Modal analysis and optimization of isothermal autocatalytic reactions

Abstract

A comprehensive and systematic analysis of optimization of autocatalytic reactions is presented in this article considering a very general objective function. The objective functions of common interest, such as maximum yield, maximum selectivity, maximum productivity and minimal reaction time, are some of the special cases of this general objective function. The following reaction scheme is considered in this work: A+ nB ⇌ ( n+1)B, A ⇌ B, B→C. The problem of maximization of the general objective function is posed as that of the optimal control of a variable-volume reactor with time-variant feed and effluent flow rates as the control variables. The systematic ‘modal’ analysis enables identification of feasible modes of reactor operation and conditions under which each feasible mode is admissible. Of all the possible modes, only a few are proved to be feasible and only certain transitions are permissible from each feasible mode. The policy for programmed manipulation of the feed and effluent flow rates is obtained employing the optimal control theory. Candidate optimal policies are identified as individual feasible modes or certain sequences of these. A singular fed-batch operation in which the feed rate is decided exclusively by the state of the reactor is part of some of the candidate optimal policies. Conditions for admissibility and feasibility of singular control and the feeding policy during singular control interval are obtained analytically for arbitrary kinetics of these reactions. The concentration trajectories in a fed-batch operation converge to or diverge from one or more limit points. Operation at each limit point is characterized by invariance of concentrations of all species, the feed rate being an exponential function of time. A linearized stability analysis, supported by numerical illustrations, reveals that some or all of the limit points may be accessible during fed-batch operation. The optimal operation strategy is linked to the disposition of batch reactor trajectories in the space of concentration variables. The rich behavioral patterns of autocatalytic reactions are revealed by specific numerical illustrations.