Ind. Eng. Chem. Res. 2009, 48, 8037–8047
8037
Effect of Competing Reversible Reactions on Optimal Operating Policies for Plants with Recycle† Derek W. Griffin, Duncan A. Mellichamp, and Michael F. Doherty* Department of Chemical Engineering, UniVersity of California, Santa Barbara, California 93106-5080
A simple classification procedure for determining optimal steady-state operating policies for plants with recycle has recently been developed by Griffin et al. to cover complex process chemistries. (Griffin, D. W.; Mellichamp, D. A.; Doherty, M. F. AIChE J. 2008, 54, 2597). This procedure classifies a process chemistry into one of two groups of operating policies based solely on the reaction kinetics. The optimal operating policy for a bounded chemistry is to operate the reactor completely full for all production rates while a nonbounded chemistry may have a variable reactor volume subject to system constraints. The current work focuses on process chemistries with reversible reactions and demonstrates that the optimal operating policy can change depending on which reaction(s) is(are) reversible and the magnitude of the equilibrium constant(s). Process chemistries with a reversible desired reaction also exhibit multiple steady-states, but only one of the steadystates corresponds to feasible operating conditions. 1. Introduction Chemical process design involves the selection of unit operations, process flow diagrams, operating policies, and control strategies that give optimal economic performance. One current line of research focuses on the optimal utilization of the excess capacity inherently included in every process design. Heuristics exist for the conceptual design of chemical processes, and methods are available to improve the flexibility of these designs. When equipment is designed with excess capacity there is the potential to use this extra capacity under all operating conditions. Process economics may be greatly affected by the percent utilization of capacity at which process units are operated; an optimal operating policy exists for this equipment capacity usage that yields optimum economic performance. The conventional process configuration of reactor subsystem, separation subsystem, and recycle(s) (RSR process) is a widely studied system that provides insight into how the optimal operating policy for plants with recycle is selected. Several authors have considered this conventional plant with an irreversible elementary reaction A f B that has no intrinsic selectivity issues. Operating policies that have been suggested to handle production rate changes in such a plant include maintaining a constant recycle flow rate while permitting the reactor holdup to vary,1-5 operating the reactor at maximum holdup with the recycle flow rate allowed to change,6,7 or allowing both the reactor holdup and recycle flow to vary.8,9 A more general approach to this problem developed by Ward et al.10 considers different classes of irreversible process chemistries that include selectivity losses, employing recycle flow rates as the degrees of freedom. The chemistries are classified on the basis of whether the optimal operating policy is to operate the reactor completely full at all times, bounded, or with a variable reactor holdup operating policy, nonbounded. This work was extended by Griffin et al.11 to include multiple undesired irreversible reactions of equal overall order. More recently Griffin et al.12 developed a generalized procedure to † This paper is dedicated to Professor J. B. Joshi, in recognition of his many outstanding contributions to multiphase reaction engineering, chemical engineering education, and practice. * To whom correspondence should be addressed. E-mail: mfd@ engineering.ucsb.edu. Tel.: (805) 893-5309. Fax: (805) 893-4731.
classify a much broader class of process chemistries which may include reversible chemistries with multiple undesired reactions of equal/unequal overall order. However, the optimal operating policy for a process chemistry with at least one reversible reaction can change depending on which reaction is reversible and the magnitude of the equilibrium constant; these issues provide the focus of this paper. Many industrial chemical processes involve reversible reactions that exhibit the problem of limiting chemical equilibrium conversion. Numerous published works discuss methods to overcome the effect of this equilibrium conversion while at the same time minimizing capital and operating costs. For the conventional RSR process, attainable region theory has been applied to many series-parallel reactor network configurations to maximize conversion.13,14 Kapilakarn and Luyben have developed several “on-demand” control structures to achieve desired production rates and desirable dynamic responses for the RSR process with competing reversible reactions.15,16 One common approach to overcome the limitations of equilibrium is to remove product from the reaction medium as fast as it is formed. Combining several reaction and separation steps in series (as shown in Figure 1) represents one method to remove product and increase conversion. A simple reversible chemistry A + B a C is demonstrated on this ternary composition diagram. In an idealized process configuration, the feed to the first reactor consists of an excess of one reactant (B), that reacts with A to form an equilibrium mixture which is fed to the first separation unit where the product C is removed. The remaining mixture of reactants (point 2 on the triangular diagram, Figure 1) is then fed to another reactor, and this process is repeated until all of the limiting reactant (A) is reacted. The overall composition of the mixture containing only the excess component (B) and product (C) at the end of such a reaction/separation chain is represented by the asterisk (*) on the diagram. In principle, given an infinite number of reactor/separator units, 100% conversion can be achieved, but at very large capital expenditure. Thus, an inherent tradeoff exists: one can increase conversion by adding unit operations, but that increases capital and operating costs. In certain situations, it may be possible to combine both steps of reaction and separation in the same unit, which is the
10.1021/ie801482z CCC: $40.75 2009 American Chemical Society Published on Web 05/26/2009
8038
Ind. Eng. Chem. Res., Vol. 48, No. 17, 2009
Figure 2. Reactor/separator/recycle structure.
