Toward a Theory of Process Synthesis - American Chemical Society

Ind. Eng. Chem. Res. 2002, 41, 3751-3761

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Toward a Theory of Process Synthesis Martin Feinberg† Department of Chemical Engineering, The Ohio State University, 140 W. 19th Avenue, Columbus, Ohio 43210

Some recent progress toward a general theory of process synthesis is reviewed. The discussion is centered around attempts to understand what F. J. M. Horn called the attainable regions roughly, the full set of outcomes (in some suitable state space) that might be realized from all possible designs within a broad class. The boundary of the attainable region is of special importance because it carries information about the outermost limits of what can be realized. We consider two major problems: the pure reactor synthesis problem, in which one tries to assess what might be attained from a specified feed in all steady-state processes that involve only reaction and mixing, and the reactor-separator synthesis problem, in which one tries to assess the full set of effluents attainable in broadly constrained processes when separators are also brought into play. For the pure reactor synthesis problem, we review how one can say a great dealsin remarkably specific quantitative termssabout designs of classical reactors that shape the attainable region’s boundary, even when that boundary is unknown. At the same time, we indicate why these very same results, however striking, point to serious obstacles along the road to future progress. The reactor-separator synthesis problem has a brighter outlook. For the purpose of finding outer bounds on the set of effluents attainable from all possible constraintconsistent steady-state designs, the Continuous Flow Stirred Tank Reactor (CFSTR) Equivalence Principle reduces the bewildering spectrum of reactor configurations that need be considered (including exotic, unimagined ones) to a surprisingly narrow class: those consisting of a small number of CFSTRs, the number depending in a simple way on the number of independent chemical reactions. To the extent that one can realize arbitrary separations, the principle serves to provide exact bounds on the set of effluents attainable from all steady-state constraintconsistent reactor-separator designs. 1. Introduction Jim Douglas spent a good part of his career advancing both the art and the science of chemical process synthesis. Although his objectives were always practical, Jim encouraged an uncompromisingly intellectual view of the subject. He was a friend and mentor when I joined the University of Rochester as an Assistant Professor, but at the time, I was not very much interested in design. That I am now writing a paper about recent developments in process synthesis is not entirely ironic, for it is traceable in two ways to my early association with Jim: First, our conversations made the subject far more interesting to me than it had been before, and, second, he was instrumental in bringing F. J. M. Horn to the Rochester faculty. Horn was, in the 1960s, greatly interested in design and had initiated seminal work on periodic processing and on the attainable region approach to process synthesis. There is now a resurgence of interest in both. My focus here will be exclusively on attainable region ideas, but I do want to mention that, with respect to periodic processing, it is often forgotten that Jim Douglas and Fritz Horn were true pioneers. Indeed, Jim’s landmark paper with Gaitonde1 was an early indicator of what was to come many years later. By the time Horn arrived in Rochester his interests had turned away from design, and Jim Douglas had already been lured to the University of Massachusetts. Still, there was a flow of design traffic through Rochester, and many of the visitors became my friends. One †

E-mail: [email protected]. Phone: 614-688-4883.

of these was David Glasser, who, in the late 1980s, was instrumental in resurrecting Horn’s attainable region ideas and who caused me to think about things that my colleagues Jim Douglas and Fritz Horn spoke of in an earlier time. This, then, is a paper in Jim Douglas’s honor about recent currents of thought in process synthesis, currents that flow from early attainable region ideas and their recent resurrection. It is probably not too soon to arrive at some tentative conclusions about where these currents are likely to lead. (Jim himself was never uncomfortable with opinion.) Certain recent results are cause for optimism, but others are not. It should be understood that results pointing to difficulties ahead are not, in themselves, bad ones. Indeed, results of this kind are often deep and have the virtue of revealing unpleasant truths clearly. It should also be understood that this is not intended to be a literature survey, nor will there be much technical explanation of how certain results come about. For this, readers should see the original references. Rather, this paper is meant to isolate just a few ideas that have emerged recently, to indicate how these ideas interact, and to draw some tentative conclusions about why some avenues of future research seem more likely than others to be fruitful. 2. The Attainable Region Idea The story begins with a 1964 paper by Horn,2 in which he advocated that process design be studied by geometric methods. In particular, he advocated the study of the attainable region in some suitable state space. By

10.1021/ie010807f CCC: $22.00 © 2002 American Chemical Society Published on Web 02/09/2002

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this, he meant the set of all possible output states that might be attained from a specified feed by means of all possible designs within some prescribed but fairly general class. (More precisely, he meant the closure of the set of attainable states, including those states, such as equilibrium points, that can be approached in the limit by a sequence of designs.) If the attainable region could be known, then the designer would have means to assess which output targets were realizable and which were, in fact, beyond reach relative to the class of designs admitted for consideration. In my judgment, Horn’s thoroughly geometric viewpoint, set in state space, was in itself a great (and easy-to-underestimate) contribution, reminiscent of the cognitive shift toward the study of phase portraits in the theory of differential equations. The boundary of the attainable region is of special significance, for states on the boundary represent the outermost limits of what any design within the specified class might achieve. For a very narrow class of admissible designs (e.g., all designs using only a single plugflow reactor with recycle), determination of the attainable region’s boundary is perhaps not so formidable a problem. When the class of admissible designs is more expansivesin particular, when chemical reactors of arbitrary configuration are brought into playsit is far less clear how the boundary of the attainable region might be elucidated. If, however, the boundary could be assessed for extremely general classes of designs, including configurations beyond current imagination, then the designer would have means to know how close the product of an existing design is to the limits what might be achieved by any design within the class. Better still, if one could articulate details of critical designs that do, in fact, yield outputs on the general attainable region’s boundary, then one could know just how, at least in principle, those limiting states might be achieved. I shall discuss two very expansive classes of designs, both having emerged in the literature as archetypes of what might be studied from an attainable region perspective. In each case, the feed will be presumed specified, as will the network of chemical reactions (along with its kinetics). In the pure reactor synthesis problem, I shall consider what might be attained from the feed in steady-state designs that involve only chemical reaction and mixing; no use is made of separators. In the reactor-separator synthesis problem, I shall consider what might be attained in steady-state designs that involve chemical reaction, mixing, and separations. The pure reactor synthesis problem is, to some extent, an artificial one. Why, after all, should one refrain from the use of separators if they are advantageous? There is, of course, truth in this, but the pure reactor synthesis problem can, nevertheless, provide clues to the circumvention of perhaps costly separations. When the formation of undesired molecules can be partially suppressed by means of judicious reactor design, their separation from the desired ones becomes a less formidable problem. Even so, it is the reactor-separator synthesis problem that is by far the more compelling one. Indeed, a design consisting solely of reactors is a special case of a reactor-separator system, and so an assessment of outer bounds on what might be attained by means of reactors and separators serves to bound what might be attained by means of reactors alone. (This is a little bit

