Importance of Process Chemistry in Selecting the Operating Policy for

Ind. Eng. Chem. Res. 2004, 43, 3957-3971

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Importance of Process Chemistry in Selecting the Operating Policy for Plants with Recycle† Jeffrey D. Ward, Duncan A. Mellichamp, and Michael F. Doherty* Department of Chemical Engineering, University of California, Santa Barbara, Santa Barbara, California 93106-5080

A novel methodology using an economic objective function and employing recycle flow rates as independent degrees of freedom is proposed for the selection of a steady-state operating policy for chemical plants with recycle streams. Recycle flow rates are chosen as degrees of freedom instead of traditional decision variables such as conversion and molar ratio because the analysis is considerably simplified, allowing for greater insight into the nature of the solution of the problem. The methodology makes it clear that the process chemistry is of paramount importance in selecting a plantwide operating policy. Based on the kinetics of the process reactions, plants can be assigned to one of two equivalence classes of operating policies. One class requires constant maximal reactor holdup for optimal operation, and the other class requires a variable reactor holdup (assuming that the process has sufficient flexibility). Results are illustrated with simple examples and a case study based on a process to produce chlorobenzene. 1. Introduction Engineers have long recognized that it is necessary to incorporate a certain amount of flexibility in the design of chemical processes.1-4 Once a flexible process has been built, another important question becomes apparent: What, if anything, should be done with the excess capacity? Should it be used immediately to reduce costs or increase production, or should it be reserved for use in disturbance rejection under abnormal circumstances? The manner and extent to which all of the unit operations in a plant are utilized is here called the “operating policy” of the plant. One approach to selecting an operating policy is to formulate an objective function and use a mathematical program to find the optimum. This can be done offline, as with nonlinear programming (NLP),5,6 mixed-integer NLP,7-12 or even mixed-integer dynamic optimization,13,14 or online, as with real-time optimization. These are powerful methods that are well established in the petroleum industry and are finding increasing application in the chemicals industry. Whether or not a mathematical program is used to determine the optimum operating policy, it is important to develop some intuition about the nature of the optimization problem: What are the dominant tradeoffs? What are the expected trends as constraints and disturbances change? Why is a particular design or operating policy the optimal one? Many researchers have published heuristics for selecting an operating policy. Fisher et al.15 suggested heuristics including minimizing the loss of reactants, setting the production rate by manipulating the fresh feed rate of the limiting reactant, keeping gas recycle flow rates at their maximum value, and operating any flash drums at the coldest possible temperature. Luyben † This paper is dedicated to Professor Art Westerberg for his many outstanding contributions to the field of process systems engineering. * To whom correspondence should be addressed. E-mail: [email protected].

et al.16 suggested heuristics including flow control of a stream somewhere in each recycle loop and no flow control of a fresh reactant stream unless there is essentially complete one-pass conversion of that species. There can be pitfalls associated with the use of heuristics, however. First, they are sometimes contradictory. Second, the motivation behind heuristics is not always sufficiently emphasized: some authors are concerned about the dynamic behavior of the process, others intend to maximize the flexibility of the process, and still others aim to maximize an economic objective function that may have only steady-state components. Finally, because many heuristics stem from individual case studies or groups of case studies, their generality may be difficult to determine. However, an advantage of heuristics is that they are simple to understand and interpret. These methods (heuristic and mathematical programming) need not be thought of as being in competition, and the prudent engineer will usually utilize both methods in the selection of an operating policy. The optimization of reactor networks in isolation from the rest of the plant is a well-studied problem that has been approached using a variety of methods; it has also given rise to the concepts of attainable region theory. Furthermore, conceptual design of reactors and reactor networks based on information about the process chemistry is covered in the standard undergraduate textbooks on reactor design. Generally, the objective in the optimization of an isolated reactor network is to maximize the yield of the desired product. This objective is appropriate if the reactants will not be separated and recycled back to the reactor. In this case, regardless of the operating policy, unconsumed reactants are wasted along with the byproducts. The appropriate goal should be to maximize production of the valuable product. In many industrial processes, however, it is possible and even economically necessary to separate and recycle unconsumed reactants. In this case, maximizing the perpass yield of the desired product is not the appropriate objective, and generally a strategy with that objective will produce far more byproduct than is economically optimal. When separation and recycle are employed, it

10.1021/ie034125z CCC: $27.50 © 2004 American Chemical Society Published on Web 06/15/2004

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Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004

is necessary to take a plantwide perspective for process optimization and to consider the total economic potential of the plant as the objective function. Consideration of the reactor network alone cannot give the optimal answer because the optimal answer will depend on a tradeoff between the performance of the reactor and the separation system and it will specify simultaneously the operating policy for both the reactor network and the separation system. For many chemical plants, the most important tradeoff for the optimization of both flowsheet design and operating policy is between selectivity losses at high reactant conversion and recycle costs at low reactant conversion. This paper explores the influence of process chemistry on the nature of this tradeoff. The optimization problem is cast as the minimization of an objective function that captures the dominant tradeoff between selectivity losses and recycle penalties, but it is simple enough to allow analytical interpretation. The analysis makes it clear that the kinetics of the desired and undesired reactions are of paramount importance in choosing the optimal operating policy. As we look ahead briefly to the results, recycle plants can be classified and assigned to one of two classes of operating policies on the basis of the kinetics of the desired and undesired reactions in the process. One policy class requires the reactor to be kept completely full at all times (constrained control); the other policy requires the reactor level to be varied with changes in the production rate or disturbances such as deactivation of the catalyst (unconstrained control). For example, we show that the best policy for a plant with the elementary reactions A + B f C (desired) and A + A f D (undesired) is to operate with constant maximal reactor holdup, while a plant with the elementary reactions A + B f C (desired) and C + C f D (undesired) should operate with variable reactor volume within the limits of process flexibility. Once the optimum operating policy has been identified, it can serve as both a guide and benchmark for the design and evaluation of a control structure. If the operating policy is relatively simple, then a decentralized control structure consisting only of proportionalintegral-derivative loops may be adequate. For example, if the optimal policy is to operate with certain process variables fixed at constant values at all times, then a decentralized, “self-optimizing” control structure17,18 can be used. If the operating policy is more complex, then a multivariable control structure such as model predictive control may be better suited. Candidate control structures can be compared on the basis of how close they come to actually implementing the optimal operating policy. An ongoing discussion in the literature concerns the best operating policy and control structure for a plant consisting of a reactor, separator, and recycle (RSR process), with the elementary reaction A f B. Luyben and co-workers16,20-23 recommended that the recycle flow rate be kept constant and the reactor holdup be permitted to change. Skogestad and co-workers24,25 recommended that the reactor holdup be kept constant and the recycle flow rate be permitted to change. Yu and co-workers26,27 recommended that both the recycle flow rate and the reactor volume be permitted to change. This paper places that discussion in a larger context: the elementary chemistry A f B is just one example of a chemistry that can be assigned to an equivalence class of operating policies using the procedure suggested here.

Figure 1. Process flow diagram with two recycle streams and two product streams.

