General Kinetic Bounds on Productivity and Selectivity in Reactor

Ind. Eng. Chem. Res. 2001, 40, 3181-3194

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General Kinetic Bounds on Productivity and Selectivity in Reactor-Separator Systems of Arbitrary Design: Principles Martin Feinberg* and Phillipp Ellison Department of Chemical Engineering, The Ohio State University, Columbus, Ohio 43210

For a specified set of feed streams and a specified network of chemical reactions, there is an almost limitless variety of reactor-separator designs that might be employed in enhancing the production rates of certain desired molecules while suppressing the production rates of undesired ones. Of special significance is the vast spectrum of very different reactor configurations available to the designer. Here we seek to determine sharp kinetic bounds on what can be achieved in steady-state reactor-separator systems of arbitrary design, subject perhaps to certain natural physical constraints. A primary conceptual tool is the continuous flow stirred tank reactor (CFSTR) equivalence principle, proven here, which asserts that the effluent of any steady-state reactor-separator design can be achieved arbitrarily closely by another steady-state design involving perhaps arbitrarily sharp separations but in which the only reactor components are s + 1 ideal CFSTRs, where s is the rank of the underlying network of chemical reactions. Thus, for the sole purpose of assessing bounds on the set of attainable effluents, it suffices to consider a surprisingly narrow and simple class of reactor configurations. 1. Introduction Our interest is in means to determine sharp kinetic bounds on what might be achieved in steady-state reactor-separator systems of arbitrary design, subject perhaps to certain physical constraints. Given a kinetics, for example, we might want to know the highest possible steady-state production rate of a certain desired species from prescribed feed streams if there is a specified availability of catalyst, if it is agreed that temperatures and pressures within any reactor component will not exceed certain levels, and if there are bounds set on the effluent rates of certain toxic side products. That is, we might want to know the highest production rate that can possibly be achieved as we consider all designs consistent with the constraints imposed. From a different perspective, we might want to know, for a specified production rate of some desired species, the absolutely smallest effluent rate of undesired side product that can be achieved in any steady-state reactor-separator design consistent with the constraints. Even when complete separations can be effected, it should not be supposed that undesired side products can be totally recycled at steady state without violating the stipulated temperature-pressure bounds within reactor components. This is obvious when undesired side products are produced irreversibly, for then the recycle would accumulate without bound. On the other hand, even when there are reactions that consume the undesired species, temperature-pressure contraints might preclude those reactions from proceeding at rates sufficient to keep pace with the recycle. In broader terms, it is important to know the best that can be hoped for in a reactor-separator design and to understand when no amount of configurative innovation will produce further gains, so long as certain design constraints are respected. At the very least, knowledge * To whom correspondence should be addressed. E-mail: [email protected]. Phone: 614-688-4882. Fax: 614-2923769.

of this kind provides important benchmarks against which existing designs can be measured. Throughout this paper it will be understood that we have in mind reactor-separator systems in which all mixtures are fluids or can at least be sensibly modeled as fluids. (It is common, for example, to invoke pseudohomogeneous models to describe solid-fluid catalytic systems such as packed or fluidized beds.) Although some assertions we make do depend strongly on a fluidlike picture, others (such as proposition 5.1) do not. Nevertheless, our preference is to presume fluid mixtures throughout in order to avoid a more fussy narrative. As a rough rule, the presumption of fluidlike mixtures enters not so much in establishing bounds on what might be produced but, rather, in demonstrating that those bounds are sharp. The demonstration is usually effected by construction of an idealized reactorseparator system that invokes fluidlike behavior and that achieves the bounds arbitrarily closely. To begin, we consider a reactor-separator system of unspecified design that operates at steady state. (The term “reactor-separator system” is not intended to preclude the possibility that, within the system, streams might be mixed.) For the moment, we envision the system as a “black box” into which certain feed streams enter and from which certain product streams exit. In particular, suppose that species A1, A2, ..., AN are fed into the system at molar flow rates M10, M20, ..., MN0. (That is, MI0 g 0 represents the total rate at which moles of AI are carried into the reactor-separator system by virtue of the presence of all feed streams.) Similarly, let M1, M2, ..., MN denote the total molar rates at which A1, A2, ..., AN are carried from the system by virtue of the presence of all effluent streams. It will be convenient to represent by M0 the molar feed rate vector [M10, M20, ..., MN0] and by M the molar effluent rate vector [M1, M2, ..., MN]. For a specified M0, our interest, roughly stated, is in determining the range of effluent possibilitiessthat is, the full range of values for Ms

10.1021/ie000697x CCC: $20.00 © 2001 American Chemical Society Published on Web 06/07/2001

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Figure 1. Varieties of reactor-separator systems.

that might emerge from our (unspecified) reactorseparator system as we consider a very broad spectrum of designs. For the purposes of this introduction, we presume that the reactions occurring among the species and the kinetic rate functions for those reactions are specified. (See, however, remark 7.4.) The reaction network and its kinetics are, of course, only part of what determines the effluent. There remains the dominant influence of design: of how streams are contacted for reaction, of how separators are employed in support of reactor units, of how recycle is implemented, and so on. Thus, our problem can be framed, at least tentatively, in the following way: Given a specified network of chemical reactions (with a specified kinetics) and given a specified molar feed rate vector M0, what is the full range of molar effluent rate vectors that can result from our reactorseparator system as we consider all possible steady-state designs? Still, there is a need for more refinement in what we mean by “all possible” designs. Should we admit for consideration reactors of arbitrary size? Should we contemplate reactor designs that require in the reactor’s interior arbitrarily high temperatures, arbitrarily high pressures, or arbitrarily high molar concentrations? Suppose, at least temporarily, that we agree to consider only designs in which the reactor volume does not exceed a specified value, perhaps very large, and in which local reactor conditions are such that the temperature, pressure, and molar concentrations are required to lie within certain ranges, again perhaps very large. Even with physical constraints such as these in place, there remains a vast variety of reactor-separator designs that can be imagined and still more that will almost certainly remain unimagined. This we illustrate schematically in Figure 1. [Note: We use the word “volume” as a concrete surrogate for the more general “capacity” or “size”. For example, in a pseudohomogeneous model, one might wish instead to measure a reactor’s capacity not by its total volume but, rather, by the total mass of available catalyst.] Thus, the problem of determining which values of M are attainable from M0 (subject to the physical con-

straints imposed) is, on its surface, daunting. Nevertheless, the problem itself and offshoots of it, such as the pollution abatement problem posed in the opening paragraph, are of clear importance. The apparent difficulty is in the breadth of designsssome transcending current imaginationsthat we are admitting for consideration. It will be useful to focus more sharply on the source of the difficulty, the crucial role played by reactor units. Their importance becomes evident once we consider an extreme situation: For a steady-state design in which there are no reactor units at all, there can be no distinction between the molar effluent rate vector and the molar feed rate vector (i.e., M ) M0). In this respect, when reactor units are absent, innovation in how separation units are employed has absolutely no effect on enlarging the spectrum of effluent possibilities nor does the sharpness of the separations achieved. Thus, what ultimately determines the extent to which M can depart from M0 is what happens in reactor units, with separator units in which no reactions occur playing only a supporting role. It would appear, then, that it is the great variety of conceivable reactor configurations that gives the set of effluent possibilities its breadth. And it is that very same great variety of reactor configurations that would seem to make so intimidating the problem of describing (or at least finding sharp bounds for) the full set of molar effluent rate vectors. In view of our interest in finding outer bounds for the set of effluent possibilities, we shall presume temporarily the capacity to make whatever separations we wish so long as the demands of material balances are met. That is, we presume the existence of a separator facility with the capacity to separate any given stream into distinct streams of specified composition and flow rates, so long as the specifications are consistent with material balances. Clearly, the set of molar effluent rate vectors attainable in the presence of this presumed capacity is larger than the set of effluents obtainable in its absence. In this sense, to know the larger set is to know an outer bound on the “true” set of effluent possibilities. That bound becomes sharp to the extent that any desired separation can, in fact, be effected for the mixture under study. It should be understood, then, that when we speak of a reactor-separator system design in the next section, we shall actually have in mind a specific reactor design supported by a separator facility of unspecified detail (apart from the nature of the material-balance-compliant streams entering and leaving it). It is not our intention to preclude the possibility that, in such a separator facility, streams might be both separated and mixed; we require only that entering and exiting streams comply with steady-state material balances. Remarks on Related Work. The problem of assessing kinetic bounds on what can be realized in the presence of omnipotent separators is complemented by another, somewhat different, problem on which there is an evolving body of work: assessing what can be attained without any separators at all, that is, by employing only reaction and mixing.1-12 The two problems are, in a sense, at opposite poles, one more relevant when separation costs are pivotal and the other when separation costs are substantially dominated by considerations of product yield and selectivity. Recent studies have attempted to extend, in interest-

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ing ways, ideas emerging from the pure reactor synthesis problem to the synthesis of reactor-separator systems.13-16 In varying degrees, these have employed special (but generally flexible) reactor-separator structures, some influenced by “attainable region” ideas in the pure reactor synthesis problem. 2. The CFSTR Equivalence Principle for Reactor-Separator Systems Even with the presumed capacity for arbitrarily sharp separations in force, the problem (described in section 1) of assessing the resulting range of effluent possibilities over all constraint-consistent reactor-separator designs would seem to be extremely formidable, for the variety of reactor configurations available to the designer remains vast. We want to argue that, for the narrow purpose of assessing what is, in fact, attainable, the class of reactor configurations that need actually be considered is surprisingly small. Before we can elaborate on this assertion, a small amount of preparation will be required: In work by Feinberg,17 the rank of a chemical reaction network was defined and means were provided to determine that rank for any given network. These ideas will be reviewed in section 4. For the moment, it suffices to say that the rank of a reaction network is a nonnegative integer that is easily computed and that indicates the maximum number of linearly independent reactions the network contains. (The rank of network (1), for example, is readily determined to be 3.)

