrate uptake / biomass growth
- Substrate decay / biomass decay
The uptake processes depend on the pH value, which is accounted for using inhibition functions, while the calculation of the pH value relies heavily on acid-base processes. The gas transfer processes describe the transition of dissolved gas from the liquid phase to the gas phase.
In the differential mass balance for a soluble (i=1-12) or particulate (i=13-24), the reaction term for biochemical processes is expressed as the sum across all processes \(j\), which sums up the products of the stoichiometric coefficients \(\nu_{ij}\) and the process rates \(\rho_j\). For the soluble components \(S_{h2}\), \(S_{co2}\) and \(S_{ch4}\) the gas transfer rate \(\rho_{T,i}\) is included.
The determination of the pH value involves a differential mass balance for cations and anions (i=4-7, 10-11), along with differential equations for ion states that incorporate the acid-base process rates \(\rho_{A,i}\).
For the gas phase there are differential mass balances for hydrogen gas, methane gas and carbon dioxide gas, that use the gas transfer rates \(\rho_{T,i}\). For dynamic state modeling, these equations are solved using numerical integration techniques.
Equations¶
Components¶
| \(i\) | Component | Symbol | Unit |
|---|---|---|---|
| 1 | Monosaccharides | \(S_{su}\) | kg(COD)\(\cdot\)m-3 |
| 2 | Amino acids | \(S_{aa}\) | kg(COD)\(\cdot\)m-3 |
| 3 | Long chain fatty acids (LCFA) | \(S_{fa}\) | kg(COD)\(\cdot\)m-3 |
| 4 | Total valerate | \(S_{va}\) | kg(COD)\(\cdot\)m-3 |
| 5 | Total butyrate | \(S_{bu}\) | kg(COD)\(\cdot\)m-3 |
| 6 | Total propionate | \(S_{pro}\) | kg(COD)\(\cdot\)m-3 |
| 7 | Total acetate | \(S_{ac}\) | kg(COD)\(\cdot\)m-3 |
| 8 | Hydrogen gas | \(S_{h2}\) | kg(COD)\(\cdot\)m-3 |
| 9 | Methane gas | \(S_{ch4}\) | kg(COD)\(\cdot\)m-3 |
| 10 | Inorganic carbon | \(S_{IC}\) | kmol(C)\(\cdot\)m-3 |
| 11 | Inorganic nitrogen | \(S_{IN}\) | kmol(N)\(\cdot\)m-3 |
| 12 | Soluble inerts | \(S_I\) | kg(COD)\(\cdot\)m-3 |
| 13 | Composites | \(X_c\) | kg(COD)\(\cdot\)m-3 |
| 14 | Carbohydrates | \(X_{ch}\) | kg(COD)\(\cdot\)m-3 |
| 15 | Proteins | \(X_{pr}\) | kg(COD)\(\cdot\)m-3 |
| 16 | Lipids | \(X_{li}\) | kg(COD)\(\cdot\)m-3 |
| 17 | Sugar degraders | \(X_{su}\) | kg(COD)\(\cdot\)m-3 |
| 18 | Amino acid degraders | \(X_{aa}\) | kg(COD)\(\cdot\)m-3 |
| 19 | LCFA degraders | \(X_{fa}\) | kg(COD)\(\cdot\)m-3 |
| 20 | Valerate and butyrate degraders | \(X_{c4}\) | kg(COD)\(\cdot\)m-3 |
| 21 | Propionate degraders | \(X_{pro}\) | kg(COD)\(\cdot\)m-3 |