Figure 2. Process flow diagram with two recycle streams and three product streams. Table 1. Stoichiometry and Kinetics for Each Chemistry no.

stoichiometry

kinetics

flowsheet

1

A+BfC A+AfD A+BfC A+CfD A+BfC C+CfD A+BfC CfD+E A+BfC A+AfD C+CfE AfC CfD

r0 ) k0[A][B] r1 ) k1[A]2 r0 ) k0[A][B] r1 ) k1[A][C] r1 ) k1[A][B] r1 ) k1[C]2 r0 ) k0[A][B] r1 ) k1[C] r0 ) k0[A][B] r1 ) k1[A]2 r2 ) k2[C]2 r0 ) k0[A] r1 ) k1[C]

Figure 1

2 3 4 5 6

Figure 1

Figure 2 Figure 2 Figure 3

2. Byproduct Production Rates in Terms of Recycle Flow Rates In this paper, the phrase “process chemistry” refers to the stoichiometry and kinetic rate expression of each reaction that can occur in the process. The scope of this work is limited to plants that produce a single product and to homogeneous reaction rate laws with possibly noninteger exponents. We argue that the process chemistry is of paramount importance in the steady-state optimization of a plant. To illustrate this point, several hypothetical plants are considered, most with the same flowsheet topology but with different process chemistries. Although the same flowsheet topology is utilized in most of the examples presented here, it is important to note that the methodology readily generalizes to processes with an arbitrary number of recycled species and an arbitrary number of undesired byproducts. Table 1 shows the process chemistry for every process that is considered as an example; Figures 1-3 show the corresponding process flow diagrams. Fresh feed flow rates are designated as F, recycle streams are designated as R, and product streams are designated as P. In every case, the reactor is a single isothermal continuous stirred tank reactor (CSTR). The nature of the separation system is not specified, but we assume that it can separate the reactor effluent into essentially pure streams. The level of detail considered for each process corresponds to level 3 in Douglas’29 hierarchy.

Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3959

can be replaced by recycle flow rates by introducing the reactor effluent volumetric flow rate q:

V r˜ k,0 ) k0RARB 2 q

r˜ s,0 ) PC

V r˜ k,1 ) k1RA2 2 q

r˜ s,1 ) PD Figure 3. Process flow diagram with one recycle stream and two product streams.

It is important to bear in mind the number of operational degrees of freedom associated with each plant. If the production rate of the desired product is fixed, the recycle streams are pure and the reactor temperature is specified but the reactor volume can be varied; then the number of operational degrees of freedom of the plant is equal to the number of recycle streams. Traditionally, these would be specified by fixing quantities such as the per-pass conversion of the limiting reactant or the molar ratio of reactants at the reactor inlet. Here we pursue an alternative approach, in which the degrees of freedom are fixed by specifying the recycle flow rates themselves, or, more precisely, the dimensionless ratio between the recycle flow rates and the production rate of the desired product. This new method has the advantage that it makes more apparent the influence of the process chemistry on the nature and solution of the optimization problem. Furthermore, it is more readily generalizable to complex chemistries with an arbitrary number of recycle streams. Note that choosing every recycle stream flow rate as a degree of freedom ensures that exactly all of the available degrees of freedom are specified. 2.1. One Undesired Reaction with the Same Reaction Order. As a first example, consider chemistry 1, which has the process flow diagram shown in Figure 1:

A+BfC

r0 ) k0[A][B]

A+AfD

r1 ) k1[A]2

desired (1) undesired

The plant has two recycle streams and therefore two operational degrees of freedom. The most important unknowns in the process are the byproduct production rate PD (which will influence the process economics) and the reactor volume requirement V (which may constrain process operation). Therefore, it is desired to express PD and V in terms of the degrees of freedom: the recycle flow rates of species A and B, RA and RB. The analytical approach for finding expressions for PD and V is to express the overall rate of reaction r˜ with units of moles per time, in two different ways: one is based on the stoichiometry of the reaction and material balances; the other is based on the kinetics of the reaction. A double-subscript notation is employed: the first subscript is a letter, either s or k, reflecting whether the reaction rate is calculated using stoichiometry or kinetics, and the second subscript is an arabic numeral identifying the reaction number. For chemistry 1, the equations are

r˜ s,0 ) PC r˜ s,1 ) PD

r˜ k,0 ) k0[A][B]V r˜ k,1 ) k1[A]2V

(2)

The concentrations of species A and B in the reactor

(3)

Clearly, for all species, rk ) rs. Therefore

V PC ) k0RARB 2 q

(4)

V PD ) k1RA2 2 q

(5)

Dividing the expression for the undesired reaction by the expression for the desired reaction gives

k1RA2 k1RA PD ) ) PC k0RARB k0RB

(6)

Flow rates are made dimensionless by dividing them by the production rate of the desired product, PC. For the case of an equal reaction order, reaction rate constants are made dimensionless by dividing by the rate constant of the desired reaction. Therefore

R′A PD′ ) k′1 R′B

(7)

where P′D ) PD/PC is the dimensionless production rate of species D and k′1 ) k1/k0 is the dimensionless reaction rate constant for the undesired reaction. For this chemistry, PD f 0 as RA f 0, reflecting the fact that the byproduct production rate becomes small in the limit of high per-pass conversion of species A, or as RB f ∞, because a large excess of species B suppresses the undesired reaction. Note that the method does not require the calculation of the fresh feed flow rates of species; however, these can be determined in a straightforward manner from global material balances. Also, conventional dimensionless parameters such as selectivity, conversion, and molar ratio are not employed, but these also are easily calculated. Finally, for the case where both reactions are of the same order, terms involving the reactor volume V and volumetric reactor effluent flow rate q cancel and it is not necessary to solve for these in order to express the byproduct production rate in terms of the recycle flow rates. The volumetric reactor effluent flow rate is discussed in more detail in section 2.3 and the reactor volume in section 2.5. For comparison, now consider chemistry 3, which has the same flowsheet (Figure 1):

A+BfC

r0 ) k0[A][B]

C+CfD

r1 ) k1[C]2


Following the same methodology, it is found that the expression for PD for chemistry 3 is

PD′ )

(

k′1 k′1 1-2 RA′ RB′ RA′ RB′

)

-1

(9)

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Table 2. Exact and Approximate Expressions for the Byproduct Flow Rate in Terms of Recycle Flow Ratesa no.

exact

1

RA′ k′1 RB′

2

k′1 k′1 1R′B R′B

3

k′1 k′1 1-2 RA′ RB′ RA′ RB′

(

(

)

approximate RA′ k′1 R′B

-1

k′1 RB′

)

-1

k′1 RA′ RB′

a A and B are reactants, C is the desired product, and D and E are the undesired byproducts.

Figure 5. Production rate of undesired byproduct vs recycle rates for chemistry 3.

Figure 4. Production rate of undesired byproduct vs recycle rates for chemistry 1.

