Consecutive Reaction System Research Articles

Optimal cold-shot cooling of an adiabatic tubular reactor

Optimal design of multi-bed adiabatic reactors with cold-shot cooling is considered for the case of several independent chemical reactions. The necessary equations for optimality are derived by applying Pontryagin's methods to the variational problem. A computational scheme based on Newton's method in the control-parameter space is devised and applied to the consecutive reaction system A → B → C in a tubular reactor with a single cold-shot injection. Upper and lower bounds in the performance index of such reactors are computed by analyzing limiting configurations. A sensitivity analysis leads to a scheme for devising an optimal feedforward controller that compensates for small perturbations in initial conditions.

The effect of dispersion on chemical reactor optimum residence times using perturbation expansion techniques

This work is a mathematical study of the effect of dispersion on the optimum residence times of chemical reactors. Attention is confined to the range of very small and very large dispersions so that perturbation expansions can be used to obtain asymptotic solutions to the equations for the axial dispersed plug flow tubular reactor. Only the consecutive reaction system A → B → C, where B is a desired intermediate product is considered. The optimum residence time is then defined as that which maximizes the yield of B in each isothermal reactor model. It is shown that the expansions for optimum residence time are excellent approximations over a wide range of operating conditions for this reactor model. An interesting maximum effect in the optimum residence time for the axial dispersed tubular reactor is discussed.

Catalytic particle stability studies—II Lumped thermal resistance model

A problem of asymptotic stability of a monomolecular reaction system in a catalyst particle is studied using a mathematical model with lumped thermal resistance on the surface of the particle and with intraparticle diffusion of the chemical species. The possible existence of multiple steady states is demonstrated and a necessary and sufficient condition for stability of a steady state is given. A particular case of a general irreversible consecutive reaction system is treated calculations are made for a reaction A 1 → A 2 → A 3 in a spherical particle.

Catalytic particle stability studies—I. Lumped resistance model

The problem of asymptotic stability of a monomolecular reaction system in a single catalyst particle is studied lumped mass and heat transfer resistance at the surface of the particle. The existence of multiple steady states is demonstrated. A linearization technique is used to analyze the asymptotical stability of the steady state and necessary and sufficient conditions for stability are found. A treatment for a particular system of general irreversible consecutive reactions is given. A numerical example of a reaction A 1 → A 2 → A 3 in a spherical particle is used to illustrate the technique developed.

Internal mass and heat transport effects on catalytic behavior—activity, selectivity, and yield in a complex reaction system

The effects of internal mass and heat diffusion on catalytic activity and on the selectivity and yield of intermediate product in a consecutive reaction system are investigated. It is shown that, over a wide range of conditions, the selectivity property is annihilated and there is a net consumption rather than production of intermediate. In other regions, selectivity is greatly decreased and the yields of intermediate are small. For consecutive reactions in which the activation energy of the first reaction is greater than that of the second, however, improved selectivity is possible and the yield of intermediate may surpass that realized in the absence of internal gradients. Expressions for estimation of activity over a large range of thermal effects are given, and a convenient numerical technique for solution of the nonlinear equations involved in description of the system is discussed.

Kinetics of Reactions in the Conversion of Na- or Ca-Saturated Clay to H-Al Clay

AbstractH-ion was added to Na or Ca bentonite suspensions. The H-ion was added either in solution, as HCl, or adsorbed on a cation exchange resin. The variation with time of the quantities of H and Al in various phases of the system was determined.In the first stage of the reaction both exchangeable H and exchangeable Al on the clay increased rapidly. Several facts showed that the exchangeable Al was not liberated by crystal structure decomposition. It is postulated that its appearance is the result of a rapid dehydroxylation, of hydroxy Al groups located at plate edges or adsorbed on basal planes, producing water and trivalent Al ions.During the second stage of the reaction, spontaneous self-decomposition of the clay structure takes place. It was found that when the H-clay in alone in suspension or with low concentration of acid, the decomposition stops after the liberation of a limited quantity of Al. Most of this Al is adsorbed on the clay and it becomes the prevailing exchangeable ion. On the contrary, when the H-clay is aging in the presence of H-resin, a system of consecutive reactions is operating. In this system, Al liberated from the structure is transferred to the adsorbed state in the clay and from there to the resin. As a result the decomposition reaction does not stop as in the absence of the resin, but is carried further.

An application of the discrete maximum principle to a stagewise biochemical reactor system

AbstractIt is a well‐known fact that choosing a sequence of optimal control variables in each reactor tank of a multistage reactor system can result in a considerable increase of the desired product for a consecutive reaction system. The discrete maximum principle is applied to a generalized optimization problem of first order consecutive biochemical reaction system. A detailed numerical solution as an example is also presented for illustration.

A study of consecutive competitive reaction systems

AbstractExperimental kinetic data are most conveniently correlated by the integrated form of the differential rate equations which the reactions are presumed to obey. A new method of obtaining an approximate integral solution of the differential equations is described and applied to a set of three consecutive competitive reactions.The approximate integral solution is used to correlate experimental data on systems whose stoichiometry would indicate a consecutivecompetitive mechanism. The compositions of the reaction mixes are predicted with a standard error of estimate of less than 4% of the original concentration of the initiating reactant, in most cases less than 2%. The estimates of the rate constants, found by fitting the approximate solution to the data, are within experimental error of the values obtained by differentiation of the published results.

Optimum temperature gradients in tubular reactors—I: General theory and methods

In this paper the mathematical techniques necessary for the determination of the optimum temperatures profile in a tubular reactor to insure maximum yields or minimum contact times are developed, and applications are made to reversible and consecutive reaction systems. The problem is shown to be reducible to a system of ordinary non-linear differential equations. The solution of these differential equations can be made by conventional numerical methods, and will allow the specification of the temperatures in the reactor. In a succeeding paper numerical calculations obtained with an analogue computer (REAC) will be presented. The problem of two consecutive reactions A → B → C, in which the reactions are of first or second order, is discussed in detail. The method of attack on more complicated problems is sketched. It is shown in general that appreciable gains in the yield may be obtained if the optimum temperature distribution is used.

The Induction Period in Chain Reactions

The conditions for the establishment of a stationary state in a system of consecutive reactions can be investigated quantitatively in a sufficient number of cases to permit precise criteria to be formulated generally for the length of the induction period (i.e., prestationary state period) in chain reactions. These are given as functions of the individual rate constants for the separate reactions, the extent of the reaction and the percent of stationary state concentrations achieved. Such relations provide a means for the evaluation of the self-consistency of experimental data with proposed mechanisms and also provide numerical limits on the values of individual rate constants. Various chain reactions are examined in the light of these criteria. Of the hydrogen-halogen reactions, the H2–Cl2 reaction mechanism does not fulfill the stationary state requirements, and it is probable that many of the discordant results found in this reaction arise from this difficulty.

The steady state kinetics of some biological systems: I

Steady state kinetic relations previously developed forclosed, homogeneous systems are extended toopen systems consisting of geometrically confined regions of arbitrary shape. The generalized system of consecutive reactions is considered to occur within the cell, and the cell plus environment are treated as an open system. The diffusion condition is imposed upon the kinetic solution for various special cases and the method ofgeneral solution when all products, reactants and intermediates, may enter and leave the cell is indicated.