Application of the Capacity-Based Economic Approach to an Industrial

Plant capacity is reduced by the fraction of time that the product is outside the predefined upper or lower specification limits. When product is off-...
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Ind. Eng. Chem. Res. 1997, 36, 1727-1737

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Application of the Capacity-Based Economic Approach to an Industrial-Scale Process Timothy R. Elliott and William L. Luyben* Chemical Process Modeling and Control Research Center and Department of Chemical Engineering, Lehigh University, Iacocca Hall, 111 Research Drive, Bethlehem, Pennsylvania 18015

Michael L. Luyben DuPont Central Science & Engineering, Experimental Station, P.O. Box 80101, Wilmington, Delaware 19880-0101

In a previous paper (Elliott, T. R.; Luyben, W. L. Ind. Eng. Chem. Res. 1995, 34 (11), 39073915) we outlined a generic methodology called the capacity-based economic approach that can be used to compare or screen preliminary plant designs by quantifying both steady-state economics and dynamic controllability. The method provides an analysis tool that explicitly considers variability in product quality. A simple reactor/stripper recycle system was used to demonstrate the method. A more complex ternary process with two recycle streams was studied in a subsequent paper (Elliott, T. R.; Luyben, W. L. Ind. Eng. Chem. Res. 1996, 35, 34703479). In this paper, we consider an even more complex process consisting of two reaction steps, three distillation columns, two recycle streams, and one purge/makeup stream. The essential contribution of this paper is to demonstrate that the capacity-based economic approach can be successfully applied to a large industrial-scale process. The system is described by approximately 750 ordinary differential equations and has 18 design and control degrees of freedom. Introduction The goal of the capacity-based economic approach is to quantitatively combine considerations of both steadystate economics and dynamic controllability. The methodology can be used to compare alternative process flowsheets and process flowsheets that differ in several design parameters. For rapid screening of a large number of alternatives, the dynamics of the process are approximated by linear models. However, the method may also be applied using rigorous nonlinear simulations of the process when subjected to preformulated disturbance sequences. When linear dynamic models are used, closed-loop frequency response techniques predict the “peak” or worst disturbance conditions. In general, Bode plots of closed-loop regulator transfer functions exhibit a maximum value of log modulus at some peak frequency where the magnitude ratio of the controlled variable to load disturbance is greatest. By noting the peak frequency and phase angle for each load disturbance to the plant, we can perturb the plant with load disturbances entering the process as sinewaves occurring at their respective peak conditions. Although sinusoidal disturbances do not resemble actual disturbance sequences that may occur in the plant, they do represent a worst case disturbance scenario that is inherent to the plant. The method is by no means limited to linear analysis. Applying the method using linear analysis may be used as a first pass screening tool. Designs that exhibit unacceptable results may be removed from consideration, and a more rigorous study using nonlinear models may be performed. Nonlinear simulation of expected disturbance sequences may also be incorporated within the framework of this methodology. Having specified limits for product quality variability, plant designs are screened on their ability to maximize * To whom correspondence should be addressed. Phone: (610)758-4256. Fax: (610)758-5297. Email: [email protected]. S0888-5885(96)00376-4 CCC: $14.00

annual profit in the presence of their associated peak disturbances. Plant capacity is reduced by the fraction of time that the product is outside the predefined upper or lower specification limits. When product is off-spec, profits are being lost and reprocessing or disposal costs are incurred. Thus, the method deals explicitly and quantitatively with the question of product quality variability, which is an increasingly important criterion of control performance (Downs et al., 1994). Annual profit is calculated by determining the amount of onspec product produced annually and the annual raw material, reprocessing, capital investment, and operating costs for the plant. If every design alternative considered is producing all on-spec product, the design that has the lowest combined capital investment and operating costs will maximize annual profit. However, depending on the difference in steady-state economic design costs, the possibility exists that a more expensive design alternative producing all on-spec product may, in fact, be more profitable than a less expensive design alternative producing some off-spec product. The capacity-based approach can be applied at two different levels. For conceptual design, where little is known about the frequency content of the disturbances and many alternative processes and parameters need to be evaluated, a linear dynamic model can be used together with the worst-case frequency. For final design selection among a small number of cases, rigorous dynamic simulations can be used together with the best estimate of the types, magnitudes, and frequency content of disturbances. It is important to note that use of the worst-case frequency is only intended to provide comparisons among alternative conceptual process designs (flowsheets, parameters, and/or control systems). We feel this is a legitimate approach in the absence of any information on the real frequency content of the disturbances that the plant must handle. The absolute profitability clearly depends on these disturbances. If they are known, which is seldom the case for a new © 1997 American Chemical Society

