Carnot-like Limits to Steady-State Productivity - Industrial

The classical Carnot analysis provides means to calculate, relative to a specified heat supply and relative to specified temperature bounds, a limit o...
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Carnot-like Limits to Steady-State Productivity Yangzhong Tang and Martin Feinberg* Department of Chemical and Biomolecular Engineering, Ohio State UniVersity, Columbus, Ohio 43210

The classical Carnot analysis provides means to calculate, relative to a specified heat supply and relative to specified temperature bounds, a limit on the work that can be obtained in any cyclic process consistent with those specifications. Moreover, the Carnot analysis indicates that the calculated limit is sharp, to the extent that it can be attained by a special process (a Carnot cycle) which cannot itself be realized but which can be approximated arbitrarily closely by actual processes. We indicate how, in the spirit of the Carnot analysis, theoretical limits to the productivity of steady-state reactor-separator systems can be calculated relative to specified commitments of resources and relative to specified constraints. We also indicate a sense in which those limits are sharp, to the extent that they might be attained by certain idealized reactor-separator systems which, like Carnot cycles, cannot be realized in practice but which might be approached by actual processes. 1. Introduction Process synthesis remains largely an art, guided by engineering judgment and past experience. The associated problems are difficult ones, for the production of a desired species often proceeds by way of an intricate network of competing chemical reactions. The challenge is to encourage the creation of prized molecules while simultaneously discouraging the creation of unwanted ones. At the end of a design cycle, questions always remain: How much more productiVity might haVe been achieVed with a different configuration that utilizes the same material resources (e.g., the same feed streams and the same reactor size) and that respects the same process constraints (e.g., temperature and pressure bounds in reactor units and strictures on production rates of disagreeable side-products)? In particular, what is the maximum productiVity that might haVe been achieVed oVer all steady-state designs, eVen unimagined ones, that utilize the same material resources and respect the same broad process constraints? And, how sensitiVe is that maximum to additional commitment of resources or to the loosening of constraints? The fact is that there are theoretical limits to productivity, and it is important to understand just what these are, for they provide benchmarks against which candidate designs might be judged. For a prescribed set of feed streams available at certain rates, the stoichiometry of the underlying chemical reaction network gives a coarse limit to the rate at which a target species might be produced, and thermodynamic considerations generally give a more refined one.1 Both serve as useful benchmarks in assessing a candidate design’s efficacy. But there are absolute productivity limits, relative to the resources allocated, that are substantially sharper and more pertinent to design assessment than are those offered either by stoichiometry or by thermodynamics. To make the point in an extreme way, imagine the availability of a fixed set of reactant streams with large flow rates (say on the order of 100 kg/min), and imagine also that, when mixed in the presence of a catalyst, these reactants combine to form a desired product, but not extremely rapidly. Clearly, the limiting steady-state formation rate of product obtainable from all reactor-separator designs employing 10 kg of catalyst will be substantially less than that obtainable from designs employing 1000 kg of catalyst (subject * To whom correspondence should be addressed. E-mail: [email protected]. Phone: 614-688-4883. Fax: 614-292-3769.

to the same pressure and temperature bounds within reactor units). The different productivity limitsswhatever those limits might beshave their roots neither in stoichiometry nor in thermodynamics but, rather, in kinetics. Yet, the general understanding of kinetic limits to process productivity remains poor. Textbooks teach that, for a specified (usually small) reaction network (with kinetics) and a specified feed stream, one reactor type might have a greater steady-state production rate than another (for the same reactor size), and such lessons often extend to comparisons of designs that invoke separation and recycle. But these are only comparisons of specific designs, not assessments of the maximum possible steady-state production rate over all possible steady-state reactor-separator designs consistent with the same reactor size and the same broad operating constraints (e.g., temperaturepressure bounds in reactor units). Although stoichiometric limits to productivity inevitably play a role in guiding design decisions, theoretical kinetic limits to productivity are more subtle, more apt, but also far more elusive. (It should be understood that our concern here is with limits to productivity for an existing kinetics, not limits to what might ever be achieved with stilluninvented catalysts.) The aim of this article is to indicate how absolute kinetic limits to productivity can actually be computed. The theory, and arguments underlying the theory, are contained in an article by Feinberg and Ellison.2 Our emphasis here, however, is on implementation of the theory, stripped of its technical underpinnings. Even more, we seek to demonstrate the kind of striking information that computations often yield and to indicate directions for future research. We shall ask very precisely framed questions about maximum theoretical yield rate relative to a particular commitment of resources (e.g., reactor size) or particular process constraints (e.g., pressure-temperature bounds in reactor units and limits on the production rate of unattractive side-products). It should be understood that having means for answering such questions will permit us to explore the sensitivity of the maximum yield both to resource commitments and to constraint strength. Thus, we can ask questions such as these: (i) For a specified feed and for specified temperaturepressure bounds in reactor units, what is the smallest reactor size that must be committed, regardless of design, before a specified target yield becomes feasible in principle? If one seeks to increase the yield modestly, might it happen that the minimum