The following proposition asserts that, for the sole purpose of ascertaining bounds on the set of possible molar effluent rate vectors, it is sufficient to study only reactor-separator systems in which the reactor component consists of a (usually) small number of CFSTRs. Moreover, an upper bound on the number of CFSTRs that need be considered is readily determined from a simple study of the underlying network of chemical reactions. (An argument in support of the proposition is given in section 5.) The 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.

Figure 2. Illustration of the CFSTR equivalence principle.

The principle is illustrated in Figure 2. It should be clearly understood that there is no claim here that the second (CFSTR-only) design is economically equivalent to the first design, only that the second design can achieve an effluent arbitrarily close to that achieved by the first with the same reactor capacity and without invoking in its reactors significantly higher (or lower) temperatures, pressures, or molar concentrations than were present in the first design’s reactor units. Indeed, the second design might require inordinately expensive separations, perhaps to almost pure components, and it might also require improbable internal pumping rates. (It should be remembered that our ultimate aim is the assessment of outer bounds on what can be attained, and with this in mind, we have assumed the capacity to implement whatever separations we wish. Otherwise, suppositions upon which the CFSTR equivalence principle rests are actually quite mild.) What the principle does say is that, for the sole purpose of assessing bounds on the set of molar effluent rate vectors attainable from steady-state reactor-separator systems (subject to the aforementioned physical constraints on reactor units), it suffices to consider only those designs in which the reactor component consists of s + 1 or fewer CFSTRs. Although such designs might indeed be entirely impractical when the costs of separation and recycle are taken into account, they can often be realized to excellent approximation (if at great expense). To the extent that the required separations can, in fact, be realized, the family of CFSTR-only designs provides sharp outer bounds on the productivity and selectivity that can be achieved by any steady-state design (subject to the stipulated physical constraints on reactor units). Although the CFSTR equivalence principle holds interest in its own right, it will serve for us primarily as a headline, one that tells part of the story but not all of it. In fact, ideas behind the principle will turn out to be more important than the principle itself. Certain of those ideas are essential in establishing bounds, while other ideas and presumptions (e.g., the presumed capacity to make arbitrary separations) play a role only in ensuring that the bounds are sharp. It will be useful, therefore, to keep track of how the various ideas and presumptions exert themselves and, in particular, to understand which conclusions depend on which hypotheses. With this in mind, we have organized the remainder

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of this paper in the following way: Section 3 contains some mathematical preliminaries. Section 4 provides a framework for addressing stoichiometric and kinetic issues. An argument in support of the CFSTR equivalence principle is given in section 5. In section 6 we extract various parts of the argument to provide general tools for assessing bounds on productivity and selectivity. Section 7 contains some remarks about numerical implementation, and in section 8 we provide an illustrative example. Concluding remarks are contained in section 9. Remark 2.1. In our statement of the CFSTR equivalence principle, we wrote M* > 0 to require, for technical reasons, that M* have strictly positive components. This is to say that, for the given design, we required all species to be represented in the effluent, perhaps in vanishingly small amounts. If some of the components of M* are zero, then the same conclusion obtains at least when the given (original) design can be perturbed slightly to produce a new steady-state design having a molar effluent rate vector M+ > 0 arbitrarily close to M*. Such a perturbed design might, for example, result from permitting a vanishingly small efflux of those species formerly subjected to total recycle. Remark 2.2. Figure 2 suggests, as is often the case, a clear distinction between reactor elements and separation elements of a reactor-separator system. When certain hybrid components such as reactive distillation columns are present, the distinction is not so clear. In this case, the reactor elementssand, in particular, the total reactor volumesshould be understood to embrace all mixtures present within which reactions occur. Thus, for example, if a reactor-separator system consists only of a single reactive distillation column and if reactions are deemed to occur only in the liquid phases on the various trays, then the totality of all such liquid phases would constitute the reactor elements of the system. 3. Mathematical Preliminaries Here we introduce only the most basic ideas, terminology, and notation. Additional mathematics will be introduced in subsequent sections as the need arises. The usual vector space of N-tuples of real numbers is denoted RN. The symbol RN + is reserved for the set of vectors in RN with entirely nonnegative components. The standard basis for RN is denoted e1, e2, ..., eN. That is,

e1 ) [1, 0, 0, ..., 0], e2 ) [0, 1, 0, ..., 0], ..., eN ) [0, 0, 0, ..., 1] Recall that a set of vectors {v1, v2, ..., vk} in RN is linearly dependent if there are numbers {R1, R2, ..., Rk}, not all zero, such that

R1v1 + R2v2 + ... + Rkvk ) 0 Otherwise, the set {v1, v2, ..., vk} is linearly independent. A set Ω of vectors in RN is said to have rank r (where r is an integer) if in Ω there is a linearly independent set of r vectors but no linearly independent set of r + 1 vectors. A nonempty set U of vectors in RN is a linear subspace if, for all vectors u and u′ in U and all real numbers R and R′, the vector Ru + R′u′ is a member of U. When U is a linear subspace, the rank of U is called the dimension of U.

The span of a set {v1, v2, ..., vk} of vectors in RN is the linear subspace of RN consisting of all vectors of the form

R1v1 + R2v2 + ... + Rkvk

(2)

where R1, R2, ..., Rk are real numbers. The dimension of the span of {v1, v2, ..., vk} is identical to the rank of the set {v1, v2, ..., vk}. A vector v* in RN is a convex combination of the vectors {v1, v2, ..., vk} if there are numbers {λ1, λ2, ..., λk} such that

v* ) λ1v1 + λ2v2 + ... + λkvk λi g 0, i ) 1, 2, ..., k and

λ1 + λ2 + ... + λk ) 1 Let Ω be a set of vectors in RN. Then Ω is convex if, for all x and y in Ω and for each number 0 e λ e 1, the vector λx + (1 - λ)y is a member of Ω. If Ω is a (not necessarily convex) set of vectors in RN, then the convex hull of Ω, denoted conv(Ω), is the smallest convex set in RN that contains Ω. In fact, conv(Ω) is the set of all (finite) convex combinations of elements of Ω. Remark 3.1. Note that if a vector v* is a convex combination of the vectors {v1, v2, ..., vk}, then v* can, in a sense, be viewed as an “average” of {v1, v2, ..., vk} (with weighting factors λ1, λ2, ..., λk). In this sense, the convex hull of a set Ω in RN is the set of all possible averages of vectors in Ω. The norm of a vector x ) [x1, x2, ..., xN] in RN is the nonnegative number, denoted ||x||, defined by

||x|| :) (x12 + x22 + ... + xN2)1/2 If x is a vector in RN and r is a positive number, then the open ball of radius r with center x is the set of vectors

Br(x) :) {y ∈ RN : ||y - x|| < r} Let Ω be a set of vectors in RN. Then Ω is open if, for each x ∈ Ω, there is a positive number (depending perhaps on x) such that the open ball B(x) is contained entirely in Ω. The boundary of Ω is the set of all x ∈ RN having the property that, for every > 0, the open ball B(x) contains elements in Ω and elements not in Ω. We say that Ω is closed if Ω contains its entire boundary. Whether or not Ω is closed, the closure of Ω, denoted as cl(Ω), is the set consisting of Ω taken together with the boundary of Ω. A set Ω in RN is connected if there is no partition of Ω into two subsets, Ω1 and Ω2, such that Ω1 and Ω2 are contained, respectively, in open sets O1 and O2, where O1 and O2 are disjoint. Otherwise, Ω is disconnected. In this case, Ω is the disjoint union of maximal connected “pieces” called the (topological) components of Ω.18 4. Stoichiometric and Kinetic Preliminaries Consider a chemical reaction network involving N species, A1, A2, ..., AN. With each reaction in the network we associate a reaction vector in RN by means of a


procedure we illustrate with network (3) (for which N R

A1 98 A2 + A3 β

γ

2A2 98 A3 a A 4 + A5 δ

(3)

