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Kinetic Bounds on Attainability in the Reactor Synthesis Problem Thomas K. Abraham and Martin Feinberg* Department of Chemical Engineering, The Ohio State University, 140 West 19th Avenue, Columbus, Ohio 43210
Given a chemical reaction network (with kinetics) and given a specified feed, we ask which product compositions are attainable from the feed by means of processes that employ only reaction and mixing. Although there has been considerable attention to this question recently, much of the “attainable region” research has focused on building the set of realizable compositions “from the inside”, that is, by attempting to extend a set of already realized compositions with further reaction and mixing. Here we seek to bound the set of attainable compositions “from the outside” by construction of a bounding polygon in composition space, within which all attainable compositions must reside. The polygon construction proceeds by what we call the method of bounding hyperplanes. When the number of hyperplanes employed becomes large, the resulting polygonal bound can, at least in some instances, approximate the actual attainable region sharply. 1. Introduction: Seeking Outer Bounds on the Attainable Region In 1964, Horn1 showed how questions of optimal process design, in particular of optimal reactor design, could be approached geometrically through consideration of the attainable region. For a prescribed chemistry, for a prescribed set of feed streams, and for a prescribed set of process constraints, the attainable region is the entire set of output states (in some suitable state space) that might be realized by all possible constraint-consistent designs. In this sense, the attainable region gives information about the full range of process outcomes that the designer might hope to attain. Knowledge of the attainable region then provides a basis for assessing the relative merits of various design candidates. If, for example, a particular design is of reasonable cost and it is known that its yield is already close to the most desirable yield that might be achieved in any constraint-consistent design, then it is probably unwise to invest effort in changing that design in substantial ways. If, on the other hand, the design is an expensive one and it is known that its yield falls very short of the best attainable yield, then serious reconsideration of the design is almost certainly warranted. In any case, to know the best possible outcome is to have some sense of what is attainable and what is not. At least in part, this was Horn’s motivation for framing design questions from the attainable region viewpoint. Horn’s geometric approach to design received little attention just after its initial advocacy. In the late 1980s, however, Glasser and co-workers2-4 initiated a substantial revival of interest in attainable region ideas, and by the late 1990s, there was already the beginning of an upsurge in attainable region research. Much of the focus was on reactor synthesis, but some of the research was also directed toward the study of reactorseparator systems. Our focus here is on the pure reactor synthesis problem. That is, our interest is in describing the set of attainable outcomes for systems employing reaction and mixing but no separators. (For very general results on * To whom correspondence should be addressed. Tel.: 614-688-4883. Fax: 614-292-3769. E-mail: feinberg@ che.eng.ohio-state.edu.
attainability in reactor-separator systems, see work by Feinberg and Ellison.5) Discussion here will be limited to steady-state designs involving mixtures that, to a good approximation, have fixed density, independent of composition. For the sake of simplicity in the presentation, we shall, for the most part, also restrict our attention to isothermal systems, but it will not be difficult to see how the methods we describe can be extended to consideration of nonisothermal designs as well (see remark 4.3). Much of the emphasis in the steady-state reactor synthesis problem has been on attempts to build the attainable region “from the inside”. That is, beginning with the feed stream, one tries to extend the set of composition states that can be attained by bringing into play new reactors fed with streams having compositions that have already been attained. The idea then is to proceed in this way until it appears that no additional compositions can be realized (see, for example, refs 2-4, 6, and 7). A well-studied example, which originated with van de Vusse,8 has as its underlying chemistry the massaction system shown in eq 1. 1
1
A1 98 A2 98 A4 10
2A1 98 A3
(1)
In the feed stream, the composition is presumed to be described by the vector of molar concentrations cf ) [1, 0, 0, 0] (in units consistent with those of the rate constants). Suppose, for the moment, that our concern is only with the output concentrations of species A2 and unconverted A1, without regard to the relative amounts of A3 and A4 produced as byproducts. From the attainable region viewpoint, it suffices in this case to consider the set of all [c1, c2] states that can be realized in steadystate designs that begin with the specified feed composition. In Figure 1, we show what is generally regarded to be the (two-dimensional) attainable region of [c1, c2] states for the van de Vusse example. (Similar figures appear much earlier. See, for example, work by Hildebrandt and Glasser.4) The dashed curve represents the locus of compositions that can be realized in a continuous-flow stirred tank reactor (CFSTR) with feed cf )
10.1021/ie030497w CCC: $27.50 © 2004 American Chemical Society Published on Web 11/27/2003
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Figure 2. Some attainable compositions for the three-dimensional van de Vusse problem. Figure 1. Putative attainable region for the two-dimensional van de Vusse example.
