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

De Crescenzo, Carmen; Marzocchella, Antonia; Karatza, Despina; Molino, Antonio; Ceron-Chafla, Pamela; Lindeboom, Ralph E. F.; van Lier, Jules B.; Chianese, Simeone; Musmarra, Dino

doi:10.1186/s13068-022-02117-x

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
Process j	\({\mathbf{S}}_{{{\mathbf{ac}}}}\) 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	\({\uprho }_{j},\mathrm{ rates}\) kg COD m⁻³ d⁻¹
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 X_ac				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			\({\text{k}}_{{{\text{A}}/{\text{BCO}}_{2} }} { }({\text{S}}_{{{\text{HCO}}_{3}^{ - } }} \left( {{\text{K}}_{{{\text{a}},{\text{CO}}_{2} }} + {\text{S}}_{{{\text{H}}^{ + } { }}} } \right)\) \(- {\text{ K}}_{{{\text{a}},{\text{CO}}_{2} }} \cdot {\text{S}}_{{{\text{IC}}}} )\)
T_CH4 Liquid–gas CH₄ transfer		− 1					1		\({\text{k}}_{{\text{L}}} {\text{a }} \cdot \left( {{\text{S}}_{{{\text{CH}}_{4} }} - 64{\text{ K}}_{{{\text{H}},{\text{CH}}_{4} { }}} \cdot {\text{p}}_{{{\text{CH}}4}} } \right)\)^*
T_CO2 Liquid–gas CO₂ transfer			− 1				1		\({\text{k}}_{{\text{L}}} {\text{a }} \cdot \left( {{\text{S}}_{{{\text{IC}}}} - {\text{S}}_{{{\text{HCO}}_{3}^{ - } }} - {\text{K}}_{{{\text{H}},{\text{CO}}_{2} { }}} \cdot {\text{p}}_{{{\text{CO}}_{2} }} } \right)\)

^*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 S_CH4 as compared to the molar basis of K_H [29]; \(S_{{H^{ + } }}\) is the H⁺ concentration.

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