Linear stability analysis of high- and low-dimensional models for describing mixing-limited pattern formation in homogeneous autocatalytic reactors

Gupta, Ankur; Chakraborty, Saikat

doi:10.1016/j.cej.2008.08.025

Cited by 14 publications

(38 citation statements)

References 27 publications

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“…2a) and the observations from these cases reaffirms the conclusions drawn from the bifurcation or hysteresis diagrams. Besides, our neutral stability studies [1] also show that higher eigen modes are excited only from significant transverse mixing limitations (large p) in the reactor (Fig. 2a).…”

Section: Steady State Resultsmentioning

confidence: 96%

“…The formation of patterned states may often reduce the product yield by deactivation of the catalyst or reactor runaway. Using a regularized low dimensional model, our previous works [1,2,3] focused on the symmetric patterns that develop due to the difference in mixing and reaction time scales, while our present paper simulates the formation of asymmetric patterns, typically observed in experimental studies [4].…”

Section: Introductionmentioning

confidence: 99%

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Dynamic Simulation of Mixing-Limited Symmetric and Asymmetric Patterns in Homogeneous Autocatalytic Reactors

Chaudhury

Chakraborty

2010

Ind. Eng. Chem. Res.

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SummaryIn this paper we study the dynamic evolution of mixing-limited patterns in homogeneous autocatalytic reactors. Using a regularized low dimensional model, our previous works [1, 2, 3] mainly focused on the symmetric patterns that develop due to the difference in mixing and reaction time scales, while our present paper simulates the formation of asymmetric patterns, which have been observed in experimental studies [4]. We find that higher eigen modes of initial perturbation lead to patterns that are more stable and more asymmetric than those formed from lower modes.

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Section: Steady State Resultsmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Dynamic Simulation of Mixing-Limited Symmetric and Asymmetric Patterns in Homogeneous Autocatalytic Reactors

Chaudhury

Chakraborty

2010

Ind. Eng. Chem. Res.

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“…[18][19][20][21][22][23] For RDA systems, only few studies are available in the literature including our recent analytical study of transversal instability in a PBR 24 (see below), and several numerical studies using either a PBR model 25,26 or an isothermal autocatalytic reactor. 27 We review these results to explain the improved condition derived here. Several analytical results are available for a generic pseudo-homogeneous PBR model that accounts for a single exothermic reaction with Arrhenius kinetics and is described by temperature (T) and limiting species concentration (C) as its state variables.…”

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confidence: 73%

“…36,37 (iv) The criterion derived here applies for 3D fixed beds, as we will verify elsewhere by numerical simulations. The linear stability analysis of a 3D model and a cylindrical shell model with respect to transversal perturbations of the form of eigenfunctions [J k (l kn r)exp(ik/) where J k is the Bessel function of the first kind, l kn are the transversal eigenvalues which are defined by boundary conditions dJ k (l kn r)| r¼0 ¼ 0 [25][26][27] ] is similar to that of the 2D shell model. Thus, the LSA of the (kn)-mode of the 3D model completely coincides with that of the k-mode in the 2D shell model under condition l kn ¼ k 2D .…”

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confidence: 99%

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

Nekhamkina

Sheintuch

2011

AIChE Journal

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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...

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Transversal patterns in three‐dimensional packed bed reactors: Oscillatory kinetics

Nekhamkina¹,

Sheintuch²

2010

AIChE Journal

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in Wiley Online Library (wileyonlinelibrary.com).Formation of transversal patterns in a 3D cylindrical reactor is studied with a catalytic reactor model in which an exothermic first-order reaction of Arrhenius kinetics occurs with a variable catalytic activity. Under these oscillatory kinetics, the system exhibits a planar front (1D) solution with the front position oscillating in the axial direction. Three types of patterns were simulated in the 3D system: rotating fronts, oscillating fronts with superimposed transversal (nonrotating) oscillations, and mixed rotating-oscillating fronts. These solutions coexist with the planar front solution and require special initial conditions. We map bifurcation diagrams showing domains of different modes using the reactor radius as a bifurcation parameter. The possible reduction of the 3D model to the 2D cylindrical shell model is discussed.

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Linear stability analysis of high- and low-dimensional models for describing mixing-limited pattern formation in homogeneous autocatalytic reactors

Cited by 14 publications

References 27 publications

Dynamic Simulation of Mixing-Limited Symmetric and Asymmetric Patterns in Homogeneous Autocatalytic Reactors

Dynamic Simulation of Mixing-Limited Symmetric and Asymmetric Patterns in Homogeneous Autocatalytic Reactors

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

Transversal patterns in three‐dimensional packed bed reactors: Oscillatory kinetics

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