Figure 1. Composition diagram for idealized multiple reactor/separator units in series for a reversible reaction.
motivation and basis for reactive distillation. In practice, the number of reaction stages is clearly finite and each reaction stage need not achieve chemical equilibrium. A recent review by Sharma and Mahajani reports a large number of industrial chemical processes that can benefit from reactive distillation.17 However, not every process is a candidate for this type of reactive distillation technology. Energy costs can dominate and make the reactive distillation process feasible but uneconomical (e.g., ethylene glycol process);18 also it may be impossible to obtain a pure product from a reactive distillation column. For such processes with reversible reactions, a conventional reactor/separator train, as discussed above, can be more economical, subject to the tradeoff noted. If adding an extra reactor does not increase the conversion enough to offset the extra equipment costs, such as with the toluene disproportionation process,19 then the process must be operated with a single reactor/separation/recycle system. For reversible reactions, reactive distillation is sometimes the obvious choice but feasibility and economics must be considered before abandoning a conventional process configuration. The work described here focuses on the operation of a conventional plant that involves equilibrium reactions. The operating policy depends first on the classification of the irreversible process chemistry, then which reaction(s) is(are) reversible, then the numerical value(s) of the equilibrium constant(s). Disturbances such as variation in reactor temperature can change an equilibrium constant and thus shift the operating policy based on the extent of reaction reversibility. The approach presented here is quite general for equilibrium processes and thus is useful for making quick estimates of the appropriate optimal operating policy. 2. Methodology for System Analysis The current work determines optimal operating policies for the RSR process depicted in Figure 2 when equilibrium process chemistries are involved. The operating policy for elementary process chemistries depends on the stoichiometry and kinetics of the coupled reactions. The methodologyswhich includes the derivation of system unknowns, the economic potential model, the optimization problem, and the classification proceduresis not described in detail here but was developed for the general case by Griffin et al.12 The reader is referred to that article and
the article by Ward et al.10 for the fundamentals of the methodology. The reactor network depicted in Figure 2 is assumed to consist of a single liquid-phase, isothermal, perfectly mixed reactor. The separation system is not specified but it is assumed that perfect separation of the reactants and products can be achieved (this assumption is made only for pedagogical purposes and is easily relaxed). Flow rates of products are denoted as P. All chemistries analyzed here are assumed to include a single desired product (always referred to as species C) with a fixed production rate. Flow rates of recycle streams are denoted as R; any unreacted reactant is separated and recycled back to the reactor. Fresh feed flow rates are denoted as F; these are mixed with the recycle stream(s) and fed to the reactor. A degrees of freedom analysis is a necessary step in the optimal design of a chemical plant. Conventionally, variables such as the molar ratio of reactants at the reactor inlet and the conversion of limiting reactant are chosen as design variables. The analysis developed by Ward et al. deviates from this conventional procedure by using recycle flow rates as design degrees of freedom.10 This approach allows for easier understanding of the optimization problem when determining an optimal operating policy. With the production rate of the desired product fixed, the reactor temperature fixed (isothermal operation), and the reactor holdup allowed to vary, the remaining control degrees of freedom are simply the recycle stream flow rates. Choosing the recycle stream flow rates as design and control degrees of freedom not only specifies all degrees of freedom but also presents a clear connection between the optimal design and operation of a plant. 2.1. Process Chemistry Classification. Previous work on optimal operating policies has focused mainly on irreversible process chemistries with elementary kinetics.10,11 The generalized methodology by Griffin et al.12 for the first time allows much more complex chemistries to be classified as bounded or nonbounded. This classification procedure is valid for determining the optimal operating policy of any irreversible process chemistry. Once reaction reversibility is introduced, the operating policy may change based on which reaction(s) is (are) reversible and the magnitude of the equilibrium constant(s). However, this potential shift in operating policy fundamentally depends on the classification of the original irreversible chemistry; thus it is vital first to classify the process chemistry as if it were irreversible. The reader is referred to the paper by Griffin et al.12 for detailed definitions and rules for classifying reactant species and/or process chemistries as bounded or nonbounded. 3. Optimal Operating Policies for Process Chemistries with Reaction Reversibility The general method for determining optimal operating policies developed by Griffin et al.12 is valid for chemistries with multiple undesired reactions of equal or unequal overall order. The following sections utilize simpler chemistries consisting of a single undesired reaction of equal overall order to
Ind. Eng. Chem. Res., Vol. 48, No. 17, 2009
demonstrate the effect of the equilibrium constant magnitude on optimal operating policies. This approach is motivated by simplicity of presentation, but the results do not change if the reactions do not have equal overall reaction order. This section describes two different cases where a possible shift in operating policy can occur depending on which reaction (either desired or undesired) is reversible and the magnitude of the single equilibrium constant. These cases include a bounded chemistry with a reversible desired reaction and a nonbounded chemistry with a reversible undesired reaction. The other two cases of interest do not result in a potential change of operating policy and are covered in the Supporting Information; those cases include a bounded chemistry with a reversible undesired reaction and a nonbounded chemistry with a reversible reaction. The results for all four cases are summarized at the end of this section. 3.1. A Bounded Chemistry with a Reversible Desired Reaction. Chemistry 1, given below by eq 1, possesses a single undesired reaction in parallel with the main reaction where both reactions are irreversible. The nondimensional expression for the byproduct formation rate is also given. A + B f C r0 ) k0[A][B] desired A + A f D r1 ) k1[A][A] undesired P′D ) k′1
R′A (1) R′B
Following the classification procedure by Griffin et al.,12 chemistry 1 is determined to be a bounded chemistry, and the predicted optimal operating policy for this chemistry is to operate the reactor completely full at all times. However, this policy can change when the desired reaction is reversible (eq 2); this chemistry is referred to as chemistry 1A. A + B a C r0 ) k0[A][B] - k-0[C] r1 ) k1[A][A] A+AfD
Figure 3. ABC composition diagram on a D-free basis.
