Prediction of 3D Transversal Patterns in Packed-Bed Reactors Using a Reduced 2D Model: Oscillatory Kinetics Olga Nekhamkina* and Moshe Sheintuch Department of Chemical Engineering, Technion-IIT, Haifa, Israel We have recently showed the formation of transversal patterns in a 3D cylindrical reactor in which an exothermic first-order reaction of Arrhenius kinetics occurs with variable catalytic activity. Under these oscillatory kinetics, the system exhibits a planar front (1D) solution, with the front position oscillating in the axial direction, while in the 3D case, three types of transversal patterns can emerge: rotating fronts, oscillating fronts with superimposed transversal (nonrotating) oscillations, and mixed rotating-oscillating fronts. In the present study, we analyze the possible reduction of the 3D model to a 2D cylindrical shell model to predict patterns. We map bifurcation diagrams showing domains of different modes using the reactor radius (R) as a bifurcation parameter and show that the front divergence and the domains of the kn-mode pattern in the 3D model [corresponding to the transversal eigenfunction J k (µ kn r) exp(ikθ), in which J k is the Bessel function of the first kind] can be predicted by those of the one wave in the 2D model using the linear transformation R 3D ) µ kn R 2D . Introduction Heterogeneous catalytic packed-bed reactors (PBRs) are extensively used in the chemical and petrochemical industry and for the reduction of environmental pollution. Several ex- perimental studies reported formation of hot spots in PBRs. 1-5 IR imaging revealed thermal pattern formation in various laboratory reactors including the exterior surface of a radial flow reactor 6 and the top of shallow PBRs 7 or the surface of a catalytic cloth 8 under conditions that induce spontaneous oscillations. Obviously, tracking such symmetry breaking is difficult in large commercial reactors. Thus, analytical criteria and direct numerical simulations are the main tools to study the emergence of patterns. Numerical solutions of PBRs subject to constant conditions are usually limited to steady axial patterns in adiabatic systems and steady two-dimensional (2D) patterns in nonadiabatic systems. The PBRs are known to exhibit stationary or moving thermal fronts propagating in the axial direction in the case of sufficiently exothermic reactions (e.g., oxidation, hydrogenation) and in response to changing input conditions. Simulations of oscillatory kinetics, which are typical of carbon monoxide, hydrogen, or hydrocarbon oxidation, have produced time-depend- ent axial patterns in one-dimensional (1D) PBR models. 9-11 The cross-sectional temperature (and conversion) distributions, which are assumed to be uniform in an adiabatic 1D PBR model, may undergo symmetry breaking in the transversal (normal to the flow) direction under certain conditions. Recently, we have simulated 12 patterns in a three-dimensional (3D) bed, subject to constant feed conditions using oscillatory kinetics (see below). Significant modeling efforts have been directed to predict the formation of transversal patterns during the past decade (see a recent review 13 ). Analysis of a 3D two-phase reactor model with complex kinetics is very intricate. Thus, previous studies have been conducted with 2D simplified models, which can be derived from the general model under several assumptions concerning the geometrical symmetry assumed. The main 2D geometries used are as follows: (i) A short monolith 14 or shallow reactor (SR) model 15,16 defined in the radial-azimuthal (r-φ) plane. [SR models can be obtained using the Liapunov-Schmit reduction for a case of small axial gradients if we average the state variables in this direction while accounting for the boundary conditions.] These are essentially reaction-diffusion (RD) systems, which are amenable to linear stability analysis (LSA). (ii) A 2D model that ignores the azimuthal dependence operating in the axial-radial (z-r) plane. 17 (iii) A thin annular cylindrical shell reactor defined in the axial-azimuthal (z-φ) plane. 11 The first model obviously allows the prediction of transversal inhomogeneity in the radial direction, but it does not account for high axial gradients and fronts. The two latter models may admit front solutions propagating in the axial direction. Model ii accounts for radial gradients but eliminates rotation. The cylindrical shell model iii is a simple 2D geometry that captures both rotating and nonrotating patterns and, as we show below, allows the prediction of the radial pattern structure in the 3D case, while it is formally eliminated by the model statement. We show that by proper scaling the domain of operation, the frequency of front oscillations, and even the front divergence of the 3D model can be estimated by the 2D one. The kinetic model used to study the instability in the 3D PBRs accounts for a single exothermic reaction with Arrhenius kinetics coupled with slow and reversible changes of a catalytic activity assuming that deactivation (activation) occurs at higher (lower) temperatures. It is described by three state variables: the temperature (T), the concentration of a limiting species (C), and the catalytic activity (θ). For a 1D system, such kinetics predicts, within appropriate domains of the parameters, an oscillating front solution in which the front position varies with time. Previous direct simulations using the simplified 2D PBR models with oscillatory kinetics show the following: (i) In the SR model, moving nonrotating patterns of the form of transversal eigenfunctions (eq 1) emerge. The complexity of the patterns increased with R. 18 (ii) In the 2D azimuthally symmetric model (for a case of a stationary planar front solution), complex transversal patterns * To whom correspondence should be addressed. E-mail: aermwon@ tx.technion.ac.il. Ind. Eng. Chem. Res. 2010, 49, 10558–10564 10558 10.1021/ie100531e  2010 American Chemical Society Published on Web 08/13/2010