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.     

Mathematical modelling of fluidized bed reactors—III: Axial dispersion model

An axial dispersion model has been developed for a continuous fluidized bed catalytic reactor with a cocurrent flow of the emulsion phase gas and the catalyst particles. The influence of some parameters on multiplicity of steady states has been reported. Several examples illustrating the transient behavior of the system are presented. In cases where three steady states are possible it appears that the intermediate steady state is unstable, while the lower and the upper steady states are locally stable. It was noted that the initial temperature of the emulsion phase is a predominant factor in determining which steady state will be approached.     

Parametric sensitivity of a CSTR

It is shown that the generalized sensitivity criterion recently developed in the context of thermal explosions and tubular reactors can be easily applied in the case of a CSTR as well. An illustrative example concerning sensitivity analysis of a single nth order irreversible exothermic reaction in a nonadiabatic CSTR is presented. A generalized region of parametric sensitivity is identified where the reactor temperature is parametrically sensitive simultaneously to all the input parameters. Asymptotic analysis for large heats of reaction is performed to investigate limiting behaviour, which leads to the classical Semenov limit in the case of large activation energies. It is shown that parametric sensitivity can occur even when unique steady states exist for all Damköhler number (Da) values. Furthermore, if operating conditions are chosen so as to avoid completely the possibility of parametric sensitivity for all Da, then the possibility of steady-state multiplicity is automatically avoided.     

Performance of an adiabatic controlled cycled stirred tank reactor

A set of general differential equations was used to describe the dynamics of an adiabatic controlled cycled stirred tank reactor (CCTR). Exothermic reaction between sodium thiosulfate and hydrogen peroxide was carried out in a CCTR under several conditions. The observed dynamic and pseudo steady state behavior agreed very well with predictions from theory. Complete information about the system could be thus ascertained from measurement of tempe only. Conversion and capacity for the three types of reactors, CCTR, CSTR and PFR, were also compared for the model reaction used here.     

Non-ideal reactor network synthesis through IDEAS: Attainable region construction

The attainable region (AR) for non-ideal reactor networks is constructed for the first time using the Infinite DimEnsionAl State-space (IDEAS) framework. The axial dispersion model is used to represent a non-ideal reactor, and it is shown that the IDEAS framework and associated Shrink-wrap algorithm are applicable to this model. A case study demonstrates that the AR for a reactor network featuring non-ideal dispersion models is larger than the AR for a reactor network featuring only ideal CSTR/PFR models.     

Modeling and stability of catalytic fixed bed reactors

The existence of a multiple or oscillatory condition in a packed bed reactor, with a strongly exothermic reaction, is reviewed. The following cases are discussed: simultaneous heat and mass transfer within and outside a catalyst particle, axial mixing in tubular reactors and a recycle reactor. It is shown that the condition describing multiplicity can be estimated solely on the basis of the CSTR analysis. A correspondence between stability and parametric sensitivity is presented.     

反应级数对理想反应器组合方式效率的研究

对于不同级数的不可逆反应,研究了其在全混流反应器(CSTR)与平推流反应器(PFR)不同组合方式中的最终转化率。对两种理想流动反应器的不同组合方式,如CSTR+PFR串联、PFR+CSTR串联及CSTR与PFR并联,反应的最终转化率随反应级数的不同而变化。当反应级数为负值时,CSTR与PFR并联组合方式的最终转化率最高;当反应级数为0~1之间的数值时,CSTR+PFR串联组合方式的最终转化率最高;而当反应级数大于1时,PFR+CSTR串联组合方式的最终转化率最高;当反应级数为0时,三种组合方式具有相同的最终转化率;但当反应为一级不可逆反应时,两种反应器串联时的最终转化率相同,并高于两种反应器并联时的最终转化率。     

Uniqueness and multiplicity criteria for an nth order chemical reaction

New and very strong criteria are presented for a priori prediction of the conditions for which the steady-state lumped parameter model of an nth order chemical reaction (n ≥ 0) in an adiabatic CSTR has either a unique or multiple solutions. The criteria show that the higher the order of the reaction the smaller is the region in the parameters space for which multiplicity can occur.New uniqueness and multiplicity criteria are developed also for an nth order reaction in a porous catalyst using a model, which accounts for intraparticle concentration gradients, while assuming a uniform intraparticle temperature different from the ambient one. The region in the parameters space for which steady state multiplicity can occur for this model is smaller than that for a corresponding lumped model, which ignores the intraparticle concentration gradients.     