A continues to react to form D until all C has back-reacted and only species B and D remain, as demonstrated in Figure 3. For chemistry 1A given in eq 2, expressions for the desired production rate, PC, the undesired byproduct formation rate, PD, and nondimensional quantities are derived using the methodology outlined in Griffin et al.12 (The reader is referred to the Supporting Information to this paper for a simplified derivation of process unknowns for the chemistry A f C f D.) Recycle flow rates are denoted as R, the reactor volumetric holdup as V, and the reactor volumetric effluent flowrate as q, which is given in eq 5. PC ) k0RARB
[
k-0 qPC V 1q k0 RARB
(2)
For chemistries whose side reactions are all in parallel with the main reaction, all reactant species represent bounded species, and the unconstrained optimal operating point is at zero recycle (100% conversion) of every reactant species.12 An important characteristic of any chemistry with a reversible desired reaction is that it is impossible to achieve complete per-pass conversion in finite time due to equilibrium limitations. For the main reaction to be near equilibrium, a finite amount of unreacted reactant must be present with the desired product at the reactor outlet. Therefore recycle flow rates of zero cannot be achieved. At a fixed production rate of the desired product, the byproduct production rate and the required reactor volume go to infinity as the conversion of the limiting reactant approaches the equilibrium conversion. And as the byproduct production rate approaches infinity, the fresh feed rate of species A must also approach infinity in accordance with the overall mass balance, FA ) PC + 2PD. Another way to consider this equilibrium problem is if the production rate, PC, is not fixed and there is a fixed initial amount of A and B fed to the reactor. If the main (desired) reaction is at equilibrium, any further conversion of the limiting reactant will produce only undesired product. This result occurs because as the reaction path follows the stoichiometric line from the AB initial condition toward the equilibrium curve, shown in Figure 3, the main reaction slows down. The undesired reaction continues at the same rate consuming A which disturbs the equilibrium in the main reaction causing C to back-react to re-form A and B as a new equilibrium is established. Species
8039
PD ) k1RA2
]
(3)
V q2
(4)
q ) RAνA + RBνB + PCνC + PDνD
(5)
The concentration-based reaction equilibrium constant for an ideal solution can be written as K0 )
k0 [C] ) [A][B] k-0
(6)
Equilibrium constants are made dimensionless by use of the following relation: K′i )
Ki
(7)
νmC
where m is the reaction order of the forward reaction minus the reaction order of the backward reaction. For the reversible reaction A + B a C, m ) 1, and K′0 ) k0/(k-0νC). Now dividing the expression for the undesired reaction (eq 4) by the expression for the desired reaction (eq 3) and forming nondimensional quantities (see Griffin et al.12 for appropriate methods to nondimensionalize) gives
[
k1RA2 qPC PD ) 1PC k0RARB K0RARB
]
-1
f
P′D ) k′1
[
R′A q' 1R′B K′0R′AR′B
]
-1
(8)
8040
Ind. Eng. Chem. Res., Vol. 48, No. 17, 2009
Figure 4. Chemistry 1A: Production rate of undesired byproduct vs recycle flow rates for different values of the equilibrium constant.
In the limit as the desired reaction becomes irreversible, K′0 f ∞ and P′D f k′1(R′A/R′B), that is, the same expression as given in eq 1 for the irreversible chemistry. The equilibrium constant for the desired reaction is written as follows: K0 )
PC q2 qPC [C] ) ) ) [A][B] q RARB RARB [RAνA + RBνB + PCνC + PDνD]PC (9) RARB
and made dimensionless K′0 )
[R′Aν′A + R′Bν′B + 1 + P′Dν′D] q' ) R′AR′B R′AR′B
(10)
Substituting the expression for the equilibrium constant given in eq 10, K′0 ) q′/(R′AR′B), into the right-hand side of eq 8 makes the bracketed term, 1 - q′/(K′0R′AR′B), go to zero and the byproduct formation rate and fresh feed flow rate of A both go to infinity as expected for situations in which the desired reaction is at equilibrium as discussed above. Note that the equilibrium constant is truly a constant equaling k0/k-0, and only when the reaction is at equilibrium does [C]/([A] [B]) ) k0/k-0. Figure 4 shows the byproduct production rate given in eq 8 as the recycle flow rates are varied for both high and low values of the equilibrium constant. The dashed lines represent the physicochemical boundary on recycle flows created by the equilibrium reaction. The dashed lines are determined by rearranging eq 10 for P′D as follows: P′D )
K′0R′AR′B - R′Aν′A - R′Bν′B - 1 ν′D
(11)
With one recycle flow fixed, eq 11 represents a straight line whose slope is a function of the equilibrium constant. As
K′0 f ∞ (in the limit of irreversibility) the slope of P′D versus R′A or R′B goes to infinity and the dashed equilibrium lines move onto the ordinate, eliminating any equilibrium limitations. Figure 4a shows P′D versus R′A for different values of R′B and a low Value of the equilibrium constant. The general results reported for the irreversible chemistry still hold: low recycle flow rates of the bounded species A and high recycle flow rates of species B suppress the undesired reaction and reduce P′D. However, the lower-bound on the recycle flow rates is not zero (in contrast to the irreversible chemistry) but some finite value related to the value of the equilibrium constant. For a fixed value of R′B, the plot of P′D versus R′A turns on itself as R′A reaches its minimum value, thus producing the theoretical possibility of multiple steady-states. In the limit as the desired reaction approaches irreversibility (K′0 f ∞), as R′A f 0 or R′B f ∞ then P′Df 0; the turning point is eliminated, and multiple steadystates cannot exist. This turning point is due to the quadratic nature of the expression for byproduct formation, P′D, given in eq 8 noting that the volumetric reactor effluent flow rate, q′, is also a function of P′D as shown in eq 10. This steady-state multiplicity will be discussed later in section 4. Plots of P′D versus R′A for different values of R′B approach the equilibrium limitation shown by the dashed lines but can never cross this boundary. A higher equilibrium constant, close to the limit of irreversibility, is shown in Figure 4b. The minimum values of R′A for a fixed R′B are smaller, corresponding to higher equilibrium conversion. Less byproduct formation occurs with the larger equilibrium constant because a lower concentration of species A is present in the reactor to form species D. This feature is also demonstrated mathematically by eq 8: as K′0 increases, the bracketed term, 1 - q′/(K′0R′AR′B), also increases thus decreasing the overall expression for byproduct production. Figure 4 panels c and d are plots of P′D versus R′B for different values of R′A. Similar observations follow from these plots: maintain R′A low and R′B high to suppress the formation of D.
Ind. Eng. Chem. Res., Vol. 48, No. 17, 2009
8041
Figure 5. Chemistry 1A: Economic optimization contour plots for (a) K′0 ) 10 and (b) K′0 ) 100.