of an oversimplification because, as we shall see, the two problems are formulated in slightly different ways.) It will turn out that, for the pure reactor synthesis problem, there are results that are surprising, beautiful, and, at the same time, very troubling. On the bright side, there are results that permit one to make extraordinarily detailed statements about the design of reactors that shape the attainable region’s boundary. Moreover, these statements can be made solely on the basis of the kinetic rate laws and the available feed, even when the boundary of the attainable region remains unknown. (For example, one can assert that a CFSTR with a specified feed composition can give rise to an attainable region boundary point only if its residence time takes one of certain discrete, distinguished values, something like eigenvalues.) On the darker side, however, is the fact that certain critical designs (of sidestream reactors) that shape the attainable region’s boundary depend in an inexorably complicated way on the rate laws and, when there are more than two or three reactions, on higher derivatives of the rate laws. This dependence is not an artifact of flawed methodology but, rather, is intrinsic to the problem itself. As we shall see, the situation for the reactorseparator synthesis problem is substantially better. The somewhat surprising CFSTR Equivalence Principle provides a means to calculate an outer bound on the productivity that can be attained in any steady-state reactor-separator system for which the reactor components have a specified capacity and are constrained to operate within specified composition-temperature limits. To the extent that one can realize arbitrary separations, these bounds are exact. It is gratifying that the more compelling of the two problems is the one with the brightest outlook. 3. The Pure Reactor Synthesis Problem Here we consider a chemistry of N species, say A1, A2, . . . , AN, and we suppose that transformations among these species proceed through the occurrence of perhaps several chemical reactions. We presume also the availability of one or more feed streams, each of a specified (time-invariant) composition. The problem we consider is this: What is the full set of mixture compositions attainable from the primary feed streams by means of steady-state designs involving only chemical reaction and bulk mixing? To keep the discussion simple we shall restrict our attention to isothermal designs, and we shall suppose also that the mixture under consideration has a fixed density. By a composition, we mean a vector c ) [c1, c2, ..., cN] in RN, where cI is the molar concentration of species AI and RN is the usual vector space of N-tuples of real numbers. Thus, our problem can be viewed geometrically: We are asking about the set of all composition vectors in RN attainable from feed stream(s) of specified composition(s) cf1, cf2, ..., cfK by means of (isothermal) steady-state designs involving only reaction and mixing. Clearly, that set will contain cf1, cf2, ..., cfK, but it will almost always be larger. For example, a stream of composition cf1 can be used as the feed to a steady-state plug-flow reactor (PFR) to produce a curve of compositions in RN beginning at cf1, with each composition along the curve corresponding to a new residence time. Similarly, a stream of composition cf1 can be used as the feed to a steady-state continuous flow stirred tank reactor (CFSTR) to produce an effluent composition that

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Suppose that, for the chemistry and the primary feed set under consideration, C is a set of compositions in RN that we believe might be the full attainable region. For C to be the attainable region, it is necessary (but not sufficient) that C satisfy three conditions articulated by Glasser et al.:3-5 AR1. The set C must be convex. If c* and c** are members of C, then any composition on the line segment joining c* and c** could be attained by mixing streams of compositions c* and c** in suitable proportions. In preparation for the statement of the two remaining conditions, we denote by r(‚) the species-formation-rate function for the chemistry under consideration. That is, for a mixture of local composition c, r(c) ) [r1(c), ..., rN(c)] is the local vector of molar formation rates (per unit volume) of the various species due to the occurrence of all chemical reactions. Note that the composition along a PFR is governed by the differential equation Figure 1. Some possibilities for the pure reactor synthesis problem.

will depend on the CFSTR’s residence time. By considering all possible values for the residence time, we can imagine the locus in RN of all possible CFSTR effluents that might derive from the feed composition cf1. Note that the PFR composition trajectory beginning at cf1 will generally be different from the CFSTR steady-state locus beginning at cf1. If c* is a composition along the PFR trajectory and c** is a composition along the CFSTR locus, then any composition along the line segment in RN joining c* and c** can also be realized by mixing streams of compositions c* and c** in suitable proportions. Similarly, any composition in the interior of a polyhedron with vertexes taken from c*, c**, cf1, ..., cfK can be attained by mixing. It is clear, then, that new compositions can evolve from the primary feed streams in a variety of ways, all involving only reaction and mixing. Indeed, any of the compositions realized by the conventional means described can be used as the feed for yet another reactor of conventional or even exotic type to expand the set of realized compositions still further. Relative to the class of steady-state isothermal designs involving only reaction and mixing, Horn’s attainable region would be the full set in RN of compositions that might be realized from the primary feed streams by all such processes, including exotic ones. (More precisely, the attainable region is the closure of the full set of realizable compositions, including compositions that might not be realizable but that can be realized arbitrarily closely.) To know the attainable region in this case is to know the full palette of target compositions that are feasible (at least in the limit) in the absence of separators. Without knowing what effluents are, in fact, feasible, it is difficult to see how a complete search for design optima might be implemented. But how, for a given chemistry and given feed compositions, can the boundary of the attainable region be determined? True, one can, in an ad hoc way, begin with the primary feed streams and consider a great variety of designs that serve to enlarge the set of compositions that are realizable. But how can one know when no further enlargement can be made, especially if all possible steady-state designs involving only reaction and mixing are admitted for consideration, even designs utilizing exotic reactors that remain unimagined? A few possibilities are shown in Figure 1.