[1, 0, 0, 0] as the residence time increases from zero to infinity. The solid curve PO represents the locus of compositions that can be realized along a plug-flow reactor (PFR), with the feed having the composition shown at point P (c1 ) 0.2236, c2 ) 0.0868). Thus, each composition along the solid curve PO can be realized by means of a two-reactor series that begins with a feed stream of composition cf: The first reactor in the series is a CFSTR that has as its effluent a stream of composition P, and the second is a PFR having as its feed the exit stream from the CFSTR. (The CFSTR residence time required to attain composition P is 0.6345.) Note that each composition along the straight line FP is attainable by mixing, in suitable proportions, a stream of composition cf and a CFSTR effluent stream of composition P. In fact, each composition in the convex region FPO, bounded from above by the straight line FP and the solid PFR trajectory PO, is attainable by mixing a stream of composition cf with either the effluent of a CFSTR having a feed of composition cf or the effluent of a two-reactor CFSTR-PFR series having a feed of composition cf. (Compositions along the line FO are attainable in the limit.) It has been generally presumed that the convex region FPO shown in Figure 1 is, in fact, the full set of [c1, c2] states attainable for the van de Vusse chemistry from the feed composition cf ) [1, 0, 0, 0]. Note that, in this case, compositions on the boundary can be realized by bringing into play only two archetypical reactor types, the CFSTR and the PFR, and one can actually specify in detail the designs that realize those compositions. The situation gets substantially more complicated in higher dimensions, even for the simple van de Vusse chemistry shown in eq 1, again with a feed stream of composition cf ) [1, 0, 0, 0]. Suppose, for example, that the effluent concentrations of A2 and A3 (as well as the concentration of unconverted A1) are all of interest. From the attainable region perspective, focus would be on the set of composition states [c1, c2, c3] attainable by means of all steady-state designs that begin with a feed stream of composition cf. We depict in Figure 2 a set of attainable compositions for the three-dimensional van de Vusse problem. (More precisely, these are compositions attainable in the limit for reactors with unbounded residence times.) In this case, the boundary of the composition set depicted is
shaped not only by PFRs and CFSTRs but also by differential sidestream reactors (DSRs). As illustrated in the figure, DSRs are similar to PFRs, except that they are fed along the reactor’s extent. There is no claim here that the shaded region in Figure 2 is the full attainable region for the threedimensional van de Vusse system. Nevertheless, arguments given by Feinberg and Hildebrandt9 suggest that the boundary of the attainable region will always be shaped, as in Figure 2, by combinations of PFRs, CFSTRs, DSRs, the three archetypes playing distinctive roles. In two subsequent papers,10,11 it is shown how, even when the boundary of the attainable region is unknown, one can still write precise and detailed formulas that must be respected by CFSTRs and DSRs that give rise to compositions on the attainable region’s boundary. For CFSTRs, there are ways to determine very special discrete values (something like eigenvalues) that the residence time must have in order for the effluent composition to be an attainable region boundary point. For DSRs, there are ways to determine precisely how the sidestream addition rate must be distributed along the reactor’s extent in order that compositions produced within the reactor reside on the attainable region’s boundary. These design formulas, however, provide only necessary conditions for CFSTR or DSR designs to realize compositions on the boundary of the attainable region; there is no assurance that, when the design formulas are respected, the resulting compositions will indeed reside on the boundary. Despite all that is now known, it remains difficult to be certain, for example, that a particular DSR composition trajectory of the kind depicted in Figure 2 contains a locus of attainable region boundary points. More generally, it remains difficult to be certain of precisely which combinations of PFRS, CFSTRs, and DSRs actually give rise to the outermost limits of what is attainable. Indeed, it even remains difficult to be certain of the precise location of the boundary. There are two aspects of the attainable region’s boundary that are under discussion here: first, knowledge of precisely which CFSTR-DSR-PFR designs give rise to compositions on the boundary and, second, knowledge of the boundary’s precise location. It will be useful to draw distinctions between the two. Although there is certainly utility in precise knowledge of designs that give rise to the outermost limits of the attainable region, recent work10-13 suggests that, from a practical viewpoint, those designs are unlikely