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required reactor size will increase greatly, regardless of design? (An example will demonstrate how modest increases in yield specifications can sometimes require distressingly large increases in reactor size, increases that design innovations cannot circumvent.) (ii) What are the trade-offs between the tightening of constraints on production rates of undesired substances and the maximum possible production rate of desired ones? In particular, with the same commitment of material resources and the same bounds on temperature and pressure in reactor units, is there a constraint level on the steady-state production rate of undesired substances at which a target production rate of a desired substance becomes impossible to achieve? We shall limit our considerations to systems in which reacting mixtures can be deemed to be fluidlike. That is, they are either fluid mixtures or are solid-fluid mixtures that, to good approximation, can be well-modeled as fluid mixtures. In the latter case, reactor volume is often taken as a surrogate for solid catalyst mass. For some other treatments of attainability in reactor-separator systems, largely focused on prespecified design configurations, see for example refs 3-8. 2. In the Spirit of Carnot Our interest is in kinetic, not thermodynamic, limits on the rate at which certain molecules can be converted to more valuable ones in steady-state processes. Nevertheless, it will be useful to recall the spirit of the Carnot analysis,9 which places limits on the extent to which heat can be converted to work in cyclic processes. Although it is already a consequence of the Second Law that, in a cyclic process, not all of the heat absorbed by a body can be converted into work, the Carnot analysis goes further by placing quantitative limits on the extent to which that conversion can take place. The fraction of heat absorbed by a body that can be converted to work is given by the so-called Carnot efficiency,

ηeff ) 1 -

TLow THigh

(1)

where TLow and THigh are the low and high temperatures experienced by the body during the course of the cycle. Note that the Carnot analysis serves to calculate the maximum amount of work that might be obtained relative to allocated resources (the heat supplied) and relative to specified process constraints (TLow and THigh.). Recall that the Carnot cycle not only plays a role in establishing the efficiency bound, but it also serves to indicate that the bound is sharp. That is, to the extent that the Carnot cycle can actually be realized in the limit, it provides an indication that no efficiency bound could be more restrictive, for otherwise, the Carnot cycle could exceed it. Carnot cycles are, of course, fictions: As they are usually depicted, traversing even a single cycle would take an infinite amount of time. Nevertheless, they serve as invaluable conceptual tools that, at least in principle, can be approximated in the limit by real processes. The Carnot analysis tells us that, for specified temperature bounds, eVen if we were permitted the use of perfectly reVersible processes, the Carnot efficiency cannot be exceeded. In what follows, we shall indicate that, for a specified commitment of resources and for broadly specified process constraints, there is a certain category of quite simple but

Figure 1. Schematic illustration of the CFSTR equivalence principle.

highly idealized reactor-separator designs that inevitably suffice to produce the best possible yield. In particular, these designs will invoke omnipotent separators (with omnipotent pumps)s that is, devices that can produce whatever separations one desires and at whatever rates one wishes, consistent with material balances. Like Carnot engines, such devices are fictions, but the spirit will be the same: EVen if we were permitted the use of omnipotent separators, certain yield rates of prized species cannot be exceeded. 3. The Underlying Idea The calculation of absolute kinetic bounds on steady-state productivity will rest on the CFSTR equiValence principle, an argument for which appears in ref 2. In preparation for its statement, we shall first need the idea of the rank of a reaction network, at least informally. By the rank of a reaction network, we mean the maximum number of linearly independent reactions taken from the network. For example, network 2 has a rank of three: The three reactions 2A1 f A2 + A3, A4 f A2 + A3, and A1 + A4 f A5 are linearly independent, and each of the remaining reactions can be generated from these by linear combination, including reversal. (This is a rough, intuitive formulation of the notion of reaction network rank. A more precise formulation, invoking standard ideas in linear algebra, can be found in ref 10.)