The optimization problem to be solved involves minimizing the operating costs of a plant that has already been designed or built. The main factors that affect the plant economics in terms of operating costs are selectivity losses from the production of undesired byproduct and the operating costs associated with separating and recycling the unreacted reactants. The dimensionless cost objective function that is to be minimized is derived by Griffin et al.12 (and discussed in the Supporting Information) and is shown below for a process chemistry with two reactant species. This cost function is minimized (to maximize profit) in order to find the economically optimal operating point, subject to the reactor volume and recycle flow constraints. The term C′D is the dimensionless cost, that is, ratio of costs of producing one mole of undesired byproduct to those for separating and recycling one mole of reactant species; C′D is normally on the order of 100. C' ) C′DP′D + R′A + R′B
(12)
The operational optimization problem, that is, minimizing operating costs using the two recycle flows as the two degrees of freedom, can be represented as the contour plot shown in Figure 5. The contours for the dimensionless cost function are represented by solid lines (s). Physical constraints imposed on the system by the reactor volume and recycle capacities are shown as dashed lines (- - -) that enclose the feasible operating region. The dotted line (...) represents the equilibrium limitations, eq 10, on the recycle flows and also the limit for infinite reactor volume. No cost or volume contours can cross this equilibrium envelope. A volume contour corresponding to V′max ) ∞ would lie exactly on the dotted equilibrium line. The optimal operating policy for an irreversible bounded chemistry is to operate the reactor completely full at all times to maximize conversion of the bounded species by suppressing byproduct formation. Because an irreversible bounded chemistry always exhibits open cost contours, the minimum cost always lies on the reactor volume constraint. By contrast, Figure 5 shows two contour plots that have closed cost contours, characteristic of a nonbounded chemistry. In this case, the cost contours are closed because the equilibrium envelope represented by the dotted line prevents any contour from reaching a recycle flow rate of zero, thus causing the cost contours to turn back and close on themselves. At small values of K′0, this feature results in an optimal operating point away from the reactor volume constraint as shown in Figure
5a. As the desired reaction nears equilibrium, any further increase in the conversion of the limiting reactant, species A, will only increase byproduct formation as discussed earlier. Thus, under these circumstances it is optimal to operate away from the reactor volume constraint and not attempt to maximize conversion. For a large value of the equilibrium constant, Figure 5b, the optimal operating point can be on the reactor volume constraint, as with the irreversible bounded case. The cost contours are still closed, but the equilibrium constant is large enough that the optimal operating point falls between the reactor volume constraint and the equilibrium envelope. Because this situation corresponds to an optimal reactor holdup larger than the available reactor volume, the reactor should be operated completely full. In the limit of K′0 f ∞, the equilibrium envelope merges with the axes, the cost contours open, and the optimal operating point is on the reactor volume constraint as predicted for an irreversible chemistry. Figure 5 panels a and b demonstrate the characteristics discussed earlier. Optimal operation for this chemistry is at a low recycle flow rate of species A and high recycle flow rate of species B; the optimal values of the recycle flow rates are found by minimizing the cost objective function. A large equilibrium constant lowers byproduct formation, thus reducing the cost penalty. This point is illustrated by comparing the values of the optimal cost contours (the solid lines that cross through the “X”) in Figure 5a,b where the figure with the larger equilibrium constant, Figure 5b, has a lower cost penalty at the optimal operating point. A bounded chemistry can actually exhibit an optimal operating point away from the reactor volume constraint when the desired reaction is reversible and the equilibrium constant is small enough; thus it is functionally equivalent to an irreversible nonbounded chemistry. Because of its interesting characteristics, a case study of a bounded chemistry with a reversible desired reaction is developed at the end of this paper to demonstrate this unusual or “unexpected” shift in operating policy. 3.2. A Nonbounded Chemistry with a Reversible Undesired Reaction. The other interesting case that may lead to a change in operating policy arises when a nonbounded chemistry has a reversible undesired reaction. Chemistry 3, considered with only the undesired reaction reversible, is referred to as chemistry 3B:
8042
Ind. Eng. Chem. Res., Vol. 48, No. 17, 2009
V q2 k-1 qPD 2V C + C a D PD ) k1PC 2 1 - k 2 q 1 PC A + B f C PC + 2PD ) k0RARB
[
]
(13)
The dimensionless expression for the byproduct production rate is given below using the low byproduct production approximation; PC . PD (in dimensionless terms, P′D , 1).
[
P′D k′1 q'P′D ≈ P′D ) 11 + 2P′D R′AR′B K′1
]
(14)
As the undesired reaction becomes irreversible, K′1 f ∞, and the expression for the byproduct production rate given in eq 14 approaches that of the irreversible chemistry, P′D f K′1/(R′AR′B). The reversibility of the undesired reaction lowers the byproduct production: as the equilibrium constant becomes lower, the backward reaction that consumes D becomes more important. This feature also can be seen from eq 14; the bracketed term 1 - q′P′D/K′1 is less than unity and progressively decreases for lower values of equilibrium constant, thus reducing byproduct formation. Mathematically it is possible for the bracketed term to be negative if q′P′D/(K′1) > 1, but this condition corresponds to infeasible recycle flows for a given equilibrium constant. To illustrate the effect of the equilibrium constant, plots of byproduct production P′D versus R′A are shown in Figure 6 for different values of the equilibrium constant. Note: plots of P′D versus R′B are identical to those presented in Figure 6 because species A and B are interchangeable in this chemistry (assuming they have the same molar volume). From Figure 6a, the byproduct production rate goes to zero as the recycle flow rate of species A goes to infinity as expected for a nonbounded species in a reversible or irreversible chemistry. Here both recycle flow rates can reach zero because there are no equilibrium limitations so long as the main reaction is irreVersible. From Figure 6a P′D does not approach infinity as R′A f 0 because a side reaction with a finite equilibrium constant produces a finite amount of species D that is in equilibrium with species C (that has a fixed production rate) for all finite values of recycle flow rates. On the other hand, the plot for K′1 ) 100 in Figure 6b provides a close approximation to the expression for the irreversible chemistry where P′D f ∞ as R′A f 0. The economic optimization for chemistry 3B is shown via the contour plots in Figure 7. The cost contours are closed as for all nonbounded chemistries, indicating an optimal operating point away from all system constraints, as shown in Figure 7b. The case of a small equilibrium constant is demonstrated in Figure 7a indicating less byproduct formation and improved economics; hence the minimum cost contours exhibit lower values than the minimum shown in Figure 7b. Because low values of the equilibrium constant improve selectivity, optimal reactant conversion is increased, thus decreasing the recycle flow rates. If the equilibrium constant is low enough the side reaction becomes nearly insignificant and the optimal reactant conversion (reactor holdup) can be increased until it is bounded by the reactor volume constraint. Figure 7a displays the effect of a very low equilibrium constant: the optimal operating policy is to operate the reactor completely full to maximize reactant conversion. Summarizing: a nonbounded chemistry with a single reversible undesired reaction can exhibit a change in operating policy, depending on the magnitude of the equilibrium constant. If the side reaction has a strong back reaction, reactant conversion