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Figure 1. Conceptual flowsheet for the process.

plant design, they can be imposed on the rigorous dynamic simulation, and the real optimum economics can be determined quantitatively. The proposed method analyzes a specific process and a specific control system. Since we are evaluating the closed-loop disturbance rejection performance of the plant, a control system must be specified. In this paper we use conventional multiloop PI controllers, but the method is applicable to any type of control structure. When we use the term “plant” in this paper, we are referring to both the process and the control system, with all tuning constants specified.

6. Obtain magnitude ratio, phase angle, and frequency that correspond to the peak log modulus of the Bode plot of the closed-loop regulator transfer function for each load disturbance. 7. Obtain closed-loop linear time response to “peak” sinusoidal disturbances. 8. For a desired specification range, estimate the percent capacity of the design from the linear time response. 9. Calculate annual profit accounting for capital costs, operating costs, and the reprocessing of off-spec material.

Capacity-Based Economic Approach

Process Studied

To apply this approach, the process flowsheet must be sufficiently defined in order to generate a process model. Design alternatives are compared based on their ability to maximize annual profit in the presence of load disturbances. The type of disturbance entering the process is specified by the user. For this demonstration of the methodology, we allow each disturbance to enter the process at the frequency where it has the greatest effect on the process outputs. The peak disturbance conditions are obtained by generating Bode plots of the closed-loop regulator transfer functions for each load disturbance. Applying the method using the peak disturbance conditions provides a lower bound on the performance of alternative plant designs. This approach ensures that the plant will minimally achieve the annual profit corresponding to the worst-case conditions. The use of linear models enables the capacity-based economic approach to be performed without dynamic simulation. Using the linear time response, the plant capacity is then calculated for a given product quality specification range. Previous papers have presented the details of the procedure (Elliott and Luyben, 1995). A list of the steps is given below for the case when linear analysis is used. 1. Perform steady-state design. 2. Linearize the nonlinear dynamic process model. 3. Select a control structure for the process. 4. Tune controllers. 5. Derive closed-loop regulator transfer functions.

A conceptual flowsheet is shown in Figure 1. The first reaction step A + B f C + D produces one desired product C (reaction rate R1).

R1 ) VR1k1z1,Az1,B

(1)

The reactants are fed to one isothermal CSTR with a specific reaction rate of 10.0 h-1. Component A is fed to the first reactor from a fresh feed stream and a recycle stream. Component B, a product from the second reaction step, is recycled back to the reactor from the top of column 3 and column 1. Since A and B are not completely converted to C and D, the reactor effluent stream contains a mixture of reactants and products. This mixture is fed to the first of two separation sections in the process. Primarily unreacted A and B are recycled back to the reactor from the top of column 1. Ideal mixtures are assumed with volatilities of RA > RB > RC > RD > RE > RF. Two distillation columns are used to separate the reactor 1 effluent stream. In the first column, A and B are separated from C and D for recycle back to the first reaction step. The second column recovers product C at 99% purity from D. The second reaction step D + E f F + B occurs in two isothermal CSTR’s in series with a specific reaction rate of 50.0 h-1 (reaction rates R2 and R3).

R2 ) VR2k2z2,Dz2,E

(2)

R3 ) VR3k2z3,Dz3,E

(3)

Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 1729

Figure 2. Process flowsheet diagram with control strategy 1c implemented.

Byproduct D from the first reaction step is consumed, B is regenerated, and desired component F is produced. The second reaction step is fed primarily D from the bottom of the second column and E coming in as fresh feed. Since E and F are similar in volatility, the process is designed so almost all E is consumed in the second reaction step. This approach circumvents the potential difficulty in separating E from F. The third column separates primarily product F (at 99% purity) from B. Component B is then recycled back to the first reaction step. The following assumptions are made in modeling the process: constant density, relative volatilities and molecular weight, equimolal overflow, theoretical trays, total condensers, and partial reboilers. Tray holdups and the liquid hydraulic time constants are calculated from the Francis weir formula assuming a 1-in. weir height. The reflux drums and column bases were sized to provide 5 min of holdup resulting from the respective steady-state flowrates into each. The fresh feed, reactor effluent, reflux, bottoms, and distillate flows are assumed to be saturated liquid. The reactors are assumed to operate isothermally. A detailed schematic of the process flowsheet is given in Figure 2. There are 12 degrees of freedom for the design and control of this process. Two design methods were used to generate design alternatives: (1) an approximate heuristic steadystate design method and (2) a nonlinear economic optimization method.

