(583e) Slow-Manifold Order Reduction of Reaction-Diffusion Equations with Dirichlet Boundary Conditions

Authors: 
Picardo, J. R., Indian Institute of Technology Madras
Pushpavanam, S., Indian Institute of Technology, Madras

The derivation of low-dimensional averaged models of reaction-diffusion equations greatly facilitates the analysis, optimization and control of dynamic reactive systems. Examples of systems where order reduction has been successful are catalyst pores, wall coated reactors, single phase tubular reactors, catalytic solid slabs etc. In most past studies of these systems, the boundaries are either impermeable (no-flux conditions) or admit a slow flux, such that the boundary condition may be considered as a small deviation from a no-flux condition. In the limit of very fast diffusion, these systems admit a manifold of equilibria which have a zero eigenvalue, corresponding to the null-space of the diffusion operator. This manifold of equilibria is the set of constant concentration modes without transverse variations. Application of slow (center) manifold reduction  (SMR) results in a reduced model in terms of the transverse average concentration. However, if the same problem is analyzed with Dirichlet boundary conditions (homogeneous or non-homogeneous), as in cases when the concentration at the boundary is fixed, then the equilibria of zero eigenvalues do not exist. This occurs because the diffusion operator with Dirichlet boundary conditions does not have a null space. Therefore the reduction techniques used on no-flux problems cannot be applied to problems with Dirichlet boundary conditions.

In this work, we devise techniques for applying SMR to Dirichlet problems. The key idea is to embed the Dirichlet problem within a larger family of problems by the introduction of a suitable parameter (α). This embedding is carried out such that a manifold of equilibria with zero eigenvalues exists in the limit of a critical value of the parameter α. We use the model system of reaction-diffusion in a 1-D slab to demonstrate two different order reduction techniques. We examine the validity and accuracy of the reduced order models for the averaged concentration. Finally, we apply the reduced models to analyze the dynamics of some systems with Dirichlet boundary conditions commonly encountered in reaction engineering.

Keywords: center manifold reduction, averaging, reaction-diffusion, Dirichlet boundary conditions