Uniqueness criteria of the steady state in automotive catalysis

Uniqueness and multiplicity criteria for the steady state are derived following previous analyses for a bimolecular Langmuir—Hinshelwood reaction occurring in a porous catalyst, in both the isothermal and non-isothermal cases. The criteria compare well with exact numerical computations in all cases, except when the mass and heat Biot numbers are unequal. Recently reported criteria for the isothermal case are shown to be always much more restrictive than the ones developed here. Also, numerical calculations show that up to five distinct steady state solutions exist for the sphere, in both the isothermal and non-isothermal cases, while only up to three solutions are obtained for the slab catalyst.     

MSMPR连续结晶器的多定态

对混合悬浮混合排料(MSMPR)连续结晶器的定态方程组进行了剖析,应用非线性方程组的延拓求解法,求得了定态方程组的分岔图,即MSMPR连续结晶器状态参数与操作参数的依赖关系.利用分岔理论对MSMPR连续结晶过程的定态数目及相应的参数范围进行了预测,导出了连续结晶过程存在多定态或唯一定态的判据。该判据仅与过程的动力学参数和操作参数有关,而与过程的状态参数无关。最后应用Routh-Hurwitz准则,确定了连续结晶过程各定态的稳定性。     

Uniqueness criteria of the steady state in automotive catalysis

For the automotive catalytic reaction system, uniqueness criteria for the steady state are obtained by use of the contraction mapping principle and the Hammerstein uniqueness theorem. Both criteria show that if φ(1 + σ) is appropriately small, where φ is the Thiele modulus and σ is the adsorption parameter, there is only one steady state. Numerical results are also given to predict the particle geometry effect on the uniqueness criteria.     

A study on the states of the non isothermal tubular wall reactor

The thermal behaviour of a cooled tube wall reactor is analyzed.The main goal of this work is the characterization of the steady state. Three different steady state solutions, in which the control through the reactor differs from each other, are presented. The possibility of simultaneous existence of these states is analyzed as a function of the parameters of the model.For the zeroth order reaction here treated, it is possible to get isothermal operating conditions. Minimum thermal levels at the wall reactor for a given conversion are discussed and they result closely related to the isothermal conditions.An analytical solution is developed for the so called “highest steady state” (HISS). This expression makes possible to predict the “quenching” conditions of the reactor.     

Multiplicity,stability, and oscillatory dynamics of the tubular reactor

Numerical bifurcation techniques were developed for studying the multiplicity, stability, and oscillatory dynamics of the nonadiabatic tubular reactor with a single A → B reaction. The techniques illustrate the existence of one, three, five, or seven steady states and bifurcating periodic solutions. We present numerical procedures for computing the Hopf bifurcation formulas which can determine the stability and location of the oscillation without integrating the parabolic partial differential equations. The combination of our Hopf techniques with steady state bifurcation methods enables us to determine all possible steady and stable oscillatory solutions exhibited by distributed parameter models such as the tubular reactor.     

Substrate-inhibited enzyme reaction in a tubular reactor with axial dispersion

For a fixed set of physico-chemical parameters, an isothermal substrate-inhibited enzyme reaction in a tubular reactor with axial dispersion can give rise to multiple steady states. Criteria previously developed for non-isothermal reactions are applied to this case to develop analytic conditions which ensure uniqueness of steady states. The criteria compare well with values obtained from exact numerical calculations.     

MULTIPLICITY IN CONTINUOUS CRYSTALLIZERS: ADIABATIC REACTIVE PRECIPITATION

The application of the theory of both concentration and temperature multiplicities is extended to the case of a reactive precipitation system in a continuous mixed suspension mixed product removal (MSMPR) crystallizer. A process involving elementary homogeneous chemical reaction with first order reaction kinetics with respect to each of the reactive components and subsequent crystallization described by conventional power law growth and power law magma dependent nucleation models is considered; the temperature dependency of each of these kinetics is described by Arrhenius relations. Exact uniqueness and multiplicity (i.e. multiple steady states) criteria are developed. The stability of these steady states is analysed using small perturbations around the steady state     

Modelling of fluidized bed reactors—II: Uniform catalyst temperature and concentration