Approximate Heuristic Steady-State Design Method The approximate heuristic steady-state design method presented in Luyben and Luyben (1995) is given below. The method is slightly modified to explicitly fix the purities in the product streams, D2 and B3. The following variables were fixed for each design: (1) the fresh feed flowrate of A (FoA ) 100 lb mol/h), (2) the fresh feed flowrate of B (FoB ) 0.5 lb mol/h), (3) the amount of C leaving the bottom of column 2 (Crecycle ) 0.5 lb mol/h), (4) the composition of C in the distillate of column 1 (XD1(C) ) 0.05 mole fraction), (5) the product purity in the distillate of column 2 (XD2(C) ) 0.99 mole fraction), (6) the product purity in the bottom of column 3 (XB3(F) ) 0.99 mole fraction), (7) the product impurity of E in the bottom of column 3 (XB3(E) ) 0.004 mole fraction), (8) the composition of F in the distillate of column 3 (XD3(F) ) 0.0001 mole fraction), (9) the holdup in reactor 3 is equal to the holdup in reactor 2 (VR3 ) VR2). The remaining 3 degrees of freedom will be used to perform parametric studies on the process. The variables selected for this study are the holdup in reactor 1, VR1, the column 1 distillate flowrate, D1, and the amount of D recycled back to reactor 1 from column 3, Drecycle ) D3XD3(D). The same three design parameters were used to determine the economic optimum plant for the approximate heuristic design method given in Luyben and Luyben (1995). The design procedure is described in detail in the appendix.

1730 Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 Table 1. Summary of Design Parameters and Economics for Heuristic Designs

Table 2. Summary of Reactor Sizes and Vapor Boilups for Heuristic Designs

design case

VR1 [lb mol]

D1 [lb mol/h]

Drecycle [lb mol/h]

TAC [103 $]

design case

VR2 ) VR2 [lb mol]

VS(1) [lb mol/h]

VS(2) [lb mol/h]

VS(3) [lb mol/h]

H1, base case H2, large VR1 H3, small VR1 H4, large D1 H5, small D1 H6, large Drecycle H7, small Drecycle

1000 2000 750 1000 1000 1000 1000

80 80 80 160 65 80 80

40 40 40 40 40 80 20

1581.8 1642.0 1579.6 1635.4 1590.0 1689.0 1574.3

H1, base case H2, large VR1 H3, small VR1 H4, large D1 H5, small D1 H6, large Drecycle H7, small Drecycle

390 390 390 390 390 276 615

458.51 424.03 487.04 541.24 460.19 494.55 443.33

382.52 382.47 382.50 382.74 382.49 436.15 354.67

274.51 277.46 274.44 275.28 274.42 355.59 235.82

Approximate Nonlinear Optimization Design Method The approximate nonlinear economic optimization problem (NLP) formulated in Luyben and Luyben (1995) is used as an additional design method. Each NLP was solved with the following constraints: (1) XD2(C) ) 0.99, (2) XB3(F) ) 0.99, and (3) FoA ) 100.0 lb mol/h. Additional constraints were placed on various design variables to evaluate their effects on the economics and dynamic controllability of the system. The solution set of design variables that minimize the total annual cost is obtained using an economic objective function comprised of the design cost correlations given in the appendix. The optimization problem was formulated in GAMS (Brooke et al., 1992) and solved using MINOS (Murtagh and Saunders, 1988). The nonlinear optimization design method alleviates the rule of thumb design assumptions necessitated in the heuristic design to use up the degrees of freedom. The NLP uses the Eduljee design equation, the Underwood equations for minimum reflux ratio, and the Fenske equation for minimum number of stages to determine the approximate operating conditions for the columns. In solving the NLP, the values of the following design variables are determined: (1) flowrates and compositions of all process streams; (2) holdups and compositions for each reactor; (3) number of stages in each column; (4) fresh feed flowrates of B and E. A detailed description of the method is given in Luyben and Luyben (1995). Results of Approximate Steady-State Designs Heuristic Design Results. The design that minimized the total annual cost for the heuristic design method given in Luyben and Luyben (1995) is used as a base case for determining the effects of design parameters on steady-state economics and dynamic controllability. The values of the three design parameters that minimize the total annual cost for the process are VR1 ) 1000 lb mol, D1 ) 80 lb mol/h, and Drecycle ) 40 lb mol/h. Each of the three design parameters was set at one value above and below the base case design value, giving seven design cases including the base case. These cases are described in detail in Elliott (1996) and correspond to the actual rigorous process steady state. These conditions are analogous to the initital conditions for the dynamic simulation studies presented in Luyben and Luyben (1995). The steady-state conditions were obtained by converging the dynamic model of the process to a steady-state condition. Table 1 gives the values of the design parameters considered in this study. The design alternatives were generated by doubling and halving the values of the base case design parameters with the exception of case H3 (small VR1) and H5 (small D1). For both cases, halving the value of the base case design variable resulted in an infeasible design. The values were increased until a feasible design was reached.