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to be implemented: DSRs, in particular, play an inexorable role in shaping the attainable region’s boundary, but formulas governing the sidestream addition rate in such critical DSRs become intractably complex. Also, as we have indicated, these formulas provide only the necessary conditions for composition yields on the attainable region’s boundary, so it remains difficult to know which designs actually do shape the boundary. Even when those boundary-shaping designs remain unknown, however, knowledge of the location of the attainable region’s boundary still has great utility, for then one can assess just how close an existing design has come to the limits of what can be achieved. Methods that attempt to construct the attainable region by building it “from the inside” are helpful in this respect because they indicate a range of compositions that can be achieved, at least by the means considered. On the other hand, those same methods leave open the question of whether there is a design, different from those considered, that could produce an effluent composition outside the range currently known to be within reach. For example, recent work by Kauchali et al.6 addresses the question of whether a reachable set of compositions could be further extended with a reactor network consisting of a fixed (perhaps large) number of interconnected CFSTRs at a discrete preselected set of operating compositions. The question then finds expression in a linear programming problem. When the answer is negative, one still cannot be certain that an affirmative answer would not derive from consideration of an appropriately chosen DSR or from an even larger number of CFSTRs operating at still other composition states. In any case, work by Kauchali et al. is just one example aimed at building the attainable region “from the inside”, that is, by attempting extension of the currently reachable states with newly added reactors fed by streams of previously realized compositions. Our aim here is to supplement such methods by providing means to determine bounds on the attainable region “from the outside”. While methods of the first kind enable one to describe certain compositions that are indeed attainable from the specified feed by means of constraint-consistent designs, we seek means to assert that certain other compositions are not attainable by any constraint-consistent design, no matter how imaginative. That is, we seek to construct a (convex) bounding set B of composition vectors such that any composition outside of B is unattainable. If AR denotes the attainableregionsthatis,thefullsetofattainablecompositionssthen we would have the inclusion AR ⊆ B. Moreover, if C is a “candidate” for the attainable region built “from the inside” by a collection of designs that begin with the prescribed feed, then we would have the inclusions C ⊆ AR ⊆ B. Thus, if C and B are reasonably “close”, then one can be confident that B is a sharp bound for the attainable region and that designs giving rise to the boundary of C take one close to the limits of what is attainable. It is perhaps the current absence of means to construct good outer bounds on the attainable region that most frustrates attempts to assert, with reasonable confidence, that a reachable set of compositions constructed “from the inside” is, indeed, a good approximation to the attainable region. Moreover, knowledge of outer bounds on the attainable region has a clear practical utility, even apart from help it provides in assessing the efficacy of attainable