In rough terms, the CFSTR equivalence principle says this: Consider a reactor-separator system that, at steady state, achieves certain molar effluent rates of the various species, and suppose that the rank of the underlying network of chemical reactions is s. Then, those same molar effluent rates can be realized arbitrarily closely from the same feed with a reactorseparator system that invokes omnipotent separators but in which the only reactors are perfectly mixed continuous flow stirred tank reactors (CFSTRs), not exceeding s + 1 in number. Moreover, the total volume of the CFSTRs in the second design is identical to the total volume of reactors in the original design, and mixture states in the CFSTRs are identical to (local) states extant in reactors of the original design.

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The CFSTR equivalence principle is illustrated in Figure 1. In the lower design, invoking an arbitrary reactor configuration, the molar feed rates of the N species are M01, M02, ..., M0N and the steady-state molar effluent rates are M/1, M/2, ..., M/N, all in moles per time. There is no presumption of homogeneity of pressure, temperature, or composition in the reactor of the lower design. In particular, the variation of local states within the reactor can be of the dispersed kind characteristic of diffusive and thermally conducting mixtures. In the upper part of the figure, we show schematically a CFSTR-only design, perhaps invoking omnipotent separators, that replicates the effluent of the lower design. The total volume of the CFSTRs in the upper design is identical to the volume of the reactor in the lower design, and the uniform mixture states (e.g., pressure, temperature, and composition) in the CFSTRs are identical to certain local states (denoted A, B, and C) in the reactor of the lower part of Figure 1. For our purposes, the CFSTR equivalence principle will serve as a special conceptual instrument: In considering what effluents might be attained from any steady-state reactor-separator system that invokes a particular reactor volume and that respects certain temperature-pressure constraints within reactor units, it suffices to consider only processes that invoke omnipotent separators but only a small number of CFSTRs, the maximum number required bearing a simple relationship to the underlying network of chemical reactions. In particular, consideration of only such processes suffices for calculation of the maximum attainable yield of some desired species, relatiVe to a specific commitment of material resources. It is worth emphasizing again that, like Carnot cycles, such processes are gross idealizations but useful conceptual ones: By focusing only on such processes, we can calculate limits to productivity that would otherwise require consideration of a virtually limitless range of reactor configurations. Thus, the problem of calculating productivity limits shifts from one of configurational imagination to one of concrete optimization over an extremely simple reactor arrangement, supplemented by an omnipotent separator facility. Once that optimum is calculated, no other steady-state process, consistent with the same commitment of material resources, could do better, even one that invokes omnipotent separators. (Clearly, the maximum yield attainable in the presence of omnipotent separators will serve to bound yields attainable in the presence of constrained separators.) Before turning to the details of how calculations might be executed, we shall find it useful to first present, in sections 4 and 5, two examples that indicate the kind of incisive results that such calculations can actually deliver. (The second example is an instructive modification of the first in that it indicates ways in which newly added environmental constraints can affect the maximum attainable productivity.) Our emphasis in the examples is on the results themselVes, not on the means to calculate them or on the underlying conceptual framework. In consideration of the examples, then, the CFSTR equivalence principle will play its role only in the background. (In a similar way, the Carnot formula (1) serves to calculate bounds on heat engine efficiency without making explicit reference to Carnot cycles.) In section 6, we will consider how calculations of maximum attainable yields can be implemented in practice. There, the role of the CFSTR equivalence principle will be much more evident. In section 7, we discuss software implementation, and in section 8, we make some brief concluding remarks.

Figure 2. Schematic picture of a constraint-consistent design.

4. Limits to Productivity: An Example Here, we consider a chemistry involving six species, designated A, B, C, D, E, and F, which are presumed to experience the reactions shown in (3). The kinetics is mass-action, with temperature-dependent rate constants indicated symbolically in (3) alongside the corresponding reaction arrows. The presumed temperature dependences R(T)

β(T)

A 98 B 98 D (desired) γ(T)

δ(T)

(T)

2A 98 C A + B 98 E C + E 98 F

(3)

are shown in (4), where the activation energies in the exponents have units of kilocalories per mole, the temperature is in degrees Kelvin, and R ) 0.0019872 kcal/(mol‚K). For the purposes of the example, the six species are presumed to constitute an ideal gas mixture.