(reactor holdup) can be increased without concern for selectivity losses. Increasing reactant conversion and analogous reactor holdup reduces the required recycle flows and lowers the operating costs of the separation system. 3.3. Summary of Results. The previous two sections have demonstrated cases where a shift in operating policy can occur for chemistries with a single undesired reaction. The Supporting Information describes the other two cases where the classification of the process chemistry does not change when reaction reversibility is introduced. These results are summarized below in Table 1. The entries in Table 1 that appear as bold represent a potential change in the operating policy of the original irreversible chemistry as described in the previous two sections. More complicated chemistries with multiple reversible reactions will result in an operating policy that can be a combination of the effects described in this paper. These results are for process chemistries with a single undesired reaction and only one reversible reaction to demonstrate the effect of reaction reversibility. This process chemistry classification not only provides insight on how to optimally operate the reactor holdup but also into potential plantwide control structures. Bounded chemistries correspond to a reactor operating policy that maintains the reactor holdup at the maximum constraint. A simple decentralized control structure could be designed to maintain the reactor holdup completely full under all operating conditions. Nonbounded chemistries have a variable reactor holdup operating policy that may shift with system throughput changes. If one wants to follow this operating policy of continuous optimal setpoint calculation and control then a centralized control structure may be required. For either class of chemistries, the recycle flow rate(s) need to be optimized. However, some insight can be gained by classifying the reactant species; bounded species generally have a small optimal recycle flow rate because of the improved selectivity and lowered operating costs, whereas nonbounded species have a larger optimal recycle flow rate due to the tradeoff between selectivity and operating costs. A centralized control structure is required for such chemistries with continuous optimal setpoint calculation and control; either class of chemistries may require such a structure around the separation system to determine optimal recycle flow rates, but for nonbounded chemistries the reactor and separation system control structures should be coupled. 4. Multiple Steady-States The term steady-state multiplicity refers to a process that exhibits several steady-states at the same operating conditions. Usually one or more of these steady-states are unstable which can lead to suboptimal operation of a process, especially during start up. Well-known examples for single units include temperature multiplicity in a stand-alone CSTR with an exothermic reaction,20 or a two-phase reactive flash unit where multiplicity depends on heats of reaction and vaporization and the compositions in the two phases.21 Several authors have studied steadystate multiplicity for the RSR process. Nazanskii et al. reported that for the RSR process with a single bimolecular reversible reaction, steady-state multiplicity depends on which reactant (light or heavy) is in excess at the reactor inlet.22,23 Blagov et al. considered competing irreversible reactions and discovered multiple steady-states for different limiting separations.24 Kienle et al. studied the multiplicity and dynamic behavior of various configurations of the RSR process with a first-order exothermic reaction A f B including a two-phase reactor under boiling conditions.25 Other works of the Kienle group include com-
Ind. Eng. Chem. Res., Vol. 48, No. 17, 2009
8043
Figure 6. Chemistry 3B: Production rate of undesired byproduct vs recycle flow rates for different values of the equilibrium constant.
Figure 7. Chemistry 3B: Economic optimization contour plots for (a) K′1 ) 0.1 and (b) K′1 ) 100. Table 1. Process Chemistry Classification for Chemistries with a Single Undesired Reaction and Only One Reversible Reaction. one reversible reaction irreversible classification
desired reaction low K0
high K0
undesired reaction low K1
high K1
bounded chemistry nonbounded bounded bounded bounded nonbounded chemistry non-bounded non-bounded bounded nonbounded
parisons of the stability, feasibility, and multiplicity when different control structures are considered for the reactor or for the separator when modeled as a flash unit.26,27 It is important to identify and classify the potential presence of multiple steadystates for a given system so proper control systems can be implemented. A chemistry involving competing reactions can exhibit multiple steady-states when the desired reaction is reversible. Figure 4 above shows an example of steady-state multiplicity for a bounded chemistry. As discussed in section 3.1, the reversibility of the main reaction forces a required minimum on the reactant recycle flow rate that is nonzero. With the production rate of the desired product fixed, a finite amount of reactant is in equilibrium with the desired product in the reactor effluent and it must be recycled. The issue then becomes how the recycle flow rate behaves as it approaches this minimum value if it cannot actually reach zero. Consider Figure 4a for R′B ) 1 initially located at the righthand side of the curve on the lower steady-state branch. Moving from right to left along the curve, the conversion of the limiting reactant to the desired product initially increases and the recycle flow rate of reactant decreases. As the main reaction approaches