In this case, species A and B are symmetric (interchangeable). PD f 0 if either RA f ∞ or RB f ∞. The per-pass conversion of both species should be kept low to dilute the product species C and suppress the undesired reaction. Table 2 shows the expressions for the production rate of the undesired species in terms of the recycle flow rates for all plants whose chemistries have the same overall reaction order in both reactions. The case of different reaction order is discussed in section 2.3, where it is seen that the symbolic results are not as compact; therefore, they are not listed in the table. Figures 4 and 5 show how the production rate of the undesired product changes as a function of the recycle flow rates for chemistries 1 and 3. Figures 4a and 5a show PD versus RA with curves parametrized by RB; Figures 4b and 5b show PD versus RB with curves parametrized by RA. From the theoretical developments and plots, it is clear that the process chemistry has a profound influence on the relationship between the recycle flow rates and the production rate of the undesired byproduct. If the order of a reactant species is greater in the undesired reaction than in the desired reaction, then the production rate of the undesired species will increase

with increasing recycle flow rate of that species. In eq 6, because species A appears to the first power in the rate of the desired reaction but to the second power in that of the undesired reaction, the recycle flow rate of species A appears to the first power in the expression for the byproduct production rate. Such species are said to be “reactor volume bounded” or simply “bounded” because no matter how large the reactor is, an operating policy that seeks to minimize the production rate of the undesired species will be constrained by the reactor volume. Byproduct production will be minimized when the recycle flow rate of species A is minimized. Thus, the reactor holdup should be as large as possible so that the per-pass conversion of species A is as high as possible. The reason that the reactor volume constraint is encountered is that a large residence time (holdup) in the reactor is required to achieve high per-pass conversion of the bounded species. The expression for the reactor holdup as a function of the recycle flow rates, as well as the trends in required reactor holdup as recycle flow rates are varied, is discussed in section 2.5. This definition can also be understood in terms of the classical chemical reactor engineering considerations of selectivity and conversion. Because the production rate of the desired product is fixed, if the byproduct production rate increases, the selectivity will decrease. Therefore, a reactor volume bounded species is a reactant species for which the selectivity to one or more undesired byproducts increases with increasing conversion of that species. Conversely, if the order of a reactant species is smaller in the undesired reaction than in the desired reaction, the production rate of the undesired species will decrease with increasing recycle flow rate and will approach infinity at a low recycle rate. Such a species is called “nonbounded” because the reactor volume will not constrain the minimum production rate of the undesired product (although the capacity of the recycle system will). A nonbounded species is a reactant species


for which the selectivity to all undesired products decreases with increasing conversion of that species. From eq 6, species B is nonbounded because it appears to the first power in the desired reaction, to the zero power in the undesired reaction, and, therefore, to the minus one power in the expression for the byproduct production rate. These differences have a profound influence on the optimum steady-state operating policy, as discussed in section 3. Chemistry 2

A+BfC A+CfD

r0 ) k0[A][B] r1 ) k1[C][C]

desired undesired

(10)

has a byproduct production rate given by

PD′ )

(

k′1 k′1 1RB′ R′B

)

-1

(11)

In the special case where one species appears to the same power in both the desired and undesired reactions, the recycle flow rate of that species will not affect the byproduct production rate; therefore, it will not affect the selectivity. Such “border-line” species are also classified as reactor volume bounded species because, as will be shown later, it is optimal to minimize the recycle flow rate of these species (maximize the per-pass conversion by maximizing the reactor holdup) in order to minimize separation costs. 2.2. Simplification for a Low Byproduct Production Rate. In many cases, an undesired reaction will consume some of the desired product, such as in chemistries 2 and 3. In such cases, the expression for the byproduct production rate will not be as simple as eq 7 because the production rate of the undesired byproduct will appear in the expression for r˜ s,0. For example, for chemistry 2

r˜ s,0 ) PC + PD r˜ s,1 ) PD

r˜ k,0 ) k0[A][B]V r˜ k,1 ) k1[A][C]V

(12)

Hence

k1RAPC k1PC PD ) ) PC + PD k0RARB k0RB

(13)

(

k′1 k′1 1RB′ RB′

)

This result is accurate at reasonable recycle flow rates (low conversion), where the production of the undesired byproduct is low, but inaccurate at low recycle flow rates, where the production rate of the undesired byproduct is high. Figure 6 shows the difference between the exact and approximate results for chemistry 2. 2.3. Different Overall Reaction Orders and Noninteger Reaction Orders. This method of analysis can be generalized to the case in which the undesired reaction is not of the same order as the desired reaction. Consider chemistry 4 (with the process flow diagram shown in Figure 2), in which the desired product can decompose irreversibly to form two undesired products:

A+BfC

r0 ) k0[A][B]

CfD+E

r1 ) k1[C]


As with chemistries 1 and 3, there are two recycle streams and, therefore, two operational degrees of freedom, RA and RB. It is desired to express the byproduct production rates, PD and PE, in terms of RA and RB. Again we begin by writing expressions for r˜ k and r˜ s:

r˜ s,0 ) PC + PD r˜ s,1 ) PD

r˜ k,0 ) k0[A][B]V r˜ k,1 ) k1[C]V

-1

(14)

A further simplification is possible if the production rate of the undesired byproduct is small compared to the production rate of the desired product, as is usually the case for profitable process operation. If PD , PC, then PC + PD ≈ PC and eq 14 reduces to

PD k1RAPC k1PC ≈ ) PC k0RARB k0RB

(15)

1 PD′ ≈ k′1 R′B

(16)

Therefore

(18)

By stoichiometry, PD ) PE. Equating the reaction rates and substituting recycle flow rates for reactant concentrations as before gives

V PC ) k0RARB 2 q

and

PD′ )

Figure 6. Exact and approximate byproduct production rates for chemistry 2.

PC + PD ) k1PC

V q

(19) (20)

Dividing the undesired reaction by the desired reaction gives

k1PCq PD ) PC + PD k0RARB

(21)

In this case, the volumetric reactor effluent flow rate q does not cancel completely from the equation. Furthermore, the ratio k1/k0 is not dimensionless. The total volumetric flow rate out of the reactor is given by the sum over species of the molar flow rate multiplied by the molar volume:

q ) RAvA + RBvB + PCvC + PDvD + PEvE

(22)

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Flow rates are made dimensionless by dividing by the production rate of the desired product, and molar volumes, by dividing by the molar volume of the desired product. Therefore, the dimensionless volumetric flow rate q′ is defined as

q′ )

q ) R′Av′A + RB′ v′B + 1 + PD′ v′D + PE′ v′E (23) PCvC

The reaction rate constant k1 is made dimensionless by dividing by k0 and multiplying by the molar volume of the desired product raised to the appropriate power. For chemistry 4, k0 has units of L/(mol h) and k1 has units of 1/h. Therefore

k′1 ) k1/k0vC

(24)

