Exact criteria for uniqueness and multiplicity of an nth order chemical reaction via a catastrophe theory approach

The development of a simple, generalized technique for the exact determination of regions of unique and multiple solutions to certain nonlinear equations via a catastrophe theory-implicit function theorem approach, is presented. The application of this technique to the nth order chemical reaction in the nonadiabatic and adiabatic CSTR yields exact, explicit bounds for all n ≥ 0. To our knowledge, this is the first report of exact, explicit bounds for these systems, except for n = 0, 1 for the adiabatic CSTR, and n = 1 for the nonadiabatic CSTR. For the nonadiabatic CSTR, these bounds show that the higher the reaction order, the smaller the region in parameter space for which multiplicity can occur for all γ and x_2c, (dimensionless activation energy and coolant temperature, respectively). This behavior is similar to that reported by Van den Bosch and Luss[1] for the adiabatic CSTR. The zeroth order reaction in the nonadiabatic CSTR exhibits more complex behavior and assumes characteristics of both high and low reaction orders insofar as increasing and/or decreasing the uniqueness space, in comparison to all other n > 0.An exact implicit bound between regions of uniqueness and multiplicity is also derived for the nth order reaction in a catalyst particle with an intraparticle concentration gradient and uniform temperature, and is fully demonstrated for the first order reaction. In addition, explicit criteria, sufficient for uniqueness and multiplicity of the catalyst particle steady state, stronger than those of Van den Bosch and Luss, are also developed by combining the present technique with bounds suggested by these authors.     

Exact universal uniqueness criteria for the adiabatic tubular packed bed reactor

Exact, universal a priori bounds and regions of multiplicity for the entire tubular packed bed reactor are developed by application of a technique reported in Chang and Calo, Chem. Engng Sci. 1979 34 285 to a cascade two-phase cell model for an nth order chemical reaction with interphase resistance to mass and heat transport, Le ≠ 1 (or Le = 1) and either lumped parameter catalyst particles or with intraparticle concentration gradients with uniform temperature. Both the more common case of interphase heat transfer greater than interphase mass transfer rate and the inverse case of particle over-temperature have been considered. In all cases it has been shown that the reactor conservation equations can be decoupled at certain points along the bed determined by the Lewis number, and that questions of multiplicity and uniqueness reduce to consideration of a single algebraic equation which is actually a form of the two-phase adiabatic CSTR. Also as for the CSTR, the topology of the adiabatic packed bed reactor is shown to be the simple cusp catastrophe. The application of the resultant criteria is quite simple and represents a practical step in the design procedure for highly exothermic reactions in packed beds. A flow chart of a suggested procedure is included.     

Multiplicity and uniqueness for binary,exothermic reaction in a non-adiabatic continuous stirred tank reactor

Exact multiplicity and uniqueness criteria for steady state in a non-adiabatic continuous stirred tank reactor are studied through simple tangent analysis for binary, exothermic reaction of the type A + bB → Products with rate expression r_A = kC_A^m C_Bⁿ, where A is the limiting reactant. Important parameters for multiplicity criteria are reaction orders m and n, stoichiometric coefficient b, the ratio p of feed concentration of A to that of B, dimensionless activation energy α, dimensionless heat of reaction β, dimensionless heat transfer coefficient γ and dimensionless coolant temperature. Necessary conditions for the system to have multiple exit conversions (temperatures) are defined in the (m, n, b, p, α, β, γ) space. Multiplicity is guaranteed by limiting the dimensionless space time ? in a proper range in addition to the necessary conditions. Effects of various parameters on multiplicity and uniqueness are numerically calculated and graphically represented. Theoretical prediction for multiplicity are further compared with multiplicity data reported in literatures.     

Exact steady-state multiplicity criteria for two consecutive or parallel reactions in lumped-parameter systems

Exact criteria are derived for predicting the conditions under which steady-state multiplicity occurs in lumped parameter systems in which either two consecutive or two parallel reactions occur. Most of the qualitative multiplicity features can be determined by considering four simple limiting cases in which only one reaction takes place in the system. The structure of all the possible multiplicity regions in the (Da₁, Da₂) plane is determined for both reaction networks.     