c3 ) r(c)

(1)

where the overdot represents differentiation with respect to the residence time. Note also that, for a steadystate CFSTR with residence time θ and feed composition c0, the effluent composition is governed by the equation

1 r(c) + (c0 - c) ) 0 θ

(2)

AR2. At no composition c* on the boundary of C should r(c*) point out of C. Were this not the case, then a PFR with feed composition c* could be used to produce compositions not in C. In fact, a solution of the PFR equation, subject to the initial condition c(0) ) c*, would yield compositions outside of C for slightly positive residence times. AR3. For no composition c* outside of C should there be a composition c0 in C and a positive number θ such that the equation r(c*) + (1/θ)(c0 - c*) ) 0 is satisfied. Otherwise, a CFSTR of residence time θ and with (attainable) feed composition c0 could be used to realize a composition outside of C, the putative attainable region. Glasser et al. invoked AR1-AR3 to formulate procedures aimed at guiding the construction of candidates for the attainable region in the context of particular examples. The examples studied involved a small number of species, for then visualization in R2 or R3 became possible. Typically, the idea was to evolve a set of compositions through use of PFRs, CFSTRs, and mixing until AR1-AR3 are satisfied. It should be kept in mind, however, that AR1-AR3 are necessary conditions that a composition set must satisfy in order that it be the attainable region. When AR1-AR3 are satisfied for a candidate composition set, the possibility remains that the attainable region is still larger and that the enlargement might be accomplished by further use of conventional reactors. Moreover, one must confront the prospect that, in some instances, the boundary of the attainable region might be shaped by designs involving exotic reactors that employ reaction and mixing in highly unconventional ways. However, it was shown in a paper by Feinberg and Hildebrandt6 that conventional reactorssPFRs, CFSTRs, and differential sidestream reactors (DSRs)s invariably shape the attainable region’s boundary, with the three reactor types playing distinctive roles: In very

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Figure 2. Manifold of extreme points meeting a straight line section roughly.

Figure 3. Manifold of extreme points meeting a straight line section smoothly.

rough terms, PFR composition trajectories provide a system of “highways” along the attainable region’s boundary that serve to access that attainable region’s extreme points, while CFSTRs and DSRs provide “service roads” that access the various PFR highways. Moreover, it was shown in two subsequent papers by Feinberg7,8 that CFSTRs and DSRs that shape the attainable region’s boundary must conform to highly detailed design formulas that derive in a precise way from the rate functions for the underlying network of chemical reactions. These results have some complications and subtleties that do not lend themselves to concise exposition. We can, however, review them in a cursory way, meant to be more suggestive than precise. At the end, perhaps it will be clear not only why there is great beauty in what can, in fact, be said generally but also why there are dark clouds on the horizon. 3.1. Universal Properties of the Attainable Region. Recall that an extreme point of a closed convex set is a member of the set that does not lie in the interior of a line segment contained in the set. Thus, the boundary of a closed convex set is the union of extreme points and straight line segments. In very rough terms, then, one can envision the boundary of a closed convex set (and of the attainable region in particular) as being made up of straight line sections and of manifolds of extreme points. Figures 2 and 3 depict two hypothetical fragments that might serve as parts of the boundary of a closed convex set in R3. In the first instance a twodimensional manifold of extreme points meets a straight line segment “roughly” along a one-dimensional manifold (i.e., a curve) of extreme points. In the second instance a two-dimensional manifold of extreme points meets a straight line segment “smoothly” along a onedimensional manifold of extreme points.

Figure 4. Manifold of extreme points as the union of PFR trajectories.

Feinberg and Hildebrandt6 argued that, on the boundary of the attainable region, manifolds of extreme points, called “protrusions”, are inevitably made up entirely of PFR composition trajectoriessthat is, of trajectories corresponding to solutions of eq 1. Moreover, when a protrusion joins a straight line section “roughly”, then the border at which the two sections join is again the union of PFR composition trajectories. Thus, were the fragment depicted in Figure 2 on the boundary of the attainable region (for some concrete chemistry and primary feed set), the two-dimensional protrusion would necessarily be striated by PFR composition trajectories, as shown schematically in Figure 4, and the curve along which the protrusion meets the straight line section would itself be a PFR composition trajectory. (Although discussion of the figure makes reference to its threedimensional setting, the more general assertions obtain in higher dimensions as well.) When, on the boundary of the attainable region, a protrusion meets a straight line section smoothly, the situation is different: Feinberg and Hildebrandt6 define a connector on the boundary of the attainable region to be a border between a protrusion and a straight line section at which the two sections join smoothly and at which the PFR trajectories making up the protrusion point away from the straight line section. They show that a connector is invariably the union of DSR composition trajectories and CFSTR operating points, with feeds to the DSRs and CFSTRs coming from elsewhere in the attainable region. Thus, were the fragment depicted in Figure 3 on the attainable region’s boundary, the situation might be something like that shown in Figure 5. (The connector shown in the fragment consists of the two DSR trajectories and the CFSTR operating point c*.) Figure 5 will help illuminate assertions made earlier about roles that PFRs, CFSTRs, and DSRs invariably play in shaping that attainable region’s boundary. As depicted in the figure, PFR trajectories traverse protrusions on the boundary of the attainable region and, in that sense, provide a highway system with which the various extreme points might ultimately be accessed. DSRs and CFSTRs, on the other hand, provide a means to navigate along the connectors between protrusions and straight line sections so that a PFR trajectory of choice might be accessed. This, of course, is a qualitative picture of the way things invariably work, but it is, nevertheless, one that can help guide incisive thinking about process synthesis. We turn now to some remarkably detailed quantitative


statements that can be made about the design of CFSTRs and DSRs that shape the attainable region’s boundary. 3.2. Critical CFSTRs. Consider a CFSTR, such as that shown in Figure 5, for which the feed composition is c0 and for which the effluent composition is c*. In the figure, c* is depicted as an extreme point on the boundary of some putative attainable region. (In the boundary fragment depicted, c* is on the border of the set of extreme points, but it is an extreme point nonetheless.) In fact, only very exceptional CFSTR designs can ever produce attainable region extreme points: For a specified (attainable) CFSTR feed composition c0 and a specified species-formation-rate function r(‚), an argument in ref 8 indicates that, typically, there are only certain exceptional residence times (reminiscent of eigenvalues) such that a corresponding effluent composition might be an attainable region extreme point. Moreover, these exceptional residence times and effluent compositions can be computed from c0 and r(‚) without knowing what the attainable region’s boundary is or, for that matter, what the primary feed compositions are that give rise to the attainable region! Indeed, for the specified r(‚), there is a distinguished set of compositions having the property that a member of the set can be simultaneously an extreme point of some attainable region (corresponding to some set of primary feed compositions) and the effluent of a CFSTR (corresponding to some attainable CFSTR feed composition).8 Consider, for example, the well-studied van de Vusse reaction network, taken with the mass-action rate constants indicated in eq 3. For this chemistry, we ask 1

