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Ind. Eng. Chem. Res. 2006, 45, 8056-8062
Steady-State Operating Policies for Plants with Multiple Reactions of Equal Overall Order Derek W. Griffin, Jeffrey D. Ward, Michael F. Doherty, and Duncan A. Mellichamp* Department of Chemical Engineering, UniVersity of California, Santa Barbara, California 93106-5080
A simple methodology for determining optimal steady-state operating policies for plants with recycle has been developed by Ward et al. (Ward, J. D.; Mellichamp, D. A.; Doherty, M. F. Ind. Eng. Chem. Res. 2004, 43, 3957). Heuristics were developed to classify irreversible process chemistries into two groups, bounded and nonbounded. This classification has important implications for how to design a control system and operate a chemical plant that contains excess equipment capacity. In this paper, formal rules are developed and presented for classifying a common special case of process chemistries that have multiple undesired irreversible reactions with equal overall reaction order. This new classification procedure is essential for determining the optimal steady-state operating policy for process chemistries of multiple parallel or series/parallel reactions. Introduction Current research in the field of plantwide design and operation addresses issues such as appropriate equipment sizing, the choice of control structure for available control degrees of freedom, and capacity-based operation of chemical processes with recycle. Shortcut methods are available for the conceptual design of chemical processes, including proper equipment sizing and evaluation of capital and operating costs.2,3 Incorporating extra capacity in equipment design is one method used to improve process flexibility and accommodate system disturbances.4 Once a process has been appropriately designed with excess capacity, the potential exists to use this extra capacity even during nominal operation. In some cases, it is economically desirable to utilize the overdesign under all conditions; in other cases, it is not. Selecting the correct strategy for equipment capacity usage is part of the optimal steady-state operating policy. The conventional process configuration of reactor subsystem, separation subsystem, and recycle(s) is a widely studied system that provides insight into selecting the optimal operating policy. Operating policies that have been suggested to handle production rate changes in such a plant include maintaining constant recycle flow rates,5 operating the reactor at maximum holdup,6,7 or allowing both the reactor holdup and recycle flows to vary.8,9 A more recent and more general approach to this problem developed by Ward et al.1 considers the recycle plant with different classes of irreversible process chemistries, using recycle flow rates as the degrees of freedom. Chemistries are classified as either bounded or nonbounded on the basis of whether the optimal operating policy is to operate the reactor either completely full or away from the reactor volume constraint, respectively. Ward et al. developed heuristics to provide insight into the optimal operating policy based on the relationship of selectivity versus conversion of limiting reactant. If the rate law of a process chemistry is known, the formalized rules presented in this paper can be used to precisely predict the classification of a chemistry as bounded or nonbounded. The earlier heuristics provide quick guidance to select the appropriate operating policy and highlight the potential economic loss from using a nonoptimal operating policy; such insight is not always apparent by * To whom correspondence should be addressed. E-mail: dmell@ engineering.ucsb.edu. Tel.: (805) 893-2821. Fax: (805) 893-4731.
inspection of the stoichiometry alone (see Figure 5), which is why heuristics or formal rules are needed. For example, consider the following isothermal, irreversible process chemistry where the first reaction is desired and the second reaction is undesired. Here, high conversion of the limiting reactant A clearly results in low reactor concentration of species A, thus suppressing the formation of the undesired product D and increasing the selectivity to the desired product, C. The heuristics suggested by Ward et al. would classify this chemistry as “bounded” because the selectivity increases with increasing conversion of the limiting reactant, and thus, the reactor should be operated at the maximum holdup.1
A+BfC A+AfD However, consider another irreversible process chemistry where the concentration of species A in the reactor does not affect the selectivity to the undesired product.