) 5). With the reaction A1 f A2 + A3 we associate the vector e2 + e3 - e1 ) [-1, 1, 1, 0, 0]; with 2A2 f A3 we associate the vector e3 - 2e2 ) [0, -2, 1, 0, 0]; with A3 f A4 + A5 we associate the vector e4 + e5 - e3 ) [0, 0, -1, 1, 1]; and with A4 + A5 f A3 we associate the vector e3 - (e4 + e5) ) [0, 0, 1, -1, -1]. The rank of a reaction network is the rank of the network’s set of reaction vectors. Thus, for example, network (3) has rank 3: The set of three reaction vectors

{[-1, 1, 1, 0, 0], [0, -2, 1, 0, 0], [0, 0, -1, 1, 1]} is linearly independent, and there is no linearly independent set of four reaction vectors for the network. Similarly, the rank of network (1) is also 3. (The rank of a reaction network is identical to the rank of the matrix obtained by stacking the network’s reaction vectors. See ref 17 and Appendix A in ref 9 for discussions of how the rank of a reaction network can be determined systematically.) The stoichiometric subspace for a reaction network is the span of the reaction vectors for the network. That is, for a reaction network with N species, the stoichiometric subspace is the linear subspace of RN consisting of all linear combinations of the network’s reaction vectors. The dimension of the stoichiometric subspace coincides with the rank of the network. For network (3) the stoichiometric subspace is the set of all vectors in R5 expressible in the form

xR[-1,1,1,0,0] + xβ[0,-2,1,0,0] + xγ[0,0,-1,1,1] + xδ[0,0,1,-1,-1] (4) where xR, xβ, xγ, and xδ are arbitrary real numbers. A kinetics for a reaction network is an assignment to each reaction network of a (volumetric) occurrence rate function that indicates how, for the reaction, the local molar occurrence rate per unit volume depends on the local composition and the local temperature. (Hereafter, we denote by c the local composition vector [c1, c2, ..., cN], where cI is the local molar concentration of species AI.) Thus, a kinetics for network (3) would consist of functions KR(‚,‚), Kβ(‚,‚), Kγ(‚,‚), and Kδ(‚,‚), where, for example, KR(c,T) would give the local molar occurrence rate per unit volume of reaction A1f A2 + A3 when c is the local composition and T is the local temperature. (Greek letters alongside the reactions in network (3) serve only as indices; they do not indicate rate constants.) Note that occurrence rate functions take nonnegative values. When, for example, the kinetics is of the usual mass-action kind, the occurrence rate functions for network (3) take the form

KR(c,T) ) kR(T) c1, Kβ(c,T) ) kβ(T) c22 Kγ(c,T) ) kγ(T) c3, Kδ(c,T) ) kδ(T) c4c5 where kR(T), kβ(T), kγ(T), and kδ(T) are (temperaturedependent) rate constants for the corresponding reac-

tions. In any case, we shall hereafter assume that occurrence rate functions are continuous. A kinetic system is a reaction network taken together with a kinetics. The (volumetric) species formation rate function for a kinetic system is the vector-valued function [generally denoted r(‚,‚)] formed by taking the sum of the reaction vectors, each multiplied by the corresponding occurrence rate function. Thus, for a kinetic system in which (3) is the underlying network of chemical reactions, the species formation rate function is formulated as follows:

r(c,T) ) [r1(c,T), r2(c,T), r3(c,T), r4(c,T), r5(c,T)] :) KR(c,T) [-1, 1, 1, 0, 0] + Kβ(c,T) [0, -2, 1, 0, 0] + Kγ(c,T) [0, 0, -1, 1, 1] + Kδ(c,T) [0, 0, 1, -1, -1] When the local composition is c and the local temperature is T, rI(c,T) is the local molar production rate per unit volume of species AI. For our example,

r1(c,T) ) -KR(c,T) r2(c,T) ) KR(c,T) - 2Kβ(c,T) r3(c,T) ) KR(c,T) + Kβ(c,T) - Kγ(c,T) + Kδ(c,T) r4(c,T) ) r5(c,T) ) Kγ(c,T) - Kδ(c,T) For any kinetic system, it is evident that, for each composition and temperature, the value of the species formation rate function is a linear combination of the reaction vectors for the underlying network of chemical reactions. This is to say that the species formation rate function takes values in the stoichiometric subspace for the underlying network of chemical reactions. 5. The CFSTR Equivalence Principle: Sketch of a Proof Our aim in this section is to provide an argument in support of the CFSTR equivalence principle. In at least certain places, the argument, done properly, has an inherently technical character. With this in mind, we prefer to convey here only a rough, intuitive sense of how those parts proceed. To compensate for this, we have provided some supplementary discussion in the appendix. The argument will proceed by way of a proposition. It should be noted that the proposition itself is independent of any presumption about our capacity to make sharp separations. By isolating the proposition in this way, we shall have a better understanding of what remains true even in the absence of any such presumption. Throughout the argument we consider a fixed steadystate reactor-separator system in which there are N species, in which the underlying network of chemical reactions gives rise to a stoichiometric subspace S ⊂ RN, and in which the species formation rate function is r(‚,‚). As before, the dimension of the stoichiometric subspace is s, the molar feed rate vector is M0, the molar effluent rate vector is M*, and the total reactor volume is V*. We presume that r(‚,‚) is continuous. For the moment, we presume about M* only that it has nonnegative components. In preparation for the statement of the proposition, we consider the totality of all local composition-tem-

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perature pairs existing within reactor elements of the system under consideration, and we denote by Γ the closure of all such (c, T) states. This is to say that Γ is containing all (c, T) the smallest closed set in RN+1 + states exhibited within the system’s reactor elements. Hereafter, we call Γ the reactor state set for the system. For example, if in our reactor-separator system the only reactor element is of the classical plug-flow variety, consisting of all (c, then Γ would be the curve in RN+1 + T) states exhibited along the reactor’s entire length (including those at the reactor’s end points). If the only reactor element is a perfectly mixed CFSTR, then Γ would be a single point in RN+1 consisting of the CFSTR operating state, (c†, T†), where c† and T† are the composition and temperature within the reactor vessel. (Were the CFSTR not perfectly mixed, then Γ would be a more diffuse set in RN+1 + .) Note that the set Γ can have two or more topological components: Suppose that in our system the reactor elements are an ideal plug-flow reactor (PFR) followed by an ideal CFSTR. In this case, Γ would consist of the (consisting of (c, T) pairs union of a curve in RN+1 + exhibited along the PFR) and a single point in RN+1 + (consisting of the CFSTR operating state). Apart from some degenerate situations, the point will be disjoint from the curve, in which case Γ would consist of two topological components. Should the reactor elements consist of, say, 10 CFSTRs, then Γ would consist of the 10 operating points in RN+1 + ; if these were distinct, Γ would have 10 topological components. We return now to consideration of the general reactor-separator system described earlier in this section. Whatever its design, we want to argue that the following proposition holds true: Proposition 5.1. Consider an arbitrary steady-state reactor-separator system, and for the system, let r(‚,‚), s, M*, M0, V*, and Γ be as described above. For some p e s + 1, there are p distinct composition-temperature states

{(c1, T1), (c2, T2), ..., (cp, Tp)}

We denote by R the vector [R1, R2, ..., RN] ∈ RN of total molar production rates for our reactor-separator system. Here RI is the total molar rate of production of species AI summed over all reactor units. Thus, a (steady-state) material balance over the reactorseparator system takes the form

M* - M0 ) R

Recalling that V* is the total reactor volume, we let

r† :)

(V*1 )R

(8)

We call r† the global-average volumetric molar production rate vector. Here, on intuitive grounds, we make an assertion that is developed more fully in the appendix: The vector r† is, in some sense, an average of all local volumetric molar production rate vectors exhibited within the reactor elements. (In very rough terms, we expect r† to be a weighted average of all r(c,T) exhibited within reactor elements, each such vector weighted by the volume fraction of mixture in state (c,T).) Because the set r(Γ) contains all volumetric species production rate vectors exhibited in reactor elements and because the convex hull of r(Γ) contains all possible weighted averages of vectors in r(Γ), we assert that

r† is contained in the convex hull of r(Γ)

(9)