A model based on the simple two phase theory of fluidization including the catalyst particles as a third phase has been developed for a nonisothermal fluidized bed catalytic reactor with continuous circulation of catalyst particles. The dilute phase is assumed to be in plug flow, the emulsion phase gas is considered to be perfectly mixed and the particles are assumed to be perfectly mixed and uniform.Exact criteria for uniqueness and multiplicity of the steady state solutions are presented and some conclusions derived therefrom. Several examples illustrating the influence of some parameters on the steady state multiplicity are reported. The steady states are analyzed for local asymptatic stability using Liapunov's direct method, but the sufficient conditions for stability are found to be rather conservative.Numerical examples illustrating the transient behavior of the system are presented, and it has been found that the initial temperature of the catalyst particles is a predominant factor in determining which steady state will be approached.     

State space formulation of ammonia reactor dynamics

The specific objective of this paper is to develop a state space model of a tubular ammonia reactor which is the heart of an ammonia plant in a fertiliser complex. A ninth order model with three control inputs and two disturbance inputs is generated from the nonlinear distributed model using linearization and lumping approximations. The lumped model is chosen such that the steady state temperature at the exit of the catalyst bed computed from the simplified state space model is close enough to the one computed from the nonlinear steady state model. The model developed in this paper is very useful for the design of continuous/discrete versions of single variable/multivariable control algorithms.     

The effect of homogeneous gas phase oxidations in char particle gasification

Two steady state models have been developed for the simultaneous oxidation and gasification of a single coal char particle. Because of the uncertainty in the relative importance of solid phase and gas phase reaction kinetics in a gasifier environment, the models assume that the velocities for the gas phase oxidations of hydrogen and carbon monoxide are either zero or infinitely fast. The global rates and steady state multiplicities are discussed as functions of particle size, film thickness, gas composition and ambient temperature. The restriction of infinite hydrogen diffusivity assumed by previous investigators is removed. The two models lead to considerably different predictions about the maximum temperature of char particles in coal gasifiers.     

Steady state multiplicity depending on coalescence in liquid—liquid continuous stirred reactors with reaction in the dispersed phase

The influence of drop coalescence and breakup on the existence of multiple steady states is studied for a two-phase stirred isothermal reactor where the chemical reaction in the d?ispersed phase obeys the rate expression ? r = kC/(1 + KC)². The random coalescence model developed by Curl was simulated using a modified Spielman and Levenspiel Monte Carlo technique.For certain range of the coalescence rate, Damköhler number, and dimensionless feed concentration, multiple steady states have been investigated.A special case has also been considered wherein the existence of multiple steady states for finite values of the coalescence rate is contrasted to the unique steady state solution for an infinite coalescence rate.     

MULTIPLICITY AND STABILITY ANALYSIS OF AGGLOMERATION CONTROLLED PRECIPITATION

The possibility of multiplicity in continuous isothermal MSMPR precipitators has been explored for agglomeration controlled conditions and general criteria, independent of nucleation kinetics, are developed for stability and multiplicity of the steady states. For the Volmer model of primary nucleation and the magma dependent power law model of secondary nucleation, parameter regions are determined in which multiple steady states exist, and their linear stability is analyzed. The analysis holds in general for all types of agglomeration kernel. For the Volmer nucleation kinetics three steady states exist in the region of multiplicity with the “middle” one always being unstable. The analysis for magma dependent power law model showed multiplicity regions having as many as four steady states, the number depending on magma and kinetic order. Unlike the case of molecular growth control, limit cycle behaviour is not possible, and the approach to the steady state is always asymptotic     

MULTIPLICITY AND STABILITY ANALYSIS OF AGGLOMERATION CONTROLLED PRECIPITATION

The possibility of multiplicity in continuous isothermal MSMPR precipitators has been explored for agglomeration controlled conditions and general criteria, independent of nucleation kinetics, are developed for stability and multiplicity of the steady states. For the Volmer model of primary nucleation and the magma dependent power law model of secondary nucleation, parameter regions are determined in which multiple steady states exist, and their linear stability is analyzed. The analysis holds in general for all types of agglomeration kernel. For the Volmer nucleation kinetics three steady states exist in the region of multiplicity with the “middle” one always being unstable. The analysis for magma dependent power law model showed multiplicity regions having as many as four steady states, the number depending on magma and kinetic order. Unlike the case of molecular growth control, limit cycle behaviour is not possible, and the approach to the steady state is always asymptotic     

