R. ARlS
CHEMICAL REACTOR A NA LYS IS A MORPHOLOGICAL APPROACH CHfMlSTKY
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USEFUL DESIGN AND PREOlCTlON
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ui MODEL OF THE REACT01
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MATHEMATICS
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I h twractiom of cngincm'ng kmwledgc with bm'c sl;entifc infarn4tion in the comhlCIion of Q rcactor modal. Comparing fhmodel wirn reality indicaks fh8 inadaqua& and pcrmifs improununrs to be ma& in ih modal
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INDUSTRIAL AND E N G I N E E R I N G CHEMISTRY
The great interest in chemical reactor analysis, design, and simulation has generated a vast literature, much of it highly specialized and closely related to the parent sciences of chemistry, physics and thermodynamics. To aid in establishing the numerous relations between the parent sciences and the particulars of reactor technology, a general .description of the structure of reactor analysis is of prime importance.
of chemical engineering is more central to the purposes of the craft than the study of the chemical reactor, yet as a coherent disciplie reactor analysis is rather new. This is in some measure due to the fact that the discipline rests on an understanding of physical and mechanical processes which have preempted the chemical engineer’s attention. Perhaps it is also because the subject makes larger demands on the engineer’s understanding of physics, chemistry, and mathematics than could be met before the rise of so-called chemical engineering science. Whatever the reason, the last few years, and even months, have seen a rapid increase of interest in the subject and it is currently compassed by all the perils of popularity. It may be appropriate, therefore, to review the subject by proposing a morphology for it, rather than by attempting to cover its history and literature, for already this is too vast to be suitable for a single review. A morphology is offered here in the hope of promoting, or even provoking, constructive discussion and not in any sense as being the only possible form in which it should be cast. We are concerned with the theoretical aspects of the subject and with what may be learned from practice, rather than with its direct application in the details of design methods,
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for the latter can only hirfully grow h m the basic understanding. As always, the practical has led the theoretical in achievement, but the chemical engineer, unless he is content to wander in a grove of particularities, must attempt to see the overall picture; the teacher -in popular parlance, the “educator”-would be irresponsible indeed to focus first on anything else. Let us 6rst block off the discipline into rather larger areas to see the interaction of engineering knowledge with basic science and mathematics. As in all theories we are making a model in symbolic language of the reality being studied, in this case the reactor, and a model can never be wholly adequate. To be any good, a theory must discover its own inadequacies, so that the model is not an end in itself but a stage in the development of a more complete picture. This does not mean that it is valueless until perfected, for at each stage uscful products can be drawn off. A steady state model, e.g., is useless for control, but pedectly adequate for siziig equipment. Figure 1 suggests this recycle scheme. The fact that a reaction is taking place distinguishes a reactor from other parts of a chemical plant and hence an expression for the reaction rate is an essential part of the mathematical model. This involves the understanding of physical and chemical rate processes in a way that we must examine later in more detail. To the formulation of the reactor model must be brought, in addition, the physical principles of conservation and the engineering design. Of critical importance here is the engineering judgment of what is significant and what may be safely neglected. While only a modest amount of mathematics may be involved up to this point, solving the rate equations to provide a u&l evaluation of the reactor will certainly demand a much fuller use of mathematics. From the comparison of the behavior of the model with reality will come an application of its inadequacies and the d m t i o n &which improvement is needed will become clear. Figure 2 attempts to show in a little more detail what may enter into the construction of the reaction rate VOL S b
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of physics and chemistry. Classical thermodynamics is as much part of one as of the other and contributes the fundamental understanding of chemical equilibrium and its dependence on physical conditions. Irreversible thermodynamics has much to say to the steady state (in contrast to the equilibrium state) and to the interaction of chemical and physical rate processes. Within the two disciplines we have a parallel series of subjects contributing to the development of the reaction rate expression. On the chemical side the “kinematics” of chemical change is provided by stoichiometry, for it is within the framework of its principles that changes of composition may consistently take place. Here, perhaps, it may be appropriate to make a plea for a suitably algebraic notation in discussingreactions. The notation &A ,, = 0, where the summation is over all species present and for convenience v j is positive for products Of the reaction, lends itself to a proper understanding of the algebraic structure of stoichiometry; the notation VJ vBB . . e v p P uqQ . . does not. It can be generalized easily to simultaneousreactions and opens the way to the concepts of linear vector spaces which are natural to thii representation. The “dynamics” of chemical change comes from the study of chemical kinetics, which relates the reaction to the structure of the molecule and to the steps by which the reaction appears
I F CONSERVATI PRINCIPLE
MEOW OF DIFFUSION HEN CONDUblON
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F i g w 2. ConrmrCtin of zh readon rate expression MUUIHtU RtAlllUll KATE
