Shape normalization for catalytic monoliths

We analyze the coupled convection, diffusion and reaction problem for laminar flow in a washcoated channel of uniform but arbitrary cross-sectional shape with nonuniform washcoat thickness. For the case of an isothermal first-order reaction, we obtain analytical solutions and show that the reactant exit mixing-cup conversion (chi(m)) depends mainly on the transverse Peclet number (P) and the effective local Damkohler number (Phi(s)(2)) which depends on relative effective diffusivity (delta), relative washcoat thickness (lambda) and local Damkohler number (phi(s)(2)), and is a weak function of the axial Peclet number Pe and the Schmidt number Sc (or local velocity profile). We generalize the results for nonlinear kinetics and show that the chi(m) versus P curve is universal for all geometric shapes and washcoat profiles, provided P and Phi(s)(2) are defined using the shape normalized length scales (R-Omega1 and R-Omega2 for the fluid phase and washcoat, respectively) and the normalized reaction time scale presented in this work. Here, R-Omega1 is defined as the ratio of flow cross-sectional area to the fluid-washcoat perimeter and R-Omega1 is defined as the ratio of washcoat cross-sectional area to the fluid-washcoat perimeter. Furthermore, we show that the chi(m) versus P curve in the kinetic regime (Phi(s)(2) much less than 1) is independent of the velocity profile and is given by the asymptotes chi(m) = 1 for P much less than Phi(s)(2) and chi(m) = Phi(s)(2)/P for P much greater than Phi(s)(2). In the mass transfer controlled regime (Phi(s)(2) much greater than 1), the conversion (chi(m)), for the case of fully developed flow, is given by the asymptotes chi(m) = 1 for P much less than Phi(s)(2) and chi(m) approximate to P-2/3 for P much greater than 1, with a transition around a P value of unity. For the practical case of long ducts (P much less than 1 or L/R-Phi1 much greater than 1) we show that the high-conversion branch may be approximated by c(m) = 1 - chi(m) at exp{-mu(1)/P} for all cases. We present both analytical and numerical results on the first eigenvalue (mu(1)) and the Fourier weight (alpha(1)) for some cases of practical interest and show that these asymptotic constants are insensitive to the channel geometric shapes and washcoat profile but depend on the numerical values Phi(s)(2), delta and lambda. Finally, we use theoretical results to present criteria for optimal design of catalytic monoliths. (C) 2004 Elsevier Ltd. All rights reserved.