Again, this inclusion is developed more fully in the appendix. At this point in the proof, we digress briefly to state Caratheódory’s theorem, one of the cornerstones of the modern theory of convex sets.19 Carathe´ odory’s Theorem. Let Q be a set in a vector space of dimension n. If x† lies in the convex hull of Q, then it is possible to represent x† as a convex combination of no more than n + 1 vectors in Q. That is, there are vectors {x1, x2, ..., xp} in Q and numbers {λ1, λ2, ..., λp}, where p e n + 1, such that

all residing in Γ, and positive numbers {V1, V2, ..., Vp} such that

M* ) M0 + V1r(c1,T1) + V2r(c2,T2) + ... + Vpr(cp,Tp) (5)

(7)

x† ) λ1x1 + λ2x2 + ... + λpxp λ1 + λ2 + ... + λp ) 1 and

0 e λi e 1, i ) 1, 2, ..., p

and

V1 + V2 + ... + Vp ) V*

(6)

Moreover, if Γ consists of no more than s topologically connected components, then p can be chosen to be less than or equal to s. In particular, p can be chosen to be less than or equal to s if Γ is connected. Proof. Associated with each local (c, T) state in Γ is a local volumetric species production rate vector r(c,T). We denote by r(Γ) the set of volumetric species production rate vectors that derive from states in Γ. That is

r(Γ) :) {r(c,T) ∈ RN : (c, T) ∈ Γ} Thus, all local volumetric species production rate vectors exhibited within reactor elements in the system under consideration are members of r(Γ).

If Q has no more than n topologically connected components, then p can be chosen not to exceed n. Returning to the main line of argument, we examine consequences that Caratheódory’s theorem has for the inclusion (9). Although r† and all members of the set r(Γ) reside in the vector space RN, they are, in particular, members of S, the stoichiometric subspace for the underlying network of chemical reactions. Note that the stoichiometric subspace is itself a vector space of dimension s e N. Applying Caratheódory’s theorem in the vector space S, it follows from (9) that r† has a representation as a convex combination of p vectors in r(Γ), say {r(c1,T1), r(c2,T2), ..., r(cp,Tp)}, where p e s + 1. That is, there are numbers {λ1, λ2, ..., λp} such that

r† ) λ1r(c1,T1) + λ2r(c2,T2) + ... + λpr(cp,Tp) (10)


λ1 + λ2 + ... + λp ) 1

(11)

0 e λi e 1, i ) 1, 2, ..., p

(12)

After making the identification Vi ≡ λiV*, i ) 1, 2, ..., p, we can see that proposition 5.1 is a consequence of (7), (8), and (10)-(12). (The parenthetical last sentence of proposition 5.1 is a consequence of Caratheódory’s theorem and the fact that, for a continuous map, the image of a set having k topologically connected components can have no more than k topologically connected components.) This completes the proof of proposition 5.1. Our aim now is to show that, granted the capacity to make whatever steady-state separations we choose (consistent with mole balances on the input and output streams), the molar effluent rate vector M*, provided that it has strictly positive components, can be attained arbitrarily closely in a reactor-separator system in which the only reactor elements are p CFSTRs, where p e s + 1 (or p e s if Γ has s or fewer topological components). In particular, the CFSTRs will have the volumes V1, V2, ..., Vp described in proposition 5.1, and the CFSTR operating states will be arbitrarily close to the composition-temperature pairs {(c1, T1), (c2, T2), ..., (cp, Tp)} described in that proposition. We begin by supposing that, for i ) 1, 2, ..., p, the ci values given by proposition 5.1 are such that ciL > 0, i ) 1, ..., p and L ) 1, ..., N. (Note the strict positivity.) In this case, we can choose ξi, i ) 1, ..., p, sufficiently positive as to ensure that

ξiciL - VirL(ci,Ti) g 0, L ) 1, ..., N

0

0

F i :) [F i1, F i2, ..., F

0

iN],

F

ξhicjiL - VirL(c j i,Ti) g 0, L ) 1, ..., N

(18)

Moreover, we can let

i :) 1, 2, ..., p (14)

where 0

where p e s + 1 (and p e s when Γ has s or fewer topologically connected components). Recall, however, our supposition that ciL > 0, i ) 1, ..., p, L ) 1, ..., N. If we merely have ciL g 0, i ) 1, ..., p, L ) 1, ..., N, then we cannot be certain that, for each i, ξi > 0 can be chosen in such a way as to have (13) hold. In this case, we can, for i ) 1, ..., p, choose c ji arbitrarily close to ci, but with cjiL > 0, L ) 1, ..., N, and given such a choice, we can then choose ξh1, ..., ξhp sufficiently positive as to satisfy

(13)

With ξ1, ..., ξp chosen in this way, we let 0

Figure 3. CFSTR-based reactor-separator system.

h 0i1, F h 0i2, ..., F h 0iN], i :) 1, 2, ..., p (19) F h 0i :) [F where

iL :)

ξiciL - VirL(ci,Ti), L ) 1, ..., N, i ) 1, ..., p (15)

(20)

and we can also define M h * by

Thus, we have

Vir(ci,Ti) + F0i - ξici ) 0, i ) 1, 2, ..., p

(16)

We can view the ith equation in (16) as a steady-state mole balance for a CFSTR having volume Vi, molar feed rate vector F0i, effluent volumetric flow rate ξi, and operating state (ci, Ti). Now consider the reactor-separator system shown in Figure 3. The reactor elements are the p CFSTRs described by (16), and the “black box” separator facility is one having input and output streams with properties indicated in the figure. We need to show that these streams are, in fact, consistent with a steady-state mole balance. In particular, we need to verify that p

M* - M0 )

F h 0iL :) ξhicjiL - VirL(c j i,Ti), L ) 1, ..., N

(ξici - F0i) ∑ i)1

(17)

However, (17) is an immediate consequence of (5) and (16). Thus, if the separations indicated in Figure 3 could be realized, then the output M* could be realized exactly in a steady-state reactor-separator system in which the only reactor elements are p CFSTRs having properties stated in the CFSTR equivalence principle (section 2),

M h * ) M0 + V1r(c j 1,T1) + V2r(c j 2,T2) + ... + Vpr(c j p,Tp) (21) Now consider a slight perturbation of the design shown in Figure 3, taken to have the same flow sheet but with M*, ci, ξi, and Fi, i ) 1, ..., p, in Figure 3 replaced by their overbarred counterparts. The new design again satisfies the steady-state mole balances and has as its reactor elements only p CFSTRs with operating states close to the original ones in Γ. It should be clear from (5) and (21) and the continuity of r(‚,‚) that the perturbed effluent, M h *, can be made arbitrarily close to the original M* by choosing c j i sufficiently close to ci (but with all cjiL strictly positive). In particular, if (as in the statement of the CFSTR equivalence principle in section 2) we have M*L > 0, L ) 1, ..., N, then we can also arrange to have M h *L > 0, L ) 1, ..., N. This completes our argument in support of the CFSTR equivalence principle. Remark 5.1. If, for the original M*, we only have M*L g 0, L ) 1, ..., N, then we need to consider that M h *, constructed as above, might have some of its components negative. In this case, however, we can choose instead to approximate not the original M* but, rather,

3188


another attainable steady-state molar effluent rate vector, M+, itself arbitrarily close to M* but with M+L > 0, L ) 1, ..., N. There is a presumption here, although an intuitively reasonable one, that the original (given) design that yielded M* can be perturbed very slightly to produce M+ > 0 arbitrarily close to M*. See remark 2.1. Note that this technical issue becomes moot if, for the original reactor-separator system, the reactor state set Γ is such that c > 0 for each (c, T) in Γ. (Recall that if each of the ci values given by proposition 5.1 has strictly positive components, then there is no technical difficulty in constructing the equivalent CFSTR-only reactorseparator system shown in Figure 3.) Remark 5.2. Having completed the argument underlying the CFSTR equivalence principle, we think it worthwhile to point out once again that certain crucial parts of the argumentsparts that will have utility of their ownsare independent of assumptions about our capacity to make whatever separations we desire or about the positivity of M*. In particular, with r(‚,‚), M0, V*, and Γ taken as described above, the molar effluent rate vector M* for our arbitrary reactor-separator system invariably has a representation

M* ) M0 + V*r†

(22)

r† is contained in the convex hull of r(Γ)

