Methodology for the Steady-State Operability Analysis of Plantwide

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PROCESS DESIGN AND CONTROL Methodology for the Steady-State Operability Analysis of Plantwide Systems Sivakumar Subramanian† and Christos Georgakis* Chemical Process Modeling and Control Research Center and Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015

Many plants presently incorporate mass and heat integration schemes because they can offer substantial material and energy savings. In designing such integrated processes, the designer runs the risk of designing plants with operational difficulties. It is prudent to carry out a detailed operability analysis before finalizing a process design. In this article, we present an extension to the operability framework of Vinson and Georgakis (in DYCOPS-5, 5th IFAC Symposium on Dynamics and Control of Process Systems, Pergamon Press, 1998, and J. Process Control 2000, 10, 185-194) to aid us in the analysis of plantwide systems. It is shown that the feasible operating region in the production-related variables, named here as achievable production output space (APOS), can be used to compare competing process designs. We also show that the same analysis tool can be used to discriminate among several control structures for a given plant design. The proposed methodology is demonstrated with a CSTR-stripper process and the well-known Tennessee Eastman process of Downs and Vogel (Comput. Chem. Eng. 1993, 17, 245-255). 1. Introduction Even though the effect of the process design on the control qualities of a plant has been recognized for several decades,1 substantial efforts to integrate process design and process control have only been initiated during the last two decades. Modern requirements of material and energy conservation and strict environmental regulations often lead to process designs with several material and/or energy recycles. It is also known that such designs pose difficult control problems. In such a scenario, it is prudent to evaluate the operability characteristics of every prospective design. To effectively integrate process design and process control, one needs a measure for quantifying the operability of a given design. Morari and Perkins2 pointed out the lack of such a measure of the tradeoff between process design and process control. Some of the earlier attempts to bridge this gap for a nonlinear plant include: the flexibility analysis,3,4 the backoff from optimum analysis,5,6 and the bifurcation analysis.7 The flexibility analysis defines the size of the largest normalized hypercube that fits in the uncertainty region without violating the process constraints as the measure. The backoff approach, on the other hand, prepares the designer for the worst disturbance by moving the optimum operating point a safe distance away from the constraints so that the plant would be operable even * Corresponding author. Present address: Department of Chemical and Biological Engineering, Tufts University, Science & Technology Center, Medford, MA 02155. Tel.: (617) 627-3900. Fax: (617) 627-3991. E-mail: Christos.Georgakis@ Tufts.edu. † Present address: Invensys India Private Limited, Chennai, India.

when the disturbances enter the plant. The bifurcationbased approach delineates the regions of multiplicity in the parameter space, and, thus, cautions the designer to stay away from them if possible. Vinson and Georgakis8,9 proposed a simple yet powerful approach to quantify the operability of processing systems. This methodology focuses attention to the constraint range of the input and output variables. It has been proven to be effective for both linear and nonlinear processes.10 Its extension to dynamic operability analysis is discussed by Uztu¨rk and Georgakis.11,12 This approach was utilized to study both the steady-state and dynamic operability characteristics of continuous stirred tank reactor (CSTR) systems.13 The original operability framework was extended by them to consider nonsquare problems, that is, systems with extra degree(s) of freedom. They classify process outputs into two broad categories: (1) set-point controlled, with outputs to be controlled at a desired value, and (2) setinterval controlled, with outputs to be controlled within a desired range. For instance, production rate and product quality may fall into the first category, whereas process variables, such as liquid level, pressure, and temperature in different units could be in the second category. In this work, we shall utilize these concepts as well. Plantwide systems with significant material and energy integration are common. Conventionally, large intermediate storage tanks were used to attenuate disturbances and avoid their propagation from one part of the plant to another. This practice is becoming obsolete, due to both the economic forces and, in some cases, the desire to avoid the storage of hazardous chemicals. Material and energy integration of process units and the lack of intermediate storage units com-

10.1021/ie0490076 CCC: $30.25 © 2005 American Chemical Society Published on Web 09/03/2005