R(T) ) 5.12 × 103 exp(-11/RT) s-1 β(T) ) 8.46 × 108 exp(-20/RT) s-1 γ(T) ) 2.28 × 1011 exp(-25/RT) s -1 M-1 δ(T) ) 3.56 × 105 exp(-12/RT) s-1 M-1 (T) ) 4.43 × 105 exp(-12/RT) s -1 M-1

(4)

Suppose that the desired product, D, is to be produced from a feed stream that supplies A at a rate of 40 mol/s. Suppose also that a steady-state yield rate of 15 mol/s of D results from a candidate reactor-separator design that invokes an 8000 L reactor in which the temperature ranges between 100 and 400 °C and in which the pressure nowhere exceeds 5 bar. Note that the stoichiometric limit to the productivity of D is 40 mol/s, and so, the candidate design gives a conversion of A to D that is far from complete. Is the candidate reactor-separator design an unskillful one? However unsatisfactory the 15 mol/s yield might seem, it is instructive to assess the candidate design against all others consistent with the same commitment of resources (e.g., reactor size) and that respect the same operational constraints (e.g., pressure-temperature bounds within reactor units). That is, one might wish to know the maximum possible steady-state D productivity over all designs, even unimagined ones, of the kind depicted schematically in Figure 2. From the CFSTR equivalence principle, we can deduce that, eVen in the presence of omnipotent separators, no steady-state reactor-separator design that inVokes an 8000 L reactor and that respects the pressure-temperature bounds shown in Figure 2 can produce a yield greater than 17.91 mol of D/s. In this sense, the candidate design is a highly respectable one, relative to its commitment of resources.

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Figure 3. Maximum steady-state D production rate as a function of reactor size commitment.

Still, its 15 mol/s yield is disappointing relative to the 40 mol/s stoichiometric limit, and one might ask to what extent an additional commitment of reactor size might bring the D productivity rate to a level of, say, 35 mol/s, while still respecting the same pressure-temperature bounds within reactor units. Might a doubling or a tripling of reactor volume suffice, perhaps in the context of a radically different design? From the CFSTR equivalence principle, we can deduce that no reactor-separator design, even one that invokes omnipotent separators, can result in a 35 mol/s steady-state productivity rate without inVoking a reactor size of close to 100 000 L. In the absence of such information, a designer might well be frustrated by attempts to deliver a 35 mol/s steady-state productivity with a substantially smaller reactor, relying instead on clever but ultimately futile design innovations. Figure 3 depicts the calculated theoretical maximum steadystate productivity of D as a function of reactor size. Viewed differently, the figure indicates the smallest reactor size commitment that must be made in order to attain a desired level of productivity. (So that comparisons might be made with the original candidate design, it is presumed that, within reactor units, the pressure is constrained to be less than 5 bar and the temperature is constrained to lie between 100 and 400 °C.) We note in passing that even substantial relaxation of the pressure constraint does only modest good. For example, if the maximum allowable pressure in reactor units is increased to 25 bar while the temperature constraint remains as before, a steadystate reactor-separator system invoking an 8000 L reactor can produce at most 20.74 mol/s of D. For a 20 000 L reactor, the maximum productivity increases to 28.03 mol/s, still short of the 35 mol/s target. (Only for the purposes of the example, we continue to presume ideal gas mixture behavior, even at high pressures.) 5. Limits to Productivity: An Example with an Environmental Constraint It will be instructive to consider the extent to which added environmental constraints might affect theoretical limits to productivity in the hypothetical example of the preceding section. We indicated there that, relative to the resource commitment and pressure-temperature constraints indicated in Figure 2, the theoretical maximum steady-state productivity of the prized D is 17.91 mol/s. Suppose now that C is a disagreeable side-product. Again in the context of designs