The selectivity of porous catalysts: parallel reactions

The influence of intraparticle mass diffusion on the overall and relative rates of the parallel reactions A → B and A → C is examined for the case of an isothermal catalyst particle with no external concentration or temperature gradients. Calculations for three pairs of reaction orders: (0,1), (0,2) and (1,2), show that internal concentration gradients can cause the yield of the product formed in the higher-order reaction to decrease by as much as 80 percent. Extension of the analysis to other reaction orders is discussed. Criteria for close approach to various asymptotic limits are presented, together with an asymptotic expression which allows the maximum effect of pore diffusion on selectivity to be estimated for any pair of reaction orders.     

Multiplicity of conversion in a cascade of imperfectly stirred tank reactors

The multiple steady states of conversion in the model of non-ideal CSTR's in series have been studied for irreversible, homogeneous exothermic reaction of first order. The influences of the parameters p and n and those of some kinetic parameters on the occurrence of multiple conversions for a given residence time are presented. The existence of steady-state multiplicity of conversion in each CSTR is shown to be independent of n and depen largely on the values of E/RT₀ (activation energy), b/T₀ (heat of reaction), and p. For certain parametric values, more than three steady states can exist in such a model.     

Multiplicity,stability and dynamics for isothermal autocatalytic reactions in CSTR

Multiplicity, stability and dynamics for autocatalytic reactions of the type A? → rR + ? with overall rate expression r_A = kC_A^mC_Rⁿ in a continuous stirred tank reactor (CSTR) are studied. Exact multiplicity criteria are defined in the parameter space. Stability analysis shows that no periodic oscillation is possible for the system. When multiplicity occurs, some minimum of R present initially in the reactor is required in order for the high steady state to be achieved. Loci of transient extreme for product are also investigated. Multiplicity and uniqueness criteria are further compared with reported experimental data in literature.     

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 (Le_f) though the patterned states are more pronounced and exist in a wider range of parameter space when Le_f<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 Le_f=1.     

Kinetic instabilities in heterogeneously catalyzed reactions—I: Rate multiplicity with langmuir-type kinetics

The question under which conditions kinetic instabilities can be described by Langmuir type kinetics is studied using a fairly general model for the surface reactionpA + qB → C. This paper is concerned with the occurrence of rate multiplicities (ignition-extinction phenomena). It is shown that rate multiplicity can be caused by the competing chemisorption of A and B upon the same active sites of the catalyst. Rate multiplicity can occur if at least one of the chemisorption or reaction steps is of second order. No rate multiplicity can be expected, if either A or B reacts via an Eley-Rideal type mechanism.     

The effect of liquid flow distribution on the behaviour of a trickle bed reactor

A mathematical model has been formulated of the effect of flow distribution of the liquid phase carrying a dissolved reactant on the progress of an nth order, irreversible, catalytic reaction with heat effects in an adiabatic trickle bed reactor. The model has been stated in terms of the density of irrigation, temperature and concentration of the reactant in the liquid, all treated as spatially distributed variables. Provisions have been made to account for the existence of the flow down the surface of the wall, which has no catalytic effect.Local concentration and temperature have been proven to be coupled by the invariant T + Uγc = γU. The same invariant governs also local concentration and temperature of the wall flow. Mathematically, the model is represented by a coupled set of nonlinear parabolic partial differential equations enabling concentration and temperature fields to be obtained for an arbitrary type of liquid distribution and intensity of the wall flow.Numerical solutions have been obtained by the finite-difference method simulating reactors irrigated by liquid distributors as central discs of different radii, or a central annulus, and strongly exothermic reactions with the reaction order ranging between 0.1 and 2. Numerical results have shown the effect of liquid distribution on the overall reaction conversion to be very complex. Optimum initial distribution varies depending on the reaction order as well as the required degree of conversion. In general, however, the entrance region flow pattern may play a significant role in affecting especially reactions exhibiting kinetics close to zero order (hydrogenations). The effect of the wall flow has been found unambigously adverse to reaching high conversions and of increasing importance for low order reactions.     