region approximations from the inside. When the attainable region is unknown, good outer bounds for it provide means to assess the effectiveness of existing designs: If an economically appealing constraintconsistent design already realizes an effluent yield close to the most attractive yield the bounds will allow, then one can be certain that no other constraint-consistent design will produce a significantly better yield. The same cannot be said if one only has in hand an approximation to the attainable region “from within”. That is, if one knows only that a certain set of compositions can be realized, then one cannot be sure that there are not still other realizable compositions dramatically different from those already known. The bounding set, constructed by what we call the method of bounding hyperplanes, will turn out to be a polyhedron in the space of composition vectors. When there are N species, the polyhedron will be described by a system of linear inequalities of the form
a1Lc1 + a2Lc2 + ... + aNLcN e bL, L ) 1, 2, ..., P (2) where c1, c2, ..., cN are the species concentrations. The number, P, of such inequalities corresponds to the number of faces of the polyhedron and will, in rough terms, depend on how close an approximation to the actual attainable region one wishes to achieve. (An increasingly close polyhedral approximation of a convex body having curved surfaces requires an increasingly larger number of polyhedral faces.) A description of the bounding set in the form shown in eq 2 is especially convenient: If, for example, one is seeking a design that maximizes some linear combination of the species concentrations (and, in particular, the value of a certain species concentration), then solution of a simple linear programming problem will indicate bounds on what the designer can hope to achieve. While the approach here is based exclusively on detailed kinetic information, we call to the reader’s attention the existence of different work by others14,15 aimed at producing bounds on composition trajectories (in, for example, a PFR) when the kinetics is known only roughly (and might indeed change when a new catalyst is introduced along the reactor’s extent). Because very little kinetic detail is invoked, the general bounds produced are unlikely to be sharp for a particular kinetics, but that is to be expected. Our aims are different: We presume detailed kinetic information, and we seek reasonably sharp bounds on compositions that might ever be produced by any constraint-consistent design, however innovative, that involves both reaction and bulk mixing. The plan of this paper is as follows: Section 2 is primarily devoted to some preliminary ideas and to matters of terminology. In section 3 we describe, in intuitive terms, how the method of bounding hyperplanes works, and in section 4 we provide more rigorous underpinnings. In section 5 we show some examples of bounds for the van de Vusse example in both two and three dimensions. Section 6 is devoted to some brief remarks about directions for future research. 2. Preliminaries, Terminology, and Notation We denote by RN the usual vector space of N-tuples of real numbers, and by RN + we mean the set of vectors in RN with nonnegative components. We take the dot product in RN to be the usual one; that is
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x‚y ) [x1, x2, ..., xN]‚[y1, y2, ..., yN] :) x1y1 + x2y2 + ... + xNyN (3) If n is a nonzero vector in RN and γ is a number, then the hyperplane H)(n,γ) is the set of vectors
H)(n,γ) :) {x ∈ RN: n‚x ) γ}
(4)
The hyperplane H)(n,γ) divides RN into two closed halfspaces, Hg(n,γ) and He(n,γ), defined by
Hg(n,γ) :) {x ∈ RN: n‚x g γ} and e H (n,γ) :) {x ∈ RN: n‚x e γ} (5) Similarly, the open half-spaces H>(n,γ) and H(n,γ) :) {x ∈ RN: n‚x > γ} H 0. In rough terms, this means that the feed composition cf lies on one side of (or on) the hyperplane H)(n1,γ1), while for each nonequilibrium composition in W 0 on the other side of (or on) the hyperplane, the species-formation-rate vector points toward the half-space containing cf. Such a
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after p such stages the bounding set W p is the intersection of the original bounding set W 0 with the closed halfspaces Hg(n1,γ1), Hg(n2,γ2), ..., Hg(np,γp). That is p W p ) [∩i)1 Hg(ni,γi)] ∩ W 0
(11)
When W 0 is closed and bounded, it follows from eq 11 that W p is closed and bounded. When W 0 is convex (as when W 0 is the polyhedral set of compositions stoichiometrically compatible with the feed), W p is also convex. 4. Some Justification
Figure 4. Trimming the current bounding composition set again.
Figure 5. Two-stage bound-trimming sequence.
