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Table 5 Petersen matrix of model components [29, 63, 64]

From: Modelling of autogenerative high-pressure anaerobic digestion in a batch reactor for the production of pressurised biogas

Process j Component i \({\uprho }_{j},\mathrm{ rates}\)
kg COD m−3 d−1
\({\mathbf{S}}_{{{\mathbf{ac}}}}\)
kg COD m−3
\({\mathbf{S}}_{{{\mathbf{CH}}4}}\)
kg COD m−3
\({\mathbf{S}}_{{{\mathbf{IC}}}}\)
kmol C m−3
\({\mathbf{X}}_{{{\mathbf{ac}}}}\)
kg COD m−3
\({\mathbf{S}}_{{{\mathbf{ac}} - }}\)
kg COD m−3
\({\mathbf{S}}_{{{\mathbf{HCO}}_{3} - }}\)
kg COD m−3
\({\mathbf{S}}_{{{\mathbf{gas}},{\mathbf{CH}}4}}\)
kg COD m−3
\({\mathbf{S}}_{{{\mathbf{gas}},{\mathbf{CO}}2}}\)
kmol C m−3
1
Uptake of acetate
− 1 \(1 - Y_{ac}\) \(- C_{ac} + \left( {1 - Y_{ac} } \right) C_{{CH_{4} }}\)
\(+ Y_{ac} C_{ac}\)
\(Y_{ac}\)      \({\text{k}}_{{{\text{m}},{\text{ac}}}} { }\left( {\frac{{{\text{S}}_{{{\text{ac}}}} }}{{{\text{K}}_{{{\text{S}},{\text{ac}}}} + {\text{S}}_{{{\text{ac}}}} }}} \right){\text{X}}_{{{\text{ac}}}}\)
2
Decay of Xac
    1      \({\text{k}}_{{{\text{dec}},{\text{X}}_{{{\text{ac}}}} }} \cdot {\text{ X}}_{{{\text{ac}}}}\)
\({{A}}_{{{{ac}} - }}\)
Acid–base acetate
     − 1     \({\text{k}}_{{{\text{A}}/{\text{Bac}}}} { }\left( {{\text{S}}_{{{\text{ac}} - }} \left( {{\text{S}}_{{{\text{H}} + { }}} + {\text{K}}_{{{\text{a}},{\text{ac}}}} } \right) - {\text{K}}_{{{\text{a}},{\text{ac}}}} \cdot {\text{S}}_{{{\text{ac}}}} } \right)\)
\({{A}}_{{{{HCO}}3 - }}\)
Acid–base inorganic carbon
      − 1    \({\text{k}}_{{{\text{A}}/{\text{BCO}}_{2} }} { }({\text{S}}_{{{\text{HCO}}_{3}^{ - } }} \left( {{\text{K}}_{{{\text{a}},{\text{CO}}_{2} }} + {\text{S}}_{{{\text{H}}^{ + } { }}} } \right)\)
\(- {\text{ K}}_{{{\text{a}},{\text{CO}}_{2} }} \cdot {\text{S}}_{{{\text{IC}}}} )\)
TCH4
Liquid–gas CH4 transfer
  − 1      1   \({\text{k}}_{{\text{L}}} {\text{a }} \cdot \left( {{\text{S}}_{{{\text{CH}}_{4} }} - 64{\text{ K}}_{{{\text{H}},{\text{CH}}_{4} { }}} \cdot {\text{p}}_{{{\text{CH}}4}} } \right)\)*
TCO2
Liquid–gas CO2 transfer
   − 1     1   \({\text{k}}_{{\text{L}}} {\text{a }} \cdot \left( {{\text{S}}_{{{\text{IC}}}} - {\text{S}}_{{{\text{HCO}}_{3}^{ - } }} - {\text{K}}_{{{\text{H}},{\text{CO}}_{2} { }}} \cdot {\text{p}}_{{{\text{CO}}_{2} }} } \right)\)
  1. *The factor of 64 was used to convert the Henry’s law coefficient of CH4 (\(K_{{{\text{H}},{\text{CH}}_{{4}} }}\)) from mol L−1 bar–1 to kg COD m–3 bar–1, in order to account for the COD basis of SCH4 as compared to the molar basis of KH [29]; \(S_{{H^{ + } }}\) is the H+ concentration.