Rearranging and introducing these relations into eq 21 and nondimensionalizing flow rates, we have

k′1 PD′ q′ ) k′1 ) (R′ v′ + RB′ v′B + 1 + 1 + PD′ RA′ RB′ RA′ RB′ A A PD′ v′D + PE′ v′E) (25) This expression is typical of the general case: it is difficult to develop an explicit expression for the byproduct production rate in terms of the recycle flow rates because q depends on PD. However, if the assumption of section 2.2 is employed, eq 25 reduces to

k′1 (R ′ v′ + RB′ v′B + 1) PD′ ) PE′ ) RA′ RB′ A A

r0 ) k0[A][B]

A+BfC

desired

A+AfD

r1 ) k1[A][A]

undesired (27)

C+CfE

r2 ) k2[C][C]

undesired

Again, this process has two recycle streams and two degrees of freedom: RA and RB. It is desired to express the byproduct production rates PD and PE in terms of these degrees of freedom. With the assumption of section 2.2

r˜ s,0 ) PC + 2PE ≈ PC

and

r˜ k,0 ) k0[A][B]V

r˜ s,1 ) PD

r˜ k,1 ) k1[A][A]V

r˜ s,2 ) PE

r˜ k,2 ) k2[C][C]V

(28)

(29)

1 PE′ ≈ k′2 RA′ RB′

(30)

Following this approach, any number of undesired reactions can be accounted for. 2.5. Reactor Volume. The method of calculating the reactor volume is illustrated here for the case of equivalent reaction order, but the same procedure can be applied to more general cases. Returning to chemistry 3, once the production rate of the undesired species is known, the reactor volume can be found using eq 4 (which, with the assumption of section 2.2, is correct for both chemistries 1 and 3):

q2 V ) PC k0RARB

(31)

Introducing the definition of the dimensionless reactor effluent flow rate q′

V ) PC

q′2PC2vC2 k0RARB

(32)

PC2 2 q′ RARB

(33)

and rearranging

(26)

If greater accuracy is desired, eq 25 can be solved numerically. However, eq 26 retains the essential features necessary to estimate the optimum operating conditions. Using this approach, kinetic expressions that include noninteger exponents can also be accommodated, as can an arbitrary number of reactant species. Appendix A discusses the case where stable intermediate species are formed and recycled and the case where one or more reactions are modeled as equilibrium reactions. 2.4. Multiple Undesired Reactions. These procedures can also be generalized in a straightforward manner to the case of multiple undesired reactions. Consider chemistry 5 (with the flowsheet shown in Figure 2), where both reactant A and the desired product C can irreversibly dimerize:

R′A PD′ ≈ k′1 R′B

Vk0 vC2PC

)

Therefore, dimensionless reactor volume V′ is defined as

/

V′ ) Vk0 vC2PC

(34)

and

V′ )

1 1 q′2 ) (R′ v′ + RB′ v′B + 1 + PD′ v′D)2 RA′ RB′ R′AR′B A A (35)

The required reactor volume becomes large if (1) any of the reactant species’ recycle streams become small (high conversion), corresponding to a large residence time needed to deplete that reactant, (2) the byproduct production rate becomes large, corresponding to the fact that the specific reaction rate is diminished owing to dilution of the reactants by the byproduct, or (3) the recycle flow rate of one reactant becomes much larger than the other. For example, if RB is finite and RA f ∞, then

(RA′ v′A)2 )∞ RAf∞ RA′ RB′

lim V ′ ) lim

RAf∞

(36)

However, if both recycle flow rates grow together, for example, if RA/RB ) 1:

(RA′ v′A + RB′ v′B)2 ) (v′A + v′B)2 RAf∞ RA′ RB′

lim V ′ ) lim

RAf∞ RBf∞

which is finite.

RBf∞

(37)


3. Economic Optimization 3.1. Economic Potential Model. Here we are concerned with the optimization of the operation of a plant after it has been built and capital costs have been “sunk”. Therefore, only operating costs are included in the economic model for process operation. Two such costs are considered here: selectivity losses, which result from the production of the undesired byproduct, and separation costs, which represent the cost to separate the components in the reactor effluent stream and to recycle the reactants (including possibly a compressor in a gas recycle line). We assume that these represent the principal operating costs of the plant. Thus, the following model for the economic potential is employed:

Figure 7. Reactor volume vs recycle rates for chemistry 1.

C ) CCPC + CDPD + CR(RA + RB)

(38)

C /PC ) CC + CDP′D + CR(R′A + R′B)

(39)

where CC is the negative of the revenue from the production of 1 mol of the desired product (CC < 0), CD is the cost of producing 1 mol of byproduct (including raw material costs and separation costs) (CD > 0), and CR is the cost of separating and recycling 1 mol of the reactant species (CR > 0). Note that C has units of $/h and PC has units of mol/h, so that C /PC has units of $/mol. It is certainly reasonable to assume that the economic potential of the plant is a linear function of the rate of production of the desired product and undesired byproduct. The assumption that the cost of operating the separation system varies linearly with the recycle flow rates is an approximation, but it is suggested and justified as a first approximation by Malone et al.28 A more exact cost expression could be used to obtain numerical results; however, the insight from the symbolic analysis of the equations would be lost among the many cost-correlation coefficients. Furthermore, the conclusions would not change. This approximation is designed to capture the desired tradeoff and to represent a judicious balance between model accuracy and complexity. Flow costs are rendered dimensionless by dividing by CR:

C /PCCR ) CC′ + CD′PD′ + (RA′ + RB′ )

(40)

The variable C ′C is constant for fixed species and the flowsheet structure. Its value will affect the value of the objective function but not the location of the minimum point. Therefore, a dimensionless cost objective function C′ is written as follows: Figure 8. Reactor volume vs recycle rates for chemistry 3.

Figures 7 and 8 show examples of how the reactor holdup must change as the recycle flow rates are varied for chemistries 1 and 3. The most severe increase in the reactor holdup occurs as the recycle flow rate of the reactants becomes small. Therefore, the practical effect of building a plant with a finite reactor volume is to constrain the recycle flow rates away from zero. This outcome also has important consequences for economic optimization, as will be discussed in section 3.

C ′ ) C /PCCR - CC′ ) CD′PD′ + (RA′ + RB′ )

(41)

When C ′ is minimized, the plant is at its most profitable operating point. The dimensionless quantity CD′ represents the cost of producing 1 mol of the undesired byproduct relative to the cost of recycling 1 mol of the reactant species. Typically, this is a large number, CD′ ∼ 100. 3.2. Unconstrained Optimization. In any real plant, the reactor volume and recycle capacity will limit the values of the recycle flow rates. However, first consider the unconstrained optimization problem, which

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Figure 9. Unconstrained optimization problem for chemistry 1.