the equilibrium conversion and consequently approaches the minimum required recycle flow rate, the required reactor volume increases. Here it should be noted that every point on the P′D versus R′A or R′B plots corresponds to a different reactor holdup and conversion. It is well-known that a reactor of infinite volume is required for a reaction to be at equilibrium and the corresponding equilibrium conversion.20 In the vicinity of equilibrium the required reactor volume increases significantly without large increases in reactant conversion, thus causing a buildup of reactant species in the reactor. Increasing reactor holdup and concentration of reactant species without substantial increase in conversion now causes the recycle flow rate to turn around and increase, resulting in an upper steady-state solution branch. This turning point is not found with irreversible reactions because recycle flow rates of zero and 100% conversion can be achieved, at least in principle, with a reactor of infinite volume. A dynamic model and corresponding analysis has been developed for several cases of bounded and nonbounded chemistries involving a reversible desired reaction in order to analyze these multiple steady-states. It was found that the lower branch of steady-states corresponds to the situation in which all eigenvalues are negative; thus it is always stable. The middle steady-state (represented by the upper branch in Figures 4a and 4c) exhibits mixed positive and negative eigenvalues corresponding to an unstable saddle. A second stable steady-state (the high steady-state) is located where the reactor volume and byproduct production rate go to infinity. Starting conditions near the middle steady-state will result either in the system going to the low steady-state or to the high steady-state. Thus, this system can exhibit three steady-states; one feasible, always stable low
8044
Ind. Eng. Chem. Res., Vol. 48, No. 17, 2009
branch; and two infeasible steady states, one is located at infinity and the other is an unstable saddle-point. The infeasibility of the middle steady-state is important in characterizing this steady-state multiplicity. First, the middle steady-state always corresponds to a higher value of byproduct formation, thus reducing profitability. Next, the middle steadystate may not be attainable for some process designs because the required reactor holdup, and the recycle flow rates corresponding to this increased byproduct production, fall outside the system constraints. Thus, the only possible feasible points on the middle steady-state solution branch are those very close to the turning point. Any chemistry that involves competing reactions, a fixed production rate of the desired product, and a reversible desired reaction will exhibit multiple steady-states. Thus, while the existence of the middle steady-state may be academically interesting, it will seldom affect the optimal operating policy because the middle steady-state is always unstable, corresponds to higher values of byproduct production (increased costs), and normally will be infeasible owing to equipment size constraints. Thus, no benefit is provided from starting up a plant near the middle steady-state nor in implementing a control structure that in principle can maintain system operation there. 5. Conclusions This paper presents a method for determining steady-state optimal operating policies for process chemistries with reversible reactions. The system analyzed is the conventional reactor/ separator/recycle (RSR) process with a single liquid-phase isothermal CSTR. Already published work provides a classification procedure for irreversible process chemistries that can be either bounded, with an operating policy that maintains the reactor full at all times, or nonbounded, with a variable reactor holdup operating policy. If the process chemistry includes any reversible reactions the result of this classification can change. For example, a bounded chemistry with a reversible desired reaction and irreversible undesired reactions can exhibit an optimal operating point away from the reactor volume constraint, as with a nonbounded chemistry. If more than one reversible reaction is present or if a nonbounded chemistry contains more than one undesired reaction, then the operating policy will depend on the numerical values of the system parameters and thus cannot be determined by simple inspection of the chemistry’s stoichiometry and kinetics alone. This work also has shown that multiple steadystates can exist with the RSR process when operating with a chemistry that includes at least one undesired reaction and a reversible desired reaction. However, in this case both the upper stable steady-state and the middle (unstable and economically undesirable) steady-state are generally located outside the feasible operating regime. Acknowledgment The authors are grateful for financial support provided by the National Science Foundation (Grant No. CTS-0554718). Supporting Information Available: The material covered includes derivation of process unknowns described in section 3.1; the cases not discussed in section 3 which include, a bounded chemistry with a reversible undesired reaction and a nonbounded chemistry with a reversible desired reaction; and the alternate process design flowsheet and resulting economic
optimization described in the Case Studies. This material is available free of charge via the Internet at http://pubs.acs.org. Appendix. Case Study. Bounded Etherification Chemistry with Reversible Desired Reaction. A case study is now presented that demonstrates the possible shift in operating policy for a bounded chemistry with a reversible desired reaction having a low equilibrium constant. The methyl tert-butyl ether (MTBE) process has been widely studied because of MTBE’s value as a fuel additive in gasoline. However, because of the solubility of MTBE in water and its widespread detection in groundwater there is a need for higher molecular weight ethers that will exhibit significantly decreased solubility. One such alternative is 2-methoxy-2-methylheptane (MMH) produced from 2-methyl-1-heptene (MH) and methanol (MeOH), as shown below. The undesired byproducts are dimethyl ether (DME) and 2-methyl-2-heptanol (MHOH). Water (H2O) is an intermediate reaction species. MH + MeOH a MMH r0 ) k0xMHxMeOH - k-0xMMH k0 ) k0,0e-EA,0/RT
k-0 ) k-0,0e-EA,-0/RT 2MeOH f DME + H2O 2 r1 ) k1xMeOH
k1 ) k1,0e-EA,1/RT MH + H2O f MHOH r2 ) fast The third reaction is fast compared to the other reactions so the last two reactions can effectively be coupled as a reaction rate law determined by the kinetics of the second reaction. The process chemistry can then be rewritten as MH + MeOH a MMH r0 ) k0xMHxMeOH - k-0xMMH 2MeOH + MH f DME + MHOH 2 r1 ) k1xMeOH The process chemistry is now of the form A+BaC 2A + B f D + E which is a bounded chemistry because the reactions are in parallel and have the same overall forward reaction order. Krause and co-workers have reported kinetic data for these reactions;28-30 pseudohomogenous mole fraction models of the form shown above have been fitted to the kinetic data to generate the kinetic parameters given in Table 2. The equilibrium constant for the desired reaction can be determined by K ) k0/k-0. This equilibrium limited chemistry is a possible candidate for a more complicated reactor network or for reactive distillation; but in this case study we illustrate the use of a single, isothermal CSTR. The process flow diagram of Douglas’ conceptual design procedure at level 4 is shown in Figure 8 and the economic optimization is based on maximizing the economic potential (EP) as described by Douglas.31 The Table 2. Kinetic Parameters for Etherification Chemistry k0, 0 ) 6.7 × 1010 mol/(kg cat s) k-0, 0 ) 2.1 × 10-3 mol/(kg cat s) k1, 0 ) 1.3 × 1012 mol/(kg cat s)
EA, 0 ) 90 kJ/mol EA,-0 ) 0.9 kJ/mol EA, 1 ) 105.9 kJ/mol
Ind. Eng. Chem. Res., Vol. 48, No. 17, 2009
8045
Figure 8. Etherification process flowsheet.