A+BfC A+CfD Now high conversion of the limiting reactant A results in low concentration of species A in the reactor (good) but also in high reactor concentration of the desired product C (which is bad in the sense that it favors the undesired reaction). Thus, it is not clear whether any excess reactor holdup should be utilized to minimize total operating costs. Numerous other cases can be found where the optimal operating policy cannot be deduced by mere inspection of the stoichiometry alone. For example, Ward et al.10 have investigated irreversible nonisothermal chemistries to find that, for certain bounded chemistries, the optimal operating policy is to operate the reactor at the high-temperature constraint under all operating conditions. This counterintuitive result contradicts and improves on previous findings in the literature. This paper provides formal rules to determine the optimal operating policy for irreversible process chemistries with multiple undesired irreversible reactions, all with equal overall reaction order, when a kinetic model is available. These rules provide a sufficient condition to validate the heuristics developed by Ward et al. The reader is referred to the original article by
10.1021/ie051412p CCC: $33.50 © 2006 American Chemical Society Published on Web 10/24/2006
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Ward et al. for the general methodology, including derivation of process unknowns (byproduct production and required reactor volume), formulation of the economic potential model, and classification of the constrained optimization problem.1 The classification procedure and decision tree developed below are valid only when an elementary rate law is available and all reactions are isothermal, irreversible, and of the same overall reaction order. This special case of equal overall reaction order is quite common and has wide applicability in the process systems community. The new classification procedure is necessary and useful because the previous method fails to correctly classify certain process chemistries with more than one undesired reaction. Classification Procedure The methodology originally developed by Ward et al.1 classifies irreversible process chemistries into two equivalence classes, bounded and nonbounded. The optimal operating policy for a bounded chemistry is to operate the reactor completely full at all times in order to maximize conversion of any bounded species. Thus, one views operation of the plant, and conversion of reactant species, as bounded by the reactor holdup. A nonbounded chemistry exhibits an optimal operating point away from system constraints such that the optimal reactor holdup shifts with system throughput (production rate) changes. To determine the optimal operating policy of a process chemistry, it can be helpful to first classify each reactant species as bounded or nonbounded by the definitions that follow. Definition 1. A bounded species is a reactant species for which increasing conversion of that species decreases the total operating costs of the entire process. The operating costs in Definition 1 refer to selectivity losses to undesired byproduct(s) plus separation and recycle costs. For a fixed production rate, increasing the conversion of a bounded species suppresses byproduct formation and lowers the recycle flow rate of that species, thus lowering fresh feed reactant costs and separation costs simultaneously (a “win-win” situation). Reactant conversion can be directly manipulated by controlling the reactor holdup, so maximizing reactant conversion corresponds to operating the reactor completely full. The unconstrained optimal operating point for all bounded species is at 100% conversion, corresponding to zero values of the recycle flow rates, infinite reactor volume, and zero byproduct production. However, this operating point clearly is outside the feasible operating regime, which leads to the following corollary: Corollary. The unconstrained optimal recycle flow rate for any bounded species is zero. (The term unconstrained in this context means not constrained by equipment capacities.) Definition 2. Nonbounded reactant species are all other reactant species that do not follow Definition 1. Increasing the conversion of a nonbounded species decreases some operating costs (e.g., separation costs) while increasing other operating costs (e.g., raw materials usage). A tradeoff generally exists between selectivity losses and separation costs when increasing the conversion of a nonbounded species, so it can be optimal not to convert a nonbounded species completely. In contrast to the situation covered by the Corollary, the unconstrained optimal recycle flow rate of a nonbounded species can be nonzero. An unconstrained optimal operating point of zero recycle flow rate is not a sufficient condition for classifying a reactant species as bounded, but it is a necessary condition. It should be noted that there is a direct correlation between reactant conversion and recycle flow rate. In this methodology, a perfectly mixed reactor is assumed where the concentration of
a reactant species inside the reactor can be approximated as follows: [A] ) RA/q, where q is the total volumetric effluent flow rate from the reactor. For a fixed production rate of desired product, increasing the conversion of a reactant species is equivalent to decreasing the reactant concentration in the reactor, in turn decreasing the recycle flow rate of the reactant. Thus, increasing conversion lowers the amount of reactant to recycle and, hence, lowers separation costs. Referring to the Corollary, zero recycle flow rate corresponds to complete reactant conversion and zero separation costs for that reactant. Recycle flow rates serve as the degrees of freedom in the current methodology; therefore, only recycle flow rates will be used in the discussion and should be thought of as surrogates for reactant conversion (and molar ratio of reactants at the reactor inlet) and reactor concentration. This paper presents a modified classification scheme for reactions with equal overall reaction order that is based on whether each undesired reaction is in parallel or in series with the main reaction. A process chemistry can have all side reactions in parallel, all side reactions in series, or a combination of side reactions both in series and in parallel with the main reaction. The case of all undesired reactions in parallel with the main reaction is discussed first. Parallel Reactions With purely parallel (i.e., competing) reactions, the reactant species are all consumed simultaneously in more than one reaction to form different products; no product species appear as reactant species in any reaction. Consider the following set of parallel reactions with two reactant species. An elementary kinetic rate law is assumed where Ri is the stoichiometric coefficient of reactant species i in the desired reaction and Rij is the coefficient of reactant species i in the jth undesired reaction.