(23)

where

Furthermore, Caratheódory’s theorem gives the more detailed representation of M* indicated in proposition 5.1. These representations did not draw on any suppositions about our capacity to make separations. 6. Assessing Bounds on Productivity and Selectivity The CFSTR equivalence principle indicates that, despite the vast range of reactor configurations available to the designer, it suffices to consider a very narrow class of designs in order to arrive at bounds on what might be produced from a given feed. Our purpose in this section is to distill arguments underlying the principle into a single proposition that will serve to guide the actual assessment of bounds in particular problems. From a practical standpoint, it is unreasonable to admit for consideration designs that require extremely high (or low) temperatures or extremely high (or low) molar concentrations. With this in mind, we suppose that the only designs admitted for consideration are those such that, within any reactor unit, local (c, T) pairs lie within a specified constraint set Ω ⊂ RN+1 + , perhaps very large. Imagine, for example, that the only designs admitted for consideration are those such that nowhere within any reactor unit do the temperature and pressure lie outside certain extremes; thus, everywhere within the various reactor units, we have Tmin e T e Tmax and pmin e p e pmax. Suppose also that, within the permissible temperature-pressure range, the species {A1, A2, ..., AN} can be regarded to comprise an ideal gas mixture so that the local molar concentrations are required to satisfy the equation

N

p

∑ cL ) RT L)1

(24)

where p and T are the local pressure and temperature. In this case, Ω would consist of all (c, T) pairs in RN+1 + such that Tmin e T e Tmax and such that (25) is satisfied.

pmin RT

N

e

∑

cL e

L)1

pmax RT

(25)

With this as background, we consider the set of molar effluent rate vectors that might possibly emerge from steady-state reactor-separator systems for which the molar feed rate vector is M0 and for which the total volume of reactor components is not greater than some upper limit V h . Hereafter, we suppose that the chemistry is such that the rank of the underlying network of chemical reactions is s, that the (continuous) species formation rate is r(‚,‚), and that designs are restricted in such a way that local (c, T) pairs within reactor units are required to lie in a closed constraint set Ω ⊂ RN+1 + . Note that elements of Ω might lie on the boundary of N+1 is the nonnegative orthant of RN+1 + . (Recall that R+ N+1 R .) For technical reasons, we hereafter assume that, if (c, T) is in Ω, then every open ball in RN+1 centered at (c, T) contains elements of Ω that lie in the interior of RN+1 + . In particular, we assume that every member of is arbitrarily close Ω that lies on the boundary of RN+1 + to a state in Ω at which all species concentrations are strictly positive. Also as a mild technical supposition, we suppose that, among all molar effluent rate vectors attainable by steady-state reactor-separator designs consistent with the requirements indicated, there is at least one with strictly positive components. That is, we assume that, for the given feed, there is at least one steady-state constraint-consistent design for which all N species are in the effluent, perhaps at vanishingly small flow rates. Proposition 6.1 serves to characterize the set of all possible molar effluent rate vectors that might emerge from steady-state reactor-separator designs consistent with the constraints indicated. Proposition 6.1. Characterization of Attainable Effluents. An element M ∈ RN is attainable as the molar effluent rate vector of a steady-state constraint-consistent reactor-separator design only if it has a representation of the form p

M ) M0 +

Vir(ci,Ti) ∑ i)1

(26)

where

pes+1

(27)

ML g 0, L ) 1, 2, ..., N

(28)

Vi g 0, i ) 1, 2, ..., p

(29)

p

Vi e V h ∑ i)1

(30)

(ci, Ti) is contained in Ω, i ) 1, 2, ..., p

(31)

If we grant the capacity to make arbitrary separations (consistent with steady-state material balances on streams


entering and leaving separation units), then, up to closure, the converse is true as well: Each vector M ∈ RN representable in the form (26)-(31) is (or is arbitrarily close to) a molar effluent rate vector for some steadystate reactor-separator design. In fact, each such M is (or is arbitrarily close to) the molar effluent rate vector for a steady-state reactor-separator design in which the reactor units consist of p CFSTRs, where p e s + 1. (If Ω has s or fewer topological components, then, for the purposes of this entire proposition, p can be taken to be less than or equal to s.) Sketch of the Proof. Although proposition 6.1 follows in a fairly straightforward way from proposition 5.1 and other ideas underlying the CFSTR equivalence principle, it is perhaps worthwhile to fill in a few of the more subtle details, at least in a cursory way. In part (i) we indicate why each molar effluent rate vector emerging from a constraint-consistent design has a representation of the form (26)-(31). In part (ii) we indicate why, when arbitrary separations are deemed feasible, every vector representable in the form (26)(31) can be realized arbitrarily closely as the effluent of a constraint-consistent reactor-separator system in which the reactor elements consist of p CFSTRs, where p e s + 1 (and p e s when Ω has s or fewer topological components). (i) Suppose that M* ∈ RN + is a molar effluent rate vector that derives from a constraint-consistent reactorseparator design in which the total reactor volume is V*, where V h g V* > 0, and in which the reactor state set is Γ. By supposition, Γ is contained in Ω, so a representation of M* of the kind given by (26)-(31) is a straightforward consequence of proposition 5.1. Moreover, from remark 5.2 we have M* ) M0 + V*r†, where r† is some vector contained in the convex hull of r(Γ) and, consequently, in the convex hull of r(Ω). Here r(Ω) is defined by

r(Ω) :) {r(c,T) ∈ RN : (c, T) ∈ Ω}

(32)

Thus, if Ω [and, therefore, r(Ω)] has s or fewer topological components, then it follows from Caratheódory’s theorem that M* has a representation of the kind given by (26)-(31) with p e s, this despite the fact that Γ might have more than s components. (Note that (31) requires membership in Ω, not in the smaller set Γ.) (ii) Suppose, on the other hand, that M* ∈ RN has a representation of the kind given by (26)-(31). Then it is not difficult to see that, for some V*, with V h g V* g 0, we have

M* ) M0 + V*r†

(33)

r† is a vector in the convex hull of r(Ω)

(34)

where

We note that if p ) s + 1 in the presumed representation of M* and if Ω has s or fewer topological components, then (33), (34), and Caratheódory’s theorem guarantee that there is a representation for M* of the kind shown in (26)-(31) for which p e s. (In this case, we suppose that such a representation has been chosen.) Granted the capacity to make arbitrary separations and provided that M* lies in the interior of RN + , we can construct from (26)-(31) a steady-state CFSTR-only reactor-separator design that yields M* as its molar

effluent rate vector (or that approximates M* arbitrarily closely) along lines similar to those in section 5. When M* lies on the boundary of RN + , certain small technical issues require attention. (See remark 5.1.) Here we draw on the supposition that there is some constraint-consistent design yielding a molar effluent rate vector M+ in the interior of RN + . Prior arguments in part (i) ensure that M+ has a representation of the form (26)-(31), and from this it can be shown that, for every 0 < λ < 1, the vector

M**(λ) :) λM* + (1 - λ)M+ also lies in the interior of RN + and has a representation of the form (26)-(31). (Moreover, in that representation p can be taken as less than or equal to s when Ω has s or fewer components.) If we grant the capacity to make arbitrary separations, then, for each 0 < λ < 1, we can construct a CFSTR-only reactor-separator system (as in proposition 5.1) that has M**(λ) as its effluent or that has an effluent arbitrarily close to M**(λ). Note, however, that M**(λ) can be made as close as we want to M* by taking λ sufficiently close to 1. Thus, M* can be approached arbitrarily closely by the effluent of a design in which the only reactors are p CFSTRs. This completes our sketch of the argument underlying proposition 6.1. To see how, at least in principle, proposition 6.1 makes tangible the full set of attainable effluents (at least when arbitrary separations are deemed feasible), we can imagine that Ω is discretized into a great many (c, T) states and that all combinations of s + 1 of these states are enumerated systematically. Similarly, we can imagine discretizing finely the polyhedron in Rs+1 consisting of vectors of the form V ) [V1, V2, ..., Vs+1] that satisfy (29) and (30). By scanning over all points in the discretized polyhedron and all combinations of s + 1 states in the discretized Ω, we can use (26) and (28) to generate the full set of attainable effluents within very good approximation. (Here s + 1 can be replaced by s if Ω has s or fewer topological components.) We certainly do not advocate this as a practical way to study the set of attainable molar effluent rate vectors. (See instead section 7.) Rather, this conceptual calculation indicates how that attainable set becomes “knowable” even when the full range of reactor configurations (including highly innovative ones) remains unknowable. More important still is that the characterization of attainable effluents given by (26)-(31) provides a real foothold for solution of a seemingly insoluble problem: the assessment of sharp bounds on productivity and selectivity that might be realized over all possible steady-state constraint-consistent designs. In most instances it will not be the determination of all possible effluents that is of immediate interest but, rather, the solution of concrete optimization problems. Thus, for example, if A1 is a highly valued molecule whose production rate is to be maximized, if A2 and A3 are pollutants whose production rates are to be no h 3, respectively, if A4, ..., AN are greater than M h 2 and M inconsequential side products, and if design constraints such as those imposed above are to be respected, then to determine outer bounds on the productivity achievable by any steady-state reactor-separator design consistent with the constraints, one would want to solve an optimization problem of the following kind: Maxih2 mize M1 subject to the constraints (26)-(31), M2 e M