Figure 5. How classical reactor types shape the attainable region’s boundary.

k

A1 98 A2 98 A4 k′

2A1 98 A3

(3) Figure 6. Improbable attainable region boundary fragment.

whether a CFSTR with (attainable) feed composition c0 ) [1, 0, 0, 0] might ever produce, as its effluent composition, an extreme point of an attainable region corresponding to some specified set of primary feed compositions. The theory in ref 8 indicates that this can be the case only if the CFSTR residence time has (in units consistent with those of the rate constants) the exceptional value

k x 2k′ θ) k+ x2k′k 1-

(4)

Note that this value is nonnegative only if k e 2k′. Thus, if k > 2k′, no CFSTR with (attainable) feed composition c0 ) [1, 0, 0, 0] can yield an effluent composition that is an attainable region extreme point (for any set of primary feed streams). On the other hand, if k < 2k′, there is exactly one positive residence time for which an attainable region extreme point might result, and that residence time is given by eq 4. In particular, for the heavily studied rate constants k ) 1 and k′ ) 10 and for a CFSTR with feed composition c0 ) [1, 0, 0, 0], the exceptional residence time θ ) 0.6345 is the only one for which an attainable region extreme composition might emerge as the effluent. Moreover, this assertion can be made without knowing

the attainable region or even the primary feeds from which the attainable region derives! For k ) 1 and k′ ) 10 and for a single primary feed cf ) [1, 0, 0, 0], it is possible to construct an otherwise plausible attainable region candidate that apparently satisfies AR1-AR3 and that exhibits on its boundary (as extreme points) a continuous curve of CFSTR effluent compositions, corresponding to a range of residence times, all for the CFSTR feed composition c0 ) cf ) [1, 0, 0, 0]. (See, for example, Figure 1 in the paper by Feinberg and Hildebrandt.6 The figure depicts a reasonable attainable region candidate proposed earlier by Hildebrandt and Glasser, one containing on its boundary a fragment similar to that shown here in Figure 6.) Current theory, not then available, indicates that such a picture cannot be correct. Finally we note that, for the van de Vusse reaction network with arbitrary (positive) values for k and k′, it can be established8 that the only CFSTR effluent compositions that might ever serve as attainable region extreme pointssfor any (attainable) CFSTR feed composition deriving from any set of primary feed compositionssare those for which c1 ) 0 or for which c1 and c2 are precisely related by eq 5.

c2 )

(2k′c1 - k)c1 k(4k′c1 - k + 1)

(5)

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and suppose that the kinetics is mass-action with all rate constants set to 1. Consider also a DSR with fixed sidestream composition Figure 7. DSR.

c0 ) [1, 0, 0, 0, 0]

Analyses of the kind discussed above are not limited in their implementation to systems as simple as those of van de Vusse. In ref 8 there is also a worked numerical example in the context of a mass-action system having six species and four (reversible) reactions. For specified rate constant values in the example and a specified CFSTR feed composition, calculations indicate that there are precisely two positive residence times for which the effluent composition might possibly serve as an attainable region extreme point. Remark. A discussion in ref 8 suggests why, generically, it should not be expected that an extended locus of CFSTR effluent compositions (corresponding fixed CFSTR feed composition and a range of residence times) will shape the attainable region’s boundary. In particular, one should not expect the border between a manifold of extreme points and a straight line section to be of the kind depicted in Figure 6. In section 3.1 it was asserted that a “connector” on the attainable region’s boundary will invariably be the union of DSR composition trajectories and CFSTR operating points. Here we are asserting more: It is reasonable to conjecture that CFSTR operating points will arise, as in Figure 5, mainly as rest points associated with DSR trajectories. That is, DSRs rather than CFSTRs are likely to be the primary shapers of connectors. This will have important implications when we assess the outlook for the pure reactor synthesis problem. 3.3. Critical DSRs. Consider a steady-state differential sidestream reactor, such as the one depicted in Figure 7, for which the sidestream is of fixed composition. (More generally, the composition of the sidestream could vary along the reactor’s extent, but here it will be advantageous to keep the discussion simple.) Although the sidestream composition is presumed fixed, we suppose that the rate at which the sidestream is added might vary axially. Thus, in Figure 7 we indicate the local sidestream addition rate (volumetric rate of sidestream added per unit reactor volume) by R(τ), where τ denotes the local residence time.6 Sidestream addition rates are sometimes discussed in terms of an addition rate law whereby the local addition rate along the reactor is tuned to the local composition interior to the reactor. That is, the “law” is specified by a function of composition, R j (‚), such that, at each τ, R j (τ) ) (c(τ)). In ref 7 it is shown that, for a DSR composition trajectory to consist entirely of attainable region extreme points, it is necessary that the sidestream addition rate policy conform to highly detailed design equations that derive from r(‚) and c0 in a very precise way. This is to say that DSRs, like CFSTRs, that give the attainable region its shape are subject to very special conditions that can be formulated even when the attainable region itself is unknown. (The same paper contains a discussion of connections to earlier papers by Hildebrandt and Glasser5 and by Palanki et al.,9,10) A worked example from ref 7 is instructive. Consider the reaction network

A1 h A2 h A3 h A4

A 1 + A 3 h A5

(6)

(7)