consistent with the resource commitments of Figure 2, we ask about the maximum steady-state productivity of D, this time subject to the enVironmental requirement that the steady-state molar production rate of C be limited to 0.1 mol/s. The problem is a complicated one. At first glance it is tempting to suppose, especially granted omnipotent separators, that unwanted C might, at steady state, be recycled to reactor units with the aim of consuming it via the reaction C + E f F. The difficulty, however, is that this reaction must proceed sufficiently rapidly as to keep pace with the production of C via the reaction 2A f C. In turn, sufficiently rapid occurrence of C + E f F might necessitate temperatures and concentrations of C and E that are inconsistent with the pressure-temperature constraints imposed in reactor units. Moreover, steady-state generation of an adequate supply of the coreactant E would require consumption of A and B, two essential precursors of the desired D. In this case, the CFSTR equivalence principle tells us that the presence of the enVironmental constraint serVes to reduce the maximum theoretical steady-state production rate of D from 17.91 down to 7.51 mol/s in any reactor-separator system consistent with an 8000 L reactor and with the reactor unit pressure-temperature constraints indicated in Figure 2. Moreover, to achieve this maximum, 13.57 mol/s of A must pass through the system unreacted. Additional consumption of A cannot result in a greater production rate of D without also producing an additional constraint-violating production of the undesired C. 6. Calculating Limits to Productivity Motivated by the CFSTR equivalence principle, we examine in this section how limits to productivity can be calculated. In the next section, we discuss software implementations. Consider a chemistry of N species, A1, A2, ..., AN, including inerts, for which an operative network of chemical reactions has rank s. We suppose that the kinetics associated with the network gives rise to species formation rate functions, designated r1(•), r2(•), ..., rN(•), that indicate the dependence of the individual species formation rates on composition and temperature. That is, rL(c1, c2, ..., cN, T) is the net molar production rate per unit volume of species AL, due to the occurrence of all chemical reactions, at a place in a mixture at which the molar species concentrations are c1, c2, ..., cN and the temperature is T. In the case of the reaction network studied in Section 4, for example, the formation rate function for species B would be

rB(cA, cB, ..., cF, T) :) R(T)cA - β(T)cB - δ(T)cAcB (5) where R(T), β(T), and δ(T) are given by (4). We suppose also that the local mixture composition, temperature, and pressure are related by an equation of state g(c1, ..., cN, T, P) ) 0. For example, an ideal gas mixture would conform to the equation

g(c1, ..., cN, T, P) :) P - RT(c1 + c2 ... + cN) ) 0 (6) As before, we presume the availability of a steady feed with molar supply rates of the various species, in moles per time, given by M01, M02, ..., M0N. We suppose in this section that A1 is the desired species, and we consider what might be its maximum steady-state rate of production over all reactor-separator systems in which reactor components are limited to a total volume of VMax, in which the temperature in reactor units is constrained to lie between TMin and TMax, and in which the pressure in reactor units is not permitted to exceed PMax. When

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Figure 4. Optimization problem motivated by the CFSTR equivalence principle.