Table 3. Summary of Fixed Design Parameters and Economics for Optimization Designs design case

description

TAC [103 $]

O* O1 O2 O3 O4 O5 O6 O7

optimal design small VR1 ) 250 medium VR1 ) 1000 large VR1 ) 2000 small D1 ) 150 large D1 ) 450 small D3 ) 120 large D3 ) 250

1446.0 1496.5 1478.5 1528.2 1471.3 1650.1 1454.4 1586.4

In examining the total annual cost relationship in Table 1, the base case is no longer the economic optimum. Design H7 now minimizes the total annual cost. This may be attributed to the use of an approximate design technique. The effects of overdesign (large) and underdesign (small) on the tradeoffs between steady-state economics and dynamic controllability will be examined using the capacity-based economic approach later in this paper. The reactor sizes and vapor boilups for each heuristic design are summarized in Table 2. Nonlinear Optimization Design Results. Several design cases were generated using the approximate nonlinear programming method presented in Luyben and Luyben (1995). The design cases are described in detail in Elliott (1996) and correspond to the actual process steady state. The holdup in reactor 1, VR1, the D1 recycle flowrate, and the D3 recycle flowrate (analogous to Drecycle) were fixed at different values, and the remaining process variables were obtained by minimizing the total annual cost through the NLP optimization method. Table 3 summarizes the design cases and economics. For both the heuristic and optimization design methods, the total annual cost (TAC) for the process is not significantly affected by varying any of the three design parameters. Whether or not these design parameters have an effect on the closed-loop dynamics of this process will be examined later in this paper. The reactor holdups and vapor boilups for each optimization design case are listed in Table 4. Note that the values of VR2 and VR3 are not equal in the optimization design method.

Application of the Capacity-Based Economic Approach The capacity-based economic approach was applied to the different design cases to quantify design and control tradeoffs. Control scheme 1c from Luyben and Luyben (1995) was implemented on all design alternatives. This control scheme is depicted in Figure 2. The reactor 1

Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 1731 Table 4. Summary of Reactor Holdups and Vapor Boilups for Optimization Designs design case

VR1 [lb mol]

VR2 [lb mol]

VR3 [lb mol]

VS(1) [lb mol/h]

VS(2) [lb mol/h]

VS(3) [lb mol/h]

O*, small VR1 O1, small VR1 O2, medium VR1 O3, large VR1 O4, small D1 O5, large D1 O6, small D3 O7, large D3

