Shape normalization and analysis of the mass transfer controlled regime in catalytic monoliths

We present a shape normalization for solving the convection-diffusion equation for the case of laminar flow in a duct of uniform cross-section but of arbitrary shape and with a wall catalyzed reaction. We show that when the flow is hydrodynamically developed and the wall reaction is infinitely fast, the reactant mixing-cup exit conversion (chi(m)) depends mainly on the transverse Peclet number, P (=R-Omega(2)(u)/LDm, R-Omega = A(Omega)/P-Omega; where (u) and D-m arc the average velocity, and molecular diffusivity of the reactant species; and A(Omega), P-Omega, and L are the channel cross-sectional area, perimeter, and length, respectively) and is a very weak function of the axial Peclet number, Pe (=(u)L/D-m) for P much greater than 1. We also show that the curve chi(m) versus P is universal (for all common channel geometric shapes such as circular, square, triangular, etc.) and is described by the two asymptotes chi(m) = 1 for P much less than 1 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 case of developing flow with a finite Schmidt number (Sc = v/D-m), we show that chi(m) = 1 for P much less than 1 and chi(m) approximate to Sc-1/6 P-1/2 for P much greater than 1 with a transition around a P value of unity. We give formulas for estimating the conversion in any arbitrary channel geometry for finite values of P and show that the commonly used two-phase models with a constant Sherwood number can be in considerable error (approximate to 20-30%) even for the case of long channels (P much less than 1). We also extend the shape normalization to the local Sherwood number (Sh) for fully developed as well as simultaneously developing flow and compare the analytical results with numerical computations and literature correlations. The asymptotic results and formulas presented here are useful for determining an upper bound on conversion and a lower bound on the Sherwood number for a given set of flow conditions and physical dimensions of the monolith. Finally, we present simple criteria for optimal design of catalytic monoliths and packed-beds operating in the mass transfer controlled regime. (C) 2002 Elsevier Science Ltd. All rights reserved.