MeOH to DME in bubbling fluidized bed: Experimental and modelling

Methanol dehydration over a ZSM‐5 containing catalyst was studied in a fluidized bed reactor. At temperatures ranging from 250 to 325°C, methanol conversion varied from 30% at a contact times of 0.14 s and approached 100% of the equilibrium conversion at a contact time starting from 10 s. Sequential and parallel reactions were negligible at low temperatures while hydrocarbon formation became appreciable at 325°C. Online gas analysis by mass spectrometry provided real‐time measurements at a frequency of 4.4 Hz that allowed for fast determination of steady‐state conditions. Gas phase residence time distribution (RTD) measurements indicated that axial dispersion was essentially negligible at short contact times with a shallow bed of catalyst. With longer residence times, the flow pattern could be approximated by six continuously stirred‐tank reactors (CSTR) in series. Both the simple 1D hydrodynamic model and a detailed multi‐zone fluidized model were used to interpret the experimental data to derive a kinetic expression for the dehydration of methanol to di‐methyl ether (DME). The expression includes the reverse reaction that is most often neglected in the literature. The reaction data were best fit with the kinetics based on the 1D model. The fluidized bed is a viable reactor type for kinetic measurements of highly exothermic reactions where hotspots and radial and axial temperature gradients are problematic in fixed beds.     

Heterogeneous Catalytic Reactor Design with Non‐Uniform Catalyst Considering Shell‐Progressive Poisoning Behavior

A novel methodology has been developed to design an optimum heterogeneous catalytic reactor, by considering non‐uniform catalyst pellet under shell‐progressive catalyst deactivation. Various types of non‐uniform catalyst pellets are modelled in combination with reactor design. For example, typical non‐uniform catalyst pellets such as egg‐yolk, egg‐shell and middle‐peak distribution are developed as well as step‐type distribution. A progressive poisoning behavior is included to the model to produce correct effectiveness factor from non‐uniform catalyst pellet. As opposed to numerical experiment with limited type of kinetic application to the model in the past, this paper shows a new methodology to include any types of kinetic reactions for the modeling of the reactor with non‐uniform catalyst pellet and shell‐progressive poisoning. For an optimum reactor design, reactor and catalyst variables are considered at the same time. For example, active layer thickness and location inside pellet are optimised together with reactor temperature for the maximisation of the reactor performance. Furthermore, the temperature control strategy over the reactor operation period is added to the optimization, which extends the model to three dimensions. A computational burden has been a major concern for the optimization, and innovative methodology is adopted. Application of profile based synthesis with the combination of SA (Simulated Annealing) and SQP (Successive Quadratic Programming) allows more efficient computation not only at steady state but also in dynamic status over the catalyst lifetime. A Benzene hydrogenation reaction in an industry scale fixed‐bed reactor is used as a case study for illustration.     

Modeling of chemical reactors—XXXI: The one-phase backflow cell model used for simulation of tubular adiabatic reactors

The backflow cell model is used to simulate steady state operation of a tubular adiabatic reactor. The model proposed embraces different mechanism of axial dispersion of heat and mass. It will be shown that the backflow cell model may be used for approximation of the dispersion model. While there are differences in qualitative behavior of simple cell and dispersion models, the backflow cell model gives results which are in agreement with the dispersion model. The model may be used for simulation of steady-state behavior of tubular homogeneous and heterogeneous reactors.     

BEHAVIOUR OF AN ADIABATIC PACKED BED REACTOR PART 2: MODELLING

The steady state and dynamic behaviour of an adiabatic packed bed reactor for the selective hydrogenation of mixtures of ethyne and ethene is studied. A heterogeneous model with axial dispersion of heat is solved numerically by means of a fully implicit discretisation scheme. From an analysis of the inlet and outlet boundary conditions, it follows that in order to avoid an influence of the type of boundary condition on the reactor behaviour, sufficiently long inert zones before and after the active bed should be present. Using preliminary kinetic expressions adapted from Mcn'shchikov ct al. (1975), the model calculations show a reasonable agreement with the experiments performed in a laboratory scale reactor and demonstrate the importance influence of small amounts of carbon monoxide both on the steady state as well as on the transient behaviour of the reactor. However, using the rate expressions derived from our own kinetic experiments in a Berty reactor, the behaviour under runaway conditions and the transition from non-runaway to runaway conditions can not be described well. Several factors that contribute to this discrepancy are discussed. This work shows that, in particular for a complex reaction system such as the selective hydrogenation of ethyne in ethene, the limited accuracy of the kinetic model and kinetic parameters dominates the precision attainable with packed bed reactor models.     

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.     

RECIRCULATING REACTOR FOR THE STUDY OF MULTIPHASE KINETICS

A new laboratory reactor was set up to measure kinetic coefficients in a solid (catalyst)-liquid-gas reacting system.

The reactor consists of two parts: an absorber, where the liquid is partially saturated by the gas reactant and a reacting zone, where the liquid alone, containing the dissolved gas, flows through a fixed bed of catalyst.

The ricircle of the liquid in the absorber maintains a high concentration of the gas reactant in the liquid also in the zone of reaction, allowing the use of a high mass of catalyst (significative from a statistical point of view) and the achievement of sufficiently high conversion.

The tested reaction is the catalysed hydrogenation of ∝-metylstyrene: in order to consider a drastic situation and to verify the results with the literature data, the experimental conditions examined corresponded to very high chemical reaction rate (instantaneous reaction) at the surface of the pellets.