596 250 1000 2000 814 294 572 708

319 325 318 316 323 304 354 214

495 514 491 486 508 455 588 261

490.97 613.56 480.65 405.06 453.67 852.75 478.61 625.13

324.14 308.43 319.95 344.84 342.38 294.59 318.48 358.91

237.84 243.34 242.60 241.20 239.34 279.16 232.55 324.25

level is controlled by manipulating the fresh feed flowrate of A and the reactor 1 effluent F1 is flow controlled. To prevent the buildup or depletion of the intermediate reactant/product B, a split-ranged level controller is implemented on the column 3 reflux drum. When B is in excess, it builds up in the column 3 reflux drum. The accumulation of B is counteracted by purging B at the composition in the reflux drum XD3 (not pure B). Conversely, when the amount of B is depleted in the system, pure B is added to the column 3 reflux drum. The compositions of the components B and D are controlled throughout the process. The composition loops are listed below. (1) The composition of B in reactor 1 z1,B is controlled by manipulating the recycle flowrate of B, D3. (2) The composition of B in the bottom of column 1 is controlled by manipulating the vapor boilup in column, VS(1). (3) The impurity of D in the C product stream, D2, is controlled by manipulating the vapor boilup in column 2, VS(2). (4) The composition of E in the first reactor of reaction step 2, Z2,E, is controlled by manipulating the fresh feed flowrate of E, FoE. (5) The impurity of D in the F product stream B3 is controlled by manipulating the vapor buildup in column 3, VS(3). The ultimate gain and frequency for each composition loop was obtained using the linear model of the process. The linear model of the process has 16 + 6NT1 + 6NT2 + 6NT3 + 31 states. Three-minute deadtimes were assumed on each of the composition measurements. The Tyreus and Luyben settings (Tyreus and Luyben, 1992) were used to tune the loops. The reactor 1 and column 3 reflux drum level loops were tuned assuming a reset time of 30 min and a damping coefficient of 1. Perfect level control was assumed for the remaining level loops throughout the process. Using the capacity-based economic approach, alternative designs are contrasted by quantifying tradeoffs between steady-state economics and dynamic controllability. The controllability of each design is evaluated for production rate changes. The production rate changes are achieved by adjusting the reactor 1 effluent flowrate F1. The magnitude of the change in F1 corresponds to an increase in the production rate of F (B3) to 102 lb mol/h and differs from design to design. Naturally, the production rate of C also increases. The capacity for each design alternative is evaluated based on the plant’s ability to keep both product purities within a desired specification range. The same specification range is used for both purities. Design alternatives generated by the approximate heuristic design method and the nonlinear optimization method are contrasted below using the capacity-based economic approach. The results from the approach are presented and verified on nonlinear process models.

Table 5. Heuristic Design Comparison for Different Values of Reactor 1 Holdup VR1 specification range [mole fraction]

design H3 VR1 ) 750 annual profit

design H1 VR1 ) 1000 annual profit

design H2 VR1 ) 2000 annual profit

0.0090 0.0070 0.0050

1.7725 1.7725 1.7725

1.7704 1.7704 1.5269

1.7102 1.7102 1.4386

Table 6. Heuristic Design Comparison for Different Values of D1 specification range [mole fraction]

design H5 D1 ) 65 annual profit

design H1 D1 ) 80 annual profit

design H4 D1 ) 160 annual profit

0.0090 0.0070 0.0050

1.7622 1.7622 1.7622

1.7704 1.7704 1.5269

1.7167 1.7167 0.7804

After the results are presented, explanations for the controllability trends are provided. Bode plots of the closed-loop regulator transfer functions relating XD2(C) and XB3(F) to F1 were generated using a linear model of the process. The magnitude ratio and phase angle corresponding to the peak frequency for the XB3(F) to F1 transfer function is used to generate closed-loop linear time response. Alternatively, the peak frequency corresponding to the XD2(C) to F1 transfer function could be used. For this study, the peak XB3(F) conditions had more of an effect on the controllability of the process. This may be attributed to the fact that this product comes from the bottom of a column with liquid feed. As Cantrell and co-workers (Cantrell et al., 1995) have shown, bottoms composition is typically more difficult to control than distillate composition for columns with liquid feeds. The amplitude of the sinusoid corresponds to the necessary change in F1 to increase the production rate of F to 102 lb mol/h (B3). Design alternatives are compared on their ability to maximize the annual profit for a given product specification range. Designs that exhibit little variability in their product streams will be able to make more onspec product at tighter specification ranges. Heuristic Design Method. Effect of Design Parameters. Table 5 compares designs with different reactor 1 holdups VR1. For the heuristic designs considered, the controllability of the process improves slightly as VR1 decreases. Additionally, the design with VR1 ) 750 lb mols is the least expensive alternative of the three. Table 6 compares heuristic designs with different D1 recycle flowrates. The base case design (H1) maximizes the annual profit for a large product specification range (all product on-spec) because its total annual cost is the lowest. If less variability in the product purities is desired, investing in the more expensive alternative H5 may be warranted. As the D1 recycle flowrate is decreased, the controllability of the process improves. The effects of varying Drecycle on the overall economics and dynamic controllability are provided in Table 7. For

1732 Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997

Figure 3. Production rate change. Closed-loop nonlinear response for product purities to a 2% production rate increase.