DEPENDENCE OF THE MULTIPLICITY FEATURES OF AN ISOTHERMAL CATALYTIC REACTION ON EXTERNAL AND INTERNAL TRANSPORT RESISTANCES

A mathematical model (Robin) which accounts for both internal and external transport resistances is used to determine the multiplicity features of a porous catalytic pellet in which an isothermal Langmuir-Hinshelwood reaction occurs. At most three solutions exist for a slab, but an arbitrarily large number of solutions can be found for an infinite cylinder or a spherical pellet. The maximal number of solutions for any finite external mass transfer resistance is bounded between that existing for a model which ignores the external mass transfer resistance and one which ignores intra-particle concentration gradients. The approximate shape of the cross section of the bifurcation set and of the uniqueness boundary of the Robin model can be estimated from the knowledge of the multiplicity features of three simplified (lumped, Dirichlet and Neumann) models, each containing one less parameter.     

Multiple steady states in two-phase reactors under boiling conditions

In this paper, we analyze the nonlinear behavior of two-phase reactors under boiling conditions. First we focus on a simple nth-order reaction of the form A→B, which allows a rigorous analytical treatment. Three necessary conditions for the existence of multiple steady states have been identified: the reactant A has to be the light-boiling component, the difference in boiling point temperatures between the reactant A and the product B has to be sufficiently large, and the order of the reaction has to be less than some physical parameter α. This parameter α can be interpreted as a measure for the phase-equilibrium-driven self-inhibition of the reaction mechanism. Thus, we have found an elegant explanation for the occurrence of multiplicities. Analytical and therefore general quantitative criteria identifying the regions of multiplicity for the model system are presented. Practical relevance of our results is demonstrated by means of two examples, the Monsanto process for the production of acetic acid and the ethylene glycol reactive distillation system.     

A general criterion to test the relative importance of intraparticle and external heat and mass transfer resistances in porous catalysts

A general criterion is developed for testing the relative importance of intraparticle and external, concentration and temperature gradients on the observed rates, for reactions catalyzed by porous solids. The criterion is applicable to any multiple reaction system and to reactions conforming to any rate law. The applicability of such criteria is extended to reaction systems with highly nonlinear temperature and concentration dependence, by taking into-account the higher order terms in the Taylor expansion of the rate expressions in the derivation.     

Modeling breakthrough and elution curves in fixed bed of inert core adsorbents: analytical and approximate solutions

A mathematical model is developed for fixed-beds packed with inert core adsorbents. New analytical solutions to predict breakthrough and elution curves are derived for linear adsorption systems coupled with axial dispersion, film mass transfer and intraparticle mass transfer. New approximate solutions are also obtained based on the assumptions of parabolic concentration profile in the adsorbent shell and the quasi-lognormal distribution for the impulse response in order to predict breakthrough and elution curves. The applicability of these approximate solutions is suggested by comparison with the new analytical solutions. The effects of the size of inert core, sample input mode, axial dispersion, film mass transfer resistance and intraparticle diffusion resistance, on the breakthrough and elution curves are discussed. The decrease of the intraparticle mass transfer resistance by using inert core adsorbents is quantitatively analyzed by introducing the parameter 1/Θ. Furthermore, an analytical expression for the resolution of two components is derived based on the quasi-lognormal distribution approximate solution; the resolution of two components is improved with the inert core adsorbent when compared with the conventional adsorbent, especially for biomacromolecules where the intraparticle diffusion rate is slow.     

Nearly irreversible, fast heterogeneous reactions in premixed flow

When heterogeneous chemical reaction is sufficiently fast, transport of reactants becomes limiting. In a fixed bed reactor, macroscopic concentration gradients cannot be eliminated as a factor limiting the rate of reaction, possibility coupling to the mesoscopic mass transfer of reactants to the surface of the catalyst as limiting, if the reaction does not occur inside a porous support. A theory for strictly irreversible binary reaction is developed that shows the possibility of regimes of kinetic asymmetry in which a crossover point occurs internally in the reactor, demarking a region of high supersaturation in which the surface is effectively depleted of one reagent, followed by a region of rapidly decaying supersaturation, where the surface is effectively depleted of the other reagent. The parametric dependence of this crossover point is given in terms of a transcendental equation which depends on operating parameters (superficial velocity and inlet concentrations) and ratios of transport properties of the reagents. These solutions are corroborated by full nonlinear numerical computations of the boundary value problem, for the case when asymmetric mass transfer coefficients admit the possibility that the mode of operation switches from relative surface depletion of one reactant to depletion of the other in a binary reaction. It is shown that X indicates the region of greatest molecular efficiency in the reactor, and if operating parameters are so chosen such that X is large, high conversion is achieved in the precrossover region. The modified Thiele modulus analysis of the numerical solutions identifies the crossover point and the region of greatest product formation.     

