Stability of Nonstationary States of a Biotechnological Process for the Production of Lactic Acid

Abstract

Estimates are given of the stability of stationary states of a process of microbiological synthesis with the use of mathematical models of kinetics based on an unstructured approach. The object of research is a continuous biotechnological process, in which, in addition to biomass, the target product is obtained. The stability was assessed with the Hurwitz matrix. The process uses a component that forms an additional amount of the main substrate. Five stages of the solution of the problem are given: the description of a mathematical model that takes into account biological and technological limitations; the determination of stationary states, which requires assessment of the stability conditions and the calculation of process indices for the adopted stationary states; the formation of a system of equations of the first approximation (a system of linear differential equations of the first order with constant coefficients) and estimation of the coefficients corresponding to the accepted stationary states; the formation of the Hurwitz matrix and the calculation of its elements; and the calculation of indices of necessary and sufficient stability conditions based on the Hurwitz matrix. Three options are given for the calculation of elements of the Hurwitz matrix: the obtainment of the characteristic equation from a system of four differential equations of the first approximation. The calculated relations for the matrix coefficients are determined and the coefficients are calculated as a result of the solution of the equation according to the system of the first approximation with the introduction of eigenvalues. The stability of three stationary states is estimated for the flow rate D = 0.1, 0.2 and 0.3 h^–1.