| 22 | Acetate degraders | \(X_{ac}\) | kg(COD)\(\cdot\)m-3 |
| 23 | Hydrogen degraders | \(X_{h2}\) | kg(COD)\(\cdot\)m-3 |
| 24 | Particulate inerts | \(X_I\) | kg(COD)\(\cdot\)m-3 |
Process rates
Biochemical process rates¶
| \(\rho_j\) | Biochemical process rate [kg(COD)\(\cdot\)m-3d-1] | Equation |
|---|---|---|
| \(\rho_1\) | Disintegration | \(k_{dis}X_c\) |
| \(\rho_2\) | Hydrolysis of carbohydrates | \(k_{hyd,ch}X_{ch}\) |
| \(\rho_3\) | Hydrolysis of proteins | \(k_{hyd,pr}X_{pr}\) |
| \(\rho_4\) | Hydrolysis of lipids | \(k_{hyd,li}X_{li}\) |
| \(\rho_5\) | Uptake of sugars | \(k_{m,su} \left( \frac{S_{su}}{K_{S,su} + S_{su}} \right) X_{su}I_5\) |
| \(\rho_6\) | Uptake of amino acids | \(k_{m,aa} \left( \frac{S_{aa}}{K_{S,aa} + S_{aa}} \right) X_{aa}I_6\) |
| \(\rho_7\) | Uptake of LCFA | \(k_{m,fa} \left( \frac{S_{fa}}{K_{S,fa} + S_{fa}} \right) X_{fa}I_7\) |
| \(\rho_8\) | Uptake of valerate | \(k_{m,c4} \left( \frac{S_{va}}{K_{S,c4} + S_{va}} \right) X_{c4} \left( \frac{S_{va}}{S_{bu} + S_{va}} \right) I_8\) |
| \(\rho_9\) | Uptake of butyrate | \(k_{m,c4} \left( \frac{S_{bu}}{K_{S,c4} + S_{bu}} \right) X_{c4} \left( \frac{S_{bu}}{S_{va} + S_{bu}} \right) I_9\) |
| \(\rho_{10}\) | Uptake of propionate | \(k_{m,pro} \left( \frac{S_{pro}}{K_{S,pro} + S_{pro}} \right) X_{pro} I_{10}\) |
| \(\rho_{11}\) | Uptake of acetate | \(k_{m,ac} \left( \frac{S_{ac}}{K_{S,ac} + S_{ac}} \right) X_{ac} I_{11}\) |
| \(\rho_{12}\) | Uptake of hydrogen | \(k_{m,h2} \left( \frac{S_{h2}}{K_{S,h2} + S_{h2}} \right) X_{h2} I_{12}\) |
| \(\rho_{13}\) | Decay of \(X_{su}\) | \(k_{dec,Xsu}X_{su}\) |
| \(\rho_{14}\) | Decay of \(X_{aa}\) | \(k_{dec,Xaa}X_{aa}\) |
| \(\rho_{15}\) | Decay of \(X_{fa}\) | \(k_{dec,Xfa}X_{fa}\) |
| \(\rho_{16}\) | Decay of \(X_{c4}\) | \(k_{dec,Xc4}X_{c4}\) |
| \(\rho_{17}\) | Decay of \(X_{pro}\) | \(k_{dec,Xpro}X_{pro}\) |
| \(\rho_{18}\) | Decay of \(X_{ac}\) | \(k_{dec,Xac}X_{ac}\) |
| \(\rho_{19}\) | Decay of \(X_{h2}\) | \(k_{dec,Xh2}X_{h2}\) |
Biochemical process rate parameters
| Symbol | Description | Unit |
|---|---|---|
| \(k_{dis}\) | Disintegration rate | d-1 |
| \(k_{hyd,ch}\) | Hydrolysis rate of Xch | d-1 |
| \(k_{hyd,pr}\) | Hydrolysis rate of Xpr | d-1 |
| \(k_{hyd,li}\) | Hydrolysis rate of Xli | d-1 |
| \(k_{m,proc}\) | Monod maximum specific uptake rate for process proc | d-1 |
| \(k_{dec,bac}\) | Decay rate for bacteria of type bac | d-1 |
| \(K_{S}\) | Monod half saturation constant | kg(COD)\(\cdot\)m-3 |
Process inhibition¶
