I n d . Eng. Chem. Res. 1987,26, 1803-1810
chlorohydrin, 107-07-3; ethylene dichloride, 107-06-2.
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Hikita, H.; Ishikawa, H. Bull. Uniu. Osaka Pref. 1969, A18, 427. Juvekar, V. A. Chem. Eng. Sci. 1974,29, 1842. Leveque, M. Ann. Mines 1928, 13, 201. Morrison, T. J.; Billett, H. J . Chem. SOC.1948, 2033. Neidleman, S. L. Hydrocarbon Process. 1980, 59(11), 135. Onda, K.; Sada, E.: Kobayashi, T.; Kito, S.: Ito, K. J . Chem. Eng. Jpn. 1970, 3, 137. Peaceman. D. W. Sc. D. Thesis. Massachusetts Institute of Technology, Cambridge, 1951. ’ Ramachandran, P. A.; Sharma, M. M. Trans. Znst. Chem. Eng. 1971, 49, 253. Ratcliff, G. A.; Holdcroft, J. G. Trans. Znst. Chem. Eng. 1963, 41, 315. Roberts, I.; Kimball, G. E. J. Am. Chem. Sac. 1937, 59, 947. Roper, G. H.; Hatch, T. F.; Pigford, R. L. Ind. Eng. Chem. Fundam. 1962, 1, 144. Shilov, E. A. J . Appl. Chem. USSR (Engl. Transl.) 1949, 22, 734. Teramoto, M.; Isoda, T.; Hashimoto, K.; Nagata, S. Kagahu Kogaku 1971, 35, 897. Unver, A. A.; Himmelblau, D. M. J . Chem. Eng. Data 1964,9, 428. Van Krevelen, D. W.; Hoftijzer, P. J. Chem. Ind. Congr., Int. Chin. Znd., 21, 1948, 168.
Received for review February 10, 1986 Accepted June 16, 1987
A Geometric Approach to Steady Flow Reactors: The Attainable Region and Optimization in Concentration Space David Glasser* and Diane Hildebrandt Department of Chemical Engineering, University of the Witwatersrand, Johannesburg, W I T S 2050 South Africa
Cameron Crowe Department of Chemical Engineering, McMaster University, Hamilton L8S 4L7, Canada
This paper examines the following problem: for a given system of reactions with given reaction kinetics, find all possible concentrations t h a t can be achieved by using any system of steady-flow chemical reactors, that is, by using the processes of mixing and reaction only. Only isothermal systems with no volume change on reaction or mixing are examined in this paper. A geometric approach is adopted, and a set of necessary conditions is derived. In particular, the attainable region must be convex with non-zero rate vectors on the boundaries not pointing outward. Using the results, one can construct a region which satisfies the necessary conditions. Furthermore, it is demonstrated that no possible combination of plug-flow, CSTR, or recycle reactors can be used to extend this region. Once this region is known, the solution of concentration optimization problems is shown t o be relatively straightforward. A number of two-dimensional examples are examined. The problem of deciding on the best steady-flow system of chemical reactors given a set of reactions with their kinetics is an old and a difficult one. Horn (1964) showed that if one could find the attainable region for the system, that is, the region in the stochiometric subspace which could be reached by any possible reactor system, then the problem of the optimization was essentially solved. We will examine this point at a later stage. Other authors such as Chitra and Govind (1985) and Paynter and Haskins (1970) have tried simpler approaches. The former asking what series of recycle reactors and the latter asking what value of the axial mixing coefficient as a function of length would give the optimum answer for the chosen objective function. The former authors also give an extensive summary of the previous work done in optimization. A more recent article by Achenie and Biegler (1986) has a general structure consisting of constant dispersion model reactors 0888-5885/87/2626-1803$01.50/0
with sinks and sources and splitting points. The results are optimal for the class of problems consistent with the initial structure chosen and if the optimal reactor system can be described by networks of constant dispersion model reactors. When we use the terminology a steady-flow reactor system, we imply that the system will not support sustained oscillations, that is, that the flows and concentrations throughout the system are at steady-state values. The whole question of chemical oscillations and multiple equilibria has been discussed in a review by Feinberg (1980), and the techniques outlined by him could be used to examine the reaction networks to see if these difficulties are likely to arise in the systems under consideration. Shinnar (1983) and Shinnar and Feng (1985) have looked at a similar problem. They have shown that the requirements of thermodynamics place “bounds on the 1987 American Chemical Society
1804 Ind. Eng. Chem. Res., Vol. 26, No. 9, 1987
reachable composition space”. This is then just an attainable region which is subject to the constraints that the free energy for each of the kinetically independent reactions in the given reaction scheme must be nonincreasing. They also use a geometric approach, and with a given feed they are able to define a region in the concentration space which is attainable subject to these thermodynamic constraints. One might refer to this as t h e “thermodynamically attainable region”, and this must contain within it the “kinetically attainable region” which this paper examines. This is so since the former must be the region for the given reaction scheme which is attainable for all possible thermodynamically consistent kinetics. If one does not have the kinetics for such a system, then clearly the former is the best bound one can obtain. If one has the kinetics as well, then the former is of course an unnecessarily loose bound and the latter is more useful. In this paper, we will only deal with the kinetically attainable region, and we will henceforth refer to this as the attainable region. In order to solve the latter problem in its entirety, one has to have a clear understanding