Derivation of the unsteady-state interaction-by-exchange-with-the-mean model

The unsteady-state interaction-by-exchange-with-the-mean (IEM) model has been derived elsewhere. That model was derived for the study of the dynamical behavior of the imperfectly micromixed continuous-flow stirred tank reactor(CSTR). In this study we have attempted to illustrate an easier method of derivation. The unsteady-state IEM model of micromixing describes a perfectly macromixed CSTR and assumes that individual fluid elements, initially segregated at the inlet, micromix by exchanging mass linearly with all fluid elements present in the reactor. Hence, the IEM model gives a mean-field description of the system where chemical reactions occur locally and mass is exchanged globally. The mean concentration is found by weighting the local concentrations according to their age in the reactor. This formulation, as presented in this paper, leads to a nonlinear integral equation for the mean concentration. Mathematically, integral equations involve "non-Markovian" or history-dependent behavior and are known, in many cases, to exhibit dynamical behavior absent from the ordinary differential equation found in the history-independent limit.