| \(I_i\) | Process inhibition due to pH [-] | Equation |
|---|---|---|
| \(I_5 = I_6\) | Process inhibition for \(\rho_5\) and \(\rho_6\) | \(I_{pH,aa} \left( \frac{1}{1 + K_{S,IN} / S_{IN}} \right)\) |
| \(I_7\) | Process inhibition for \(\rho_7\) | \(I_{pH,aa} \left( \frac{1}{1 + K_{S,IN} / S_{IN}} \right) \left( \frac{1}{1 + S_{h2} / K_{I,h2,fa}} \right)\) |
| \(I_8 = I_9\) | Process inhibition for \(\rho_8\) and \(\rho_9\) | \(I_{pH,aa} \left( \frac{1}{1 + K_{S,IN} / S_{IN}} \right) \left( \frac{1}{1 + S_{h2} / K_{I,h2,c4}} \right)\) |
| \(I_{10}\) | Process inhibition for \(\rho_{10}\) | \(I_{pH,aa} \left( \frac{1}{1 + K_{S,IN} / S_{IN}} \right) \left( \frac{1}{1 + S_{h2} / K_{I,h2,pro}} \right)\) |
| \(I_{11}\) | Process inhibition for \(\rho_{11}\) | \(I_{pH,ac} \left( \frac{1}{1 + K_{S,IN} / S_{IN}} \right) \left( \frac{1}{1 + S_{nh3} / K_{I,nh3}} \right)\) |
| \(I_{12}\) | Process inhibition for \(\rho_{12}\) | \(I_{pH,h2} \left( \frac{1}{1 + K_{S,IN} / S_{IN}} \right)\) |
\[ I_{pH,aa} = \frac{K_{pH}^{n_{aa}}}{S_{H^+}^{n_{aa}} + K_{pH}^{n_{aa}}} \quad \text{with} \quad K_{pH} = 10^{-\frac{pH_{LL,aa}+pH_{UL,aa}}{2}} \quad \text{and} \quad n_{aa} = \frac{3.0}{pH_{UL,aa} - pH_{LL,aa}} \]
\[ I_{pH,ac} = \frac{K_{pH}^{n_{ac}}}{S_{H^+}^{n_{ac}} + K_{pH}^{n_{ac}}} \quad \text{with} \quad K_{pH} = 10^{-\frac{pH_{LL,ac}+pH_{UL,ac}}{2}} \quad \text{and} \quad n_{ac} = \frac{3.0}{pH_{UL,ac} - pH_{LL,ac}} \]
\[ I_{pH,h2} = \frac{K_{pH}^{n_{h2}}}{S_{H^+}^{n_{h2}} + K_{pH}^{n_{h2}}} \quad \text{with} \quad K_{pH} = 10^{-\frac{pH_{LL,h2}+pH_{UL,h2}}{2}} \quad \text{and} \quad n_{h2} = \frac{3.0}{pH_{UL,h2} - pH_{LL,h2}} \]
Process inhibition parameters
| Symbol | Description | Unit |
|---|---|---|
| \(K_{S,IN}\) | Inhibition parameter for inorganic nitrogen | kmol(N)\(\cdot\)m-3 |
| \(K_{I,h2,fa}\) | 50% inhibitory concentration of inhibitor H2 on LCFA uptake process | kg(COD)\(\cdot\)m-3 |
| \(K_{I,h2,c4}\) | 50% inhibitory concentration of inhibitor H2 on valerate and butyrate uptake process | kg(COD)\(\cdot\)m-3 |
| \(K_{I,h2,pro}\) | 50% inhibitory concentration of inhibitor H2 on propionate uptake process | kg(COD)\(\cdot\)m-3 |
| \(K_{I,nh3}\) | 50% inhibitory concentration of inhibitor NH3 on acetate uptake process | kmol\(\cdot\)m-3 |
| \(S_{nh3}\) | Molar concentration of ammonia | kmol\(\cdot\)m-3 |
| \(pH_{LL,aa}\) | Lower limit of pH for uptake rate of amino acids | - |
| \(pH_{UL,aa}\) | Upper limit of pH for uptake rate of amino acids | - |
| \(pH_{LL,ac}\) | Lower limit of pH for uptake rate of acetate | - |