R1A + R2B f C
r0 ) k0[A]R1[B]R2
desired
R11A + R21B f D
r1 ) k1[A]R11[B]R21
undesired
R12A + R22B f E
r2 ) k2[A]R12[B]R22
undesired
(1)
Following the methodology developed by Ward et al.,1 the production rates of the desired and undesired products can be expressed as
P C ) k0
RAR1 RBR2 q
R1 R11
PD ) k1 PE ) k 2
RA q
R11
q
R2
V ) k0RAR1RBR2
R21
RB q
R21
RAR12 RBR22 q
R12
q
R22
V R1+R2
q
V ) k1RAR11RBR21 V ) k2RAR12RBR22
V R11 + R21
q
(2)
V q
R12 + R22
The overall forward reaction order of the desired reaction, υT, is the sum of the reaction orders of the individual reactant species. In general, n
υT )
Ri ∑ i)1
(3)
The overall forward reaction order of each undesired reaction is expressed similarly as
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Ind. Eng. Chem. Res., Vol. 45, No. 24, 2006 n
υTj )
Rij ∑ i)1
(4)
For the case of two reactant species and two undesired reactions, υT ) R1 + R2, υT1 ) R11 + R21, and υT2 ) R12 + R22. The byproduct production rates, PD and PE, are made dimensionless by dividing by the fixed production rate of the desired product, PC; the resulting expressions are as follows:
P′D ) k′1 P′E ) k′2
RAR11RBR21 T
qυ1
RAR12RBR22 υ2T
q
T
qυ RAR1RBR2
(5)
υT
q RAR1RBR2
For the special case investigated in this work, every reaction has the same overall reaction order; thus υT ) υTj for j ) 1 and 2 and eq 5 can be simplified to
() () R′A R′B
(R11-R1)
R′A P′E ) k′2 R′B
(R12-R1)
P′D ) k′1
() () R′A R′B
(R2-R21)
R′A ) k′2 R′B
(R2-R22)
) k′1
(6)
Note that the reactor effluent volumetric flow rate term, q, does not affect the byproduct production rate for the special case of equal overall reaction order. Inserting the byproduct production expressions in eq 6 into the dimensionless economic potential model developed by Ward et al.1 yields
()
R′A C ′ ) 100k′1 R′B
(R11-R1)
()
R′A + 100k′2 R′B
(R12-R1)
+ R′A + R′B (7)
It is the goal of the economic optimization to minimize the operating costs of a chemical plant; these costs are represented by the economic potential model given in eq 7. For the case of increasing reactor conversion, (Ri f 0 ∀ i), it can be shown that
lim C ′ f 0
R′Af0 R′Bf0
(8)
Thus, minimization of the cost function occurs when the recycle flow rate(s) approach zero (100% conversion) for all reactant species, the condition that completely suppresses all byproduct formation. If there is complete reactant conversion, no byproduct production, and a pure product, a separation system is not required and there will be no variable operating costs corresponding to the situation C ′ ) 0 (baseline production cost). However, the operating point of Ri ) 0 ∀ i requires an infiniteholdup reactor and is outside the feasible operating region; therefore, optimal operation is bounded by the reactor volume constraint. Rule 1 (below) is a new contribution, not part of the original work by Ward et al.,1 and is necessary to correctly classify process chemistries with multiple parallel reactions that have the same overall reaction order. Rule 1 is valid for process chemistries with any number of reactant species and side reactions. The original classification scheme developed by Ward et al. is equivalent to the current one for the case of a single undesired reaction (see Appendix) but does not extend rigorously to process chemistries with multiple undesired reactions.
