Newly designed multi-stacked circular tray solid-state bioreactor: analysis of a distributed parameter gas balance during solid-state fermentation with influence of variable initial moisture content arrangements

Manan, Musaalbakri Abdul; Webb, Colin

doi:10.1186/s40643-020-00307-9

Bioresources and Bioprocessing

Table 2 Equations used for simulation data from respiratory gases

From: Newly designed multi-stacked circular tray solid-state bioreactor: analysis of a distributed parameter gas balance during solid-state fermentation with influence of variable initial moisture content arrangements

Mathematical modelling	Equations	Equation number
Oxygen uptake rate (OUR) (Sukatsch and Dziengel 1987)	\({\text{OUR}} = \frac{{F_{1} }}{{V_{m} \times V_{0} }} \left( {X_{{{\text{O}}_{ 2} \left( {\text{in}} \right)}} - \frac{{1 - \left( {X_{{{\text{O}}_{ 2} \left( {\text{in}} \right)}} + X_{{{\text{CO}}_{2} \left( {\text{in}} \right)}} } \right)}}{{1 - \left( {X_{{{\text{O}}_{ 2} \left( {\text{out}} \right)}} + X_{{{\text{CO}}_{ 2} \left( {\text{out}} \right)}} } \right) }} \times X_{{{\text{O}}_{ 2} \left( {\text{out}} \right)}} } \right)\)	1
Carbon dioxide evolution rate (CER) (Sukatsch and Dziengel 1987)	\({\text{CER}} = \frac{{F_{1} }}{{V_{m} \times V_{0} }} \left( {X_{{{\text{CO}}_{ 2} \left( {\text{out}} \right)}} \times \frac{{1 - \left( {X_{{{\text{O}}_{ 2} \left( {\text{in}} \right)}} + X_{{{\text{CO}}_{ 2} \left( {\text{in}} \right)}} } \right)}}{{1 - \left( {X_{{{\text{O}}_{ 2} \left( {\text{out}} \right)}} + X_{{{\text{CO}}_{ 2} \left( {\text{out}} \right)}} } \right) }} - X_{{{\text{CO}}_{ 2} \left( {\text{in}} \right)}} } \right)\)	2
The Gompertz model is a sigmoid function, as the logistic curve (Skiadas and Skiadas 2008)	\((\ln x)' = - b_{0}\, { \ln }\,x\)	3
Direct integration of Eq. 3	\(x = \exp {\kern 1pt} \,\left( {\ln \left( {x_{o} } \right)\exp \left( { - bt} \right)} \right)\)	4
The integrated Gompertz model-logistics-like model the product CO₂ is a function of time (t)	\(\left[ {{\text{CO}}_{2} } \right] = \left[ {{\text{CO}}_{{ 2_{ \text{max} } }} } \right]\exp \left( { - b\exp \left[ { - kt} \right]} \right)\,\)	5
Respiratory quotient (RQ)	\({\text{RQ}} = \frac{{{\text{CER}}}}{{{\text{OUR}}}}\)	6

OUR, oxygen uptake rate (mole/L h); CER, carbon dioxide evolution rate (mole/L h); F₁, air flow rate of inlet gas (L/h) at 1 atm and 30 °C; V_m, molar volume of gases = 24.88 L/mole at 1 atm and 30 °C; V₀ working volume solid phase (L); \(X_{{{\text{O}}_{{ 2 { }\left( {\text{in}} \right)}} }}\), molar fraction of oxygen at gas inlet; \(X_{{{\text{O}}_{{ 2 { }\left( {\text{out}} \right)}} }}\), molar fraction of oxygen at gas outlet; \(X_{{{\text{CO}}_{{ 2 { }\left( {\text{in}} \right)}} }}\), molar fraction of CO₂ at gas inlet; \(X_{{{\text{CO}}_{{ 2 { }\left( {\text{out}} \right)}} }}\), molar fraction of CO₂ at gas outlet; x, the function of time; \(b_{0}\), a positive constant expressing the rate of growth of the system; \(\left[ {{\text{CO}}_{{ 2_{ \text{max} } }} } \right]\), the maximum CO₂ concentration (at t > ∞) (mole); \(b\), a constant related to the initial conditions (when \(t = 0\), then \(\left[ {{\text{CO}}_{2} } \right] = \left[ {{\text{CO}}_{{2_{0} }} } \right] = \left[ {{\text{CO}}_{{ 2_{ \text{max} } }} } \right] { \exp } \left( { - b} \right)\) (dimensionless); k, the specific CO₂ evolution rate (h⁻¹); t, fermentation time (h), RQ, respiratory quotient (dimensionless)

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