3190


and M3 e M h 3. Problems of this kind are discussed in remark 7.2. 7. Remarks on Implementation Although it is our intention to address computational issues in another paper, we can indicate here the sense in which proposition 6.1 provides a basis for numerical solution of certain problems. Our purpose in this section is not to describe the most efficient numerical strategy but, rather, to illustrate briefly one of perhaps several ways in which bounds on what is attainable might be computed. We proceed by way of a series of remarks. In all that follows, M0 is the molar feed rate vector, r(‚,‚) is the species formation rate function, and Ω ⊂ is the composition-temperature constraint set. RN+1 + Remark 7.1. It is useful to observe that (26)-(31) describe a convex set. In fact, with V denoting the interval [0, V h ], the set described by (26)-(31) is identical to the set

M :) conv(M0 + V ‚ r(Ω)) ∩ RN +

(35)

where

M0 + V ‚ r(Ω) :) {M ∈ RN : M ) M0 + Vr#, V ∈ V, r# ∈ r(Ω)} (36) When Ω is closed and bounded, the set M is also closed and bounded. Remark 7.2. Finding Bounds on Yields for Processes Subject to Effluent Constraints. Suppose, as before, that A1 is a highly valued species whose production rate is to be maximized, that A2 and A3 are pollutants whose molar effluent rates are to be no h 3, respectively, and that A4, ..., greater than M h 2 and M AN are inconsequential side products. Suppose also that Ω is closed and bounded. Then, to determine outer bounds on the productivity achievable by any steadystate reactor-separator design consistent with the constraints, we would want to solve an optimization problem of the following kind: Maximize M1 subject to the constraints

M ) [M1, M2, ..., MN] is contained in M, M2 e M h 2, and M3 e M h 3 (37) Here M is as in remark 7.1. We can proceed in the following way: By taking discrete approximations to Ω and V, we can approximate the set M0 + V ‚ r(Ω) as a large number of points in RN. In turn, the set conv(M0 + V ‚ r(Ω)) can be approximated by any of several computer implementations of convex-hull-finding algorithms (given, as input, the discrete approximation to M0 + V ‚ r(Ω)). Such algorithms20 typically have the capacity to describe the resulting (approximate) convex hull as a polyhedron specified by a (sometimes large) number of bounding hyperplanes. In our case, for example, a vector M ∈ RN would be identified as a member of

conv(M0 + V ‚ r(Ω))

(38)

if and only if it satisfies a set of inequalities (given as output by the convex-hull-finding program) of the form

aθ1M1 + aθ2M2 + ... + aθNMN e bθ, θ ) 1, 2, ..., P (39) where P is the number of bounding hyperplanes. To require that M also be a member of M, we would add to (39) the constraints

MI g 0, I ) 1, 2, ..., N

(40)

Finally, to complete the constraint set (37), we would add the environmental requirements

M2 e M h2

and

M3 e M h3

(41)

In the discrete setting, then, the problem of finding productivity bounds becomes this: Find M ) [M1, M2, ..., MN] ∈ RN satisfying (39)-(41) such that M1 is maximized. Note that this is a conventional linear programming problem for which solution implementations are readily available. A good resource for advice on computational issues related to convex hulls, linear programming, and their interplay is the paper by Fukuda.21 In particular, he shows how convex-hull-finding problems are often equivalent to linear programming problems, and he suggests that such a recasting entirely in terms of linear programming can be computationally advantageous in high dimensions (in our case, when there are more than just a few species). [We are grateful to L. Biegler and his students for suggesting that a computational approach virtually identical to the one advocated by Fukuda might be more efficient than the procedure described here.] The technique described here, while perhaps not the most computationally efficient for large problems, seems more geometrical in spirit and, therefore, more readily visualized. Remark 7.3. Determining the Full Set of Attainable Yields. For a prescribed feed, kinetics, and (closed and bounded) constraint set, we can ask about the nature of the full set of molar effluent rate vectors attainable from all steady-state reactor-separator designs. Taken together, proposition 6.1 and remark 7.1 indicate that the full attainable set is contained in M and actually coincides with M, up to closure, if we are granted the capacity to make whatever separations we choose. (That the full attainable set is contained in M also follows easily from the discussion containing (7)(9).) To know M, on the other hand, is to have a means to determine whether a given vector, say M* ∈ RN + , is a member of M. In effect, this amounts to determining whether M* belongs to the set (38). Recall from remark 7.2, however, that convex-hull-finding programs can describe, to a good approximation, a set such as (38) by a system of linear inequalities such as that shown in (39). It is not a difficult matter to check whether the prescribed M* is consistent with the resulting linear system of inequalities. (Again, see the paper by Fukuda21 for a different computational approach.) Remark 7.4. Is It Important To Have Good Models or To Know Mechanistic Details? It should be noted that, in remarks 7.1-7.3, it is only the volumetric species formation rate function r(‚,‚) that is of significance, not the individual elementary reactions and their occurrence rates. At least in principle, r(‚,‚) is measurable, for example, in a CFSTR or batch reactor. More to the point, the set r(Ω) can be approximated by measuring values of r(c,T) over a variety


of (c, T) states in Ω. In this sense, knowledge of the mechanistic details giving rise to r(‚,‚) are unnecessary for the implementations described in remarks 7.1-7.3. Rate measurements alone will suffice. 8. Sharp Productivity Benchmarks in a “Toy” Example Consider the well-studied van de Vusse reaction network (42) taken with mass-action kinetics. The rate k1

k2

A1 98 A2 98 A4 k3

2A1 98 A3

(42)

constants are presumed to depend on temperature in the following way (with activation energies in kcal/mol):

k1(T) ) 4.22 × 103 exp(-10/RT) (s-1) k2(T) ) 1.02 × 105 exp(-18/RT) (s-1) k3(T) ) 5.24 × 108 exp(-22/RT) (L/mol‚s) We suppose that A4 is our desired product, that A3 is an environmentally disagreeable side product, and that we have available a steady feed stream of pure A1 at a rate of 10 mol/s. We admit for consideration all steady-state reactorseparator system designs in which the total reactor capacity is not greater than 1000 L, in which the temperature in no reactor component is less than 100 °C or greater than 300 °C, and in which the pressure in no reactor component exceeds 50 bar. Despite the high upper pressure limit, we suppose (for the purposes of this example) that A1, A2, A3, and A4 constitute an ideal gas mixture within all reactor elements. In this case Ω consists of all (c, T) pairs in R5+ such that Tmin e T e Tmax and such that (43) is satisfied. 4

0e

∑

cL e

L)1

pmax RT

(43)

Now suppose that there is an environmental constraint requiring that the molar productivity rate of the undesirable A3 be no more than 0.1 mol/s. Then, by numerical methods described in remark 7.2, it can be determined that there is no steady-state design consistent with the imposed constraints in which the desired A4 is produced at a rate greater than 6.12 mol/s. Moreover, this kinetic bound is sharp in the following sense: When the capacity for arbitrary separations is granted (at arbitrary rates), the bound can actually be achieved arbitrarily closely (if at great expense). It is interesting to note that when the maximum productivity rate of A4 is achieved, 3.68 mol/s of the A1 must pass through to the effluent unreacted. In fact, without an increase in the 1000 L reactor size limit, no steady-state reactor-separator design (consistent with the indicated constraints) can consume all 10 mol/s of A1 without exceeding the 0.1 mol/s effluent limit on A3. We expect to discuss this example at greater length in another paper. 9. Concluding Remarks In closing, we reiterate what was said at the outset: Given a feed and a kinetics, it is important to know the

best that can be hoped for in a reactor-separator design and to understand when no amount of configurative innovation will produce further gains, so long as certain design constraints are respected. Our aim has been to provide practical tools for designers, managers, and regulators, who, faced with a bewildering array of configuration choices, would do well to know in advance outer limits on what can be expected from any steadystate constraint-consistent reactor-separator configuration. We believe that ideas and techniques in this paper constitute an important step in that direction. Our work here has focused on the assessment of outer bounds for what might be attained in steady-state reactor-separator systems of arbitrary design. At the end of section 1, we indicated that there is also interest in the “attainable region” for the pure reactor synthesis problem, that is, in the full set of attainable output states for designs in which the only operative processes are reaction and mixing. Regrettably lacking for the pure reactor synthesis problem are means for assessing good kinetics-based outer bounds on what the full attainable region might be. It should be clear, though, that outer bounds for the reactor-separator problem, such as those deduced here, serve also as outer bounds for the pure reactor synthesis problem: The set of output states attainable from all reactor-separator systems, including those with no separators at all, is clearly larger than that which can be attained by means of reactors alone. As they stand, these bounds for the pure reactor synthesis problem are likely to be somewhat conservative, but perhaps not as conservative as might at first be supposed: For the pure reactor synthesis problem, the only realizable states are those attainable from the primary feed solely through the occurrence of chemical reactions and mixing; separators play no role. Thus, in invoking the ideas in this paper to obtain outer bounds for the pure reactor synthesis problem, it is natural to impose narrow stoichiometrical constraints on compositions that can come into play. The resulting state constraints (carried by the set we called Ω) will, in general, be far more stringent than those natural to systems in which separations are invoked, and so tighter outer bounds on the product are to be expected. Acknowledgment The authors are grateful for support from the United States National Science Foundation. Appendix: The Overall Production Rate Vector for Generalized Reactors In section 5, the important inclusion (9) was invoked through an intuitive appeal to the idea that the globalaverage volumetric molar production rate vector should be a suitable average of the local volumetric molar production rate vectors exhibited across all reactor components. Our aim in this appendix is to outline how, in a very general way, the inclusion (9) can be given formal mathematical underpinningssin particular, without commitment to a special reactor picture. We consider a kinetic system for which the (volumetric) species formation rate function is r(‚,‚). That is, for a local composition-temperature state (c, T) ∈ RN+1 +

r(c,T) ) [r1(c,T), r2(c,T), ..., rN(c,T)]