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PHRICAL AND CHEMICAL RATE PROCESSES
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INDUSTRIAL AND ENGINEERING C H E M I S T R Y
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to take place. In many cases the painful elucidation of the mechanism of the reaction, so necessary to a real understanding, is not available to the chemical engineer, either because it has yet to be studied by the kineticist or because there is no time to attempt a study in depth. He is, therefore, forced back on an overall description of the reaction system and on empirical or semiempirical rate laws gleaned from experimental data. These are usually entirely adequate provided the subsequent use of them does not involve great extrapolation beyond the observations, but even in the absence of the full kinetic picture certain important notions need to be borne in mind. For example, when a full description of the mechanism of a reaction is available, the overall reaction is split up into number of steps each of which represents a definite molecular event and to which a definite rate can be assigned. These steps will not be linearly independent in general and the reaction which they describe is some h e a r combination of them. In combining the rates of the elementary steps into a single rate expression involving only the concentrations of observable species entering the overall reaction, a number of hypotheses, such as extreme rapidity of the one step or the constancy of the concentrations of an intermediate, may be used. If several reactions take place simultaneously, it is certainly desirable to work with the smallest number of linearly independent reactions. Any nonsingular,
linear combination of an independent set is stoic metrically equivalent, but, unless these happen to represent elementary steps, the reaction rate is not unique and the same reaction in an equivalent set of reactions may have a different rate. This chemical avenue to the primary reaction rate expression may suffice for homogeneous reactions, but where heterogeneous catalysis or two phases are involved physical rate process may have to be considered. The physical counterparts of stoichiometry are the principles of conservation of mass and energy. The theory of diffusion, heat conduction, and physical adsorption parallels the chemical kinetics by defmiig the rates at which the physical processes take place. Again there are many processes of such complexity as to defy precise mathematical analpis and here the semiempirical correlations that have been the particular province of the engineer come into play. In the craft of combining the physical and chemical rate processes there is again a great need for skied engineering judgment as to which are the dominant modes of action. A notable example of the way in which very complicated interactions can be wrapped up in a useful approximation is the introduction of an effectiveness factor to account for the limitation of reaction rate by diffusion within a porous catalyst particle. Here, too, is scope for considerable mathematical virtuosity in the analysis of systems of nonlinear
ECONOMICS
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FLUID MECHANICS
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iNTllATlYE STUDIES
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equations. It is often mistakenly thought that the digital computer can be relied on to grind out a numerical solution of virtually any problem. In point of fact, if it is presented with an ill-defined problem, the modern computer is an incomparable producer of expcnsive waste paper. If all these resources have been drawn upon to yield an economical and accurate representation of the reaction rate, we must now ask how this can be incorporated into the reaction model. In Figure 3 some of the types of reactors are listed to which we should advert before proceeding further. The tubular reactor requires a set of ordinary differential equations for its simplest model, one for each independent reaction and one for the temperature (and for the pressure if this is an important variable). As soon as this model is sophisticated various difficulties appear. With longitudinal diffusion present, the equations are numerically unstable and must be treated with a little more care; when variations across the section of the reactor are considered we immediately have partial differential equations with two or three independent variables. The transient model is a set of partial differential equations and though in the simplest case these can be transformed to ordinary equations, in all other cases we are involved in nonlinear partial differential equations from the outset. The stability analysis of partial equations is in its infancy compared with that of ordinary equations and many general theorems that one would like to call on are not yet provided by the mathematician. The fixed bed catalytic reactor has the added complication that equations may have to be written both for the flowing reactant stream and for the processes that take place within the catalyst particle. Here a new class of stability questions opens up, for the particle is often capable of multiple steady states under the same ambient conditions. The moving bed equations are only a slight extension of those of the fixed bed, but in countercurrent cases many involve two point boundary conditions. In the fluid bed there is a random element that makes the analysis both easier and more difficult. In one class of model the fluid bed is treated much in the same way as a stirred tank and this makes for simplicity. However, if this is not done, the resulting stochastic differential equations are very difficult to handle. The hydrodynamics of the fluid bed and such things as the presence of bubbles are here of the first importance. Much interest centers too on the poisoning and aging of the catalyst and the reactor must often be considered in conjunction with the regenerator. I t is not surprising that the stirred tank reactor has received the most attention for its steady state equations are algebraic or trancendental and the transient model is only a set of ordinary differential equations. A fairly