Figure 10. Unconstrained optimization problem for chemistry 3.

is to find the minimum value of the cost function over the entire range 0 e Ri < ∞, as shown in Figures 9 and 10. The recycle cost adds a penalty that increases linearly with the recycle flow rate of each species to the penalty from the production of the undesired byproduct. For bounded species [such as species A in chemistry 1 (Figure 9a)], both components of the cost function strictly increase with increasing recycle flow rate. An important and inherent property of bounded species is

Figure 11. Constrained optimization problem for chemistry 1.

that the unconstrained optimum value of their recycle flow rate is zero. The shape of the objective function for nonbounded species, however, is qualitatively quite different (Figures 9b and 10a,b). Because the byproduct production rate decreases with increasing recycle flow rate while the recycle penalty increases, the optimal value of the recycle flow rate of nonbounded species is nonzero. This is an inherent and important property of nonbounded species. 3.3. Constrained Optimization. Now consider the problem of minimizing the cost function for a plant with finite reactor volume and recycle capacity, the situation shown in Figures 11 and 12. Each figure is constructed for a specific reactor volume. For bounded species, because the unconstrained optimum value of the recycle flow rate is zero, the constrained optimum always lies at the reactor volume constraint. This is the real reason they are referred to as reactor volume bounded. If there is even one reactor volume bounded species present in a process chemistry, it is guaranteed that the economically optimal policy is to operate with the reactor completely full at all times, regardless of the production rate. (Changing the production rate is considered in more detail in section 3.5.) Therefore, we apply the terms “bounded” and “nonbounded” to the overall process chemistries as well as to individual reactant species; a process chemistry is bounded if it has one or more bounded species. For nonbounded species, the behavior is qualitatively quite different, as shown in Figures 11b and 12a,b. Because the unconstrained optimum is nonzero, it is expected, but not guaranteed, that the optimum does not lie on the reactor volume constraint. However, depending on the size of the reactor and recycle capacity, the optimum may lie on one or the other of these constraints. Therefore, an overall process chemistry is referred to as nonbounded if and only if all of the


Figure 13. Constrained optimization in two dimensions for chemistry 1.

Figure 12. Constrained optimization problem for chemistry 3.

reactant species present in that chemistry are nonbounded. The reactor volume constraint chosen for chemistry 1 was V′ ) 20, while the reactor volume constraint for chemistry 3 was V′ ) 3. These choices reflect the fact that plant 1 is expected to have a much larger reactor than plant 3, because for chemistry 3, it is optimal from a selectivity point of view to operate with low conversion and high reactant concentrations to suppress the undesired reaction. Therefore, a much smaller reactor volume is required for good economic performance. This observation is true in general: nonbounded chemistries generally require much smaller reactor volumes than bounded chemistries because it is not necessary to achieve high conversion of any species in order to achieve good economic performance. The fact that nonbounded chemistries generally require a smaller reactor holdup than bounded chemistries is one reason that it is likely that a plant with a nonbounded chemistry will operate away from the reactor volume constraint. Another reason is that many processes are designed by first choosing a nominal operating point and then oversizing all of the process equipment by some amount (perhaps 20%). If this is done for a bounded chemistry and the nominal operating point is the economic optimum, then the extra reactor capacity is not needed and should only be used if the production rate is to be increased or if there is a disturbance that reduces the specific reaction rates (for example, catalyst deactivation or a sustained decrease in the reactor temperature). A final reason is that reactors are often relatively inexpensive to build; therefore, the prudent engineer would usually include a sufficient reactor capacity to accommodate all expected production rate increases. 3.4. Contour Plots. For the case of two degrees of freedom, the multivariable optimization can be represented as a contour plot, as shown in Figures 13 and 14, in which contours of the dimensionless cost are

Figure 14. Constrained optimization in two dimensions for chemistry 3.

shown as solid lines. The constraints imposed by the recycle capacity and reactor volume are shown as dashed lines. The “feasible region” in which the plant can be operated corresponds to the area enclosed by these constraints. As discussed above, when multiple recycle streams are present, process chemistry is said to be nonbounded if all of the recycled species are nonbounded. If one or more of the recycled species is bounded, then the process chemistry is bounded. Consider Figure 13. Chemistry 1 is a bounded chemistry because it has a bounded recycled species (species A), and as a result, the economic optimum operating point for the plant lies on the reactor volume constraint, as shown in the figure. Determining the exact location of this optimum corresponds to solving a constrained optimization problem. Because the expression for the reactor volume is a complicated function of the recycle flow rates (eq 35), it is usually necessary to solve the optimization problem numerically. The special insight from process chemistry is that, for bounded chemistries, the optimum always lies on the constraint. By contrast, Figure 14 shows that for chemistry 3, which is nonbounded because both of the recycled species are nonbounded, the cost contours are closed. Thus, the minimum operating cost is located away from the edges of the diagram. If possible, the process should be operated at this point, away from the constraints. If the feasible region does not encompass this point, then the process should be operated on the constraint closest to this point, which could be either the reactor volume constraint or the recycle capacity constraint.

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Figure 15. Constrained optimization after a production rate increase for chemistry 1.

Figure 16. Constrained optimization after a production rate increase for chemistry 3.

3.5. Accommodating a Production Rate Change. Now consider the optimal operating policy for a production rate increase. Because all flow rates and the reactor volume have been made dimensionless by dividing by the production rate, increasing the production rate corresponds to a decrease in the dimensionless recycle capacity and reactor volume, as shown in Figures 15 and 16. Correspondingly, there is a reduction in the size of the feasible region. However, the contours of the dimensionless cost do not change. Again consider chemistry 1 (Figure 15) first. Because the previous operating point was on the reactor volume constraint, that operating point is no longer in the feasible region. It is necessary to change the location of the operating point so that it lies on the new reactor volume constraint, within the new feasible region. The dimensionless cost associated with the new operating point is greater than before. Therefore, although the total profit of the plant may increase as a result of the sale of the additional product, the marginal cost of producing each mole of product must increase because constraints force the plant to operate in such a way that more byproduct is produced for each mole of product produced. A plant with bounded chemistry necessarily operates with “diminishing marginal returns”. In contrast, Figure 16 shows the results for nonbounded chemistry. Because the optimal operating point is not on a constraint, it is possible (even likely) that the new feasible region will still encompass the optimal point; hence, it is unnecessary to move the operating point. Therefore, the production rate of the undesired product, the reactor holdup, and the recycle flow rates all vary in proportion to the production rate, and the cost of producing each mole of product is unchanged.