there are unattractive side-products, designated A2, A3, ..., AK, we suppose that their steady-state production rates are not to h 3, ..., M h K (in moles per time). exceed the bounds M h 2, M Since, by virtue of the CFSTR equivalence principle, any steady-state effluent that is realizable can be realized in the limit by a reactor-separator system of the kind depicted in the upper part of Figure 1, we consider such a system, invoking an omnipotent separator facility and s + 1 ideal CFSTRs. For θ ) 1, 2, ..., s + 1, we denote with Tθ and Pθ the temperature and pressure in the θth CFSTR, with Vθ the volume of the θth CFSTR, and with cθ ) [cθ1 , cθ2 , ..., cθN] the list of molar concentrations in the θth CFSTR. (Note that certain of the Vθ values might be zero, in which case fewer than s + 1 CFSTRs would be invoked.) Motivated by this picture, we consider the optimization problem displayed in Figure 4. In Figure 4, the symbols M1, M2, ..., MN indicate the molar effluent rates, in moles per time, of the N species. [In fact, the symbols M1, M2, ..., MN appear in Figure 4 only so that the optimization problem considered might be made more transparent. Equation 7 can be eliminated once expressions for ML, L ) 1, 2, ..., N, given by eq 7 are substituted directly into eqs 8 and 13 and into the objective, M1. That is, the true variables in the optimization problem are Vθ, Tθ, Pθ, and cθ1 , cθ2 , ..., cθN, θ ) 1, 2, ..., s + 1.] Equation 7 amounts to a calculation of the molar steady-state effluent rate of species AL, deriving from the total production rate of AL across the various CFSTRs. Eqs 8-11 indicate natural and imposed constraints on variables associated with the individual CFSTRs, while eq 12 requires that the pressure, temperature, and composition within each of the CFSTRs respect the mixture equation of state. Equation 13 reflects the imposed environmental constraints (if any). is the maximum given by the optimization problem If MMax 1 in Figure 4, then, by virtue of the CFSTR equivalence principle, no steady-state reactor-separator design that respects the same bound on total reactor volume, that respects the same pressuretemperature bounds in reactor units, and that respects the same environmental discharge limits can produce a molar effluent rate of A1 that exceeds MMax 1 , no matter what design configuration is inVoked. Moreover, the bound is sharp in the following sense: Just as the theoretical limiting Carnot efficiency of heat engines given by eq 1 can be realized in the limit by (practically unrealizable) Carnot cycles, so too can the steady-state producbe realized in the limit by a design of tivity maximum MMax 1 the kind depicted at the top of Figure 1sa design that might invoke (practically unrealizable) omnipotent separators and practically unrealizable exchange rates between separators and reactors. Remark. There is a technical question about the sharpness of the bound that requires some consideration: If variables Vθ, Tθ, Pθ, and cθ1 , cθ2 , ..., cθN, θ ) 1, 2, ..., s + 1, are chosen to satisfy eqs 7-13 can there indeed exist a separator system, even

an omnipotent one, of the kind drawn schematically at the top of Figure 1 that will serve to maintain the CFSTRs at steady state? More precisely, can one invariably produce effluent flow rates from the CFSTRs to the separator system and feed supplies from the separator system to the CFSTRs (as shown in the figure) such that steady-state material balances can be maintained around the CFSTRs and around the separator system? This question is addressed in ref 2. The answer is affirmative when all of the cθL in the various CFSTRs are strictly positive, even when some are vanishingly small. On the other hand, the productivity bound given by the optimization problem in Figure 4 might require some of these concentrations to be zero. Again, such a maximum-productivity-achieving design can be regarded as a Carnot-like limiting one, approached by slightly different ones in which the corresponding concentrations are positive but vanishingly small. (At least intuitively, the constraint in eq 10 can be replaced by cθL g , θ ) 1, 2, ..., s + 1, L ) 1, 2, ..., N, where  is an extremely small positive number, with very little effect on the calculated M1 maximum.) In any case, the maximum for the problem shown in Figure 4 serves to bound the steady-state productivity of A1 over all steady-state designs subject to the same constraints. The issues addressed in this remark are only about the sharpness of the boundsthat is, whether the maximal Value of M1 can actually be realized in the limit by some reactor-separator system. It is a simple matter to calculate, through eq 1, the Carnot bound to the efficiency of all cyclic heat engines operating within specified temperature limits. The remarkable fact is that the equation serves to bound, sharply, the efficiencies of all such engines of arbitrary design. Calculation of sharp bounds on steady-state reactor-separator productivity through solution of the nonlinear optimization problem in Figure 4 is certainly not as simple. Nevertheless, the underlying idea shares a virtue of the Carnot idea: It shifts the question from one inVolVing contemplation of a limitless Variety of designs and configurations to one inVolVing a concrete, readily stated, mathematical problem. In the Carnot case, the required calculation reduces, through eq 1, to simple arithmetic. The problem in Figure 4 is much more difficult, but advances in nonlinear optimization theory and computational technology bring its solution well within reach. Even now, we can begin to envision practical software tools that would serve to make the calculation both tractable and convenient. The next section contains a discussion of one such implementation. 7. About Software Implementation The science of nonlinear optimization is a rapidly advancing one in which chemical engineers have played a significant role.5,11,12 Much can be said about the subject and, in particular, about special aspects of the problem shown in Figure 4. Indeed, for various kinds of kinetics and for various equations of state, the problem will take on special features for which one or another nonlinear programming technique might have particular strengths. Discussion of these technical matters is beyond the intended scope of this article. Nevertheless, it is instructive to consider some experiments with software, employing a commercial grade nonlinear optimization module, aimed at making computation of productivity bounds rapid, flexible, and practical. The fine details of the problem shown in Figure 4 are carried by the species formation rate functions r1(•), r2(•), ..., rN(•), the equation of state function g(•), and values of the constraint parameters TMin, TMax, PMin, PMax, VMax, M h 2, M h 3, ..., M h K. Although these might be supplied directly to an optimization module on a case-by-case basis, such

