An analysis of chemical reactor stability and control—Xa: Polymerization models in two immiscible phases in physical equilibrium

Abstract

Various physical and chemical models of free-radical addition polymerization in one- and two-phase continuous stirred tank reactors are considered. Both the steady state and the transient behaviour are studied, and the methods of solution are discussed. A simple kinetic model, consisting of first-order monomer initiation, propagation and mutual radical (combination and disproportionation) termination, is considered first. Using these kinetics, three physical models for interfacial heat and mass transfer are analysed: physical equilibrium between phases; thermal equilibrium, resistance to interfacial mass transfer; and resistance to interfacial heat and mass transfer. Also, it is shown that the physical equilibrium model is analogous to a single-phase case, so that the phenomena associated with each are studied simultaneously. More sophisticated models are then considered for systems in physical equilibrium. These models include catalyst initiation kinetics, chain transfer reactions, depolymerization reactions, temperature-dependent phase equilibrium (partition) constants and chain length-dependent partition constants; the first three apply also to a one-phase reactor. The non-linear, algebraic steady state heat and mass balances are readily solved by the Newton-Raphson iteration, although special methods are required for systems with depolymerization reactions or chain length-dependent partition constants. Typical examples are considered, and the effect of important reactor parameters (rate constants, holding time, etc.) upon the steady state behaviour (average chain length, number of steady states, etc.) is studied for a single reactor and for a series of reactors. Examples are shown with as many as twenty-five steady state solutions. The local stability of the steady states is analysed by linearizing the differential equations for the unsteady state heat and mass balances. It is found that under ordinary conditions a steady state is stable if and only if the sign of the determinant of the stability matrix (matrix of coefficients of the linearized system) is positive for a matrix of even order or is negative for an odd-ordered matrix. If the two phases are in thermal equilibrium, these conditions correspond to the condition that the slope of the heat removal function is greater than the slope of the heat generation function at the steady state point. For the analysis of the local approach to the steady state, the eigenvalues of the stability matrix are computed by the direct (power and inverse power) method and by the extraction of the roots from the characteristic polynomial of the matrix by Lehmer's method. Several computational schemes for obtaining the characteristic polynomial are tested, and the Danilevsky method is found to give very accurate results, even for a ninth-order matrix. This analysis shows all stable steady states to be basically nodes and all unstable steady states to be basically saddle points. Reaction paths are computed by solving the non-linear transient equations by a modified Runge-Kutta technique.