Averaging theory and low-dimensional models for chemical reactors and reacting flows

Abstract

A novel approach based on the Liapunov–Schmidt technique of bifurcation theory is presented for the spatial averaging of a class of convection–diffusion–reaction models. It is used to derive low-dimensional averaged models for different types of homogeneous and catalytic reactors, as well as coupled homogeneous–heterogeneous systems. For the homogeneous isothermal case, the averaged models consist of a pair of balance equations for each species A j in terms of the mixing-cup ( C j, m ) and spatially averaged (〈 C j 〉) concentrations. The first (global) equation traces the evolution of C j, m with residence time while the second (local) equation, which is independent of the reactor type, gives the local concentration gradient as a difference between C j, m and 〈 C j 〉 in terms of the local variables (such as species diffusivities, shear and reaction rates). For the wall-catalyzed reaction case, the averaged models are described by a pair of equations for each species in terms of C j, m and the surface concentration C j, s and are similar to the classical two-phase models of catalytic reactors. For the coupled homogeneous–heterogeneous case, the averaged models consist of three balance equations for each species in terms of C j, m , 〈 C j 〉 and C j, s , and contain four mass transfer or exchange coefficients. The accuracy, convergence and the region of validity of the averaged models are examined for some special cases. Finally, the usefulness of the averaged models in predicting the reactor behavior is illustrated with an example for each of the three cases, homogeneous, heterogeneous and coupled homogeneous–heterogeneous case.