The plant simply “scales up” in the same way that a design engineer would scale-up the design of the plant prior to construction if the production rate requirement were changed. A plant with nonbounded chemistry usually operates with constant marginal returns. Note that if the production rate change is sufficiently drastic or if the process does not have sufficient flexibility, then a constraint may become active during the production rate change. Each of these policies is analogous to a policy that has been proposed in the literature in conjunction with the RSR plant. For a bounded chemistry, the optimal policy as described above is analogous to what Luyben calls the “conventional method”. The reactor holdup is kept constant (completely full), and changes in the production rate are accommodated by adjusting the flow rate through the recycle system. The following analogy may be helpful: If one resource is “free” (reactor volume) and another resource is “expensive” (recycle), it is best to utilize the free resource as much as possible at all times and pick up the slack by adjusting the use of the expensive resource. By contrast, for nonbounded chemistries, the optimal policy is analogous to what Yu and co-workers call “balanced methods”. These distribute the load of a production rate change more evenly throughout the plant by scaling up both the reactor volume and recycle flow rates linearly with the production rate. The simple chemistry A f B, which has been widely considered in the plantwide control literature, is a nonbounded chemistry because there are no side reactions. Therefore, the economically optimal operating policy is the conventional method, with the reactor completely full at all times. Of course, this analysis considers only steady-state economic optimization, and other methods may offer advantages in terms of dynamic behavior. The design of dynamic control structures that are consistent with the insight provided by this methodology will be considered in a separate publication. 3.6. Another Example: Multiple Undesired Reactions. An alternative way to interpret and understand these results is to start with a single desired reaction, such as A + B f C, and consider how the optimal recycle policy changes when this chemistry is augmented with one or more undesired reactions. With no undesired reactions, both reactant species in this chemistry are reactor volume bounded. Because there are no undesired reactions, there are no selectivity losses and the optimal policy is to minimize the recycle flow rates in order to minimize energy costs. The reactor should be kept completely full at all times to achieve as high a conversion as possible of the reactants. If the undesired reaction C + C f D, r ) k[C]2, occurs as well, as in chemistry 3, the outcome changes. The optimum operating policy will be away from the reactor volume constraint, with more than the minimum recycle flow rate of both species A and B in order to suppress the undesired reaction. If, instead, the undesired reaction A + A f D, r ) k[A]2, occurs, as in chemistry 1, then species B is nonbounded but species A remains bounded. It is optimal to recycle as little of species A as possible and to recycle species B at a rate higher than the minimum possible. In any case, the reactor holdup should be kept as high as possible in order to minimize the recycle flow rate of species A.


If both of these undesired reactions can occur, as in chemistry 5, then both species are nonbounded, although the optimum value of the recycle flow rates will be different from that for chemistry 3. The general rules can be summarized as follows: (i) If an undesired reaction takes place that causes a species to be nonbounded, then the addition of other undesired reactions cannot make it bounded. (ii) If there is a species present that makes the overall process chemistry bounded, the addition of other species cannot make the process chemistry nonbounded. 3.7. Effect of k1′ and CD′ . It has been shown that the optimal unconstrained dimensionless recycle flow rate of reactants is zero for bounded species and a nonzero constant for nonbounded species. This result invites the question, what is the value of the unconstrained optimum recycle flow rate for nonbounded species? In certain cases, the answer is especially simple. Consider chemistry 6 (the flowsheet shown in Figure 3):

AfC CfD

r0 ) k0[A] r1 ) k1[C]

desired undesired

(42)

There is only one recycled species in this example (A). Because an undesired reaction occurs and the order of species A in the desired reaction (1) is greater than the order of species A in the undesired reaction (0), species A is nonbounded and the unconstrained optimal value of the recycle flow rate is not zero. The production rate of the undesired byproduct is given by

PD′ ) k′1/R′A

(43)

On substituting this result into eq 41, we obtain

1 + RA′ C ′ ) CD′ k′1 RA′

(44)

This cost function is to be minimized by varying R′A. The first two derivatives are

( ) ( )

dC ′ 1 ) -(CD′ k′1) dR′A R′A

2

d2 C ′ 1 ) 2(CD′ k′1) 2 R dR′A A′

+1 3

(45)

(46)

from which it is seen that the objective function is convex over the whole domain 0 e RA < ∞. Setting the first derivative equal to zero and solving for RA give

R h ′A ) xCD′ k′1

(47)

where the overbar indicates that this is the optimal value. Figure 17 shows how the cost function changes as the quantity k′1CD′ is varied. In the limit that k′1CD′ approaches zero, the undesired reaction becomes irrelevant, either because it occurs extremely slowly compared to the desired reaction (k′1 , 1) or because a negligible cost penalty (CD′ , 100) is incurred. In this limit, species A becomes “effectively bounded”, meaning that the unconstrained optimum value of the recycle flow rate R′A is so small that the reactor volume constraint will be active at all times unless the available reactor volume is enormous. Thus,

Figure 17. Cost function for various values of k ′CPD for chemistry 6.

the process effectively behaves as if that species were bounded. In general, if the assumption of section 2.2 is employed, then the cost function will depend only on a product of terms such as k′1CD′ and not on k′1 and CD′ individually. 4. Applicability and Expected Benefit of the Method Because the rules for selecting operating policies that are proposed in this paper have been derived analytically from first principles rather than observed or inferred from case studies, it can be determined when they are applicable and when they are likely to yield the greatest benefit. Because the methodology proposed here is simple and easy to conduct and implement, it should be worthwhile to conduct this analysis on most processes. However, the benefit in terms of cost savings will be different for different processes. In general, the cost savings from implementing the optimal operating policy will be smaller if any of the following apply: (i) The process has a limited amount of flexibility. If the process must operate near most of its constraints, then it is less likely that the economics will change considerably over the feasible operating region because the feasible operating region will be small. (ii) The process chemistry is bounded, and the reactor is large. In this limiting case, the economics may plateau in the limit of large reactor holdup and may be relatively insensitive to changes in the recycle flow rates. (iii) Capital costs far exceed operating costs. In this case, the profitability of the process will be determined primarily at the design stage. Again, none of these considerations invalidate the analysis presented in this paper; they merely influence the magnitude of the expected savings. 5. Summary and Conclusions An analytical method is presented that predicts the optimum plantwide steady-state operating policy for chemical processes with a reactor, separation system, and recycle. On the basis of the reaction kinetics, recycled species are classified as “reactor volume bounded” or “reactor volume nonbounded”, both with respect to individual undesired reactions and with respect to the complete process chemistry. Once this distinction has been made, it is straightforward to predict the optimum plantwide level/flow rate operating policy: If all species are nonbounded, then the process should usually be operated with variable reactor holdup. If one or more species is bounded, then the process should be operated with fixed maximum reactor holdup.

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The method also provides a shortcut estimate of the optimal recycle flow rate when this value does not lie on a process constraint. Appendix A: Extensions and Generalizations A.1. Equilibrium Reactions. It is straightforward to incorporate equilibrium reactions into the framework discussed in this paper. For example, consider the chemistry

A+BfC A + Af D

r0 ) k0[A][B] r1 ) k1[A][A]

A+CTE

K ) [E]/[A][C]

desired undesired (48)

Figure 18. Process flow diagram with two recycle streams and a PFR.

the rate of its generation must equal the rate of its consumption by material balance:

undesired

As before, concentrations are expressed in terms of flow rates using the total reactor effluent flow rate, so for the third reaction, we have

K ) PEq/RAPC

(49)

Introducing the dimensionless reactor effluent flow rate q′ gives

K ) PEq′PCvC/RAPC

(50)

(51)

and the dimensionless reaction equilibrium constant is defined as

(57)

kI[A][B]V ) k0[A][I]V

(58)

and

Thus, only two degrees of freedom are available because all of species I in the reactor effluent is recycled back to the reactor. Converting concentrations to flow rates and introducing the dimensionless reaction rate constant

Therefore

PE′ K ) q′ vC RA′

rI ) r0

k′IRARB ) RARI

(59)

RI ) k′IRB

(60)

and therefore

K ′ ) K/vC

(52)