Figure 9. Optimization landscape for K ) 100 and a production rate of (a) 14 mol/s and (b) 7 mol/s. Economic potential (EP) in $/yr: * > EP3 > EP2 > EP1.
equipment is sized and costed by the methods of Doherty and Malone.32 It is assumed the feed streams are pure; the reactor is operated at the upper temperature limit of its acid resin catalyst (120 °C); and both reactant species are taken off from the top of the second distillation column and recycled in a single stream. The optimization variables are the recycle flow rates of the two individual reactant species; RA is the recycle flow rate of MeOH (the limiting reactant) and RB is the recycle flow rate of MH. The goal of this study was to illustrate the shift in operating policy that is driven by high and low values of the equilibrium constant. At 120 °C, K ) 15, which is an intermediate value but low enough to demonstrate the predicted results for a low equilibrium constant. The high equilibrium constant is chosen as 100 corresponding to a reactor temperature of approximately 148.6 °C; this value does not match the kinetic data at the chosen reactor operating conditions, but is used to illustrate the case of an irreversible bounded chemistry. The economic optimization landscape for K ) 100 is shown in Figure 9. The base case design before oversizing is shown by the “X” in Figure 9a representing a nominal production rate of 14 mol/s of MMH. The plant is then oversized to handle a 20% increase in the production rate; the constraints on the reactor volume and recycle flow rates are then shown as the dashed lines that enclose the feasible operating regime. For this bounded chemistry with a reversible desired reaction, an equilibrium envelope exists, shown by the dotted line, that cost contours cannot cross. Once the plant has been oversized, the optimal operating point is shown by an asterisk (*), but this point is outside the feasible region, so the plant must be operated on the reactor volume constraint, shown by the “O”. This point
corresponds to an economic potential of $4.18 million, a 2.9% increase in profit from the base case design where the reactor is not operated completely full. Figure 9b, for a 50% production rate decrease to 7 mol/s, shows that the optimal operating point will still fall on the reactor volume constraint (again shown by the “O”) for the case of a high equilibrium constant. Figure 10 shows the contour plots for K ) 15, representing a low equilibrium constant. At the nominal production rate after oversizing (Figure 10a) the optimal operating point, shown by “*”, once again is outside the system constraints forcing operation onto the reactor volume constraint. The operating point at “O” after oversize has an economic potential of $3.64 million, which is 6.1% higher than the base case at “X” before oversizing. After a production rate decrease (Figure 10b) the feasible operating region becomes larger, and the optimal operating point actually is located away from the reactor volume constraint. The optimal operating point in Figure 10b is marked by an “*”, corresponding to a reactor holdup that is 85% of the maximum value. Thus for this bounded chemistry, operating conditions related to the equilibrium situation exist where it can be optimal to operate the reactor away from its volume constraint, just as with a nonbounded chemistry. The flowsheet shown in Figure 8 separates both reactant species and recycles them in a single stream. Consider an alternative design where the azeotrope between MeOH and MH is separated in the second column and recycled, and a third column is used to remove the remaining MH which is also recycled (the flowsheet may be found in the Supporting Information). This design requires a larger reactor volume at
8046
Ind. Eng. Chem. Res., Vol. 48, No. 17, 2009
Figure 10. Optimization landscape for K ) 15 and a production rate of (a) 14 mol/s and (b) 7 mol/s. Economic potential (EP) in $/yr: * > EP3 > EP2 > EP1.
Figure 11. Alternate process design optimization landscape for K ) 15 and a production rate of (a) 14 mol/s and (b) 7 mol/s. Economic potential (EP) in $/yr: * > EP3 > EP2 > EP1.
Figure 12. Multiple steady-states for the unconstrained optimal operating point “*” in Figure 11a: RB ) 49 mol/s, K ) 15.
the design stage and may provide greater operational flexibility having two separate recycle streams, yet, the predicted results of the classification do not change. The optimization landscape for this design for the low equilibrium case is shown in Figure 11 demonstrating that conditions still exist where the optimal reactor holdup is away from the constraint, Figure 11b. The economic optimization for the high equilibrium constant can
be found in the Supporting Information. The operating point at “O” in Figure 11a has an economic potential of $3.54 million, which is 4.2% higher than the base case at “X”, and the optimal operating point in Figure 11b at “*” corresponds to a reactor holdup that is 77% of the maximum value. To demonstrate the existence of multiple steady-states for this case study the optimization in Figure 11a is considered here in Figure 12. This is a case of two reactions of equal overall order where the main reaction has a reversible desired reaction with a low equilibrium constant, K ) 15. The unconstrained optimal operating point at “*” is chosen for demonstration in this example. A fixed recycle flow rate of species B of 49 mol/s is chosen. The equilibrium limitation is shown by the dashed line in Figure 12 and it is clear that a turning point exists beyond which multiple steady states are present. Note that the minimum in the byproduct formation rate is at a recycle flow rate of species A of 2.5 mol/s, approximately the same optimal value as obtained in the economic optimization at * shown in Figure 11a. Nomenclature C ′ ) dimensionless cost objective function CD′ ) dimensionless cost ratio F ) fresh feed flow rate (mol/h) k ) reaction rate constant
