Stability Analysis and Transient Behavior of the CSTR

The steady-state behavior of a single CSTR can be calculated using Equ. 6.3a, b, representing the long-term behavior of the model system. However, nonlinear algebraic equation systems may have more than one solution. To 

The second stationary solution is characterized by a nonvanishing biomass concentration  

Starting the second step of analyzing the long-term behavior we may ask these questions Are there two stationary states or is only one of them realizable in a practical case What kind of changes occur in the system if the system in one of these states is disturbed These questions can be answered by application of the stability theory, using analytical or numerical calculations or computer simulation. A stationary state is called stable if the system returns to this state after any kind of disturbance. Such a stable stationary state is called a steady state. For our model one can demonstrate that the washout state ( x, s) is stable for dilution rates greater than a critical dilution 

In contrast to closed discontinuous (batch) cultivation systems, in which the final concentrations depend on the initial concentrations, in an open system in continuous culture the final concentration is largely independently of the initial states, which phenomenon is called equifinality. An appropriate plot for demonstration of equifinality is the plot of so-called trajectories in the phase plane, as shown in the next section and in Fig. 6.13. 

General considerations concerning stability analysis of biochemical reaction systems have been presented by Stucki (1978). The problem of monostability, bistability, and multiplicity (multiple steady state) is further discussed in review articles and books (Aris and Humphrey, 1977 Bergter, 1972 Chi et al., 1974 Fiechter, 1982 Gutke and Knorre, 1980 Knorre, 1980 Koga and Humphrey, 1967 Prokop, 1978 Romanovsky et al., 1974 Russell and Tanner, 1978 Yang and Humphrey, 1975 Yano and Koga, 1973). 

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