Biogas production modeling: Developing a logistic equation satisfying the zero initial condition

Biogas produced by the fermentation of organic waste has emerged as a viable alternative for displacing fossil fuels. The accurate characterization of the biogas production kinetics is an important issue for management, optimization, and control purposes. The classical logistic equation (CLE) and its modifications are widely used for modeling biogas production. Although a tight-fitting can be obtained, these models have the physical inconsistency of predicting a non-zero value of initial biogas production. This work fixes the problem found with CLE by deriving a new function, named biogas logistic equation (BLE), from a simple kinetics scheme. The derivation departs from the differential equations for substrate, biomass and biogas obtained via the law of mass action to reduce these equations to a differential equation having an analytical solution. The parameters of the BLE are linked to the parameters of the kinetics scheme, having a meaningful physical interpretation. An extension to the multi-substrate case was proposed, leading to an expression with the flexibility of detecting phase transitions in the biogas production dynamics. Experimental data from the literature showed that the proposed logistic equation has superior fitting performance than the modified Gompertz equations and in most instances to the CLE.