Ind. Eng. Chem. Res., Vol. 48, No. 17, 2009 K ) equilibrium constant P ) production rate (mol/h) q ) reactor effluent volumetric flow rate (L/h) R ) recycle flow rate (mol/hr) ν ) molar volume (L/mol) V ) reactor volume (L) Subscripts 0 ) reference to desired reaction 1 ) reference to undesired reaction - ) reference to the back reaction A,..., E ) species A,..., E Superscript ′ (prime) ) dimensionless quantity
Note Added after ASAP Publication: The version of this paper that was published on the Web May 26, 2009 gave an incorrect title for the work cited as ref 11. The correct version of this paper was reposted to the Web May 28, 2009. Literature Cited (1) Luyben, W. L. Dynamics and Control of Recycle Systems. 1. Simple Open-Loop and Closed-Loop Systems. Ind. Eng. Chem. Res. 1993a, 32, 466. (2) Luyben, W. L. Dynamics and Control of Recycle Systems. 2. Comparison of Alterative Process Designs. Ind. Eng. Chem. Res. 1993b, 32, 476. (3) Luyben, W. L. Dynamics and Control of Recycle Systems. 3. Alternative Process Designs in a Ternary System. Ind. Eng. Chem. Res. 1993c, 32, 1142. (4) Luyben, W. L. Snowball Effects in Reactor/Separator Processes with Recycle. Ind. Eng. Chem. Res. 1994, 33, 299. (5) Luyben, W. L. Design and Control Degrees of Freedom. Ind. Eng. Chem. Res. 1996, 35, 2204. (6) Larsson, T.; Skogestad, S. Plantwide ControlsA Review and a New Design Procedure. Model., Identif. Control EnViron. Syst. 2000, 21, 209. (7) 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. (8) Wu, K.-L.; Yu, C.-C. Reactor/Separator Processes with Recycle-1. Candidate Control Structures 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. The Importance of Process Chemistry in Selecting the Operating Policy for Plants with Recycle. Ind. Eng. Chem. Res. 2004, 43, 3957. (11) Griffin, D. W.; Ward, J. D.; Doherty, M. F.; Mellichamp, D. A. Steady-State Operating Policies for Plants with Multiple Reactions of Equal Overall Order. Ind. Eng. Chem. Res. 2006, 45, 8056. (12) Griffin, D. W.; Mellichamp, D. A.; Doherty, M. F. Selectivity versus Conversion and Optimal Operating Policies for Plants with Recycle. AIChE J. 2008, 54, 2597.
8047
(13) Hopley, F.; Glasser, D.; Hildebrandt, D. Optimal Reactor Structures for Exothermic Reversible Reactions with Complex Kinetics. Chem. Eng. Sci. 1996, 51, 2399. (14) Glasser, D.; Hildebrandt, D. Reactor and Process Synthesis. Comput. Chem. Eng. 1997, 21, S775. (15) Kapilakarn, K.; Luyben, W. L. Plantwide Control of Continuous Multiproduct Processes: Two-Product Process. Ind. Eng. Chem. Res. 2003a, 42, 1890. (16) Kapilakarn, K.; Luyben, W. L. Plantwide Control of Continuous Multiproduct Processes: Three-Product Process. Ind. Eng. Chem. Res. 2003b, 42, 2809. (17) Sharma, M. M.; Mahajani, S. M. Industrial Applications of Reactive Distillation. In ReactiVe Distillation; Sundmacher, K., Kienle, A., Eds.; Wiley-VCH: Weinheim, Germany, 2002. (18) Okasinski, M. J.; Doherty, M. F. Design Method for Kinetically Controlled, Staged Reactive Distillation Columns. Ind. Eng. Chem. Res. 1998, 37, 2821. (19) Stitt, E. H. Reactive Distillation for Toluene Disproportionation: A Technical and Economic Evaluation. Chem. Eng. Sci. 2002, 57, 1537. (20) Fogler, H. S. Elements of Chemical Reaction Engineering, 3rd Ed.; Prentice Hall PTR: Upper Saddle River, NJ, 1999. (21) Rodrı´guez, I. E.; Zheng, A.; Malone, M. F. The Stability of a Reactive Flash. Chem. Eng. Sci. 2001, 56, 4737. (22) Nazanskii, S. L.; Solokhin, A. V.; Blagov, S. A.; Timofeev, V. S. Possible Steady States of a Reactor-Rectifying Column Recycle System for an A + B a C Reaction. Theor. Found. Chem. Eng. 1999, 33, 292. (23) Nazanskii, S. L.; Solokhin, A. V.; Blagov, S. A.; Timofeev, V. S. Steady States of the Reactor-Distillation Column System for an A + B a C Reaction. Theor. Found. Chem. Eng. 2001, 35, 479. (24) Blagov, S.; Bessling, B.; Shoenmakers, H.; Hasse, H. Feasibility and Multiplicity in Reaction-Distillation Processes for Systems with Competing Irreversible Reactions. Chem. Eng. Sci. 2000, 55, 5421. (25) Waschler, R.; Pushpavanam, S.; Kienle, A. Multiple Steady States in Two-Phase Reactors Under Boiling Conditions. Chem. Eng. Sci. 2003, 58, 2203. (26) Pushpavanam, S.; Kienle, A. Nonlinear Behavior of an Ideal Reactor Separator Network with Mass Recycle. Chem. Eng. Sci. 2001, 56, 2837. (27) Zeyer, K.-P.; Pushpavanam, S.; Kienle, A. Nonlinear Behavior of Reactor-Separator Networks: Influence of Separator Control Structure. Ind. Eng. Chem. Res. 2003, 42, 3294. (28) Kiviranta-Paakkonen, P. K.; Struckmann, L. K.; Linnekoski, J. A.; Krause, A. O. I. Dehydration of the Alcohol in the Etherification of Isoamylenes with Methanol and Ethanol. Ind. Eng. Chem. Res. 1998, 37, 18. (29) Karinen, R. S.; Krause, A. O. I. Reactivity of some C8-Alkenes in Etherification with Methanol. Appl. Catal. A 1999, 188, 247. (30) Karinen, R. S.; Linnekoski, J. A.; Krause, A. O. I. Etherification of C5 and C8 alkenes with C1 to C4 Alcohols. Catal. Lett. 2001, 76, 81. (31) Douglas, J. M. Conceptual Design of Chemical Processes; McGrawHill: NewYork, 1988. (32) Doherty, M. F.; Malone, M. F. Conceptual Design of Distillation Systems; McGraw-Hill: New York, 2001.
ReceiVed for reView October 1, 2008 ReVised manuscript receiVed February 6, 2009 Accepted February 10, 2009 IE801482Z
