David H. West

Light-off criterion and transient analysis of catalytic monoliths

A one-dimensional two-phase model is used to derive an analytical light-off criterion for a straight channeled catalytic monolith with washcoat, in which the flow is laminar. For the case of uniform catalyst loading and a first order reaction, the light-off criterion is given by Here, Tf,in is the inlet fluid temperature, ΔTad is the adiabatic temperature rise, is one-half the channel hydraulic radius (, , cross-section area, perimeter), L is the channel length, ū is the fluid velocity, De is the reactant effective diffusivity in the washcoat, δc is the effective washcoat thickness, kf is the fluid thermal conductivity and kv(Tf,in) is the first order rate constant per unit washcoat volume at the inlet fluid temperature. NuH,∞ is the asymptotic Nusselt number in the channel. The function f accounts for diffusional limitations in the washcoat and is given by f(ϕ)=1 for ϕ 0.5. The factor g(Peh) depends on the solid conductivity, or more precisely, the heat Peclet number, , where δw(kw) is the effective wall thickness (thermal conductivity). The function g(Peh) decreases monotonically from 2.718 for Peh=0 to unity for Peh=∞. We also show that if the second term is negligible and the first exceeds unity, then ignition occurs at the back-end. If the second term exceeds unity then ignition occurs at the front-end. If the sum exceeds unity with the second term less than unity and not negligible compared to the first term then ignition occurs in the middle of the channel. This analytical ignition criterion is verified by numerical simulations using an accurate transient model that uses position dependent heat and mass transfer coefficients. We show that the plot of exit concentration versus time consists of two regions: kinetically controlled transient region and the mass transfer controlled steady-state asymptote. For the case of high solid conductivity, we present an analytical expression for the transient time at which the monolith shifts from the kinetically controlled to the mass transfer controlled regime. We also determine the influence of various parameters such as the washcoat thickness, channel dimensions, catalyst loading and initial solid temperature on this transient time and the cumulative emissions. Examination of the influence of solid conduction and channel geometry on cumulative emissions showed that designs that are optimum for steady-state operation lead to higher transient emissions and vice versa. Finally, we discuss the transient and steady-state behavior of the monolith for the special case of Lef<1 (hydrogen oxidation).

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

Abstract 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 (χm) depends mainly on the transverse Peclet number, P (=R Ω 2 〈u〉/LD m , R Ω =A Ω /P Ω ; where 〈u〉 and Dm are the average velocity, and molecular diffusivity of the reactant species; and A Ω , P Ω , 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≫1. We also show that the curve χm versus P is universal (for all common channel geometric shapes such as circular, square, triangular, etc.) and is described by the two asymptotes χm=1 for P≪1 and χm≈P−2/3 for P≫1 with a transition around a P value of unity. For the case of developing flow with a finite Schmidt number (Sc=ν/Dm), we show that χm=1 for P≪1 and χm≈Sc−1/6P−1/2 for P≫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 (≈20–30%) even for the case of long channels (P≪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.

A simplified model for analyzing catalytic reactions in short monoliths

We present a new simplified model for analyzing catalytic reactions in short monoliths. The model is described by a two-point boundary value problem in the radial co-ordinate with the reaction terms appearing in the boundary conditions. For the case of linear kinetics, we compare the predictions of the new short monolith (SM) model with the more general two-dimensional model as well as other literature models such as the widely used one-dimensional two-phase model and the two-dimensional convection model (plug flow or parabolic velocity profile but without axial diffusion or conduction). For the case of monotone kinetics, we show that the steady-state behavior of the general model is bounded by the two limiting models, namely the SM model and the convection model (this is analogous to the homogeneous CSTR and PFR models bounding the behavior of the more general axial dispersion model). More importantly, for the case of an exothermic reaction, the SM model retains all the qualitative bifurcation features of the general two-dimensional model. We use the SM model to analyze and classify the steady-state bifurcation behavior of the catalytic monolith for the case of a single exothermic surface reaction and derive explicit analytical expressions for the ignition, extinction and hysteresis loci in terms of the system parameters. We show that there exist four qualitatively different types of bifurcation diagrams of exit temperature (or conversion) versus residence time when the fluid Lewis number is less than unity (Lef<1). Some of the diagrams contain isolated high-temperature branches and solution profiles on these branches show a local maximum in the surface temperature. We also show that in the practically important mass transfer controlled regime, the predictions of the SM model are close to the more general two-dimensional model. Finally, we discuss the practical implications of the results presented in this work.

Transverse concentration and temperature nonuniformities in adiabatic packed-bed catalytic reactors

Abstract Several catalytic reactions such as partial oxidations and hydrogenations are carried out in packed-bed reactors with diameter-to-length (aspect) ratios of order unity and a reactor-to-particle diameter ratios of 20 or more. In the design and operation of these reactors one objective is to prevent the side reactions that may occur due to localized hot spots or temperature variations. In such reactors, the transport coefficients in the flow and transverse directions can be quite different. In the present study, we report a new type of instability, the transverse instability , which leads to non-uniform temperature and concentration profiles in the lateral direction (across the diameter). This instability can occur even when the physical property variations are negligible and velocity is constant and unidirectional. We consider a reaction of the type A+ ν B→P with a Langmuir–Hinshelwood-type kinetic expression and use a pseudohomogeneous model. For a fixed number of catalyst particles in the direction of flow ( L / d p ), the boundary that determines the onset of transverse concentration and temperature nonuniformities is presented in the parameter space defined by the number of catalyst particles in the transverse direction ( R / d p ) and the residence time ( Da ). Our results indicate that transverse nonuniformities are likely to occur for typical values of the reaction parameters when the reactor to particle diameter exceeds about 5. For aspect ratios of order unity, this number is nearly independent of the number of particles in the flow direction.

