Quasi-Invariants of Chemical Reactions in the Ideal Displacement Reactor

Abstract

Mathematical models of nonequilibrium processes in distributed systems are described by partial differential equations with given boundary conditions and, as a rule, cannot be solved analytically. The absence of exact solutions of such models does not allow one to find exact invariants; i.e., functions are strictly constant on these solutions. This article has developed a method for determining the approximate spatiotemporal kinetic invariants (quasi-invariants) of chemical reactions that do not take into account diffusion in an ideal displacement reactor. Spacetime quasi-invariants are explicit algebraic expressions that relate the nonequilibrium values of the concentrations of the reagents and/or temperatures measured in two or more experiments with various specially selected initial and input conditions (multiexperiments). The selection of suitable initial conditions makes it possible to find such functions of solutions that weakly depend on the time and length of the reactor, i.e., remain almost constant throughout the reaction in the entire reactor volume. It is shown that the number of quasi-invariants is determined by the number of independent model variables (reagent concentrations and temperature). The application of the method is illustrated by the example of a one-stage reaction occurring in a reactor of ideal displacement. The quasi-invariant 3D surfaces found for this reaction are compared with 3D plots of concentration and temperature changes during the reaction. It is shown that quasi-invariants vary in a smaller range (along the ordinate axis) than the corresponding concentrations and temperatures in different experiments, i.e., are spacetime approximate invariants. Quasi-invariants of this kind can be observed on the dependences of the model variables (concentrations and temperatures) on the time and length of the reactor in the form of various flattened three-dimensional structures (“lying” waves). The method expands the tools for analyzing the unsteady kinetics of chemical reactions in distributed systems of ideal displacement.