(621ct) Steady State Multiplicity Patterns for Multiple Autocatalytic Reactions

Chemical and biological reactions with nonlinear kinetics when carried out in well-mixed reactors give rise to the possibility of reactor operation at multiple operating points at steady state. When multiple steady states are physically realizable, identification of various multiplicity regions in the space of reactor operating conditions is important for design, optimization and controlled operation of chemical and biological reactions. Prediction of steady state multiplicity patterns in terms of profiles of concentrations of species influencing reaction kinetics is important for better understanding of the reaction behavior and design, optimization and control of reactors used to conduct these reactions. These patterns are simpler with respect to certain operating variables, such as the feed concentration of a reactant, and complex with respect to other operating variables, such as the reactor space time. With reactor space time as the variable parameter, exotic patterns, such as isolas, mushrooms, and multi-stability, have been reported for isothermal and non-isothermal reactor operation. In prior studies, these have been obtained via extensive iterations involving sweeps through parameter spaces. In this study, a convenient procedure for analyzing and predicting steady state multiplicity patterns is developed and illustrated considering one or more autocatalytic reactions as specific examples. Autocatalytic reactions and processes are commonly encountered in growth of all living cells, processes involving free radicals, polymerization processes, many inorganic and organic reactions, and crystallization processes. A detailed analysis of steady state multiplicity of single and two parallel/series autocatalytic reactions occurring in a well-mixed reactor is presented. The generation of the autocatalyst from competing or consecutive resources by cubic autocatalysis is followed by its decay. Parallel and series autocatalysis is observed in growth of living cells on multiple substitutable resources (nutrients) and co-metabolism of primary and secondary nutrients (simultaneous utilization of multiple nutrients via independent metabolic pathways inside living cells). A single well-mixed reactor may operate at up to three steady states for a single autocatalytic reaction and at up to five steady states for two parallel/series autocatalytic reactions. Steady state multiplicity patterns are predicted in a non-iterative fashion by a judicious choice of parameter combinations. The space of ratios of the kinetic parameters for the autocatalytic reactions and ratios of supply of resources is divided into different regions depending on the maximum number of steady states admissible, which vary from three to five. The space of the remaining kinetic and operating parameters is divided into multiple regions based on the number and identity of physically realizable steady states. This division allows exact determination of appearance and disappearance of particular steady states. Continuation curves for limit point (LP) bifurcation are identified in the multi-dimensional kinetic and operating parameter space. While the LP continuation curves have a single disposition for a single autocatalytic reaction, there is a wide variety of dispositions of these for two autocatalytic reactions. Concentration profiles of species participating in the reactions are predicted conveniently using the LP continuation diagrams. These are simple when feed composition is varied and the same is the case when reactor space time is varied in the absence of autocatalyst decay. With autocatalyst subject to decay, exotic patterns such as single and multiple isolas and mushrooms of different varieties, are possible with reactor space time as the variable parameter. The LP continuation diagrams enable convenient and precise prediction of emergence and extinction of isolas and mushrooms. A reversible sweep in a parameter combination as reactor space time is varied leads to these exotic patterns. Two or more consecutive reactions are therefore necessary for admission of isolas and mushrooms. The reaction systems with two autocatalytic reactions exhibit rich varieties of steady state patterns. Some of these will be presented as specific illustrations. The present approach enables a much easier identification of the multitude of rich steady state multiplicity patterns.