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shunting of E (and the unattractive C) through the separator system to the small reactor as quickly as it produced. The smaller reactor, in turn, has as its primary purpose the consumption of the undesired C in the reaction C + E f F. (The harmless F is immediately shunted through the separator system to the plant effluent as quickly as it is produced.) The software’s somewhat odd choices of temperature, cA, and cB in the large reactor are dictated by the need to simultaneously encourage the production of D, discourage the production of C, and encourage the production of the C-consuming species E. Note also the distribution of reactor volumes, the far smaller CFSTR being allocated the secondary task of thwarting production of the undesired C. Figure 5. Limiting two-CFSTR schematic design producing the theoretical maximal yield of D in the environmentally constrained problem of Section 5. (Reactors not drawn to scale.)

a procedure would be tedious and prone to error, especially when there are many reactions in play. Far better would be the availability of a supplementary executiVe module, apart from an optimization module, that would accept more primitive information such as the reaction network, the nature of the kinetics for individual reactions (e.g., mass action), the nature of the equation of state (e.g., ideal gas mixture and SoaveRedlich-Kwong), and appropriate parameter values (e.g., mass action rate constants). Such an executive module would then automate, for example, generation of the individual speciesformation-rate functions and, finally, would export the detailed and fully formed problem described generally in Figure 4 to the optimization module. The optimization module would subsequently return the theoretical productivity limit to the executive module so that it might be reported to the user. The availability of such an executive module would permit convenient consideration of a wide variety of problems, involving a wide variety of reaction networks. Moreover, the effect of weakening or strengthening of constraints could be explored readily, as could study of which kinetic changes, perhaps through the deployment of new catalytic agents or the addition of pollutant-consuming reactants, might have most effect on the maximum attainable yield. A draft of a wide-ranging executive module, in conjunction with a commercial-grade optimization module (Frontline Systems Inc.’s Solver dynamic link library platform, version 4.5) was used to deduce productivity limits in all examples considered in sections 4 and 5. (The executive module, The Productivity Limits Calculator, is freely available for download13 and can be used in conjunction with either Frontline Systems dynamic link library (DLL) or with any other optimization module adapted to the calling and reporting conventions of the Frontline library.) We show in Figure 5 a schematic diagram of a limiting (Carnot-like) two-CFSTR design, determined by the software, that produces the maximal attainable yield of species D for the environmentally constrained problem of section 5. Although the software does little but solve the optimization problem in Figure 4 algorithmically, qualitative wisdom can be discerned in the design that emerges. The far larger reactor has two purposes: Its first purpose is to produce the desired species D by means of the reaction sequence A f B f D. Concomitant with extremely rapid interchange between the large reactor and the (omnipotent) separator system, the desired D is shunted to the plant effluent as quickly as it is produced. A second purpose of the large reactor is to produce the intermediate E by means of the reaction A + B f E. In this case, extremely rapid exchange between the large reactor and the separator system permits the