Figure 1. Economic optimization contour plot for example 1 for k′1 ) 0.4, k′2 ) 0.4
Rule 1. If all undesired reactions are in parallel with the main reaction, then the overall process chemistry is bounded (subject to all reactions having the same overall reaction order). In summary, for process chemistries with only parallel reactions of equal overall order, maximal conversion of all reactant species maximally suppresses byproduct formation and lowers separation costs, thus leading to a reactor volume bounded operating policy. Example 1:
A+BfC
r0 ) k0[A][B]
A+AfD
r1 ) k1[A][A]
P′D ) k′1
B+BfE
r2 ) k2[B][B]
R′B P′E ) k′2 R′A
R′A (9) R′B
The first step in classifying a process chemistry is to determine if each undesired reaction is in parallel or in series with the main reaction. Both side reactions in Example 1 are in parallel with the main reaction, thus making the overall chemistry bounded by Rule 1. The recycle flow rates of both reactant species should be minimized to suppress both side reactions. Figure 1 shows an economic optimization landscape of constant cost contours for Example 1 with the unconstrained optimal operating point at the origin marked with an open circle, “O”. This point is outside the feasible operating region enclosed by the dashed lines representing the physical constraints of the process. Hence, the feasible optimal operating point corresponding to lowest cost lies on the reactor volume constraint (marked with an “X”), confirming that optimal operation is bounded by the reactor holdup, as expected for this bounded chemistry. Series and Series/Parallel Reactions With series (i.e., consecutive) reactions, the reactant species form a product, which in turn reacts to form another product. Process chemistries can have all undesired reactions in series with the main (desired) reaction or have a combination of undesired reactions that are both in series and in parallel with the main reaction. General definitions for bounded and nonbounded species have been presented above; the following rules describe a procedure to determine if each reactant species is
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bounded or nonbounded for reactions in series or series/parallel when all reactions have the same overall reaction order. Rules 2 and 3 (below) have been modified from the original work by Ward et al.1 and are understood to be valid only for process chemistries of series and series/parallel reactions. Rule 2. A bounded species is a reactant species for which the selectivity to all undesired byproducts either (i) decreases with or (ii) is unaffected by increasing conversion of that species (subject to all reactions having the same overall reaction order). Explanation for Rule 2(i). If the reaction order of a reactant species is higher in an undesired reaction than in the desired reaction, then the undesired byproduct formation will decrease with decreasing recycle flow rate of that reactant. Hence, if the reaction order of a specific reactant species is higher in all undesired reactions than in the desired one, the selectivity to all undesired byproducts will decrease with decreasing recycle flow rate (increasing conversion) of that reaction species. The recycle flow rate of this type of reactant species should be minimized to suppress all byproduct formation; thus, the reactant conversion will be bounded by the reactor volume constraint. Explanation for Rule 2(ii). If the reaction order of a reactant species in an undesired reaction is equal to that in the desired reaction, the recycle flow rate will not affect the byproduct production rate nor the selectivity to this undesired byproduct. It is desirable to increase the conversion of these species to minimize the recycle flow rate(s) and lower separation costs. Once again, conversion is maximized by operating the reactor completely full; thus, reactant conversion is bounded by the reactor volume constraint. Summarizing Rule 2, if the reaction order of a reactant species in all undesired reactions is higher than or equal to the reaction order of that species in the desired reaction, the reactant species is a bounded species. Rule 3. A nonbounded species is a reactant species for which the selectivity to any undesired product increases with increasing conversion of that reactant species (subject to all reactions having the same overall reaction order). Explanation of Rule 3. If the order of a specific reactant species is lower in an undesired reaction than in the desired one, then formation of that undesired byproduct will decrease with increasing recycle flow rate of the reactant species. Thus, it is desirable to have a large recycle flow rate of this reactant species to minimize byproduct formation; however, large recycle flow rates increase the operating cost of the separation system. Hence, there is a tradeoff between selectivity losses and separation costs and it can be suboptimal to convert a nonbounded species completely. Summarizing Rule 3, if the reaction order of a reactant species is lower in any undesired reaction than in the desired one, the selectivity to that undesired byproduct will increase with decreasing recycle flow rate of that reaction species. Example 2:
A+BfC
r0 ) k0[A][B]
A+AfD
r1 ) k1[A][A]
R′A P′D ) k′1 R′B
A+CfE
r2 ) k2[A][C]
P′E ≈
(10)
k′2 R′B
The first undesired reaction is in parallel with the main reaction while the second undesired reaction is in series/parallel with the main reaction, so Rules 2 and 3 are used to classify the reactant species. The selectivity to the undesired product D
Figure 2. Economic optimization contour plot for example 2 for k′1 ) 0.2, k′2 ) 0.2.
