Quantitative Assessment of Controllability during the Design of a

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Ind. Eng. Chem. Res. 1996, 35, 3470-3479

Quantitative Assessment of Controllability during the Design of a Ternary System with Two Recycle Streams 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

In a previous paper (Elliott and Luyben, 1995), 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. In this paper, we consider a more complex recycle system consisting of a reactor and two distillation columns. The process was first studied in Tyreus and Luyben (1993). The method is used to compare alternative flow sheets, alternative choices of design parameters, and alternative plantwide control structures. Introduction The capacity-based economic approach described in this paper incorporates both steady-state economics and dynamic controllability into a simple methodology that allows conceptual plant designs to be compared quantitatively. This method is an assessment tool that may be used to screen alternative plant designs on their ability to minimize costs (capital and operating) and process variability. The method can be applied at the stage in the conceptual design where we want to compare alternative flow sheets and various choices of design parameters. It should be emphasized that this is not a synthesis tool but an analysis tool. 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. If desired, nonlinear simulation of expected disturbance sequences may be incorporated within the framework of this methodology. As we demonstrate in this paper, the results using the worst case sinusoidal disturbances were the same as those obtained when the disturbances were a sequence of typical load changes. Having specified limits for product quality variability, plant designs are screened on their ability to maximize 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 specified upper or lower specification limits. When product is off-spec, profits are being lost and reprocessing or disposal costs * To whom correspondence should be addressed. Phone: (610)758-4256. Fax: (610)758-5297. E-mail: [email protected].

S0888-5885(96)00111-X CCC: $12.00

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). Other authors have made significant contributions in assessing controllability during process design. Detailed descriptions of these efforts are described in Elliott (1996) and Morari and Perkins (1995). Capacity-Based Economic Approach 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 flow sheets and process flow sheets that differ in several design parameters. The process flow sheet must be sufficiently defined in order to generate a process model necessary for this approach. Design alternatives are compared based on their ability to maximize annual profit for a given product quality specification range. A brief description of the method using linear models is described below. The method is described in more detail in Elliott and Luyben (1995). (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. (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. Process Studied In this paper, we demonstrate the applicability of the capacity-based economic approach to a more complex process flow sheet than that presented in our first paper (Elliott and Luyben, 1995). The process is described in Tyreus and Luyben (1993) and consists of a reactor and © 1996 American Chemical Society

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Figure 1. Ternary recycle system with light-out-first separation sequence and control scheme 1: A + B f C.

two full distillation columns (Figure 1). In addition to the flow sheet considered by Tyreus and Luyben, which uses the heavy-out-first separation sequence, we also study a flow sheet that uses the light-out-first separation sequence. Two reactants A and B are fed separately and react to form product C. The reactants are fed to a CSTR, which is operated at 150 °F with a specific reaction rate k of 1.0 h-1. An activation energy Eact of 15 000 Btu/ lb‚mol is used. Each fresh feed stream is binary in the reactants and consists of 80% of the primary component. The reactor is fed by the two fresh feed streams along with two recycle streams. The reactor effluent is assumed to be saturated liquid and contains a ternary mixture of A, B, and C because some A and B remain unreacted. This ternary mixture of constant relative volatility (RA ) 4.0, RB ) 1.0, RC ) 2.0) is fed to the first column. In the light-out-first configuration (Figure 1) component A, the lightest, is recycled from the top of the column back to the reactor. Component B, the heaviest, is recycled from the bottom of the second column back to the reactor. The final product stream is the distillate stream from the second column from which the controllability of the system will be evaluated. The alternative flow sheet with the heavy-out-first separation sequence is considered later in this paper. The steady-state design procedure for the light-out-first separation sequence is described in the Appendix. Results of Steady-State Design The range of the two design parameters (D1 and B2) selected for this study are given below:

20 lb‚mol/h < D1 < 100 lb‚mol/h

(1)

20 lb‚mol/h < B2 < 100 lb‚mol/h

(2)

D1 and B2 were varied by increments at 20 lb‚mol/h, giving a total of 25 different designs. The optimum steady-state design for the range of design parameters considered (total annual cost minimized) was determined to have a light recycle flow rate D1 of 40 lb‚mol/h and a heavy recycle flow rate B2 of 40 lb‚mol/h (Figure 2).

Figure 2. Total annual cost versus D1 and B2. Optimum steadystate design: D1 ) 40 lb‚mol/h and B2 ) 40 lb‚mol/h.