A hybrid swarm optimizer for efficient parameter estimation

This paper proposes a hybrid algorithm for parameter estimation - a population-based, stochastic, particle swarm optimizer to identify promising regions of search space that are further locally explored by a Levenburg-Marquardt optimizer. This hybrid method is able to find global optimum for six benchmark problems. It is sensitive to the swarm topology which defines information transfer between particles; however, the hypothesis (Kennedy et al., 2001) that a star topology is better for finding the optimum for problems with large number of optima is not supported by this study. It is also seen that in the absence of the local optimizer, particle swarm alone is not as effective. The proposed method is also demonstrated on an identical catalytic reactor model.

Transport limited pattern formation in catalytic monoliths

We consider the problem of flow in a tube with an exothermic surface reaction and show that the azimuthally symmetric steady-state can lose stability giving rise to patterned states with nonuniform concentration and temperature profiles. The primary cause for this transport limited pattern formation is the slow relative rate of heat and mass diffusion compared to surface reaction. Patterned states in which the temperature and concentration profiles vary in the azimuthal direction, can exist (or coexist with symmetric states) for all values of the fluid Lewis number (Lef) though the patterned states are more pronounced and exist in a wider range of parameter space when Lef<1. We analyze a three-dimensional model of catalytic monolith and develop analytical criteria for identifying the parameter regions in which patterned states exist. These criteria indicate that patterned states are formed whenever the local balance equations have multiple solutions and the characteristic reaction time is much smaller compared to the heat/mass diffusion time. Examination of the numerical values of the various parameters shows that most catalytic monoliths and combustors may operate in the region in which a large number of patterned states may exist. It is also found that when nonuniform and three-dimensional temperature and concentration fields exist, there can be hot spots in which the temperature exceeds the adiabatic temperature even when Lef=1.

Evaluation of laminar mixing in stirred tanks using a discrete-time particle-mapping procedure

Abstract While much progress has been made in computing the flow field in stirred tanks, the flow field alone does not really give any direct information about a very important characteristic of the design, namely the mixing time. Efficient and accurate computational tools are needed to compute mixing time and to identify isolated mixing regions. In this paper we attempt to address both of these needs by developing a discrete-time model of the flow in a stirred tank based on a numerical approximation of the Poincare map. We start by computing the 3D flow field. Next we integrate the advection equation for more than 10 4 passive particles through one period of the flow. A mapping is defined between each particles’ initial and final radial and axial coordinates, and the elapsed time for each particle trajectory. The time evolution for tracer particles can then be continued for arbitrarily long times by iterating the map. This mapping procedure is demonstrated on four different impeller stirred tank configurations. It is shown that the error in the mapping procedure can be made to be less than the error in the time integration scheme at a significantly reduced computational effort. Furthermore, as a quantitative aid in evaluating the mixing efficiency, we compute a mixing time which is defined as the time needed for a particle to travel a prescribed distance from its starting location.

Optimal complex networks spontaneously emerge when information transfer is maximized at least expense: A design perspective

Complex networks with multiple nodes and diverse interactions among them are ubiquitous. We suggest that optimal networks spontaneously emerge when “information” transfer is maximized at the least expense. We support our hypothesis by evolving optimal topologies for a particle swarm optimizer (PSO), a population-based stochastic algorithm. Results suggest that (1) an optimum topology emerges at the phase transition when connectivity is high enough to transfer information but low enough to prevent premature convergence, and (2) Small World (SW) networks are a compromise between higher performance and resistance to mutation. The graph characteristics of the optimal PSO networks in the SW regime are similar to that of the visual cortices of cat and macaque, thereby suggesting similar design principles.

Bifurcation analysis of a two-dimensional catalytic monolith reactor model

We present a complete bifurcation analysis of a general steady-state two-dimensional catalytic monolith reactor model that accounts for temperature and concentration gradients in both axial and radial directions and uses Danckwerts boundary conditions. We show that the ignition/extinction characteristics of the monolith are determined by the transverse Peclet number (P=R2u/LDm, ratio of transverse diffusion to convection time) and the transverse Thiele modulus (φs2=2Rks(T0)/Dm, ratio of transverse diffusion to reaction time). When φs2⪡1, ignition occurs at P values of order Bφs2 and the monolith behaves like a homogeneous reactor with simultaneous ignition/extinction of the surface and the fluid phase. However, when φs2⪢1, surface ignition occurs very close to the inlet (or for very short residence times corresponding to large values of P) to a maximum surface temperature of B/(Lef)a(a=1/2 for flat velocity and a=2/3 for parabolic velocity profile) while the fluid-phase conditions are still close to the inlet values. In this fast reaction, mass transfer controlled regime, the fluid temperature reaches the adiabatic value (and the mean exit conversion is close to unity) only when the P values are of order unity or smaller. We show that the behavior of the monolith is bounded by two simplified models. One of them is the well-known convection model and the second is a new model which we call the short monolith (SM) model. The SM model is described by a two-point boundary value problem in the radial coordinate and has the same qualitative bifurcation features as the general two-dimensional model. We also show that when the fluid Lewis number is less than unity (Lef<1), there exist bifurcation diagrams of surface temperature versus residence time containing isolated solution branches on which the surface temperature exceeds the adiabatic temperature. Finally, we present explicit analytical expressions for the ignition, extinction and hysteresis loci for various models and also for the fluid phase conversion and temperature in the fast reaction (mass transfer controlled) regime.