Because both desired reactions take place at the same rate, either one can be chosen as a reference reaction for the determination of the byproduct production rate PD. For definiteness, take the reaction that actually produces the desired species C:

K′ R′ q′ A

(53)

PC ≈ k0[A][I]

(61)

Therefore, species A is reactor volume bounded with respect to the equilibrium reaction. It has already been shown that species A is bounded with respect to the kinetically limited undesired reaction:

PD ) k1[A][C]

(62)

Again dividing the undesired reaction by the desired reaction and converting concentrations to flow rates

R′A PD′ ) k′1 R′B

PD k1RAPC ) P C k 0 R AR I

so that

PE′ )

(54)

Therefore, the cost function is

R′A K′ C ′ ) RA′ + RB′ + k′1CP′D + C′PE R′A R′B q′

PD′ ) (55)

Thus, species A is bounded with respect to the overall process chemistry, and species B is nonbounded. A.2. Desired Reactions with Stable Intermediates. The method can also be applied to cases in which there are multiple desired reactions but only a single desired product. Consider the following chemistry:

A+BfI

rI ) kI[A][B]

desired

A+IfC

r0 ) k0[A][I]

desired

A+CfD

r1 ) k1[A][C]

(56)

undesired

Although there are three recycled species (A, B, and I), because species I neither enters nor leaves the plant,

k′1 1 k′I R′B

(63)

(64)

Therefore, the process is nonbounded with respect to species B, but bounded with respect to species A, as is the case for chemistry 2. Appendix B: Extension to Plug-Flow Reactors (PFRs) Consider the flowsheet shown in Figure 18, which is the same as that shown in Figure 1 except that the CSTR has been replaced with a PFR. In the case of the CSTR, variable reactor holdup policies allow the plant to achieve a specified per-pass conversion independent of the production rate. The corresponding situation of a single large-diameter PFR with a variable liquid level is rarely, if ever, encountered in industry. However, a PFR with variable holdup is mathematically analogous


to several systems that are of interest, including gasphase PFRs (where the gas pressure can be changed), batch systems (where the holdup in the batch reactor can be adjusted), a set of PFRs in parallel, where the number of reactors online at any given time can be adjusted (this is often the case when it is necessary to have some fraction of the tube bundle offline at any given time for catalyst recharging), and chains of CSTRs in series, which are meant to approximate the selectivity-conversion profile for a plug-flow system. Therefore, we imagine a PFR with variable cross-sectional holdup A and fixed length L. For chemistry 1, the evolution of the concentration of species B in a PFR reactor is given by Figure 19. Optimization for chemistry 1 with a PFR.

q dCB ) -k0CACB V dξ

(65)

where ξ ) x/L is the dimensionless distance from the reactor inlet. As before, concentrations are converted to flow rates via the volumetric flow rate q. The flow rate of each species in the reactor (which is a function of the distance down the reactor) is designated with the variable F.

dFB Vk0 ) - 2 FA FB dξ q

(66)

of variables at both the initial and final positions, it is a nonlinear two-point boundary value problem. As written, the system may appear to be overspecified because it has has four differential equations and five boundary conditions. However, the reactor holdup V′ is indeterminate; thus, this system of differential equations defines an implicit relationship that gives V ′(R′A, RB′ ). P′D(RA′ ,RB′ ) ) FD′|ξ)1 is also found implicitly as part of the solution to the differential equations. Taking a cue from the analysis of section 2.1, we seek a simplification by dividing eq 77 by eq 76, which gives

As before, flow rates F are made dimensionless by dividing by PC

dFB′ k0VPC )FA′FB′ dξ q2

(67)

and

( )

q′ dFA′ ) -FA′FB′ V′ dξ

(68)

Following this method of analysis, the complete set of differential equations describing the concentration profile in the PFR is

q′2 dFA′ ) -FA′FB′ - 2FA′2 V′ dξ q′2 dFB′ ) -FA′FB′ V′ dξ

(69)

F ′

∫01FBA′ dFC′

lim PD′ ) 0

(80)

lim PD′ ) ∞

(81)

R′Bf∞

q′ dFC′ ) FA′FB′ V′ dξ

(71)

q′2 dFD′ ) FA′2 V′ dξ

(72)

2

subject to the boundary conditions

FA′|ξ)1 ) R′A

(73)

FB′|ξ)1 ) R′B

(74)

FC′|ξ)0 ) 0

(75)

FC′|ξ)1 ) 1

(76)

FD′|ξ)0 ) 0

(77)

These differential equations are nonlinear, and, furthermore, because there are specifications on the value

(79)

Of course, FA′ and FB′ depend on FC′ in a complicated manner. Moving down the PFR, FB′ decreases mono′ + RB′ at the inlet to tonically from a value of FB′ ) FF,B FB′ ) R′B at the exit. Furthermore, the integral of a monotonic function is itself a monotonic function. Therefore R′Bf∞

(70)

(78)

and

PD′ )

2

( ) ( ) ( ) ( )

dFD′ FA′ ) dFC′ FB′

By similar reasoning, PD′ is strictly increasing with R′A. A similar line of reasoning should be able to accommodate other reactor networks, although this is by no means proven here. Figures 19 and 20 show the economic optimization problem as a contour plot for chemistries 1 and 3 (compare with Figures 13 and 14). Appendix C: Benzene Chlorination Case Study This appendix presents a summary of key results from a case study of a process for benzene chlorination. Further details concerning the case study are available in the Supporting Information. C.1. Description of the Process. Chlorination reactions are employed to introduce reactive sites on organic molecules. For example, one route in the production of phenol (C6H5OH) from benzene is via the intermediate chlorobenzene (C6H5Cl). In this case, care must be taken

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Figure 22. Operational optimization of the flexible benzene chlorination process.

Figure 20. Optimization for chemistry 1 with a PFR.

Figure 23. Operational optimization of the flexible benzene chlorination process after a production rate decrease.

Figure 21. Process flow diagram for benzene chlorination.

to minimize the production of higher chlorinated benzenes. The reactions are

benzene + Cl2 f chlorobenzene + HCl chlorobenzene + Cl2 f dichlorobenzene + HCl These reactions can be carried out in the liquid phase at a temperature of 60 °C. We assume that dichlorobenzene has no value and must be disposed of safely; i.e., the second reaction is a representative byproductforming reaction. Chlorine is introduced into a single CSTR through a sparger and dissolved in the liquid phase. The concentration of chlorine in the liquid phase is assumed to be constant. The reaction can be catalyzed homogeneously by stannic chloride (SnCl4). If the concentration of the catalyst and chlorine are constant, then the reaction rates will be given by

r0 ) k0[benzene] k0 ) 0.22 h-1 r1 ) k1[chlorobenzene] k1 ) 0.041 h-1 where the kinetic parameters are determined from data published by Silberstein et al.31 Therefore, the process chemistry is effectively of the type A f B f C, which is nonbounded. The reactor network considered in this case study is a single CSTR, although a PFR or a cascade of CSTRs would be expected to give a better selectivity-conversion profile. Stripping operations, which are not considered in this case study, are employed to remove unreacted chlorine and HCl. The separation of benzene, chlorobenzene, and dichlorobenzene is accomplished by distillation. The process flow diagram is shown in Figure 21. C.2. Design and Equipment Sizing. The distillation columns are sized and costed according to the