Figure 4. Production rate change. Closed-loop nonlinear response for production rates to a 2% production rate increase. Table 7. Heuristic Design Comparison for Different Values of Drecycle specification range [mole fraction]

design H7 Drecycle ) 20 annual profit

design H1 Drecycle ) 40 annual profit

design H6 Drecycle ) 80 annual profit

0.0090 0.0070 0.0050

1.7779 1.7779 1.7779

1.7704 1.7704 1.5269

1.6541 0.9425 0.3386

the three designs considered, as the amount of D recycled back from the third distillation column decreases, the controllability of the process improves. Design H7 exhibits the least amount of variability in both product purities and is able to produce all on-spec product for a product specification range of 0.005 mole fraction. In addition, this alternative is the least expensive of the three alternatives. The results from the capacity-based economic approach were verified through nonlinear simulation. The reactor 1 effluent flowrate was stepped to achieve a 2% production rate increase in B3. Figures 3 and 4 confirm the controllability trends for varying VR1. The heuristic design that maximizes the annual profit for the tightest specification range considered is design H7. This design corresponds to a small value of Drecycle. Optimization Design Method. Effect of Design Parameters. In this section, designs generated by the approximate nonlinear design method will be compared

using the capacity-based economic approach. Table 8 compares optimization designs with fixed reactor 1 holdups to the optimal design O*. Upon fixing the value of VR1, the remainder of the process operating conditions were obtained by minimizing the total annual cost. The variability in the product purities decreases as the holdup in reactor 1 increases. The results from the optimization design method for varying reactor holdup are the opposite of those found using the heuristic design method. The design parameters generated by the approximate optimization design method correspond to the design that minimizes the total annual cost. This is accomplished by allowing all unspecified variables to vary. There is far less variation between alternative heuristic designs because only one design variable is being changed at a time. Table 9 compares the annual profit for designs with different values of the D1 recycle flowrate to the optimal design. The design with a large recycle flowrate is unable to make enough on-spec product to gain a profit even at the larger specification range. Therefore, as D1 is decreased, the controllability of the process improves. This agrees with the results obtained using the heuristic design procedure. Table 10 compares designs differing in D3 recycle flowrate to the optimal design. Varying D3 is analogous to varying Drecycle. Although there is a slight discrepancy between designs O6 and O*, the controllability appears to improve as D3 decreases. This result coincides with the trend found using the heuristic design method. Bode plots for both product purities are shown for the VR1 design alternatives in Figure 5 (VR1 ) 596 is the economic optimum design). The trends demonstrated by the capacity-based economic approach (using peak disturbance conditions) are prevalent over a very large frequency range. To verify the controllability trends, nonlinear simulations were performed for all the design cases. Figures 6 and 7 confirm that the controllability of the process improves as VR1 increases. For the approximate optimization design method, the design that maximizes the annual profit at the tightest product specification range (0.0050 mole fraction) is design O3 (large VR1). In comparing the annual profit results obtained for the two design methods, the annual profit for designs generated by the heuristic design method falls off less rapidly than that for designs generated by the optimization method. However, design O3 maximizes the annual profit for the 0.0050 mole fraction product specification. At that specification range, several designs are able to make all on-spec product (H3, H5, H7, and O3), but because design O3 has the lowest total annual cost of the four, it maximizes the annual profit. Steady-State Disturbance Sensitivity Analysis. Comparing the controllability trends found for the two design methods, there appears to be some conflict in the results. Dynamic controllability was found to improve as VR1 decreased for the heuristic design method. Conversely, controllability was found to worsen as VR1 decreased for the optimization design method. The results obtained for varying D1 and Drecycle were consistent for both design methods (controllability improves as D1 and Drecycle decrease). As demonstrated in Elliott and Luyben (1996), steadystate disturbance sensitivity analysis (SDA) may be used to explain controllability trends. SDA provides a

Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 1733 Table 8. Optimization Design Comparison for Different Values of Reactor 1 Holdup VR1 specification range [mole fraction]

design O1 VR1 ) 250 annual profit

design O* VR1 ) 596 annual profit

design O2 VR1 ) 1000 annual profit

design O3 VR1 ) 2000 annual profit

0.0090 0.0070 0.0050