R. Aris is Projessor of Chemical Engineering University of A4innesota.
AUTHOR
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complete theory of the behavior of the stirred tank \vitli. simple reaction systems is thcrefore available and current research is directed to more complex reactions, such a s polymerization, and to two phase systems. The theory of the stability and control of the stirred tank reactor is quite well developed, but the ineluctable nonlinearity of temperature poses some severe problems in the application of some techniques; for example, Lyapunov’s direct method. The large volume requirements of a stirred tank make optimal design of some importance and thc sequence of stirred tanks is amenable to full analysis. The batch reactor leads to a set of ordinary differential equations for the extents of the several reactions and the thermodynamic variables. Whereas the optimal. temperature profile in a tubular reactor is virtually unrealizable the optimal temperature control in a batch reactor is a much more practical possibility and here the recent mathematical advances in control theory find immediate application. Let us now return to Figure 3 which is really a recycle process of the same sort as the whole endeavor. In setting up adequate models for the reactor description mass, energy and inoinentum balances must be foriiied in accordance \lit11 physical conservation laws. I n certain cases a knowledge of fluid mechanics is brought into play. Besides being classified as to type, reactors may also be classified as to mode of operation and principally as to the manner of temperaturc control. For each type of design and mode of operation a set of equations for the steady state and a model for the transient behavior may be obtained, the steady state equations being a specialization of the transient model by setting the time derivatives equal to zero. The exception to this is the batch reactor which by the nature of things works always in a transient condition. However, here there is a direct analogy between the batch reactor in time and the tubular reactor in space. From the steady state equations design procedures, fixing the size and operation of the reactor, can immediately be obtainccl. Design leads naturally to the consideration of optimal design and here the economics of the process, which can never be put far into the background, are dominant. Although the optimal design of the reactor in itself is now fairly well understood, a true optimal design of the reactor within the total context of the plant is still far from being an easy or routine matter. Even though it ma)’ he impossible to achieve in practice, the optimal design has the merit of being an objective standard, and more practical designs may be compared with the optimal instead of with themselves-which is “not wise.“ The feasibility of a steady state design, however, depends on a rather deeper consideration of the transient behavior. By careful examination of the steady state for various conditions of operation it becomes fairly clear when such a state is liable to be unstable, but precise conditions for stability must involve an analysis of the transient equations. An example of this is the early discussion of the autothermicity of a stirred tank by van Heerden. Using a balance of the heat generation and removal rates (a form of argument that has proved to be
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a basic tool in all subsequent work) it can be seen that, where three steady states are possible, the intermediate one can never be approached by a gradual change of conditions, such as bringing the feed temperature slowly up to the prescribed value. This is strong ground for presuming its instability and leads to a necessary criterion for stability, namely that at the steady state the slope of the heat removal curve should be greater than that of the heat generation curve. However it was not until Amundson applied the methods of nonlinear mechanics to the problem that the necessary and sufficient criteria for stability were obtained and the pattern of transient behavior became fully clear. The stability of the stirred tank reactor is now fairly well understood, but the stability of tubular reactors is still a field of current research and may well be so for some time. Similarly the sensitivity of a design to changes in parameters can be studied with the steady state equations, but is so closely related to the time dependent behavior as to be only half understood from this angle. For the discussion of control the transient model is needed from the outset and here again the optimal control is worth investigating. The rise of the high speed computer has opened up the possibility of an adaptive control, always striving for the ‘optimum and compensating to some degree for the inadequacy of the model. The range of inadequacies of the simpler models is quite wide, and perhaps one of the most significant is the spectrum of uncertainty as to the state of mixing. At one extreme the tubular reactor may be regarded as in plug flow with no longitudinal diffusion, so that all elements have the same residence time. Allowing for the superposition of a longitudinal flow introduces a slight spread in the probability distribution of residence time and is comparable to a sequence of a large number of small stirred tanks. The extreme case is the deliberately cultivated mixing of the stirred tank but even here the mixing may be imperfect in a variety of ways. There may be relatively stagnant