methods of Doherty and Malone.30 The nominal process is designed to produce 50 kmol/h of chlorobenzene with a reactor temperature of 60 °C. To make the process operable, it is necessary to overdesign the actual process equipment by some amount compared to the nominal design. This was accomplished by sizing the process equipment for the case where the production rate is 30% larger than the nominal production rate. C.3. Operating Policies. Figure 22 shows the optimization problem for the flexible process after it has been built. The solid line shows the operating economic potential of the process as a function of the operational degree of freedom, the recycle flow rate of benzene. The vertical dashed lines show the reactor volume constraint (on the left) and the recycle capacity constraint (on the right). Again, as expected, there is a maximum in the economic potential function. However, the location of the maximum has shifted now that capital costs are fixed and lies outside the feasible region. As is sometimes the case for nonbounded chemistries, the optimal operating point lies on the recycle capacity constraint, at 290 kmol/h. The economic potential of the process at this point is $3.1 million/year. This operating point corresponds to a reactor holdup of 72% of the maximum value. An alternative inferior operating policy would be to operate with the reactor completely full. The economic potential of the process with this inferior operating policy is $2.2 million/year, a loss of nearly 30% compared to the maximum achievable economic potential. Now consider a decrease in the production rate of 50%. The optimization problem is shown in Figure 23. Because much less product is being produced, the economic potential of the process is significantly reduced. Also, the optimal recycle flow rate has been reduced by 50% and now lies within the process constraints, at 202 kmol/h, where the economic potential of the process is $494 thousand/year, and the reactor holdup is at 34% of the maximum value. The process is within the region of operation where the optimal operating policy is to scale the reactor holdup and recycle flow rate linearly with the production rate. The process incurs a loss of $1.6 million/year if it is operated


with the reactor completely full. The economic potential of the process when it is operated on the recycle capacity constraint is $458 thousand/year, corresponding to a loss of 7.3% relative to the maximum achievable economic potential. Supporting Information Available: Further information concerning the benzene chlorination case study. This material is available free of charge via the Internet at http://pubs.acs.org. Nomenclature C ) cost ($/h) CR ) cost associated with a recycle stream ($/mol) CD ) cost associated with the byproduct stream of species D ($/mol) k ) reaction rate constant P ) production rate of the species (mol/h) q ) reactor effluent volumetric flow rate (L/h) r ) specific reaction rate [mol/(L h)] r˜ ) overall reaction rate (mol/h) R ) recycle flow rate (mol/h) R h ) optimum value of the recycle flow rate (mol/h) vA ) molar volume of species A (L/mol) V ) reactor volume (L) Subscripts 1, ..., n ) reactions 1, ..., n A, ..., E ) species A, ..., E k ) reaction rate expression based on kinetics s ) reaction rate expression based on stoichiometry

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(11) Kookos, I. K.; Perkins, J. D. An Algorithm for Simultaneous Process Design and Control. Ind. Eng. Chem. Res. 2001, 40, 4079. (12) Perkins, J. D.; Walsh, S. P. K. Optimization as a Tool for Design/Control Integration. Comput. Chem. Eng. 1996, 20, 315. (13) Bansal, V.; Sakizlis, V.; Ross, R.; Perkins, J. D.; Pistikopoulos, E. N. New Algorithms for Mixed-Integer Dynamic Optimization. Comput. Chem. Eng. 2003, 27, 647. (14) Bansal, V.; Perkins, J. D.; Pistikopoulos, E. N. A Case Study in Simultaneous Design and Control Using Rigorous, MixedInteger Dynamic Optimization Models. Ind. Eng. Chem. Res. 2002, 41, 760. (15) Fisher, W. R.; Doherty, M. F.; Douglas, J. M. The Interface between Design and Control. 3. Selecting a Set of Controlled Variables. Ind. Eng. Chem. Res. 1988, 27, 611. (16) Luyben, W. L.; Tyreus, B. D.; Luyben, M. L. Plantwide Process Control; McGraw-Hill: New York, 1999. (17) Skogestad, S. Plantwide Control: the Search for the SelfOptimizing Control Structure. J. Process Control 2000, 10, 487. (18) Zheng, A.; Mahajanam, R. V.; Douglas, J. M. Hierarchical Procedure for Plantwide Control System Synthesis. AIChE J. 1999, 45, 1255. (19) Luyben, W. L. Design and Control of Recycle Processes in Ternary Systems with Consecutive Reactions. In Interactions between Process Design and Process Control; Perkins, J. D., Ed.; Pergamon Press: Oxford, U.K., 1992; p 65. (20) Luyben, W. L. Dynamics and Control of Recycle Systems. 1. Simple Open-Loop and Closed-Loop Systems. Ind. Eng. Chem. Res. 1993, 32, 466. (21) Luyben, W. L. Dynamics and Control of Recycle Systems. 2. Comparison of Alterative Process Designs. Ind. Eng. Chem. Res. 1993, 32, 476. (22) Luyben, W. L. Dynamics and Control of Recycle Systems. 3. Alternative Process Designs in a Ternary System. Ind. Eng. Chem. Res. 1993, 32, 1142. (23) Luyben, W. L. Snowball Effects in Reactor/Separator Processes with Recycle. Ind. Eng. Chem. Res. 1994, 33, 299. (24) 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. (25) Larsson, T.; Skogestad, S. Plantwide ControlsA Review and a New Design Procedure. Model., Identif. Control Environ. Syst. 2000, 21, 209. (26) Wu, K. L.; Yu, C. C. Reactor/Separator Processes with Recycles1. Candidate Control Structures for Operability. Comput. Chem. Eng. 1996, 20, 1291. (27) Wu, K. L.; Yu, C. C.; Luyben, W. L.; Skogestad, S. Reactor/ Separator Processes with Recycless2. Design for Composition Control. Comput. Chem. Eng. 2002, 27, 401. (28) Malone, M. F.; Glinos, K.; Marquez, F. E.; Douglas, J. M. Simple, Analytical Criteria for the Sequencing of Distillation Columns. AIChE J. 1985, 31, 683. (29) Douglas, J. M. Conceptual Design of Chemical Processes; McGraw-Hill: New York, 1988. (30) Doherty, M. F.; Malone, M. F. Conceptual Design of Distillation Systems; McGraw-Hill: New York, 2001. (31) Silberstein, B.; Bliss, H.; Butt, J. B. Kinetics of Homogeneously Catalyzed Gas-Liquid Reactions: Chlorination of Benzene with Stannic Chloride Catalyst. Ind. Eng. Chem. Fundam. 1969, 8, 366.

Received for review September 12, 2003 Revised manuscript received April 6, 2004 Accepted April 12, 2004 IE034125Z

Importance of Process Chemistry in Selecting the Operating Policy for

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