decreases with decreasing recycle flow rate of species A, while the selectivity to species E is unaffected by the recycle flow rate of species A; thus, species A is a bounded species by Rule 2. The recycle flow rate of species A affects the concentration and mole fraction of the desired product, C, in the reactor; but the production rate of C, PC, at the reactor effluent is fixed (regardless of reactor concentration) because of design specifications. The byproduct formation, P′E, is independent of the recycle flow rate of species A, R′A; therefore, the recycle flow rate and the corresponding conversion of A do not affect the selectivity to the undesired product, E. Species B is a nonbounded species by Rule 3 because the selectivity to the undesired product E increases with decreasing recycle flow rate of species B. A single bounded species makes the overall chemistry bounded, and the reactor should be operated completely full at all times to maximize the conversion of the bounded species. Figure 2 shows the economic optimization landscape for Example 2. From the Corollary, the unconstrained optimal recycle flow rate of the bounded species A should be at zero, while the unconstrained optimal recycle flow rate of the nonbounded species B is nonzero. This unconstrained point is shown in Figure 2 by the open circle “O”. This point is outside the feasible operating region, so the optimal operating point, marked by “X”, once again is on the reactor volume constraint for this bounded chemistry. Example 3:
A+BfC
r0 ) k0[A][B]
A+AfD
r1 ) k1[A][A]
C+CfE
r2 ) k2[C][C]
P′D ) k′1 P′E ≈
R′A R′B
(11)
k′2 R′AR′B
The second undesired reaction is in series with the main reaction, so Rules 2 and 3 are used here. Both reactant species A and B are nonbounded species because the selectivity to the undesired product E increases with decreasing recycle flow rates of either species, as can be seen from the expression for the formation of the undesired product E. When all reactant species are nonbounded, the overall chemistry is nonbounded. Once a
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Figure 3. Economic optimization contour plots for example 3 for (a) k′1 ) 0.6, k′2 ) 0.1 and (b) k′1 ) 0.1, k′2 ) 0.1.
reactant species is classified as nonbounded, other side reactions cannot make it bounded. Therefore, it is not necessary to classify the reactant species with respect to the undesired reaction that produces D, because both reactant species are already classified as nonbounded with respect to the undesired reaction that produces E. The overall chemistry in Example 3 is classified as nonbounded. However, decreasing the recycle flow rate of species A lowers the selectivity to the undesired product D while at the same time increasing the selectivity to undesired product E. Hence, it is not clear where the optimal operating point will lie; in fact, it will depend on the relative magnitudes of the kinetic rate constants. For this chemistry, several possible limiting cases are identified, including the following: (1) the second reaction is very slow compared to the other reactions, (2) the third reaction is very slow, or (3) all reactions have comparable rates. For case (1) when the undesired reaction that produces D is very slow compared to the other reactions, the only remaining side reaction C + C f E is in series with the main reaction, the chemistry is nonbounded, and the optimization diagram is similar to that for Chemistry 3 shown in Figure 14 of Ward et al.1 For case (2) when the undesired reaction that produces E is very slow, the remaining side reaction A + A f D is in parallel with the main reaction, which is classified as a bounded chemistry. This limiting case is shown in Figure 3a for k′1 ) 0.6 and k′2 ) 0.1. The optimal operating point (marked by the “X”) is on the reactor volume constraint, even though the overall chemistry (all three reactions) is classified as nonbounded. For case (3) when the rate constants of both undesired reactions are comparable, then the optimal operating point can be away from the reactor volume constraint. This case of comparable rate constants is illustrated in Figure 3b for k′1 ) 0.1 and k′2 ) 0.1 and exhibits an optimal operating point away from all system constraints, as expected for this nonbounded chemistry. Thus, in this case, one encounters a shift in operating policy that is based solely on the relative values of the kinetic rate constants. Example 3 demonstrates that a nonbounded chemistry can have an optimal operating point on the reactor volume constraint, behaving like a bounded chemistry; this point depends on the physical parameters of the system (molar volumes, kinetic rate data, production rate, etc.) and cannot be determined simply by inspection of the reaction orders alone. However, a bounded chemistry will always have its optimal operating point on the reactor volume constraint and can never be made nonbounded simply by shifting parameters.
Figure 4. Reactant species classification system for reactions with equal overall reaction order in series or series/parallel.
Figure 5. Constrained and unconstrained operation for bounded species.