(A.1)

3192


where rI(c,T) is the local molar production rate per unit volume of the Ith species due to the occurrence of all reactions. For a fixed but arbitrary steady-state design, recall that R denotes the vector [R1, R2, ..., RN] of total molar production rates of the N species across all reactor elements. Recall also that if, for the design, V* is the total volume of all reactor elements, then

r† :)

(V*1 )R

(A.2)

is the global-average volumetric molar production rate vector. We begin by considering how both R and r† are computed in the context of some simple examples. Example A.1. Suppose that the design is such that the only reactor element is an ideal (homogeneous) CFSTR of volume V*, in which the composition is c* and the temperature is T*. In this case, the vector of the total molar production rates is given by

R ) V*r(c*,T*)

(A.3)

and r† is easily seen to be identical to r(c*,T*). Example A.2. Suppose that the only reactor elements are three ideal CFSTRs of volumes V1, V2, and V3, where

V1 + V2 + V3 ) V*

(A.4)

If the mixtures within the reactors are in states (c1, T1), (c2, T2), and (c3, T3), then R is given by the formula

R ) V1r(c1,T1) + V2r(c2,T2) + V3r(c3,T3) (A.5) From (A.2) and (A.5) it follows that

r† ) λ1r(c1,T1) + λ2r(c2,T2) + λ3r(c3,T3) (A.6) where λi :) Vi/V*, i ) 1-3. Note that

λ1 + λ2 + λ3 ) 1

(A.7)

so r† lies in the convex hull of

{r(c1,T1), r(c2,T2), r(c3,T3)}

(A.8)

Example A.3. Suppose that the only reactor element is a single ideal PFR of cross-sectional area A and length L (and with total volume V* ) AL). Suppose also that, along the reactor, the local composition-temperature state is given by (c(z), T(z)), 0 e z e L, where z describes the distance from the reactor’s entrance. In this case, we have

∫0 r(c(z),T(z)) dz ) AL∫0

R)A

L

L

dz (A.9) r(c(z),T(z)) L

Since V* ) AL, it follows from (A.2) that

r† )

∫0 r(c(z),T(z)) dz L L

(A.10)

Because

)1 ∫0Ldz L

(A.11)

it is at least intuitively clear from (A.10) that r† is a convex combination of the reaction vectors exhibited along the reactor. That is, r† lies in the convex hull of the set

{r(c(z),T(z)) ∈ RN : 0 e z e L}

(A.12)

In each of the three examples, the inclusion (9) obtains. We turn now to very general means for describing the steady-state condition of an arbitrary assembly of reactor components (not necessarily of classical design), means that will draw the three preceding examples into a common framework and that will, at the same time, permit proper computation of the total molar production rates in quite general circumstances. Essential to our calculations is, first, a precise description of how much of the total reactor volume is allocated to the various local mixture states and, second, a suitable summation procedure to assess contributions of the various local parts to the overall molar production rate vector. The natural framework for both stages of the analysis is provided by the modern theory of measure and integration. An excellent and concise exposition can be found in the first two chapters of the book by Rudin.22 (For a slightly different treatment see the book by Lieb and Loss.23) Although the theory is delicate in places, our needs here are not so technical as to preclude a rudimentary discussion of how the state of a reactor has a natural description in measure-theoretic terms and of how that description leads to a calculation of R. In the end, we want to provide justification, in a very general setting, for the inclusion (9), which played such a pivotal role in section 5. is a rule, A positive Borel measure µ(‚) on RN+1 + satisfying certain restrictions, that assigns to each a nonnegative number “nonpathological” set Ω ⊂ RN+1 + µ(Ω). (The “nonpathological” sets, called Borel sets, include all of the open sets, all of the closed sets, and all countable unions and intersections of these. Hereafter, when we speak of a set in RN+1 + , it will be understood that we mean a Borel set.) In rough terms, it will be useful to think of µ(Ω) as the “volume” that the measure µ(‚) assigns to the set Ω. (As the example below will indicate, the volumes so assigned can be very different from the standard geometric (i.e., Lebesgue) volumes normally associated with sets in RN+1 + .) The simplest example of a positive Borel measure on is the so-called Dirac measure. Suppose that RN+1 + (c*, T*) is a fixed composition-temperature state in RN+1 + . Then the Dirac measure concentrated at (c*, T*), denoted as δ(c*,T*)(‚), is defined by the requirement that, for each set Ω ⊂ RN+1 + ,

δ(c*,T*)(Ω) ) 1

if (c*, T*) ∈ Ω

δ(c*,T*)(Ω) ) 0

if (c*, T*) ∉ Ω

and

By the total volume associated with a positive Borel measure µ(‚), we mean the number µ(RN+1 + ). A set has µ-measure zero if µ(Ω) ) 0; thus, a set Ω ⊂ RN+1 + has µ-measure zero if, with respect to the measure µ(‚), that set has zero volume. By the support of the measure µ(‚), denoted as supp µ, we mean the complement in of the union of all open sets having µ-measure RN+1 + zero. Thus, the total volume associated with δ(c*,T*)(‚) is 1. not containing (c*, T*) has measure Any set in RN+1 +


zero relative to δ(c*,T*)(‚). It is not difficult to see that the support of this measure consists of the single point (c*, T*). Measures can be multiplied by positive numbers in the following sense: If V is a positive number and µ(‚) is a positive Borel measure on RN+1 + , then Vµ(‚) is the positive Borel measure that assigns to each set Ω ⊂ the number Vµ(Ω). Similarly, measures can be RN+1 + added in the obvious way: If µ(‚) and µ′(‚) are positive Borel measures on RN+1 + , then (µ+µ′)(‚) is the positive Borel measure that assigns to each set Ω the number µ(Ω) + µ′(Ω). Now consider a steady-state reactor-separator design in which the reactor components have total volume V*. With the design we can associate a positive Borel having the following interpretameasure µ(‚) on RN+1 + is a set of composition-temperature tion: If Ω ⊂ RN+1 + states, then µ(Ω) is the volume of that part of the mixture (within reactor components) in which local material points have composition-temperature states residing in Ω. Note that µ(RN+1 + ) ) V*. Note also that if O ⊂ RN+1 is an open set of states that are not exhibited + anywhere within the reactor components, then µ(O) ) 0. In rough terms, then, supp µ is (the closure of) the set of all local composition-temperature states that do appear within reactor components for the design under consideration. This is the set we called Γ in section 5. For the design in example A.1, the corresponding measure is just

mixture states are exhibited within the reactor (i.e., supp µ) but also how the total volume V* is allocated among those local mixture states. In turn, the function r(‚,‚) associates with each such local state (c, T) a vector r(c,T) that gives, for the individual species, a net production rate per unit volume due to the occurrence of chemical reactions. In terms that are meant to be only suggestive (and definitely not precise), calculation of R, the vector of total molar production rates across all reactor units, requires a summing of the various r(c,T), weighted by the volume allocated to various local composition-temperature states. The theory of integration of a function relative to a positive Borel measure serves to give the weightedsummation process precise mathematical meaning. Integration theory, well treated in both Rudin22 and Lieb and Loss,23 is beyond the scope of this appendix. Nevertheless, we can say that, for virtually any design we might consider (not necessarily involving classical reactor types), there is a positive Borel measure µ(‚) associated with its reactor components and that the formula

R)

r† )