In this case, the only sidestream addition rate “law” that might give rise to a DSR composition trajectory consisting of attainable region extreme points is shown in Figure 8. Moreover, this peculiar formula can be deduced from r(‚) and the specified c0 without knowing the attainable region or even the primary feed compositions from which the attainable region derives. Despite the relative simplicity of the reaction network in eq 7, the required sidestream addition rate law is already distressingly complicated, but it is, nevertheless, the only one that could serve to shape the attainable region’s boundary. When there are more reactions, the situation becomes far worse, as we shall see in the next section. 3.4. The Pure Reactor Synthesis Problem: Dark Clouds on the Horizon. In the context of the pure reactor synthesis problem, it is extraordinary that we can say so much in general about reactors that shape the attainable region’s boundary: Qualitatively, PFR composition trajectories universally provide a system of highways that traverse manifolds of extreme points, while connectors between manifolds of extreme points and straight line sections are invariably made up of DSR trajectories and CFSTR operating points. In a more quantitative vein, CFSTRs that serve to form those connectors can only have certain exceptional (computable) residence times, and sidestream addition rates in DSRs that serve to form those connectors must comply with highly detailed equations, derivable from the kinetics and sidestream composition even when the attainable region is unknown. All of this is remarkable, but, even so, it is no small matter to know the attainable region with certainty in any but the most trivial circumstances. Even if we accept the premise that any attainable composition can be realized with some arrangement of PFRs, CFSTRs, and DSRs, there are so many ways in which these might be configuredsrecall Figure 1sthat it becomes difficult to know whether a given attainable region candidate might be enlarged still further by invoking as yet untried and more complex combinations. As of this writing there appears to be no (nontrivial) instance in which someone has proved for the pure reactor synthesis problem that a particular set of compositions is, in fact, the attainable region. We should draw a distinction between two reasons for wanting to know the attainable region’s details: First, one might like to know how close the product of an existing design (involving only reaction and mixing) is to the outer limits of what might be achieved. For this purpose, it is enough to know the attainable region’s boundary, and means for achieving the boundary with particular designs are of no consequence; indeed, it is enough to have a good estimate of where the boundary lies. With respect to this first purpose, then, progress will depend on the extent to which ways can be developed to assess good outer bounds on the attainable region in nontrivial circumstances. Second, one might like to know just how the attainable region’s boundary is shaped by particular designs so that those designs might, in fact, be implemented to realize compositions at the outermost limits of what is


Figure 8. Critical sidestream addition rate law.

achievable. With respect to this second purpose, there aresfrom a practical standpointsalready reasons for pessimism. Although it is remarkable that we can delineate in great detail those exceptional CFSTR residence times and DSR sidestream addition rate policies that serve to shape the attainable region’s boundary, the results for DSRs (unlike those for CFSTRs) are far from comforting. We already saw in the preceding section how, even for a relatively simple four-reaction network, the required sidestream addition rate “law” is surprisingly complex. For a mass-action system in which there are five linearly independent reactions, the sidestream addition rate is governed not by an algebraic equation but, rather, by differential equations involving polynomials in the species concentrations that can have hundreds of terms. When there are six independent reactions, the sidestream addition rate is governed by differential equations that, in their barest symbolic form, fill a page.7 Yet, these are the equations that the sidestream addition rate must respect if the resulting DSR trajectory is to reside on the attainable region’s boundary. There is another aspect of these equations that is even more troubling than their inherent complexity: As the number of independent reactions increases, increasingly higher-order derivatives of r(‚) inexorably enter the equations that govern the critical sidestream addition rate policy.7 In real applications, the species-formationrate function derives from kinetic measurements that carry uncertainty, so, however artfully it might be constructed, the function r(‚) can reflect only approximately the true state of affairs. It would seem unwise, then, to place much faith in higher derivatives of the function so constructed, especially when those derivatives ultimately become the basis for sidestream-addition-rate equations that are themselves so complex as to inhibit practical consideration. However much we might dislike this situation, it is the way things are. It should be remembered, though, that the entire discussion so far was in consideration of the pure reactor synthesis problem, which, as we have said, is interesting but somewhat artificial. We turn next to the far more compelling reactor-separator synthesis problem. Like the pure reactor synthesis problem, the reactorseparator problem will yield some very pleasant surprises, but it will give rise to fewer disappointments. 4. The Reactor-Separator Synthesis Problem Here we seek to understand, for a given chemistry, limits on what might be attained from an available feed

in consideration of all possible steady-state reactorseparator designs, subject perhaps to broad constraints. Thus, for a given kinetics and prescribed feed streams, we might like to know the highest possible steady-state production rate of a certain desired species if there is a specified availability of catalyst, if certain temperature and pressure bounds are to be respected within reactor units, and if limits are set on the effluent rates of certain toxic side products. (Consideration of a reactor-separator system is not meant to preclude the possibility that, within the system, streams might be mixed.) Although we shall come to very specific conclusions about designs that can, in fact, attain such a maximum, this will not be our primary concern, for, even when they are known, such designs might be economically disadvantageous. Rather, our concern will be in computing the value of the maximum itself because it serves as a benchmark against which all designs might be measured. There is no point in expending much effort to improve the effluent of an economically attractive design whose yield is already close to the maximum attainable by any design consistent with the specified constraints. On the other hand, a design whose yield is far less than the maximum is one that should be brought into question. In any case, it is important to know just what the maximum is and, more generally, what is attainable in principle. Here we shall survey some general results in this direction, results contained in a recent paper by Feinberg and Ellison.11 For a fuller, more precise statement of those results, readers should see the original paper. Consider, then, a reactor-separator system of arbitrary design, operating at steady state, viewed as a “black box” into which certain feed streams enter and from which certain product streams emerge. We denote by M0 ) [M10, M20, ..., MN0] the vector of molar feed rates of species A1, A2, ..., AN; that is, MI0 g 0 is the total rate at which moles of AI are carried into the reactorseparator system because of the presence of all feed streams. Similarly, we denote by M ) [M1, M2, ..., MN] the vector of molar effluent rates; MN g 0 is the total rate at which moles of AN are carried from the reactorseparator system in all effluent streams. For the kinetics at hand and for the specified M0, our interest is in knowing sharp outer bounds on the set of attainable values for M as we consider all steady-state designs in which the reactor volume does not exceed a specified value, perhaps very large, and in which local reactor states (e.g., temperature, pressure, and molar concentra-