| \(pH_{UL,ac}\) | Upper limit of pH for uptake rate of acetate | - |
| \(pH_{LL,h2}\) | Lower limit of pH for uptake rate of hydrogen | - |
| \(pH_{UL,h2}\) | Upper limit of pH for uptake rate of hydrogen | - |
Acid-base rates¶
| \(\rho_{A,i}\) | Acid-base rate [kg(COD)\(\cdot\)m-3d-1] | Equation |
|---|---|---|
| \(\rho_{A,4}\) | Acid-base equilibrium of valerate | \(k_{A,Bva} \left( S_{va^-} (K_{a,va}+S_{H^+}) -K_{a,va}S_{va} \right)\) |
| \(\rho_{A,5}\) | Acid-base equilibrium of butyrate | \(k_{A,Bbu} \left( S_{bu^-} (K_{a,bu}+S_{H^+}) -K_{a,bu}S_{bu} \right)\) |
| \(\rho_{A,6}\) | Acid-base equilibrium of propionate | \(k_{A,Bpro} \left( S_{pro^-} (K_{a,pro}+S_{H^+}) -K_{a,pro}S_{pro} \right)\) |
| \(\rho_{A,7}\) | Acid-base equilibrium of acetate | \(k_{A,Bac} \left( S_{ac^-} (K_{a,ac}+S_{H^+}) -K_{a,ac}S_{ac} \right)\) |
| \(\rho_{A,10}\) | Acid-base equilibrium of inorganic carbon | \(k_{A,Bco2} \left( S_{hco3^-} (K_{a,co2}+S_{H^+}) -K_{a,co2}S_{IC} \right)\) |
| \(\rho_{A,11}\) | Acid-base equilibrium of inorganic nitrogen | \(k_{A,BIN} \left( S_{nh3} (K_{a,IN}+S_{H^+}) -K_{a,IN}S_{IN} \right)\) |
Acid-base rate parameters
| Symbol | Description | Unit |
|---|---|---|
| \(k_{A,Bsub}\) | Acid-base kinetic parameter for substance sub | m3\(\cdot\)kmol-1\(\cdot\)d-1 |
| \(K_{a,acid}\) | Acid-base equilibrium constant for acid | kmol\(\cdot\)m-3 |
| \(S_{va}\) | Total valerate, sum of acid-base pair (\(S_{va}=S_{va^-}+S_{hva}\)) | kg(COD)\(\cdot\)m-3 |
| \(S_{bu}\) | Total butyrate, sum of acid-base pair (\(S_{bu}=S_{bu^-}+S_{hbu}\)) | kg(COD)\(\cdot\)m-3 |
| \(S_{pro}\) | Total propionate, sum of acid-base pair (\(S_{pro}=S_{pro^-}+S_{hpro}\)) | kg(COD)\(\cdot\)m-3 |
| \(S_{ac}\) | Total acetate, sum of acid-base pair (\(S_{ac}=S_{ac^-}+S_{hac}\)) | kg(COD)\(\cdot\)m-3 |
| \(S_{IC}\) | Inorganic carbon, sum of acid-base pair (\(S_{IC}=S_{hco3^-}+S_{h2co3}\)) | kmol(C)\(\cdot\)m-3 |
| \(S_{IN}\) | Inorganic nitrogen, sum of acid-base pair (\(S_{IN}=S_{nh3}+S_{nh4^+}\)) | kmol(N)\(\cdot\)m-3 |
| \(S_{H^+}\) | Molar concentration of hydrogen | kmol\(\cdot\)m-3 |
Gas transfer rates¶
| \(\rho_{T,i}\) | Gas transfer rate [kmol\(\cdot\)m-3d-1] | Equation |
|---|---|---|
| \(\rho_{T,8}\) | Transfer rate of hydrogen | \(K_{L}a_{h2}(S_{h2}-16 \cdot K_{H,h2}p_{gas,h2})\) |
| \(\rho_{T,9}\) | Transfer rate of methane | \(K_{L}a_{ch4}(S_{ch4}-64 \cdot K_{H,ch4}p_{gas,ch4})\) |
| \(\rho_{T,10}\) | Transfer rate of carbon dioxide | \(K_{L}a_{co2}(S_{co_2}-K_{H,co2}p_{gas,co2})\) |
