Vemuri Balakotaiah

Global analysis of the multiplicity features of multi-reaction lumped-parameter systems

Abstract Mathematical models of lumped-parameter systems in which many chemical reactions occur simultaneously contain a large number of parameters, so that a p Theoretical guidance is needed to determine all the multiplicity features and the corresponding parameter regions. A systematic, efficient scheme is presented for finding parameter values corresponding to a specific number of solutions. A new scheme is developed for bifurcation diagrams, which describe the dependence of a state variable on a slowly changing operating variable. Some general predictions are made abou systems. Bounds on the values of the bifurcation or state variable may create bifurcation diagrams which cannot be found close to the highest order sin of solutions even when an isola variety does not exist. Several examples illustrate the application of the mathematical techniques.

Structure of the steady-state solutions of lumped-parameter chemically reacting systems

Abstract A new, powerful mathematical technique enables a systematic determination of the maximal number of steady-state solutions of lumped parameter systems in which several chemical reactions occur simultaneously. The method can predict also the different types of diagrams describing the dependence of a state variable of the reactor on a design or operating variable. The technique is applied to several reaction networks giving new results and insight. For example, it is proven that when N independent, parallel exothermic reactions with equal and high activation energies occur in a CSTR there exist N ! distinct regions of parameters in each of which 2 N + 1 steady-state solutions exist.

Multiplicity features of reacting systems: Dependence of the steady-states of a CSTR on the residence time

Abstract Singularity theory with a distinguished parameter, as developed by Golubitsky and Schaeffer, is a very useful tool for predicting the influence of changes in a control or design variable on the steady-states of lumped-parameter systems. The theory is used to construct various bifurcation diagrams describing the influence of changes in the residence time on the temperature in a CSTR in which several reactions occur simultaneously. The number of different bifurcation diagrams increases very rapidly with increasing number of reactions. The predictions of this local theory provide important theoretical guidance in the global analysis of the multiplicity features.

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).

Heat and mass transfer coefficients in catalytic monoliths

Abstract We analyze the classical Graetz problem in a tube with an exothermic surface reaction and show that the heat(mass) transfer coefficient is not a continuous function of the axial position and jumps from one asymptote to another at ignition/extinction points. We show that the steady-state heat(mass) transfer coefficient is not a unique function of position in parameter regions in which the Graetz problem with surface reaction has multiple solutions. We also analyze the more general two-dimensional model (with axial conduction/diffusion included and Danckwerts boundary conditions) and show that for fixed values of the reaction parameters, the heat(mass) transfer coefficient has three asymptotes. Unlike the Graetz problem, in this case the heat(mass) transfer coefficient is always finite and bounded at the inlet and is given by a new asymptote. We present analytical expressions for all three asymptotes for the case of flat and parabolic velocity profiles. It is also shown that in catalytic monoliths, ignition/extinction may often occur in the entry region and hence the local transfer coefficients and not the average values proposed in the literature determine the ignition/extinction behavior. Finally, we use the new results to develop and analyze an accurate one-dimensional two-phase model of a catalytic monolith with position dependent heat and mass transfer coefficients and determine analytically the dependence of the ignition/extinction locus on various design and operating parameters.

ANALYSIS OF THE MULTIPLICITY PATTERNS OF A CSTR

Continuous changes in the residence time of a cooled continuously stirred tank reactor, in which a single, exothermic, first-order reaction occurs, may lead to one of six different multiplicity patterns. A simple technique is developed for the exact prediction of the multiplicity pattern existing for any set of parameter values and of the influence of changes in the parameter values on the transition from one pattern to another.

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.

Modeling and experimental studies of wave evolution on free falling viscous films

A new simplified model is developed for describing the characteristics of free falling wavy liquid films. The model consists of a set of three partial differential equations (in x and t) for the local film thickness, volumetric flow rate, and wall shear stress. It is shown that the new model is a substantial improvement over all existing simplified models of wavy films such as the long wave equation, the Nakaya model (extended third-order long wave equation), the Shkadov model, and the Kapitza boundary layer model. These prior models predict nonphysical negative wall shear stress when the wave amplitude is large and cannot explain the experimentally observed relationship between the maximum wave amplitude and the Reynolds (Re) and Kapitza (Ka) or Weber (We) numbers. In contrast, the present model yields physically meaningful results and quantitative predictions of large amplitude waves. Local bifurcation analysis of the model for small Re gives the following analytical relations for the velocity (Ce) and ...

