Mathematical modeling of stability of biotechnological process in its optimum condition

Practical realization of microbiological synthesis in continuous operating conditions is connected with the stability of operation at steady state. Prediction of the parameters of the process, provided sustainable operation, is carried out using mathematical modeling, methodology of which includes assessment of steady states parameters and changes of the character for possible disturbances in the process. The methodology has been developed to estimate the stability of the stationary states of the process of anaerobic microbial synthesis in optimal operating conditions. These methodology based on the use of the linearized system of equations of unsteady state recorded in deviations. The relations are obtained for calculating the coefficients of the linearized equation. The system of equations is reduced to a single differential equation of the third order. Hurwitz matrix was composed and the necessary and sufficient conditions for stability were recorded. The system of equations has been simplified and possible variants of the solutions for different types of roots of the characteristic equation have been received. The analytical solutions for unsteady state have been received. The numerical example of this methodology used known kinetic relations for different processes of microbiological synthesis is presented in this report. The productivity of the desired product has been used as an optimality criterion. The results of modeling showed that the optimal condition is stable for these process parameters. The methodological approach can be recommended for the analysis of other kinetic schemes of the microbiological processes.