3758


Figure 9. Varieties of reactor-separator systems.

tions) are required to lie within certain ranges, again perhaps very large. It is clear that if, in a particular design, there are no reactors at all, then M ) M0, no matter how elaborate the system of separators (and mixers) might be. Thus, what gives breadth to the set of attainable molareffluent-rate vectors is the variety of reactor configurations that might be employed, with separators playing a largely supportive role. In assessing the full set of attainable effluents, then, our problem lies in the almost limitless spectrum of reactor arrangements that can be brought into play. (See, for example, Figure 9.) So that we might assess outer bounds on the set of attainable molar-effluent-rate vectors, we shall hereafter presume the capacity to make whatever separations we wish, so long as they are consistent with steady-state material balances. Clearly, the set of molareffluent-rate vectors that can be attained in the presence of the capacity for arbitrary separations is not smaller than the set attainable in its absence. The outer bound assessed in this way becomes sharp to the extent that any desired separation can, in fact, be realized. In the presence of the capacity for arbitrary separations, it is tempting to suppose that, at steady state, unused reactants and disagreeable byproducts can always be separated completely from the desired products, totally recycled, and ultimately consumed through further reaction. That is, it is tempting to suppose that the molar effluent rates of certain designated species (unused reactants and unwanted side products) can always be reduced to zero while at the same time maintaining the molar effluent rates of desired products at stipulated levels. In the context of steady-state designs consistent with a specified molar-feed-rate vector, a specified reactor capacity, and specified pressuretemperature constraints within reactor components, this is not generally so. Certainly, when unwanted molecules are formed as end products of irreversible reactions, total recycle of them will result in an unending transient accumulation. Even when all reactions are reversible, it might still be the case that pressure-temperature constraints within reactor components prevent recycleconsuming reactions from keeping pace with total recycle in any steady-state design consistent with those

constraints, with the specified reactor capacity, and with the specified molar-feed-rate vector. (In the reversible case, it is instructive to think about the situation in which, for the stipulated pressure-temperature range, rates of reactions that consume an unwanted side product are small.) Even granted the capacity to make whatever separations we wish, the problem of delineating the full set of molar-effluent-rate vectors attainable from all possible constraint-consistent reactor-separator designs would appear to be insoluble. After all, the spectrum of reactor configurations that might be invoked is limited only by the designer’s imagination. Nevertheless, we shall soon assert that, for the sole purpose of assessing what is attainable, it suffices to consider reactor configurations within a remarkably small class. In preparation for the assertion, we define the rank12 of a chemical reaction network to be the maximum number of linearly independent reactions in the network. Thus, for example, networks (8), (9), and (10) have respectively ranks 3, 4, and 2. Means are provided in ref 12 for calculating the rank of a network systematically.

Granted the capacity for arbitrarily sharp separations, the CFSTR Equivalence Principle asserts that, for the purpose of determining the full set of attainable molar-effluent-rate vectors, it suffices to consider only reactor-separator systems in which the reactor components consist of a (usually) small number of CFSTRs, with the maximum number depending in a simple way on the rank of the operative network of chemical reactions. The principle is stated fully below; an argument supporting it appears in ref 11. CFSTR Equivalence Principle for ReactorSeparator Systems. For a prescribed chemistry, in which the underlying reaction network has rank s and for a prescribed molar-feed-rate vector M0, suppose that a steady-state reactor-separator design, having total reactor volume V*, yields the molar-effluent-rate vector M* > 0. Then, for the same chemistry and the same molar-feed-rate vector M0, there is another steady-state design that yields a molar-effluent-rate vector arbitrarily close to M* and that has the following properties: (i) The only reactors are CFSTRs, these being no more than s + 1 in number. (ii) The total reactor volume is again V*. Moreover, in the following sense, mixture states (e.g., temperatures, pressures, and molar concentrations) within these CFSTRs are no more extreme than those within reactor units employed in the first design: (iii) Each CFSTR mixture state in the second design can be chosen to be arbitrarily close to some local mixture state within a reactor unit in the first design.


The principle asserts that any molar-effluent-rate vector that can be attained at steady state by means of some steady reactor-separator design can be attained arbitrarily closely by another steady-state reactorseparator design (perhaps invoking very sharp and rapid separations), in which the reactor components consist of no more than s + 1 CFSTRs, where s is the rank of the underlying network of chemical reactions. Moreover, the total reactor volume in the CFSTR-only design is the same as the volume in the original design, and the mixture states exhibited in the CFSTR-only design are no more unreasonable than those in the initial one. This is not to say that the CFSTR-only design is economically equivalent to the first design, for the CFSTR-only design might require extraordinary separations and recycle rates. What the principle does say is that, for the limited purpose of assessing bounds on the set of molar-effluent-rate vectors, it suffices to consider a remarkably narrow class of reactor configurations, those consisting of a small number of CFSTRs. To the extent that arbitrarily sharp and rapid separations can be realized, the bound becomes exact. In this case, the CFSTR Equivalence Principle serves to determine the “attainable region” of molar-effluent-rate vectors exactly, and, in any case, the principle serves to delineate bounds for it. We shall consider, in a very cursory way, just what the principle enables us to say generally about the attainable region; a more detailed discussion is contained in ref 11. Suppose that, for the chemistry under consideration, r(c,T) ∈ RN is the volumetric species formation rate at composition c and temperature T. That is, suppose that r(‚,‚) is the (continuous) species-formation-rate function, this time viewed as a function of both composition and temperature. (In our discussion of the pure reactor synthesis problem, we regarded the temperature as fixed.) We consider all steady-state reactor-separator designs in which (c, T) states within reactor components are constrained to lie in a closed and bounded set Ω ⊂ RN+1, perhaps very large, and in which the total reactor volume lies within the closed interval V :) [0, V h ]. Although transition from the CFSTR Equivalence Principle to the following statement is beyond the intended scope of this paper, it should at least be intuitively clear why the principle makes it feasible to actually delineate, through focus on a very narrow class of reactors, the “attainable region” of molar-effluent-rate vectors: Describing the Attainable Region. Granted the capacity for arbitrary separations, the set M ⊂ RN of all molar-effluent-rate vectors that are attainable (or that are attainable arbitrarily closely) from the molar-feedrate vector M0 is given by

