Mathematical Modeling for a Continuous Anaerobic Digester

The microbial reaction is given by S — P. This is catalyzed by X, which also grows. The rate of production growth rx of the microorganism X is given by 

An appropriate unsteady-state model is obtained from the microorganism and substrate material balances as follows  

These two coupled differential equations are subject to the initial conditions C(g(0) = Cs0 and Cxi0) = Cx0 at t = 0. 

Equations (4.23) and (4.24), together with their initial conditions, formulate the dynamic model for this microbial system. We note that the volume V of the digester is assumed to be constant. 

A steady state for this system is reached when the variations in Cs and Cx have ceased, i.e., when their derivatives with respect to time t are equal to zero in equations (4.23) and (4.24). If we set dCs/dt = dCx/dt = 0 in (4.23) and (4.24), we obtain two coupled steady-state equations for the anaerobic digester, namely 

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