Gas transfer rate parameters
| Symbol | Description | Unit |
|---|---|---|
| \(K_{L}a_{gas}\) | Transfer coefficient of gas | d-1 |
| \(K_{H,h2}\) | Henry’s law coefficient for hydrogen | kmol\(\cdot\)m-3\(\cdot\)bar-1 |
| \(K_{H,ch4}\) | Henry’s law coefficient for methane | kmol\(\cdot\)m-3\(\cdot\)bar-1 |
| \(K_{H,co2}\) | Henry’s law coefficient for carbon dioxide | kmol\(\cdot\)m-3\(\cdot\)bar-1 |
| \(p_{gas,h2}\) | Partial pressure of hydrogen | bar |
| \(p_{gas,ch4}\) | Partial pressure of methane | bar |
| \(p_{gas,co2}\) | Partial pressure of carbon dioxide | bar |
| \(S_{co2}\) | Molar concentration of carbon dioxide | kmol\(\cdot\)m-3 |
Liquid phase equations¶
Differential mass balance for soluble components:¶
\[ \frac{dS_i}{dt} = \frac{Q}{V_{liq}} \left( S_{i,in} - S_{i,out} \right) + \left( \sum_{j} \nu_{ij}\rho_j \right) - \rho_{T,i} \]
Differential mass balance for particulate components:¶
\[ \frac{dX_i}{dt} = \frac{Q}{V_{liq}} \left( X_{i,in} - X_{i,out} \right) + \sum_{j} \nu_{ij}\rho_j \]
Differential mass balance for cations:¶
\[ \frac{dS_{i,cat^+}}{dt} = \frac{Q}{V_{liq}} \left( S_{i,cat^+,in} - S_{i,cat^+,out} \right) \]
Differential mass balance for anions:¶
\[ \frac{dS_{i,an^-}}{dt} = \frac{Q}{V_{liq}} \left( S_{i,an^-,in} - S_{i,an^-,out} \right) \]
Differential equations for ion states:¶
\[ \frac{dS_{va^-}}{dt} = - \rho_{A,4} \]
\[ \frac{dS_{bu^-}}{dt} = - \rho_{A,5} \]
\[ \frac{dS_{pro^-}}{dt} = - \rho_{A,6} \]
\[ \frac{dS_{ac^-}}{dt} = - \rho_{A,7} \]
\[ \frac{dS_{hco3^-}}{dt} = - \rho_{A,10} \]
\[ \frac{dS_{nh3}}{dt} = - \rho_{A,11} \]
Stoichiometric coefficients \(\nu_{ij}\)¶
\[ \begin{array}{c|ccccccccccccccccccccccccc} \text{component} \, i \rightarrow & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 \\ \text{process} \, j \downarrow & S_{su} & S_{aa} & S_{fa} & S_{va} & S_{bu} & S_{pro} & S_{ac} & S_{h2} & S_{ch4} & S_{IC} & S_{IN} & S_I & X_c & X_{ch} & X_{pr} & X_{li} & X_{su} & X_{aa} & X_{fa} & X_{c4} & X_{pro} & X_{ac} & X_{h2} & X_I \\ \hline 1 & & & & & & & & & & \displaystyle -\sum_{i=1-9,12-24} C_i\nu_{i,1} & (N_{xc}-f_{xi,xc}N_I-f_{si,xc}N_I-f_{pr,xc}N_{aa}) & f_{S_I,xc} & -1 & f_{ch,xc} & f_{pr,xc} & f_{li,xc} & & & & & & & & f_{X_I,xc} \\ 2 & 1 & & & & & & & & & \displaystyle -\sum_{i=1-9,11-24} C_i\nu_{i,2} & & & & -1 & & & & & & & & & & \\ 3 & & 1 & & & & & & & & \displaystyle -\sum_{i=1-9,11-24} C_i\nu_{i,3} & & & & & -1 & & & & & & & & & \\ 4 & 1-f_{fa,li} & & f_{fa,li} & & & & & & & \displaystyle -\sum_{i=1-9,11-24} C_i\nu_{i,4} & & & & & & -1 & & & & & & & & \\ 5 & -1 & & & & (1-Y_{su})f_{bu,su} & (1-Y_{su})f_{pro,su} & (1-Y_{su})f_{ac,su} & (1-Y_{su})f_{h2,su} & & \displaystyle -\sum_{i=1-9,11-24 \setminus \{8\}} C_i\nu_{i,5} & -(Y_{su})N_{bac} & & & & & & Y_{su} & & & & & & & \\ 6 & & -1 & & (1-Y_{aa})f_{va,aa} & (1-Y_{aa})f_{bu,aa} & (1-Y_{aa})f_{pro,aa} & (1-Y_{aa})f_{ac,aa} & (1-Y_{aa})f_{h2,aa} & & \displaystyle -\sum_{i=1-9,11-24 \setminus \{8\}} C_i\nu_{i,6} & N_{aa}-(Y_{aa})N_{bac} & & & & & & & Y_{aa} & & & & & & \\ 7 & & & -1 & & & & (1-Y_{fa})0.7 & (1-Y_{fa})0.3 & & \displaystyle -\sum_{i=1-9,11-24 \setminus \{8\}} C_i\nu_{i,7} & -(Y_{fa})N_{bac} & & & & & & & & Y_{fa} & & & & & \\ 8 & & & & -1 & & (1-Y_{c4})0.54 & (1-Y_{c4})0.31 & (1-Y_{c4})0.15 & & \displaystyle -\sum_{i=1-9,11-24 \setminus \{8\}} C_i\nu_{i,8} & -(Y_{c4})N_{bac} & & & & & & & & & Y_{c4} & & & & \\ 9 & & & & & -1 & & (1-Y_{c4})0.8 & (1-Y_{c4})0.2 & & \displaystyle -\sum_{i=1-9,11-24 \setminus \{8\}} C_i\nu_{i,9} & -(Y_{c4})N_{bac} & & & & & & & & & Y_{c4} & & & & \\ 10 & & & & & & -1 & (1-Y_{pro})0.57 & (1-Y_{pro})0.43 & & \displaystyle -\sum_{i=1-9,11-24 \setminus \{8\}} C_i\nu_{i,10} & -(Y_{pro})N_{bac} & & & & & & & & & & Y_{pro} & & & \\ 11 & & & & & & & -1 & & (1-Y_{ac}) & \displaystyle -\sum_{i=1-9,11-24} C_i\nu_{i,11} & -(Y_{ac})N_{bac} & & & & & & & & & & & Y_{ac} & & \\ 12 & & & & & & & & -1 & (1-Y_{h2}) & \displaystyle -\sum_{i=1-9,11-24 \setminus \{8\}} C_i\nu_{i,12} & -(Y_{h2})N_{bac} & & & & & & & & & & & & Y_{h2} & \\ 13 & & & & & & & & & & \displaystyle -\sum_{i=1-9,11-24} C_i\nu_{i,13} & N_{bac}-N_{xc} & & 1 & & & & -1 & & & & & & & \\ 14 & & & & & & & & & & \displaystyle -\sum_{i=1-9,11-24} C_i\nu_{i,14} & N_{bac}-N_{xc} & & 1 & & & & & -1 & & & & & & \\ 15 & & & & & & & & & & \displaystyle -\sum_{i=1-9,11-24} C_i\nu_{i,15} & N_{bac}-N_{xc} & & 1 & & & & & & -1 & & & & & \\ 16 & & & & & & & & & & \displaystyle -\sum_{i=1-9,11-24} C_i\nu_{i,16} & N_{bac}-N_{xc} & & 1 & & & & & & & -1 & & & & \\ 17 & & & & & & & & & & \displaystyle -\sum_{i=1-9,11-24} C_i\nu_{i,17} & N_{bac}-N_{xc} & & 1 & & & & & & & & -1 & & & \\ 18 & & & & & & & & & & \displaystyle -\sum_{i=1-9,11-24} C_i\nu_{i,18} & N_{bac}-N_{xc} & & 1 & & & & & & & & & -1 & & \\ 19 & & & & & & & & & & \displaystyle -\sum_{i=1-9,11-24} C_i\nu_{i,19} & N_{bac}-N_{xc} & & 1 & & & & & & & & & & -1 & \\ \end{array} \]
Stoichiometric parameters
| Symbol | Description | Unit |
|---|---|---|
| \(f_{fa,li}\) | Yield (catabolism only) of Sfa on Xli | - |