The formula in eq 11 reduces the problem of determining the attainable region (granted the capacity for arbitrary separations) from a conceptual, designcentered one to a purely computational one. To “know” the attainable region is to have a means to determine whether a given M ∈ RN is or is not contained in M. It is not difficult to show that the right-hand side of eq 11 is equivalent to

M ) conv(M0 + V ‚r(Ω)) ∩ R+N

Suppose that a feed stream of pure A1 is available at 10 mol/s, that A4 is the desired product, and that A3 is an environmentally unfriendly side product. Our interest is in the maximum rate at which the desired A4 can be produced in a steady-state reactor-separator design subject to the following constraints: (i) The total reactor volume does not exceed V h ) 1000 L. (ii) In no reactor component does the local temperature lie below Tmin ) 100 °C or above Tmax ) 300 °C, nor does the local pressure exceed pmax ) 50 bar. (iii) The production rate of the disagreeable A3 does not exceed 0.1 mol/s.

(11)

where conv(‚) denotes the taking of the convex hull and

M0 + V ‚r(Ω) :) {M∈ RN: M ) M0 + Vr(c,T), V∈ V, (c, T)∈ Ω} (12) (Recall that the convex hull of a set Γ ⊂ RN is the smallest convex set in RN that contains Γ.) In the absence of the capacity for arbitrary separations, the set of attainable molar-effluent-rate vectors is contained in, but might be smaller than, the right-hand side of eq 11.

(M0 + conv(V h r(Ω) ∪ {0})) ∩ R+N

(13)

One way to assess attainability is as follows: The set Ω can be well-approximated by a large, finite number h r(Ω) of points {(ci, Ti) ⊂ Ω, i ) 1, 2, ..., P}, so the set V can be approximated by the finite set {V h r(ci,Ti), i ) 1, 2, ..., P}. In this way, the convex set described by eq 13 can be well-approximated discretely as a convex polyhedron in RN, having a (perhaps large) number of vertexes. Computational techniquessin particular, linear programming techniquessfor determining whether a particular M ∈ RN lies within the resulting polyhedron are readily available. In this sense, one has means to “know” the attainable region for the reactor-separator synthesis problem, at least when the capacity for arbitrary separations is presumed. (A good resource dedicated to computations involving convex hulls, polyhedra, and linear programming is the paper by Fukuda.13) An example taken from ref 12 illustrates the kinds of concrete problems that can be solved: Example. Consider once again the well-studied van de Vusse reaction network (eq 14) taken with massaction kinetics, this time with temperature-dependent rate constants as shown in eq 15. (Activation energies are in kcal/mol.) For the purposes of the example, we k1

k2

A1 98 A2 98 A4 k3

2A1 98 A3

(14)

presume that A1, A2, A3, and A4 comprise an ideal gas mixture.

(

10 (s-1) RT

(

18 (s-1) RT

k1(T) ) 4.22 × 103 exp k2(T) ) 1.02 × 105 exp -

(

k3(T) ) 5.24 × 108 exp -

) )

22 (L/mol‚s) RT

)

(15)

3760


In this case, it follows from (ii) and the presumption of ideal gas behavior that Ω consists of all (c, T) pairs for which Tmin e T e Tmax and for which

0e

4

pmax

L)1

RT

∑ cL e

(16)

Thus, for the prescribed kinetics, for M0 ) [10, 0, 0, 0], for V h ) 1000 L, and for Ω as indicated, the set M can be well-approximated as a convex polyhedron in R4 through computations along lines sketched earlier. The problem then becomes one of finding the maximum of M4 over all M ) [M1, M2, M3, M4] that lie in the resulting polyhedron and that satisfy the environmental constraint M3 e 0.1. This can be formulated as a conventional linear programming problem.12 The (numerical) analysis indicates that there is no steady-state reactor-separator design satisfying the imposed constraints that can produce A4 at a rate greater than ∼6.12 mol/s. This bound is sharp in the following sense: If arbitrary separations could be realized (at arbitrary rates), then there is a steady-state design that respects all of the constraints and that can produce A4 at a specified rate below but arbitrarily close to ∼6.12 mol/s. Moreover, without a relaxation of at least one of the constraints (i)-(iii), at least 3.68 mol/s of A1 must pass from the feed to the effluent unreacted in any steady-state design. (Total recycle of A1 cannot be realized at steady state, for reactive consumption of A1 cannot keep pace with the recycle while (i)-(iii) are respected.) 5. The Outlook in Summary Recent results for the pure reactor synthesis problem and for the reactor-separator synthesis problem have this much in common: Both are remarkable in what we can now sayssometimes in great detailsabout reactors that provide access to the outermost limits of what is attainable. The results themselves, however, tell us that the prognoses for the two problems are not the same. In the case of the pure reactor synthesis problem, we now understand the sense in which classical reactor typessPFRs, CFSTRs, and DSRssplay universal roles in shaping the attainable region’s boundary. Even more remarkable are statements that we can now make about CFSTR and DSR designs that provide access to the outermost limits of the attainable region, statements that can be made in great detail even when the attainable region is unknown. Thus, it becomes possible to assert that a CFSTR with a specified feed can have as its effluent composition an attainable region extreme point only if its residence time takes one of just a few exceptional (and computable) values. Also, it becomes possible to say that a DSR with a specified sidestream composition can give rise to a composition trajectory of attainable region extreme points only if the axial distribution of sidestream addition rates conforms to very specific equations. However surprising these results are, they foreshadow difficulties for the pure reactor synthesis problem that should not be ignored. It is worth calling to mind a sequence of ideas that lead us toward the gloom: In a sense, the “connectors” between straight line sections and manifolds of PFR trajectories are what give that attainable region its form, for once the con-