When there is at least one undesired reaction in series with the main reaction, it is simple to classify a reactant species by inspecting and comparing reaction orders in each reaction. Rules 2 and 3 (derived from Ward et al.1 with minor but important differences in wording) are summarized in the reactant species classification system presented in Figure 4. Once each reactant species is identified as bounded or nonbounded for series or series/parallel reactions according to the classification procedure described above, the following rules are used to classify the process chemistry. Rules 4 and 5 below are general rules taken from Ward et al.1 and are valid for series or series/parallel reactions of equal overall order. Rule 4. A single bounded reactant species makes the overall chemistry bounded. Rule 5. If there are no bounded species (i.e., all reactant species are nonbounded), the overall chemistry is nonbounded.
Ind. Eng. Chem. Res., Vol. 45, No. 24, 2006 8061 Table 1. Summary of Rules for Reactions with Equal Overall Reaction Order Rule 1: If all undesired reactions are in parallel with the main reaction then the overall chemistry is bounded. Rules 2-5 are for process chemistries of series or series/parallel reactions Rule 2: A bounded species is a reactant species for which the selectivity to all undesired byproducts either (i) decreases with or (ii) is unaffected by increasing conversion of that species. Rule 3: A nonbounded species is a reactant species for which the selectivity to any undesired product increases with increasing conversion of that reactant species. Rule 4: A single bounded reactant species makes the overall chemistry bounded. Rule 5: If there are no bounded species (i.e., all reactant species are nonbounded), the overall chemistry is nonbounded.
Therefore, as long as there is at least one bounded reactant species, the overall chemistry is bounded, and the reactor should be operated completely full to minimize the recycle flow rate of any bounded species. Figure 5 demonstrates the situation of one or more bounded species with the unconstrained operating points marked with an open circle, “O”, the constrained operating points marked with an “X”, and the reactor volume constraint represented by the dashed line. The unconstrained optimal operating point for any bounded species is outside the feasible operating region, causing the constrained optimal operating point to be on the reactor volume constraint. An example of this feature can be seen in Example 2, where species A is bounded and B is nonbounded. The situation in which all reactant species are nonbounded corresponds to a nonbounded chemistry in which the optimal operating point can be away from the reactor volume constraint and can shift with changes in the production rate, feed flow rate, and other system disturbances (as shown in Example 3). Using Rules 1-5 (summarized in Table 1), one can classify any process chemistry of competing/consecutive irreversible reactions. For the special case of a single reaction, there are no selectivity issues, and the optimization problem simplifies to one of minimizing the operating costs associated with the separation system. The economic optimum corresponds to lowering the recycle flow rates, which is accomplished by increasing conversion subject to the reactor volume constraint. Thus, for a plant with a fixed production rate and a single reaction, the chemistry is bounded and the reactor should be operated completely full to minimize separation costs. This result
has been identified previously in the literature.6,7 At this point, any irreversible process chemistry, with or without side reactions, can be classified as long as all reactions have the same overall order; the decision tree for this procedure is shown in Figure 6. Conclusions In this paper, formal rules have been presented and a complete decision tree classification procedure has been developed that can be used to classify chemistries with multiple irreversible undesired reactions of equal overall reaction order. A process chemistry can be classified as bounded or nonbounded by first determining if each undesired reaction is in parallel or series with the main reaction; when at least one undesired reaction is in series or series/parallel with the main reaction, one must inspect the reaction order of every reactant in every reaction. Process chemistry classification determines the steady-state optimal operating policy, which has significant implications for the plantwide control structure and operations but does not guarantee or imply optimal dynamic performance. However, Ward et al. earlier investigated the dynamic implications of these steady-state operating policies for several cases and demonstrated that individual controllers required to implement the suggested steady-state control structure can be tuned to provide satisfactory dynamic performance with superior time-averaged economics.11
Figure 6. Decision tree for irreversible process chemistry classification for reactions with equal overall reaction order.