| \(f_{va,aa}\) | Yield (catabolism only) of Sva on Saa | - |
| \(f_{bu,su}\) | Yield (catabolism only) of Sbu on Ssu | - |
| \(f_{bu,aa}\) | Yield (catabolism only) of Sbu on Saa | - |
| \(f_{pro,su}\) | Yield (catabolism only) of Spro on Ssu | - |
| \(f_{pro,aa}\) | Yield (catabolism only) of Spro on Saa | - |
| \(f_{ac,su}\) | Yield (catabolism only) of Sac on Ssu | - |
| \(f_{ac,aa}\) | Yield (catabolism only) of Sac on Saa | - |
| \(f_{h2,su}\) | Yield (catabolism only) of Sh2 on Ssu | - |
| \(f_{h2,aa}\) | Yield (catabolism only) of Sh2 on Saa | - |
| \(f_{S_I,xc}\) | Fraction of composites to SI by disintegration | - |
| \(f_{ch,xc}\) | Fraction of composites to Xch by disintegration | - |
| \(f_{pr,xc}\) | Fraction of composites to Xpr by disintegration | - |
| \(f_{li,xc}\) | Fraction of composites to Xli by disintegration | - |
| \(f_{X_I,xc}\) | Fraction of composites to XI by disintegration | - |
| \(Y_{su}\) | Yield of biomass, sugar degraders | kg(COD)\(\cdot\)(kg(COD))-1 |
| \(Y_{aa}\) | Yield of biomass, amino acid degraders | kg(COD)\(\cdot\)(kg(COD))-1 |
| \(Y_{fa}\) | Yield of biomass, long chain fatty acid degraders | kg(COD)\(\cdot\)(kg(COD))-1 |
| \(Y_{c4}\) | Yield of biomass, valerate and butyrate degraders | kg(COD)\(\cdot\)(kg(COD))-1 |
| \(Y_{pro}\) | Yield of biomass, protein degraders | kg(COD)\(\cdot\)(kg(COD))-1 |
| \(Y_{ac}\) | Yield of biomass, acetate degraders | kg(COD)\(\cdot\)(kg(COD))-1 |
| \(Y_{h2}\) | Yield of biomass, hydrogen degraders | kg(COD)\(\cdot\)(kg(COD))-1 |
| \(N_{bac}\) | Nitrogen content of biomass | kmol(N)\(\cdot\)(kg(COD))-1 |
| \(C_i\) | Carbon content of component \(i\) | kmol(C)\(\cdot\)(kg(COD))-1 |
Gas phase equations¶
Differential mass balance for hydrogen:¶
\[ \frac{dS_{gas,h2}}{dt} = -\frac{S_{gas,h2} Q_{gas}}{V_{gas}} + \rho_{T,8} \frac{V_{liq}}{V_{gas}} \]
Differential mass balance for methane:¶
\[ \frac{dS_{gas,ch4}}{dt} = -\frac{S_{gas,ch4} Q_{gas}}{V_{gas}} + \rho_{T,9} \frac{V_{liq}}{V_{gas}} \]
Differential mass balance for carbon dioxide:¶
\[ \frac{dS_{gas,co2}}{dt} = -\frac{S_{gas,co2} Q_{gas}}{V_{gas}} + \rho_{T,10} \frac{V_{liq}}{V_{gas}} \]
Source code documentation¶
mod adm1_bsm2
mod adm1init_bsm2
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Benchmarking of Control Strategies for Wastewater Treatment Plants, chap. 4.2.1 Anaerobic Digestion Model No. 1 ↩
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Benchmark Simulation Model no. 2 (BSM2), chap. 2. Modeling of the activated sludge section ↩