Neal R. Amundson

An analysis of chemical reactor stability and control—I: The possibility of local control, with perfect or imperfect control mechanisms

Abstract To design rationally a chemical reactor with its associated control equipment, it is necessary to study the transient behaviour of the system. In this series of papers a well-agitated continuous reactor either of the liquid or fluidized variety is considered under various types of control and criteria are derived enabling one to determine under what conditions different modes of control are successful. This first part, after an introduction to the whole subject, introduces the method of linear approximation and goes on to discuss the possibility of controlling a naturally unstable steady state by various systems of perfect and imperfect control.

A study of the shock layer in nonequilibrium exchange systems

Abstract Shock layer theory is extended to the nonequilibrium exchange system with axial dispersion. A moving coordinate scheme enables us to establish the existence condition and the uniqueness of the shock layer. In addition to the strategy for the numerical determination of the shock layer, an approximate solution is discussed which neglects the couple effect of the dispersion and the mass transfer resistance. This solution provides a basis for the quantitative analysis of the shock layer thickness and the departure from equilibrium. Applications are illustrated by using the Langmuir isotherm and a numerical solution of the transient equation confirms the usefulness of the shock layer theory.

Studies in the control of tubular reactors-I General Considerations

Abstract In this part the dynamical model of a jacketed non-adiabatic tubular reactor is developed and the effect of the wall heat capacity is briefly examined. The distributed stability and feed-back control problems are defined and the method of orthogonal collocations is used to obtain a discretized model. The design of the cooling section around the reactor is examined and the number of collocation points and their location is determined for an accurate approximation of the distributed model.

An analysis of chemical reactor stability and control—II: The evolution of proportional control

Abstract In this part it is assumed that the control system receives signals of the deviation of reactor temperature and uses them to give a perfect proportional control of the rate of cooling. It has already been shown that control cannot be attained until the proportionality constant is great enough but it appears here that even when this minimum value has been exceeded, the control may only extend to comparatively small disturbances. To see how this comes about, it is necessary to follow the behaviour of the reactor through the variation of the control parameter, and it is shown that the methods of non-linear mechanics make this possible without excessive labour. There comes a point, however, when some computation is necessary and this is exemplified in a complete description of a simple system.

Uniqueness of the steady state solutions for chemical reactor occurring in a catalyst particle or in a tubular reaction with axial diffusion

Abstract When chemical reactions occur within a catalyst particle, multiple steady states may occur. Sufficient conditions for the existence of a unique steady state for catalyst particles of arbitrary shape are obtained from an eigenvalue problem. However, general criteria for uniqueness can be obtained without solving for the eigenvalues or performing other computations. Under certain conditions, a unique steady state exists for catalyst particles of arbitrary geometry. A proof is given that for any reaction mixture the possibility of multiple steady states can usually be eliminated by use of small catalyst particles, or a dilute mixture. A simple criterion to estimate the particle size below which only one steady state is possible is derived for the case of a single reaction. Comparison between the approximated bound and eight examples in the literature shows that the agreement is satisfactory. The same technique can be applied to the case of a chemical reaction occuring in a tubular reactor with axial diffusion. It is shown that in this case one can usually guarantee a unique steady state by use of a dilute mixture or a short reactor.

Shock layer in two solute chromatography: effect of axial dispersion and mass transfer

Abstract A theoretical study for two solute chromatography when axial dispersion or interphase mass transfer is significant is presented. With a Langmuir isotherm the mathematical theory of the shock layer plays a key role in generating asymptotic solutions, from which one can deduce the effect of axial dispersion or mass transfer resistance. A full discussion is given for the existence and uniqueness of the asymptotic solution. If a shock layer exists, the end states satisfy the compatibility condition and it propagates at the same speed as the corresponding shock. Analytic expressions are developed for the case of equal Peclet numbers (or equal Stanton numbers) whereas for other cases the shock layer profiles are determined numerically. Comparison between transient solutions and shock layers demonstrates the validity and usefulness of the present study.

Intraparticle diffusion and conduction in porous catalysts—II: Complex reactions

Abstract The problem of intraparticle diffusion and conduction is considered for complex reactions taking place in porous catalytic spheres. It is assumed that Knudsen diffusion prevails and that heat generated in the reaction is conducted through the porous structure to the catalyst surface. Equations are written for an arbitrary number of reactions and criteria are developed to determine the number of conservation equations for complex systems. A particular numerical example is worked both approximately and exactly by the same technique as used in the previous paper; the term exactly meaning that partial pressure and temperature profiles may be calculated numerically as accurately as desired. The approximate method should be satisfactory for almost all industrial work.

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

Abstract 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.

Continuous polymerization models—I: Polymerization in continuous stirred tank reactors

Abstract This paper involves an analytical study of polymerization kinetics in a stirred tank reactor. The technique used considers the polymer chain length as a continuous variable and reduces the infinite set of algebraic equations for the reactor to a finite set of differential equations. The solutions to the equations are presented for a variety of linear polymerization mechanisms. Non-linear polymerization and block polymerization are also investigated by the same technique. The feasibility of the technique is studied by extensive computation and some parametric studies are made.

Transport in composite materials: Reduction to a self adjoint formalism

Abstract A general formalism is presented for the solution of boundary value problems arising in the description of transport of mass or energy through multiphase media, which consists in identifying physical conditions of interfaces with criteria for self-adjointness of the operators concerned. Several examples have been presented which demonstrate the advantage of the formalism.