Transient microkinetic modelling of n-heptane catalytic cracking over H-USY zeolite

A microkinetic model for the catalytic cracking of n-heptane is proposed comprising a set of significant elementary steps which generate a complex reaction network. This approach constitutes a compromise between fundamental and lumped models since the reaction scheme is detailed to the carbon atom level by considering separately olefins, paraffins and adsorbed carbenium ions with the same carbon atom number. Elementary rate constants are estimated through expressions relating species reactivity with their carbon atom number. Fitting the model against experimental data over a large parameter space was performed using a micro-genetic algorithm with binary encoding and logarithmic distribution. The advantage of this approach is that it allows modelling the reaction network without supposition of rate determining steps. The model predicts very well the observed transient activity for n-heptane cracking over H-USY zeolite at and in general, it reasonably predicts the experimental trends for the products distribution.     

Acid Black 172 dye adsorption from aqueous solution by hydroxyapatite as low-cost adsorbent

The Acid Black 172 dye adsorption on the uncalcined hydroxyapatite nanopowder was investigated. The hydroxyapatite prepared by wet coprecipitation method has high specific surface area of 325 m²/g and crystal sizes smaller than 70 nm. The batch adsorption experiments revealed that under the optimum adsorption conditions (pH 3, hydroxyapatite dosage 2 g/L, initial dye concentration 400 mg/L and temperature 20 °C) the dye removal efficiency was 95.78% after 1 h of adsorption. The adsorption kinetics was best described by the pseudo-second order kinetic model. The intraparticle diffusion model shows that intraparticle diffusion is not the sole rate-limiting step; the mass transfer also influences the adsorption process in its initial period. The Langmuir isotherm model best represented the equilibrium experimental data, and the maximum adsorption capacity (q _m ) was 312.5 mg/g.     

Gas absorption with an unusual chemical reaction: the chlorination of toluene

The rate of absorption of chlorine in a laminar jet of toluene, and the corresponding bulk liquid temperature increases, have been measured as a function of contact time. Since the reaction rate is very highly dependent on the concentration of dissolved chlorine, bulk mixing of the jet on entering the receiver effectively ‘freezes’ the reaction and enables experimental determination of the ratio

This ratio is independent of the mass diffusion coefficient for 1st order kinetics, and numerical studies show this is also true for nth order reactions. The ratio therefore should be particularly valuable for abstracting kinetic information from absorption rate data in systems where interfacial turbulence may occur. A numerical analysis, which takes into account the increasing interface temperature, indicates that the overall concentration dependency of the chlorine reaction rate can be fitted by an empirical fifth order equation.     

Thermally safe operation of liquid-liquid semibatch reactors. Part I: Single kinetically controlled reactions with arbitrary reaction order

The thermally safe operation of an indirectly cooled semibatch reactor in which an exothermic reaction occurs corresponds to conditions of potentially very high effective reaction rate compared to the dosing rate of the coreactant, whose accumulation in the reaction system is consequently small. On this basis it is possible to build boundary diagrams in terms of suitable dimensionless parameters, which summarize all the possible thermal behaviors of the reactor and can be used for safe scaleup purposes.In this work the influence of reaction kinetics on the shape and location of boundary diagrams for a single liquid-liquid reaction in the slow regime is discussed. First of all, the theory of boundary diagrams, originally developed for (1,1) reaction order, is extended to a generic (n,m) rate of reaction expression. Then it is shown that in many practical systems, using boundary diagrams based on (1,1) reaction order can lead to both unsafe and unnecessary (from a safety point of view) low-production operating conditions. New boundary diagrams for a few (n,m) reaction orders are presented. Some rules-of-thumb are also discussed to identify in which cases a boundary diagram developed for a given (n,m) reaction order can be reasonably used to approximate the real kinetic behavior of the system of interest.Moreover, since building a boundary diagram for the specific kinetics considered can be necessary, a simple and general procedure for building such diagrams that can be easily implemented in a computer code is also presented.