back waters within the reactor or there may be a state of segregation such that, although the probability of residence time is the same for all molecules, those molecules near to one another tend to be of the same age. For anything but first order reactions these effects are not completely described by the residence time distribution alone. The interaction of the flow profile and diffusion across it may be represented by an effective longitudinal dispersion. This is an interesting example of the interchangeability of models, as is also the relation between the tubular reactor with diffusion and the stirred tank sequence. The scale of turbulence and its interaction with chemical reaction is also of great importance and more than a little difficult to represent adequately. In control studies it is often necessary to start by assuming an unusual degree of perfection in the controller and to come back later and examine the effect of imperfections. Here there are falltraps for the unwary, as, for example, when the criteria for controllability in the limit of zero time lag are not the same as those obtained when the time lag is set equal to zero from the first. In two
phase systems the breakup and coalescence of drops plays an important role, but it is extraordinarily difficult to represent this at all completely and models assuming one aspect to be rate determining are often used. Two features save the day for the reactor analyst. One is that it is often possible to estimate the effect of some process whose full analysis is ineluctable and either to dismiss it as insignificant or to make a safe, if perhaps clumsy, correction for it. The other is the happy principle that a sound quantitative study of an unfortunately oversimplified model may yet give valuable qualitative information about the real process. It may not be possible to predict the exact changes that will take place, but the indicated trends can be of the greatest value. Estimated corrections may be fed back into the model to make its predictions more adequate without improving the model in essence or, better yet, the model may be refined to give a still deeper understanding of the reactor. If this is a valid description of the morphology of chemical reactor analysis, it is small wonder that it holds the attention of a number of researchers. For there is, on every hand, scope for the use of the most penetrating tools available from the scientific handmaidens of engineering. Above all, there is an intellectual structure: not the structure of a lifeless skeleton, but of an evolving organism. This is worthy of the curiosity of any intelligent human being, which, it is to be hoped, the chemical engineer will be found to be. BIBLIOGRAPHY As explained preuiously, no uttempt has been made t~ reuiew literature o/ chcmical reactor analysis, However. the following chronological list of some of the nvuilable and forthcoming books and major reuiew articles may be of ualuc. 1. Hougen, 0. A,, Watqon, K. M., “Chemical Process Principles,” Vol. 3, John Wiley and Sons, Inc., New York, 1947. 2. Hougen 0. A. “Reaction Kinetics in Chemical Engineering,” Chem. Eng. Prog. Monogrndh Ser., k 7 , 1951, 3. Smith, J. M., “Chemical Engineering Kinetics,” McGraw-Hill Book Co., Inc., New York, 1956. 4. Bri)tz, W., “Grundriss der Chemischen Reaktionstechnik,” Verlag Chemie, Weinheim, 1958. 5. Dialer, K., Horn, F., Kiichler, L., “Chemische Reaktions Technik,” in “Chemische Technologie,’’ Vol. 1, Carl Hanser Verlag, Munchen, 1958. 6. Jungers, J. C., “CinCtique Chimique Appliqude,” Technip, Paris, 1 9 5 8 . 7 . Rietema, K., ed., “Chemical Reaction Engineering,” Pergamon Press, London, 1958. (See a l s o : Chcm. Eng. Sci., Vol. 14; Proc. of 2nd European Symposium o n Chemical Reaction Engineering, 1960; Proc. of 3rd European Symposium on Chemical Reaction Engineering to be held in Sept. 1964) 8. Walas, S.M.,“Reaction Kinetics for Chemical Engineers,” McGraw-HillBook 1 Co., Inc., New York, 1959. 9. Bognar, E., de Bernardi, X.,Gardy, H., “Reacteurs Chimiques Industrielles,” in “Precis de Genie Chimique,” Vol. 2, Berger-Levrault, Nancy, 1961, 10. Aris, R., “The Optimal Design of Chemical Reactors,” Academic Press, New York, 1961. 11. Beek, J., “Design of Packed Catalytic Reactors,” in “Advances in Chemical Engineering,” Vol. 3, Academic Press, New York, 1962. 12. Wilhelm, R. H., “Progress Toward the a Priori Design of Chemical Reactors,” Pure and Appl. Chem. 5 , 403, 1962. 13. Levenspiel, O., “Chemical Reaction Engineering,” John Wiley and Sons, Inc., New York, 1963. 14. Kramers, H., Westerterp, K . R., “Elements of Chemical Reactor Design and Operation,” Academic Press, New York, 1963. 15. Davidson, J. F., Harrison, D., “Fluidized Particles,” Cambridge Univ. Press, Cambridge, 1963. 16. Satterfield, C. N., Sherwood, T. K., “The Role of Diffusion in Catalysis,” Addison-Wesley Pub. Co., Reading, 1963. 17. Levenspiel, O., Bischoff, K . B., “Patterns of Flow in Chemical Process Vessels,” in “Advances in Chemical Engineering,” Vol. 4, Academic Press, New York, 1963. 18. Denbigh, K. G., “Chemical Reactor Theory,” Cambridge Univ. Press, Cambridge, to appear in 1964. 19. Boudart M. “The Kinetics of Chemical Processes,’’ Prentice-Hall, Englewood Cliffs, to appe)ar 1964/65. 20. Petersen E. E. “Chemical Reaction Analysis,” Prentice-Hall, Englewood Cliffs, to a’ppear 1\64/65, 21. Aris R. “An Introduction to the Analysis of Chemical Reactors,” PrenticrHall, Engiewood Cliffs, to appear in 1964/65.
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