hyperplane is depicted in Figure 3, where the smaller arrows are intended to represent rate vectors associated with compositions in W 0 ∩ He(n1,γ1) (the light gray region). In compliance with the n1‚r(c) > 0 requirement, each small arrow shown makes an acute angle with n1, the normal to the hyperplane H)(n1,γ1) that points outward from He(n1,γ1). At least in an intuitive way, Figure 3 suggests that no steady-state reactor design, having cf as its feed composition, can realize a composition residing in the interior, W 0 ∩ H 0. If so, then we can again trim the current outer bound on the set of compositions attainable from cf to produce the smaller bounding set W 2 (see Figure 4). The two-stage bound-trimming sequence is summarized in Figure 5. The process can be continued sequentially: Given a current bounding set of compositions W i we can ask if there is a hyperplane H)(ni+1,γi+1) such that (i) cf lies in the half-space Hg(ni+1,γi+1) and such that (ii) ni+1‚ r(c) > 0 for all nonequilibrium c in W i ∩ Heni+1,γi+1). In this case, we can trim W i ∩ H 0. In this case, we want to argue that no composition in H 0. If so, we can “push the hyperplane H)(n*,γ*+) forward” by increasing , checking again, and pushing forward still further until is such that the check is no longer satisfied. The resulting corner section can then be removed from W 0 to produce a new polyhedron W 1 (with a different set of vertices). For a new choice of vertex, the process can be repeated to remove a section of W 1, and so on. In Figures 6-8, we show successively more refined
Figure 7. 20-hyperplane bound for the two-dimensional van de Vusse problem.
Figure 8. 110-hyperplane bound for the two-dimensional van de Vusse problem.
bounding polygons for the two-dimensional van de Vusse problem described in section 1. In the figures, the sides of the polyhedra derive respectively from 10, 20, and 110 hyperplanes. We include in each figure the boundary of the realizable composition set constructed as in section 2. This is generally believed to be the boundary of the attainable region for the problem at hand. Note that the bounding polygons shown in Figures 6-8 were constructed without reference to the putative attainable region, nor were any specific reactor
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Figure 9. 58-hyperplane bound for the three-dimensional van de Vusse problem (magnified).
configurations invoked. Because the (realizable) boundary of the putative attainable region approaches so closely the outer limits to attainability given by the polygon in Figure 8, that figure provides powerful support for the contention that the putative attainable region is indeed the full attainable region. In Figure 9, we show a constructed polygonal bound, formed from 58 hyperplanes, for the three-dimensional van de Vusse attainable region, and in the same figure, we reproduce the realizable composition set depicted earlier in Figure 2. It is unknown whether the realizable composition set in Figure 2 is the full attainable region for the problem at hand, but it is clear that it resides within the constructed polygonal bound. If the Figure 2 composition set is the full attainable region, it is apparent that a great many hyperplanes would be required to approximate closely the curvature of its outer boundary. For practical purposes, however, a less refined approximation should be adequate if the designer needs only a good sense of which effluent compositions are kinetically infeasible. For such a purpose, we believe that the 58-hyperplane polygonal approximation in Figure 9 already gives surprisingly incisive information. This can be seen more clearly in Figure 10, which reproduces Figure 9 but at full scale, with the full range of stoichiometrically feasible compositions depicted. (The stoichiometrically feasible compositions fill out the tetrahedron bounded by the dashed triangle and the coordinate axes.) It is apparent from Figure 10 that the 58-hyperplane polygonal bound severely circumscribes, on kinetic grounds, the set of compositions that the designer might hope to realize from the specified feed. The bound indicates that the set of compositions attainable from designs that invoke only reaction and bulk mixing is far smaller than what might be inferred from stoichiometric considerations alone. 6. Directions for Future Research The examples in section 5 suggest that the method of bounding hyperplanes can provide practical (and sometimes very sharp) bounds on compositions that any reactor configuration, however innovative, might realize. This paper was intended only to provide a conceptual basis for bound construction; it was not intended to take up issues of implementation extensively.
Figure 10. 58-hyperplane bound for the three-dimensional van de Vusse problem (unmagnified).
The bounds constructed for the van de Vusse example were generated automatically (in Maple) through a crude procedure, discussed only briefly at the beginning of section 5. Certainly there is a need for study of intelligent algorithms that automate the judicious selection and forward movement of hyperplanes (in the sense of the discussion at the beginning of section 5). Even apart from questions of implementation, there are interesting theoretical questions associated with the method described in this paper for the construction of bounds on attainability: Suppose, for example, that a certain “irreducible” polygonal bound W has been constructed in the manner of section 3. We say that W is “irreducible” if no additional hyperplane can give rise to further “bound trimming”. In more precise terms, we say here that W is irreducible if, for any choice of hyperplane H)(n,γ) such that cf resides in Hg(n,γ) and such that H(n,γ), H