Application of Capacity-Based Economic Approach to Ternary Recycle System The capacity-based economic approach was applied to quantify tradeoffs between steady-state economic design and dynamic controllability for (1) different choices of design parameters (2) alternative conceptual designs (3) alternative control structures. Effect of Design Parameters. For this study, the controllability of the ternary recycle system will be evaluated for production rate changes. Production rate changes were achieved by changing the set points of both recycle streams equally (same magnitude). Lyman (1995) showed that the process has greater rangability in terms of production rate when both recycle flow rates are manipulated. Both recycle flow rates were varied sinusoidally at the magnitudes necessary to achieve a 2% decrease in production rate. These magnitudes will vary for each design. The magnitudes were obtained by solving the set of nonlinear equations describing the process with additional equations for the controllers. Larger production rate changes were examined; however, many designs

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Figure 3. Bode plots of closed-loop regulator transfer functions relating XD2(B)/(D1 + FoA) for different light recycle flow rates with B2 ) 60.

were unable to attain the new steady-states dynamically (controllers in automatic). This problem is discussed by Lyman (1995). The product stream for this process is the distillate of the second column (Figure 1). The impurities of components A and B are controlled in the D2 stream. The controllability of each design is evaluated based on its ability to keep the compositions of both impurities (XD2(A) and XD2(B)) within a desired specification range. The same specification range will be used for both impurities. Control scheme 1 from Tyreus and Luyben (1993) was implemented on all designs. This structure is depicted in Figure 1. The reactor temperature is controlled using the jacket cooling water flow rate, and the reactor level is maintained by manipulating the effluent flow rate. Both reflux flow rates in the columns are fixed. The vapor boilup in the first column is used to control the impurity of component A in the final product stream through a composition/composition cascade strategy. The impurity of component B in the product stream is controlled by manipulating the vapor boilup in the second column. The fresh-feed flow rate of A controls the level in the reflux drum in column 1, and the freshfeed flow rate of B is used to control the level in the base of column 2. Conventional proportional-integral controllers were implemented on all loops. The controllers were tuned using the Tyreus-Luyben settings (Tyreus and Luyben, 1992) with the exception of the reactor temperature loop, which was tuned with the Ziegler-Nichols settings. A nonlinear model of the process was linearized about the steady-state operating point using a finite difference approximation of the Jacobian. The state space model contained 13 + 3NT(1) + 3NT(2) states. The linear model of the process was used to obtain the ultimate gains and frequencies for each loop. Bode plots of the closed-loop regulator transfer functions relating the impurities in the product stream to the each flow controller set point were generated. Figure 3 compares the closed-loop frequency response of XD2(B) to (D1 + FoA) for designs with varying light recycle flow rates (D1 ) 20, 60, 100). The frequency responses for (D1 + FoA) and (B2 + FoB) are the same due to the symmetry of the design for this process. Each curve is adjusted by the magnitude required to achieve

Figure 4. Bode plots of closed-loop regulator transfer functions relating XD2(B)/(D1 + FoA) for different heavy recycle flow rates with D1 ) 60. Table 1. SDA for 2% Production Rate Decreasea D1/B2

20/60

60/60

100/60

VR SDA ∆ recycle

2527.3 2.4 3.23

1300.1 3.8 5.2

1096.8 4.4 5.8

a ∆ recycle is the necessary change in the recycle flow rate in lb‚mol/h. D1 ) 20, 60, and 100. B2 ) 60 in lb‚mol/h.

Table 2. SDA for 2% Production Rate Decreasea D1/B2

60/20

60/60

60/100

VR SDA ∆ recycle

2527.3 2.4 3.2

1300.1 3.8 5.2

1096.8 4.3 5.9

a ∆ recycle is the necessary change in the recycle flow rate in lb‚mol/h. B2 ) 20, 60, and 100. D1 ) 60 in lb‚mol/h.

the 2% production rate decrease. As the light recycle flow rate is decreased, the controllability of the process improves. This trend holds true over a fairly large frequency range. Figure 4 shows the effect of varying the heavy recycle flow rate (B2 ) 20, 60, 100) on product impurity B. Again, each curve was adjusted by the magnitude required to achieve a 2% production rate decrease. As the heavy recycle flow rate is decreased, the process exhibits less variability in the D2 product stream in the presence of production rate changes. Frequency response analysis for production rate changes achieved through simultaneous manipulation of both recycle flow rates demonstrates that the inherent controllability of the process can vary significantly with the choice of the steady-state recycle flows. Decreasing one or both recycle flow rates improves the controllability of the process. As the recycle flow rates are decreased, the required reactor holdup VR necessary to meet the desired production rate increases (Tables 1 and 2). The improvement in process controllability may in part be attributed to the additional dynamic filtering provided by the larger reactor. Steady-state disturbance sensitivity analysis (SDA) may be used to explain controllability trends. The basic idea is to calculate the distances that manipulated variables must move to achieve the new steady state when a load disturbance occurs. Generally, the smaller the required move, the more controllable the plant. Lyman (1995) applied SDA to a similar ternary recycle