Search for all steady-states of chemical kinetic equations with the modified method of elimination—II: Application

A steady state kinetic model is considered on the basis of the algorithm of elimination, described in Part I, and corresponds to the general scheme of hydrogen catalytic oxidation. A resulting polynomial of one variable is clearly expressed (through kinetic parameters). It is shown that reversibili of a monomolecular step of water formation is a source of the steady state multiplicity.A modified method of elimination has been fully realized for the general system of three nonlinear equations—the resulting polynomial is expressed in     

Investigation of the temperature distribution and material limits of membrane water wall tubes of circulating fluidized bed boilers

Assuming steady state, constant bed temperature and constant tube thermal conductivity, the three‐dimensional heat conduction equations were numerically solved. It was found that the temperature distribution can be expressed as dimensionless temperature in terms of Froude number, particle to furnace diameter ratio, and local to total furnace height ratio. The models for predicting the dimensionless wall temperature distribution and dimensionless mean wall temperature were also developed in this work. Finally, the limits of common tube materials used as water wall tubes based on maximum allowable stress and oxidation criteria were also investigated.     

Analysis of multiple steady states of complex biochemical reactions

A general theory for the identification of criteria within which an enzymic reaction can exhibit multiplicity (i.e. multiple steady states) is reviewed. Application of the theory to four types of kinetic models encountered in biochemical reactions is illustrated and figures are presented to delineate the region of multiplicity. The stability of the steady states is analysed for small perturbation about the steady state.     

Isothermal oscillations in surface reactions with coverage independent parameters

A procedure for obtaining necessary and sufficient conditions for the existence of periodic solutions in surface reactions with constant temperature and coverage independent parameters is described. An analytic method for the analysis of bifurcation to periodic solutions is developed. This formulation, based on the fundamental matrix of a system of ordinary differential equations, provides a simple means of examining the behavior of system in the neighborhood of a singular point. This method is applied into a reaction model which can yield oscillatory solutions with both single and multiple steady states. When a unique steady state becomes unstable, a limit cycle develops. With three steady states, a state may develop into a stable or unstable limit cycle or may exhibit only periodic solutions without a limit cycle.     

A pseudo‐steady state model of the kinetics of laccase‐catalysed oxidation of aqueous phenol

A model has been developed to describe the kinetics of the oxidation of phenol by oxygen using laccase (EC 1.10.3.2) from Trametes versicolor as catalyst. The model incorporates an assumption, referred to as the pseudo‐steady state assumption, that at any instant during the reaction the enzyme achieves an approximate steady state distribution of its various forms around the catalytic cycle. The model was validated by comparing model predictions with experimental observations of phenol transformation and oxygen consumption over time. Excellent agreement was found between experimental data and model predictions. The pseudo‐steady state assumption had the advantage of reducing the complexity of the model equations to be solved while simultaneously maintaining the model's predictive ability. The pseudo‐steady state model was further simplified in accordance with specific initial conditions to provide a simplified method for the estimation of the model constants. Copyright © 2006 Society of Chemical Industry     

Stability of the steady states and transient behavior for a non-isothermal bimolecular langmuir-hinshelwood reaction

The steady state stability and transient behavior of a catalytic reaction following the bimolecular Langmuir-Hinshelwood expression, phenomenologically adequate for carbon monoxide and hydrocarbon oxidation on noble metal catalysts, is describe. The corresponding steady state problem was discussed earlier. Some unique features of the isothermal and non-isothemal transient behavior with uniform initial conditions are identified.     

Some problems concerning the non-adiabatic tubular reactor a-priori bounds,qualitative behavior,preliminary uniqueness and stability considerations

An investigation of the non-adiabatic tubular reactor, in which a single chemical reaction occurs, is presented. The steady state mass and heat balance equations are shown to be a pair of coupled, nonlinear differential equations, and are shown to uncouple into a single functional-differential equation for the case Le = 1. A-priori bounds on the steady state concentration and temperature are developed, and some remarks about the shape of the steady state profiles made. Sufficient conditions ensuring uniqueness of steady states and their asymptotic stability are derived.