Singularity Theory Approach for Calculating the Runaway Boundaries of Heterogeneous Reactor Models

This work presents a systematic procedure based on singularity theory for determining the runaway boundaries of catalytic reactors. The procedure is illustrated using a well-mixed cooled heterogeneous reactor (continuous stirred tank reactor) model. Initially, a comprehensive steady-state analysis is presented and the different possible bifurcation diagrams of the solid phase temperature versus residence time are evaluated using singularity theory and the continuation technique. These results are then used to determine the different possible shapes of runaway boundaries, and a methodology is presented to calculate them. It is observed that the particle Lewis number, Le p, and the particle Damköhler number, Da p, are the two important parameters that determine the conditions under which the pseudohomogeneous model predictions break down. Depending on the values of these parameters, the runaway boundary is either given completely by the boundary limit set (corresponding to ignition of catalyst particles at zero residence time) or completely by the isola variety (corresponding to the coalescence of low- and high-temperature solution branches) or partially by the isola and the hysteresis (corresponding to the coalsecence of an ignition and extinction point) varieties. Finally, the influence of the reactor model type, various parameters, and time dependent perturbations on the runaway boundary is discussed. The results of this work have industrial significance because of the increasing trend of using highly active catalysts. In such cases, the interphase heat and mass transfer and intraparticle diffusion are the rate limiting steps and heterogeneous models are essential for accurate prediction of the runaway boundary. For example, unlike the pseudohomogeneous models which predict that reactor operation can be made safe by using sufficiently high cooling, heterogeneous models predict that beyond a certain critical point (boundary limit or isola variety) no amount of cooling can bring the reactor from runaway to safe region. In such cases, runaway conditions are solely determined by the transport processes occurring at the particle level.