µ(Ω) ≡ AL (f-1(Ω)) consisting of In this case, supp µ is the curve in RN+1 + composition-temperature states exhibited along the reactor; that is,

supp µ ) {f(z) ∈ RN+1 : 0 e z e L} + With this as background, we consider an arbitrary reactor-separator design in which the total volume associated with reactor components is V* and in which the Borel measure associated with those components is µ(‚). In rough terms, µ(‚) indicates not only what local

r(c,T) dµ

(A.13)

∫R

N+1 +

r(c,T) dν

(A.14)

where ν(‚) is the positive Borel measure defined by

ν(‚) :)

V1δ(c1,T1)(‚) + V2δ(c2,T2)(‚) + V3δ(c3,T3)(‚) Its support is the triplet {(c1, T1), (c2, T2), (c3, T3)}. The measure associated with the design in example A.3 is a little harder to describe: We denote by f(‚): [0, L] f the function that assigns to each axial position RN+1 + along the reactor the corresponding mixture state; that is, f(z) ) (c(z), T(z)). Now suppose that Ω is a set in -1 RN+1 + . Then f (Ω) is the set of axial positions along the reactor at which mixture states within Ω are exhibited. (This set can be empty.) If L (f-1(Ω)) is the total length associated with such axial positions (in the sense of Lebesgue measure), then AL (f-1(Ω)) is the corresponding reactor volume. Thus, for example A.3, the pertinent associated with the design is Borel measure on RN+1 + given by

N+1 +

interpreted in the sense of the modern theory of integration gives the proper calculation of R. Here it is understood that µ(RN+1 + ) ) V*, where V* is the total volume of all reactor components. For our arbitrary design, it follows from (A.2) and (A.13) that

V*δ(c*,T*)(‚) Its support is the single composition-temperature state (c*,T*). For the design in example A.2, the corresponding measure is

∫R

1 µ(‚) V*

(A.15)

Note that

ν(RN+1 + ) ) 1

(A.16)

Because any set disjoint from supp ν has ν-measure zero, (A.14) and (A.16) can actually be replaced by

r† )

∫supp νr(c,T) dν

(A.17)

and

ν(supp ν) ) 1

(A.18)

Taken together, (A.17) and (A.18) suggest that

r† is contained in conv{r(supp ν)}

(A.19)

(In fact, a formal proof in a different setting is provided in Noll.24) The support of ν(‚), however, is identical to the support of µ(‚), which, in turn, is what we called Γ in section 5. Thus, this appendix indicates how, for arbitrary designs, the important inclusion (9) in section 5 might be given formal and general underpinnings. Remark A.1. The aforementioned paper by Noll is, in part, about foundations of Gibbs’s phase rule. Arguments in support of the phase rule generally begin with the presumption of a finite number of phases and then make assertions (sometimes on specious grounds) about the maximum number of phases that can coexist in a

3194


Gibbs-stable equilibrium. Noll’s argument admits the possibility of a diffuse continuum of local states at the outset and then deduces that, at a (strictly) Gibbs-stable equilibrium, the number of such states (phases) is finite and consistent with the classical Gibbs upper bound. Noll’s argument goes to the heart of what might be the most remarkable part of the phase rule assertion: that, at equilibrium, one should not expect a diffuse continuum of local states; rather, one should expect that the number of distinct local states exhibited should be finite and small. A different argument, close in spirit to Noll’s but based instead on Caratheódory’s theorem, is given in Feinberg.25 In turn, the mathematics there has much in common with the mathematics underlying this paper. Indeed, the CFSTR equivalence principle, which asserts that the effluent of a steady-state reactor-separator system can be achieved arbitrarily closely with a small, finite number of homogeneous CFSTRs, is reminiscent of the phase rule’s discrete character. Readers of both this paper and Feinberg’s25 will perhaps see an analogy between the small (bounded) number of distinct local states in the phase rule and the small (bounded) number of homogeneous CFSTRs in the equivalence principle. List of Symbols A1, ..., AN ) species names c ) local composition vector cI ) local molar concentration of species AI c† ) CFSTR operating composition cl(‚) ) closure conv(‚) ) convex hull eI ) Ith standard basis vector in RN F0i ) molar feed rate vector in the ith CFSTR FiL0 ) molar feed rate of species L in the ith CFSTR kR, ..., kδ ) rate constants KR, ..., Kδ ) kinetic functions 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 pmax ) maximum allowed pressure r(‚,‚) ) volumetric species formation rate function r† ) global-average volumetric molar production rate r(Ω) ) rate vectors corresponding to states in Ω R ) total molar production rate vector RI ) total molar production rate of species AI RN ) vector space of N-tuples RN + ) set of all nonnegative N-tuples s ) rank of the reaction network S ) stoichiometric subspace supp(‚) ) support T ) temperature Tmax ) maximum allowed temperature Tmin ) minimum allowed temperature T† ) CFSTR operating temperature T(z) ) temperature at axial position z V h ) maximum allowed total volume V* ) specified reactor volume V ) vector of CFSTR volumes Greek Letters Γ ) reactor state set λi ) weighting factor

Ω ) composition-temperature constraint set ξi ) effluent volumetric flow rate in the ith CFSTR

Literature Cited (1) 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. (2) 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. (3) Achenie, L. K. E.; Biegler, L. T. A Superstructure-based Approach to Chemical Reactor Network Synthesis. Comput. Chem. Eng. 1990, 14, 23. (4) Hildebrandt, D.; Glasser, D. The Attainable Region and Optimal Reactor Structures. Chem. Eng. Sci. 1990, 45, 2161. (5) Kokossis, A. C.; Floudas, C. A. Optimization of Complex Reactor NetworkssI. Isothermal Operation. Chem. Eng. Sci. 1990, 45, 595. (6) Kokossis, A. C.; Floudas, C. A. Optimization of Complex Reactor NetworkssII. Nonisothermal Operation. Chem. Eng. Sci. 1994, 49, 1037. (7) Feinberg, M.; Hildebrandt, D. Optimal Reactor Design from a Geometric ViewpointsI. Universal Properties of the Attainable Region. Chem. Eng. Sci. 1997, 52, 1637. (8) Feinberg, M. Recent Results in Optimal Reactor Synthesis via Attainable Region Theory. Chem. Eng. Sci. 1999, 54, 2535. (9) Feinberg, M. Optimal Reactor Design from a Geometric ViewpointsII. Critical Sidestream Reactors. Chem. Eng. Sci. 2000, 55, 2455. (10) Feinberg, M. Optimal Reactor Design from a Geometric ViewpointsIII. Critical CFSTRs. Chem. Eng. Sci. 2000, 55, 3553. (11) Lakshmanan, A.; Biegler, L. T. Synthesis of Optimal Chemical Reactor Networks. Ind. Eng. Chem. Res. 1996, 35, 1344. (12) Schweiger, C. A.; Floudas, C. A. Optimization Framework for the Synthesis of Chemical Reactor Networks. Ind. Eng. Chem. Res. 1999, 38, 744. (13) Kokossis, A. C.; Floudas, C. A. Synthesis of Isothermal Reactor-Separator-Recycle Systems. Chem. Eng. Sci. 1991, 46, 1361. (14) Balakrishna, S.; Biegler, L. T. A Unified Approach to the Simultaneous Synthesis of Reaction, Energy and Separation Systems. Ind. Eng. Chem. Res. 1993, 32, 1372. (15) Lakshmanan, A.; Biegler, L. T. Synthesis of Optimal Chemical Reactor Networks with Simultaneous Mass Integration. Ind. Eng. Chem. Res. 1996, 35, 4523. (16) Nisoli, A.; Malone, M.; Doherty, M. Attainable Regions for Reaction with Separation. AIChE J. 1997, 43, 374. (17) 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. (18) Kelley, J. General Topology; D. Van Nostrand: New York, 1955. (19) Eggleston, H. G. Convexity; Cambridge University Press: Cambridge, 1958. (20) Barber, C. B.; Dobkin, D. P.; Huhdanpaa, H. The Quickhull Algorithm for Convex Sets. ACM Trans. Math. Software 1996, 22, 469. (21) Fukuda, K. Frequently Asked Question in Polyhedral Computation, available at http://www.ifor.math.ethz.ch/ fukuda/ polyfaq/polyfaq.html, 2000. (22) Rudin, W. Real and Complex Analysis, 2nd ed.; McGrawHill: New York, 1974. (23) Lieb, E. H.; Loss, M. Analysis; American Mathematical Society: Providence, 1997. (24) Noll, W. On Certain Convex Sets of Measures and on Phases of Reacting Mixtures. Arch. Ration. Mech. Anal. 1970, 38, 1. (25) Feinberg, M. On Gibbs’s Phase Rule. Arch. Ration. Mech. Anal. 1979, 70, 219.

Received for review July 28, 2000 Accepted April 16, 2001 IE000697X

General Kinetic Bounds on Productivity and Selectivity in Reactor

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