Dispersion of Chemical Solutes in Chromatographs and Reactors

The dispersion of a chemically active solute in unidirectional laminar flow in a channel of constant cross-sectional area is considered. Adsorption/desorption of the solute at the wall or the presence of a bulk or surface chemical reaction introduce additional timescales, in addition to the diffusive and convective ones, such that, under certain conditions, the asymptotic evolution of the cross-sectional mean concentration cannot be described by a one-dimensional Taylor-Aris model. We use the centre and invariant manifold theories to establish the proper time and length scale separations necessary for the existence of an effective transport equation and to determine the dependence of the effective transport coefficients on the kinetics of adsorption/desorption and reaction. For the case of classical Taylor-Aris dispersion with no reaction, we derive the effective transport equation to infinite order in the parameter, p, representing the ratio of the characteristic time for radial molecular diffusion to that for axial convection. We show that the infinite series in the effective transport model is convergent provided p is smaller than some critical value, which depends on the initial concentration distribution. We also examine the spatial evolution of time dependent inlet conditions and show that the spatial and temporal evolutions differ at third and higher orders. It is shown that, except for slow reactions with a kinetic timescale of the same order as the transverse diffusion time, fast bulk reaction does not allow an asymptotic axial dispersion description. Slow bulk reactions do not affect dispersion but a correction to the apparent kinetics may arise due to nonlinear interaction among reaction, diffusion and convection. It is also shown that with a slow bulk reaction, steady-state dispersion due to a coupling of reaction and transverse velocity gradient can arise. Although this mechanism is distinct from the transient Taylor—Aris mechanism, the dispersion coefficient is identical to the classical unreactive Taylor—Aris coefficient. Surface reaction of any speed yields the proper asymptotic behaviour in time because the species still needs to diffuse slowly to the conduit wall. In the limit of fast surface reaction, the Taylor-Aris dispersion coefficient is reduced by a factor of 4.2, 7.1 and 4.0 for pipe, plane Poiseuille and Couette flows, respectively, as the slow-moving solutes near the wall are depleted. For the case of a linear surface reaction, we use the invariant manifold theory to derive the effective transport equation to infinite order. We also show that the radius of convergence of the invariant manifold expansion is approximately three times that of the no reaction case. We demonstrate that if adsorption/desorption is as slow as transverse diffusion an adsorption-induced dispersion, distinct from the Taylor-Aris shear dispersion, exists. While the total dispersion may increase because of the contribution of both, the Taylor-Aris component is reduced by a physical mechanism similar to surface reaction. The adsorption/desorption induced dispersion coefficient is shown to have a maximum when the adsorption equilibrium constant is exactly 2. Nonlinear Langmuir type adsorption at large concentration is shown to introduce a nonlinear drift term which causes non-Gaussian pulse responses with long tails These tails are detrimental to separation chromatography since they cause overlaps which increase with the length of the chromatograph

Low-dimensional models for describing mixing effects in laminar flow tubular reactors

Abstract The Liapunov–Schmidt (LS) technique of bifurcation theory is used to average the convective-diffusion equation in the transverse direction and obtain low-dimensional two-mode models that describe mixing effects in laminar flow tubular reactors. For the isothermal case, these models are described by a pair of equations involving two modes, namely, the spatially averaged (〈 C 〉) and the mixing-cup ( C m ) concentration vectors. The first equation traces the evolution of C m with residence time, while the second is a local balance equation that describes local mixing as an exchange between the reaction scale (represented by 〈 C 〉) and the convection scale (represented by C m ) in terms of the local mixing time . The LS method also shows that such low-dimensional description is possible only if the local Damkohler number (ratio of local mixing time to reaction time) satisfies the convergence criteria of being less than 0.858. It is shown that the two-mode models have the same accuracy as the infinite (radial) mode convection model, within the range of validity of the latter. The two-mode models for homogeneous reactors have many similarities with the classical two-phase models for heterogeneous catalytic reactors, with the transfer coefficient concept (between surface and mixing cup concentrations, C S and C m , respectively) being replaced by that of an exchange coefficient (between 〈 C 〉 and C m ). Examples are presented to illustrate the usefulness of the two-mode models in predicting the effects of non-identical local mixing times, non-uniform reactant feeding and non-linear kinetics on conversion and yields of products for single and multiple reactions in laminar flow tubular reactors.