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Appendix The original classification scheme developed by Ward et al.1 for the case of a single byproduct-producing reaction is equivalent to the current method for parallel reactions of equal overall order. Consider the following two general reactions with n arbitrary reactant species, n
R1A1 + R2A2 + ... + RnAn f C
r ) k0
[Ai]R ∏ i)1
i
n
R11A1 + R21A2 + ... + Rn1An f D
(12)
[Ai] ∏ i)1
r ) k1
Ri1
n and the special case of equal overall reaction order, ∑i)1 Ri ) n ∑i)1 Ri1. Clearly, a chemistry of this type would be classified as bounded according to the present methodology, because the undesired reaction is in parallel with the desired reaction. However, it can also be shown that such a process chemistry will always be classified as bounded according to the original methodology of Ward et al.1 When Ri ) Ri1 for all i, every reactant species has the same reaction order in both reactions; therefore, each species would be classified as bounded by the original methodology, causing the overall chemistry to be bounded. Now consider the case where Ri * Ri1 for some value(s) of i. Let j be the smallest number such that Rj * Rj1. If Rj < Rj1, then species Aj is a bounded species. If Rj > Rj1, then there n must exist some k for which Rk < Rk1 because ∑i)1 Ri ) n ∑i)1 Ri1. In this case, species Ak is a bounded species. In either case, the above process chemistry must contain a bounded species; therefore, it will always be classified as a bounded chemistry according to the original methodology.
Acknowledgment The authors (D.W.G., M.F.D., and D.A.M.) are grateful for financial support provided by the National Science Foundation (Grant No. CTS-0554718) and by the Fulbright Program for U. S. Students (J.D.W.). Nomenclature r ) specific reaction rate [mol/(L hr)] k ) reaction rate constant k′j ) dimensionless kinetic rate constant (k′j ) kj/k0) P ) production rate (mol/hr) P′i ) dimensionless production rate (P′i ) Pi/PC)
R ) recycle flow rate (mol/hr) R′i ) dimensionless recycle flow rate (R′i ) Ri/PC) V ) reactor volume (L) q ) reactor effluent flow rate (L/hr) C ′ ) dimensionless operating costs Ri ) stoichiometric coefficient of species i in desired reaction Rij ) stoichiometric coefficient of species i in jth undesired reaction νT ) overall forward reaction order of desired reaction νTj ) overall forward reaction order of jth undesired reaction Subscripts 0 ) reference to desired reaction 1, ..., i ) reactant species 1, ..., i 1, ..., j ) undesired reactions 1, ..., j A, ..., E ) species A, ..., E Superscripts ′ (prime) ) dimensionless quantity T ) total for all species Literature Cited (1) Ward, J. D.; Mellichamp, D. A.; Doherty, M. F. Importance of Process Chemistry in Selecting the Operating Policy for Plants with Recycle. Ind. Eng. Chem. Res. 2004, 43, 3957. (2) Doherty, M. F.; Malone, M. F. Conceptual Design of Distillation Systems; McGraw-Hill: New York, 2001. (3) Douglas, J. M. Conceptual Design of Chemical Processes; McGrawHill: NewYork, 1988. (4) Fisher, W. R.; Doherty, M. F.; Douglas, J. M. The Interface between Design and Control. 2. Process Operability. Ind. Eng. Chem. Res. 1988, 27, 606. (5) Luyben, W. L. Snowball Effects in Reactor/Separator Processes with Recycle. Ind. Eng. Chem. Res. 1994, 33, 299. (6) Larsson, T.; Govatsmark, M. S.; Skogestad, S.; Yu, C.-C. Control Structure Selection for Reactor, Separator and Recycle Processes. Ind. Eng. Chem. Res. 2003, 42, 1225. (7) Larsson, T.; Skogestad, S. Plantwide ControlsA Review and a New Design Procedure. Model., Identif. Control EnViron. Syst. 2000, 21, 209. (8) Wu, K.-L.; Yu, C.-C. Reactor/Separator Processes with Recycle. 1. Candidate Control Structure for Operability. Comput. Chem. Eng. 1996, 20, 1291. (9) Wu, K.-L.; Yu, C.-C.; Luyben, W. L.; Skogestad, S. Reactor/ Separator Processes with Recycle. 2. Design for Composition Control. Comput. Chem. Eng. 2003, 27, 401. (10) Ward, J. D.; Mellichamp, D. A.; Doherty, M. F. Novel Reactor Temperature and Recycle Flow Rate Policies for Optimal Process Operation in the Plantwide Context. Ind. Eng. Chem. Res. 2005, 44, 6729. (11) Ward, J. D.; Mellichamp, D. A.; Doherty, M. F. Insight from Economically Optimal Steady-State Operating Policies for Dynamic Plantwide Control. Ind. Eng. Chem. Res. 2006, 45, 1343.
ReceiVed for reView December 19, 2005 ReVised manuscript receiVed July 7, 2006 Accepted August 30, 2006 IE051412P