nectors are specified, the PFR trajectories emanating from them are determined. The connectors, on the other hand, are made up of CFSTR operating points and DSR trajectories. We indicated earlier that the CFSTR operating points are likely to be few and isolated, so in the typical situation a connector will be filled out largely by DSR trajectories. (See the Remark section at the end of section 3.2.) But we also indicated in section 3.4 that the equations governing those DSR trajectories are, even in dimensions 4 or 5, extremely complex and inexorably tied to higher derivatives of kinetic rate functions that are usually not known with great precision. Thus, it seems unlikely that precise designs for DSRs that shape the attainable region’s boundary could be articulated either with confidence or with an eye toward practical implementation. Even apart from the DSR dilemma, it is worth stating again that, for the pure reactor synthesis problem, there is as yet no systematic and certain way to piece together those precise combinations of DSRs, CFSTRs, and PFRs that shape the attainable region’s boundary, even in three dimensions when visualization is possible. There is much of great beauty and value in what has been learned so far, but for the pure reactor synthesis problem, it is difficult to be optimistic about “knowing” reactor configurations that give rise to the attainable region’s boundary in any but the most simple circumstances. The situation for the reactor-separator synthesis problem is different and very much brighter. Granted the capacity for arbitrary separations, the CFSTR Equivalence Principle begins to make “knowable” the attainable region of molar-effluent-rate vectors, for it narrows a virtually limitless variety of reactor configurations that need to be considered down to configurations comprised of a small collection of CFSTRs. Thus, a seemingly intractable conceptual problem is reduced to one that is primarily computational. Moreover, meaningful constraints are not difficult to introduce. The pure reactor synthesis problem and the reactorseparator synthesis problem (with the capacity for arbitrary separations) are, in a sense, at opposite poles. In one case, we allow omnipotent separators and, in the other, no separators at all. This is not to say that the two problems, at the two extremes, are equally compelling. The reactor-separator synthesis problem, successfully resolved, provides limits on what can be attained from any steady-state design (with or without separators, omnipotent or not). Those limits become exact to the extent that one can effect (and is willing to pay for) whatever separations are required. The pure reactor synthesis problem, even if it could be solved systematically, tells us only what might be attained if separations were to be avoided at any cost. It is fortunate that the road to solution of the reactor-separator problem is the one with far fewer obstacles. Acknowledgment The author gratefully acknowledges support from the United States National Science Foundation. List of Symbols A1, ..., AN ) species names c ) local composition vector c0 ) CFSTR feed composition vector cI ) local molar concentration of species AI

Ind. Eng. Chem. Res., Vol. 41, No. 16, 2002 3761 cf1, cf2, ..., cfK ) feed composition vectors conv(‚) ) convex hull k′, k′′, k1, k2, k3 ) rate constants M ) molar-effluent-rate vector MI ) molar effluent rate of AI M0 ) molar-feed-rate vector MI0 ) molar feed rate of AI N ) number of species pmin ) minimum allowed pressure r(‚,‚) ) volumetric species-formation-rate function r(Ω) ) set of rate vectors corresponding to states in Ω RN ) vector space of N-tuples R+N ) set of all nonnegative N-tuples s ) rank of the reaction network T ) temperature Tmax ) maximum allowed temperature Tmin ) minimum allowed temperature V h ) maximum allowed total volume V* ) specified reactor volume Greek Letters R, R j ) sidestream addition rate (time-1) θ ) CFSTR residence time τ ) DSR and PFR residence time Ω ) composition-temperature constraint set

Literature Cited (1) Douglas, J. M.; Gaitonde, N. Y. Analytic Estimates of the Performance of Chemical Oscillators. Ind. Eng. Chem. Fundam. 1967, 6, 265. (2) Horn, F. J. M. Attainable and Non-attainable Regions in Chemical Reaction Technique. In Chemical Reaction Engineering (Proceedings of the Third European Symposium); Pergamon Press: London, U.K., 1964.

(3) Glasser, D.; Hildebrandt, D.; Crowe, C. A Geometric Approach to Steady Flow Reactors: The Attainable Region and Optimization in Concentration Space. Ind. Eng. Chem. Res. 1987, 26, 1803. (4) Hildebrandt, D.; Glasser, D.; Crowe, C. M. Geometry of the Attainable Region Generated by Reaction and Mixing: With and Without Constraints. Ind. Eng. Chem. Res. 1990, 29, 49. (5) Hildebrandt, D.; Glasser, D. The Attainable Region and Optimal Reactor Structures. Chem. Eng. Sci. 1990, 45, 2161. (6) Feinberg, M.; Hildebrandt, D. Optimal Reactor Design from a Geometric ViewpointsI. Universal Properties of the Attainable Region. Chem. Eng. Sci. 1997, 52, 1637. (7) Feinberg, M. Optimal Reactor Design from a Geometric ViewpointsII. Critical Sidestream Reactors. Chem. Eng. Sci. 2000, 55, 2455. (8) Feinberg, M. Optimal Reactor Design from a Geometric ViewpointsIII. Critical CFSTRs. Chem. Eng. Sci. 2000, 55, 3553. (9) Palanki, S.; Kravaris, C.; Wang, H. Y. Synthesis of State Feedback Laws for End-point Optimization in Batch Processes. Chem. Eng. Sci. 1993, 48, 135. (10) Palanki, S.; Kravaris, C.; Wang, H. Y. Optimal Feedback Control of Batch Reactors with a State Inequality and Free Terminal Time. Chem. Eng. Sci. 1994, 49, 85. (11) Feinberg, M.; Ellison, P. General Kinetic Bounds on Productivity and Selectivity in Reactor-Separator Systems of Arbitrary Design: Principles. Ind. Eng. Chem. Res. 2001, 40, 3181. (12) Feinberg, M. Chemical Reaction Network Structure and the Stability of Complex Isothermal Reactors: I. The Deficiency Zero and Deficiency One Theorems. Chem. Eng. Sci. 1987, 43, 2229. (13) Fukuda, K. Frequently Asked Question in Polyhedral Computation, available at http://www.ifor.math.ethz.ch/ fukuda/ polyfaq/polyfaq.html, 2000.

Received for review September 27, 2001 Revised manuscript received January 4, 2002 Accepted January 7, 2002 IE010807F

Toward a Theory of Process Synthesis - American Chemical Society

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