Transversal thermal patterns in packed‐bed reactors with simple kinetics: Bifurcation criterion and simulations

Abstract

We derive a new criterion for transversal instability of planar fronts based on the bifurcation condition dV f /dK| K¼0 ¼ 0, where V f and K are the front velocity and its curvature, respectively. This refines our previously obtained condition, which was formulated as a ¼ (DT ad Pe T )/(DT m Pe C ) [ 1 to a [ 1 þ |d|, where DT ad and DT m are the adiabatic and maximal temperature rise, respectively, Pe C and Pe T are the axial mass and the heat Pe numbers, respectively, and d is a small parameter. The criterion is based on approximate relations for DT m and V f , which account for the local curvature of a propagating front in a packed bed reactor with a first-order activated kinetics. The obtained relations are verified by linear stability analysis of planar fronts. Simulations of a simplified 2D model in the form of a thin cylindrical shell are in good agreement with the critical parameters predicted by dispersion relations. Three types of patterns were detected in simulations: ''frozen'' multiwave patterns, spinning waves, and complex rotating-oscillating patterns. We map bifurcation diagrams showing domains of different modes using the shell radius as the bifurcation parameter. The possible translation of the 2D cylindrical shell model results to the 3D case is discussed. V V C 2010 American Institute of Chemical Engineers AIChE J, 57: 735-748, 2011 Keywords: reactor analysis, simulation, process, fronts, bifurcations, transversal patterns IntroductionHeterogeneous catalysts are extensively used in the chemical and petrochemical industry and for pollution abatement purposes. It is usually assumed that the temperature and concentrations of the reactants at any transversal cross section of an adiabatic packed-bed reactor are uniform. However, various industrial and laboratory experimental observations revealed formation of nonuniform temperature in the reactor cross section. Such patterns may pose severe safety hazards by altering the wall strength and inducing cracks. Ignoring such patterns in design and simulations may lead to erroneous results. Understanding which kinetics may lead to spatiotemporal pattern formation is essential for the development of design and control strategies that circumvent them.Symmetry-breaking leading to either spatial and/or spatiotemporal pattern formation has been observed in many biological, physiological, chemical, and electro-chemical systems [1][2][3][4][5][6][7][8] governed by the interaction of diffusion and reaction processes. Majority of the studies of pattern formation in chemically reacting systems are of homogeneous systems. Pattern formation in heterogeneous catalytic and electrochemical systems, like wires, pellets, and packed bed reactors has been known for 20 years. Recent reviews of the and Krischer. 12 The current knowledge and understanding about the formation of transversal temperature patterns has been reviewed by Viswanathan et al. 13The obtained results are kinetics-dependent 14 and in the present study we focus on kinetics that admits multiple steady sta...

Are 3‐D models necessary to simulate packed bed reactors? analysis and 3‐D simulations of adiabatic and cooled reactors

“…[19][20][21][22] Equations 13 and 14 can be converted (using finite-difference approximations of differential operators), for a specified 1-D solution preliminary defined on N spatial points, into a [2(N À 2) Â 2(N À 2)) eigenvalue matrix problem. [19][20][21][22] Equations 13 and 14 can be converted (using finite-difference approximations of differential operators), for a specified 1-D solution preliminary defined on N spatial points, into a [2(N À 2) Â 2(N À 2)) eigenvalue matrix problem.…”

“…22 Initial conditions (IC) are described at the end of this section. Below, we describe the moving front solution, the bifurcation diagram with varying R and compare them to those of the 2-D cylindrical shell model.…”

“…(R) are quite similar and they terminate at large R by switching to another mode. 22 The 3-D system exhibits extended domains of multiple ''frozen'' modes, and the transition on varying a parameter occurs from one stable branch to another; it does not follow the sequence of l kn (see below). In the 2-D shell model, the stable branches of ''frozen'' modes are not overlapping and transition from the m to (m þ 1) mode solution occurs via formation of spinning patterns and complex rotating-oscillating patterns (marked by stars in Figure 4d).…”

“…25 Even if the bifurcation condition (2) is satisfied, we need to check the width of the system that can sustain the patterns. 22 Analysis and simulations of transversal patterns in PBRs using a full 3-D model, to the best of our knowledge, were not published yet. The latter were constructed analytically using matched asymptotic expansions coupled with certain assumptions on the nonlinear source functions for gas-phase RD thermokinetic 14 and for abstract two variable RD systems.…”

“…22 Within that domain, we conduct LSA around the 1-D solution and verify these by simulations. 22 Within that domain, we conduct LSA around the 1-D solution and verify these by simulations.…”