of what constitutes all possible reactor systems. In essence, one means that the only processes that one allows in the reactor system are mixing and chemical reaction, and for our purposes, we will also assume that the initial feed and its flow rate are given. A general model for such a reactor system which covers all these cases has recently been presented and is the General Mixing Model of Glasser and Jackson (1985,1986). In deriving this model, it was assumed that there is no volume change on reaction and mixing. With this assumption, it was shown that all steady-flow reactors can be characterized by a residence time distribution and a micromixing function. Given the model, the system of reactions, the kinetics, and the objective function, one can indeed perform the required variational optimization. However, because of the complexity of the model and the subsequent difficulty of the analysis using the calculus of variations, it has not as yet proved to be possible to obtain any useful results. In this paper an entirely different approach will be taken, using essentially a geometric approach. This will mirror Horn’s (1964) original work except that instead of using optimization techniques to obtain the boundary of the attainable region, geometric techniques will be used throughout. Geometry of Reaction and Mixing Let us begin by assuming that we are given a set of reactions with known kinetics such that if we have a concentration vector c , we can write the rate of formation of each species as the vector r ( c ) . In particular, we will assume that our reactor system is isothermal, that there are no volume changes on mixing or reaction, and that we are given a feed with known composition and flow rate. We will also assume that the reaction network, the kinetics, and the reactor systems are of such a nature that we do not have sustained oscillations. As mentioned before, we might be able to use the methods described by Feinberg (1980) to exclude these possibilities. The overall mass balance on the system and the requirement of the positivity of the concentrations will constrain them to remain within a region of the nonnegative orthant of the concentration space. Because the rates of reaction need not be linearly independent, the attainable region will be constrained to lie in a restricted region of this space which is called the stoichiometric subspace (called the stoichiometric compatibility class by Feinberg (1980)). It may be convenient to work with ex-
,, , , /,,,, ,, , ,,,, -_-_~---~-, , ,,,,,,,,,-,, ,,/ /,,-.-----,; ,, ,~ ,, ~-~ - - , , , , , ,,_ , ,, -~ , ; / / / ,,----,, ,, ,, ,,,,,~~ _--- --,,---.-.----_
~-
/,,,---------
I
,
l l
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l l
/
l
, ,
~
___-. ---.-..- -.-.-~ -.---.----.--_---. -.-. .--.
,,___.-
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~
,,,~-_-.-~--I-
i
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1’
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Figure 1. Rate vector and mixing in concentration space.
tents of reactions and linearly independent reactions rather than concentrations, in which case we will be working in the subspace itself. This latter approach will be taken where it proves to be convenient. The reaction kinetics that we are given will further ensure that we will not be able to attain all possible points in this subspace. Thus, for example, for the reaction k
k
A-B-C with first-order kinetics with equal rate constants, we can only achieve a maximum value of B relative to the feed concentration of pure A of 0.368, while the maximum value defining the boundary of the subspace is unity. If the optimization is of the form that the objective function is only a function of concentrations and not the volume of the reactor system as well, then we can limit ourselves to looking only at the concentration space (or its stoichiometric subspace). We will for the purposes of this paper restrict ourselves to this situation. If we have values of all the concentrations, we can calculate the reaction rates so that we can assign to each point in the concentration space a vector which is the rate vector. This vector will have a direction and, when multiplied by a time, a length. A two-dimensional example showing the directions of the rate vector in the stoichiometric subspace is illustrated in Figure 1. Thus, at any point in the space, we know that, locally, reaction will move the concentration vector in the direction of the rate vector. The only other operation that we are allowed to perform is the operation of mixing. When we mix two streams with composition c and cotrespectively, it is clear that for constant density, the resultant must lie on the straight line between c and co; that is, we have the point c * given by c* = ac
+ (1 - a)cO
0 Ia I1
(1)
This is usually referred to as the lever-arm rule and is shown in Figure 1,the point c * being on the mixing vector which is given by c - co. Let us consider a situation in which we have a combination of mixing and reaction. The resultant must be the appropriately weighted vector sum of the rate vector and the mixing vector and thus must always lie in the angle (which is less than 180O) between them. Thus, locally the operations of mixihg and reaction are such that if we are at a point c and we are mixing in material of concentration c oand allowing reaction to take place simultaneously, the net direction will be in the convex combination of r ( c )and c - cos
The important conclusion from this is that locally no combination of mixing and reaction can take us in the direction that is not between the two vectors, that is, g = ar(c)+ (1 - a)(c- co)
where g is the resultant direction.
0 -