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process and defined the distance the plant must move to the new steady state using the vapor boilups of the columns. The move was defined using a sum of squares approach

x(∆VS(1))2 + (∆VS(2))2 where ∆VS(1) and ∆VS(2) are the vapor boilup changes in columns 1 and 2, respectively. The results are given in Tables 1 and 2. For both light and heavy recycle flow rates, the value of SDA decreases as the recycle flow rate is decreased. Also, the change in recycle flow rate required to meet the 2% production rate change decreases as the recycle flow rate is decreased in both cases. This trend is probably attributable to the reactor holdup. Equation 9 in the appendix shows that, for a given production rate change, the necessary change in the product of reactor compositions decreases as the reactor holdup increases. The magnitude ratio and phase angle corresponding to the peak frequency for each load variable were used to generate closed-loop linear time responses to the two load disturbances occurring simultaneously. Results obtained from the linearized model of the process were verified through nonlinear simulation. Relay feedback tests were performed on each loop, and closed-loop sinewave tests were performed on each load variable. While the controllability trends are prevalent over most frequencies, the use of the peak disturbance conditions in this approach provides a worst-case prediction of the overall controllability for each design. Plant designs were then compared on their ability to maximize annual profit in the presence of their peak disturbances for a given specification range. The use of annual profit as a measure allows us to incorporate both steady-state economics and dynamic operability into our methodology. For this analysis, the peak disturbance conditions for XB2(A) were used. The peak disturbance conditions corresponding to XB2(B) or XB2(C) may also be used in this analysis. Quantitative assessments of the tradeoffs between steady-state economic design and dynamic controllability were made for production rate changes. For a given specification range, the linear responses to sinusoidal variation of both recycle flow rates were then scanned to determine the percent of product that was off-spec. The capacity of the plant was reduced by periods of off-spec production. The results were then prorated to encompass 1 yr of time, and the annual capacity of the plant was estimated. Several specification ranges were analyzed to determine if a process controllability trend existed for the two design parameters (D1 and B2) and if it could be quantified. As the specification range decreases, the amount of off-spec product increases (annual profit decreases) because the plant is unable to hold the product impurities within the desired limits. Less controllable systems will produce more off-spec product for a given specification range because their peak disturbance conditions produce more variability in the second column’s distillate stream. Figures 5-7 demonstrate the effects of varying both recycle flow rates on the economics and the controllability of the system. Clearly the capacity-based economic approach demonstrates a trend in controllability for the two design parameters. Alternative designs with larger recycle flow rates are less profitable at lower specification ranges because there is more variability in their product

Figure 5. Production rate change. Specification range of 0.02 mol fraction.

Figure 6. Production rate change. Specification range of 0.01 mol fraction.

Figure 7. Production rate change. Specification range of 0.005 mol fraction.

impurity compositions in the presence of a production rate change. For this study, the steady-state economic optimum design (D1 ) 40 lb‚mol/h and B2 ) 40 lb‚mol/ h) maximizes the annual profit for all three specification ranges considered. This demonstrates that in some

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Figure 8. Production rate change. Closed-loop nonlinear response for XD2(i) to a 2% production rate decrease for B2 ) 60.

Figure 10. Production rate change. Closed-loop nonlinear response for XD2(i) to a 2% production rate decrease for D1 ) 60.

Figure 9. Production rate change. Closed-loop nonlinear response for D2 to a 2% production rate decrease for B2 ) 60.

Figure 11. Production rate change. Closed-loop nonlinear response for D2 to a 2% production rate decrease for D1 ) 60.

cases the optimum steady-state economic design may provide very acceptable closed-loop dynamics, thus alleviating any need for overdesign. To verify the controllability trends found from the linear dynamic capacity-based economic approach, nonlinear simulations for a 2% production rate decrease were performed. The recycle flow rates were stepped in order to achieve a 2% production rate decrease. Figures 8 and 9 confirm the trends for different values of the light recycle flow rate, and Figures 10 and 11 confirm the trends for different values of the heavy recycle flow rate. It should be noted that the nonlinear production rate step changes affect each design less than their respective peak disturbances (worst possible case).

Comparison of Alternative Conceptual Designs. The capacity-based economic approach may also be used to screen alternative conceptual designs. For this process, another potential conceptual design may be generated by reversing the separation sequence and taking the heavy component B out first (Figure 12). Component B is recycled from the bottom of the first column, and component A is recycled from the top of the second column. The product stream now becomes the bottom of column 2. The design of the process is essentially the same, except we now specify B1 (heavy recycle) and D2 (light recycle). Also, the control structure changes slightly. The vapor boilup in the first column is used to control the impurity of component B in the final product stream through a composition/

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Figure 12. Ternary recycle system with heavy-out-first separation sequence and control scheme 1: A + B f C. Table 3. Annual Profit ($106/yr) Comparison for Alternative Conceptual Designs spec range (mole fraction)

light-out-first annual profit

heavy-out-first annual profit

0.010 0.008 0.005

3.2038 3.2038 3.2038

3.1728 1.5593