The tests were carried out with the reactor working both in batchwise and in continuous operative mode (steady state); the results show the reliability of the new reactor above all when the steady state operation is considered. For the use of the reactor in batchwise condition, the accumulation of the product inside the catalyst particles must be considered for an accurate measurement of the kinetic parameters,     

OPTIMUM REACTOR DESIGN IN METHANATION PROCESSES WITH NONUNIFORM CATALYSTS

A generic methodology is developed to design a heterogeneous catalytic reactor for methanation processes. For the optimization of a heterogeneous catalytic reactor, nonuniform catalyst pellets such as a layered catalyst are considered with respect to reaction type, reactor performance, and component distribution inside the catalyst. Heterogeneous uniform and nonuniform catalyst models were developed to analyze the effect of mass and heat transfer between both bulk phase and catalyst surface and inside a catalyst pellet. Then, concentration profiles of hydrogen and carbon monoxide in the catalyst pellet and along the reactor axis were obtained by analyzing simulation results. It was shown that the application of different types of nonuniform catalyst pellets at a certain number of separate zones within a reactor could produce higher catalyst performance than a reactor with uniform catalyst. Furthermore, it proved a significant decrease of catalyst deactivation behavior such as coking and sintering. Layered catalysts were optimized to maximize an overall reactor performance over the catalyst lifetime, achieving capital cost reduction characterized by reactor size, catalyst amount, and degree of catalyst deactivation. Last, temperature control throughout the reactor operating periods was strategically planned for a reactor operation with distribution of nonuniform catalyst pellets. This methodology can also be usefully applied to the design of heterogeneous catalytic reactors for other processes such as hydro-treating process and cracking process.     

Differentiation criteria study for continuous stirred tank reactor and plug flow reactor

All reactors in reality are not ideal plug flow reactor (PFR) or ideal continuous stirred tank reactor (CSTR). They are difficult to differentiate. This study was to investigate the reactor analysis of PFR and CSTR through tracer response curves, residence time distributions (RTD) and several hydraulic performance indexes. We set up the differentiated value of each index. The tracer response curve showed that our lab-scale CSTR was close to ideal CSTR and got 99.9% recovery. In the RTD curves, the results could significantly recognize the PFR nature of high rate pond (HRP). With hydraulic performance indexes study, every selected index demonstrated that the studied HRP was closer to PFR than the studied CSTR. Based on the lab-scale study results, this study established the cutting point between the PFR and CSTR in each index; we were looking through the different types of reactors in literature and we confirmed the criteria with all literature reactors with the “graphic method”. The method helped us to establish those important values to help us to differentiate the reactor types in practice and to understand the designs better.     

State multiplicity in PFR-separator-recycle polymerization systems

This article explores the non-linear behaviour of isothermal and non-isothermal plug-flow reactor (PFR)-separator-recycle systems, with reference to radical polymerization. The steady-state behaviour of six reaction systems of increasing complexity, from one-reactant first-order reaction to chain-growth polymerization, is investigated. In PFR-separator-recycle systems feasible steady states exist only if the reactor volume exceeds a critical value. For one-reaction systems, one stable steady state is born at a transcritical bifurcation. In case of consecutive-reaction systems, including polymerization, a fold bifurcation can lead to two feasible steady states. The transcritical bifurcation is destroyed when two reactants are involved. In addition, the thermal effects also introduce state multiplicity. When multiple steady states exist, the instability of the low-conversion branch sets a lower limit on the conversion achievable at a stable operating point. A low-density polyethylene process is presented as a real plant example.The results obtained in this study are similar to CSTR-separator-recycle systems. This suggests that the behaviour is dictated by the chemical reaction and flowsheet structure, rather than by the reactor type.     

反应器网络综合分区法中的简捷计算

针对定态、等温、恒容、单股进料的复杂反应过程,首先将反应器网络综合可得区分区法由浓度空间拓展至瞬时选择性-关键反应物未转化率（S-x）空间,然后根据分区法中两条重要曲线的特性（选择性最大曲线dS/dx=0,单程收率最大曲线S=0）建立了以废料最少为目标的反应器网络基本组成单元（PFR和CSTR）的数学模型,并以Van de Vusse反应模式为例将其应用于由分区法确定的最优反应器网络结构的模拟计算.计算结果与文献值一致,表明本文所用方法具有一定的简捷性和实用性.     

Continuous emulsion polymerization of vinyl acetate. II Operation in a single Couette–Taylor vortex flow reactor using sodium lauryl sulfate as emulsifier

Continuous emulsion polymerizations of vinyl acetate were conducted at 50°C in a single continuous Couette–Taylor vortex flow reactor (CCTVFR) using sodium lauryl sulfate as emulsifier and potassium persulfate as initiator. The polymerization can be carried out very smoothly and stably, but the steady‐state monomer conversion attained in a CCTVFR is not as high as that in a plug flow reactor (PFR), but only slightly higher than that in a continuous stirred tank reactor (CSTR), even if the Taylor number is adjusted to an optimum value. Also, the effects of operating variables, such as the emulsifier, initiator, and monomer concentrations in the feed and the mean residence time on the kinetic behaviors were almost the same as those observed in a CSTR. © 2002 Wiley Periodicals, Inc. J Appl Polym Sci 86: 2755–2762, 2002