8. Concluding Remarks The classical Carnot analysis indicates that, for cyclical engines respecting specified temperature constraints, there is a theoretical bound on the amount of work that can be obtained from a specified amount of heat. At the same time, the Carnot analysis indicates that the calculated bound is sharp by indicating a fictitious process, the Carnot cycle, that can achieve the boundsa process which, while not realizable, might be approached by actual processes in the limit. In a similar way, the CFSTR equivalence principle provides means to calculate, over all steady-state reactor-separator processes consistent with specified constraints, a bound on the extent to which a desired product might be obtained from a specified feed stream. In the Carnot spirit, the principle indicates that a fictitious process, involving just a few CFSTRs but perhaps omnipotent separators, will serve to achieve that bound. Calculation of the theoretical maximum yield is not simple, but the problem is concrete and readily stated. It is divorced from any search over the infinite number of qualitatively distinct reactor-separator configurations, some exotic, that might otherwise be considered as maximum-producing candidates. The CFSTR equivalence principle shifts the problem from one of configurational imagination to one of nonlinear programming. Like the Carnot bound, we think that calculation of the maximum attainable yield is one worth making. Ultimately, the value of a reactor-separator design will be judged in economic terms, but often, one will want to understand the extent to which a yield (and profit) might be improved through configuration changes or through additional commitments of resources. In this regard, the example of section 5 is instructive. Although the stoichiometric limit on the productivity of D is 40 mol/s, calculations indicate that no steady-state reactor-separator system, of any configuration, that respects specified pressuretemperature constraints and invokes an 8000 L reactor can produce more than 17.91 mol of D/s; configuration changes alone will not help. Moreover, to increase the yield to 35 mol of D/s, one needs a reactor size of close to 100 000 L. Without some understanding of the relationship between productivity bounds and resource commitments in general, independent of any particular design, attempts to improve yield with specific new designs might reduce to an exercise in frustration. We believe that the ideas presented here provide some avenues for future research. At the very least, we would hope that fast, robust, and reliable nonlinear optimization algorithms dedicated specifically to the problem in Figure 4 might be developed and disseminated. (We have in mind, for example, algorithms dedicated specifically to mass action kinetics with Arrhenius-type rate constants.) Moreover, examples in sections 4 and 5 indicate that there is much to be learned about the sensitivity of the maximum attainable yield rate to changes in

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resource commitments and to the tightening or loosening of constraints: In which situations will a Very large additional commitment of reactor size fail to result in a greatly improVed yield, no matter what new design configuration might be tried? In which situations will a slight tightening of an enVironmental constraint result in a dramatic loss of productiVity that, for a giVen commitment of resources, cannot be mitigated by any design? These are all avenues of inquiry suitable for experts in nonlinear programming. Finally, a maximizing CFSTR-only design will almost always be very far removed from practical implementation, for the maintenance of steady state will generally require infinite separation and pumping rates. Still, it is worth exploring whether quantitative aspects of the maximizing process (such as those depicted in Figure 5) might perhaps provide initial guidance in the creation of designs that are both practical and economically attractive. Acknowledgment This work was supported by grant CTS-0002800 from the United States National Science Foundation. Literature Cited (1) Shinnar, R.; Feng, C. A. Structure of Complex Catalytic Reactions: Thermodynamic Constraints in Kinetic Modeling and Catalyst Evaluation. Ind. Eng. Chem. Fundam. 1985, 24, 153. (2) 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.

(3) Kokossis, A. C.; Floudas, C. A. Synthesis of Isothermal ReactorSeparator-Recycle Systems. Chem. Eng. Sci. 1991, 46, 1361. (4) Balakrishna, S.; Biegler, L. T. A Unified Approach for the Simultaneous Synthesis of Reaction, Energy, and Separation Systems. Ind. Eng. Chem. Res. 1993, 32, 1372. (5) Floudas, C. A. Nonlinear and Mixed-Integer Optimization: Fundamentals and Applications; Oxford University Press: New York, 1995. (6) Nisoli, A.; Malone, M. F.; Doherty, M. F. Attainable Regions for Reaction with Separation. AIChE J. 1997, 43, 374. (7) Gadewar, S. B.; Malone, M. F.; Doherty, M. F. Feasible Region for a Countercurrent Cascade of Vapor-Liquid CSTRs. AIChE J. 2002, 48, 800. (8) Nisoli, A.; Doherty, M. F.; Malone, M. F. Feasible Regions for StepGrowth Melt Polycondensation Systems. Ind. Eng. Chem. Res. 2004, 43, 428. (9) Fermi, E. Thermodynamics; Dover Publications, Inc.: New York, 1956. (10) 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, 42, 2229. (11) Biegler, L. T.; Grossmann, I. E.; Westerberg, A. W. Systematic Methods of Chemical Process Design; Prentice Hall: Englewood Cliffs, NJ, 1997. (12) Biegler, L. T.; Grossmann, I. E. Retrospective on Optimization. Comput. Chem. Eng. 2004, 28, 1169. (13) Tang, Y.; Feinberg, M. The ProductiVity Limits Calculator (2006); available at http://www.chbmeng.ohio-state.edu/∼feinberg/ProductivityLimitsCalculator.

ReceiVed for reView November 21, 2006 ReVised manuscript receiVed June 4, 2007 Accepted June 11, 2007 IE061489T