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pound the control problem of such plantwide systems. The dynamics and control of integrated plantwide systems is an active area of research in the chemical engineering literature. Gilliland and co-workers14 studied a reactor-separator process. They found that the presence of material recycle increases the response time and sensitivity to disturbances. Verykios and Luyben15 studied a similar process and showed that the integrated processes can exhibit underdamped behavior. Denn and Lavie16 studied the processes with dead times in the recycle path. They showed that the plant transfer function may contain resonant peaks that are comparable in magnitude to the plant steady-state gain. Papadourakis and co-workers17 underscored the importance of studying integrated processes as a whole by showing that the interaction measure RGA (relative gain array) of process units in integrated processes can be quite different from that when the processes were isolated. In a series of papers, Luyben studied the effect of recycle on the process dynamics, starting from simple processes to complex ones.18-21 Morud and Skogestad22 discussed the impact of different types of process integrations with some basic flowsheets. Their analysis, drawing comparison to linear systems, provided some useful insights. Seferlis and Grievink23 presented an optimization-based approach to rank competing plant designs. They used an integral of weighted input and output functions as the disturbance varies along a predetermined trajectory as their performance index. More recently, Groenendijk and co-workers24 presented a systematic approach utilizing the time-domain simulations and frequency-domain analysis for the design of process flowsheets with an emphasis on controllability. The main message from these studies is that the operability characteristics of chemical plants as a whole are not obvious from the operability characteristics of the individual units. Solovyev and Lewin25,26 extend the disturbance cost (DC) measure of Lewin27 to nonlinear systems. Their search for the worst-case disturbance in assessing the DC bears some similarity to the bounding-box technique of Subramanian et al.13 The bounding-box technique evaluates the required ranges of input variables one-by-one; Solovyev and Lewin26 have formulated the same problem as a solution of a single optimization problem. A good body of literature exists on the challenging problem of plantwide control, and the interested reader is referred to refs 28-30. The objective of the present work is to extend the steady-state operability methodology of Vinson and Georgakis and Subramanian et al.8,9,13 and, thus, to be able to address large-scale plantwide problems. The main challenge in directly applying the original operability methodology to plantwide problems is the high dimensionality of any system of reasonable complexity. It is addressed here by limiting the focus to a few selected process outputs related to the production, while maintaining all the other critical variables within their constraints. For this purpose, we introduce an operating space called achievable production output space (APOS). Here the main focus is on steady-state operability. The complementary issues related to dynamic performance are not addressed here. The outline of the paper follows. In Section 2, APOS is discussed along with a computational technique for obtaining it in two dimensions. In Section 3, this APOS approach is applied to a reactor-stripper process with

Figure 1. Classification of variables into exogenous and endogenous.

recycle. This methodology’s effectiveness in studying the effects of (a) design variables and (b) the decentralized control structure on the operability are demonstrated. In Section 4, this methodology is used to characterize the operability of the well-known Tennessee Eastman (TE) process of Downs and Vogel.31 The effect of an important disturbance on the operability of this process is also studied. Further, we also analyze and compare the performance of a decentralized control structure. Section 5 concludes this article. 2. Achievable Production Output Space The input-space-based operability methodologies8,9,13 face an important limitation under one condition. The calculation of the desired input space (DIS) will not be possible if some of the operating points in the desired output space (DOS) or in the expected disturbance space (EDS) or their combinations are infeasible for the current design. No input manipulation within the present selection of inputs will be able to correct the infeasibility. If one encounters such a problem in the DIS calculation, the defined DOS and EDS should be reevaluated to make them as reasonable as possible without compromising the real demands. If the infeasibility persists, one has to explore alternate design options, declaring the current design inoperable. Clearly, an adaptation which could directly deal with infeasibility is preferred. It can be argued that an approach based on the output space could overcome this deficiency. In the output space, one might want to include a large set of variables, such as liquid levels, pressure, and temperature, and realize that the dimensionality of the corresponding space is indeed overwhelming in any realistic process with many units. Though it is essential to maintain many of these outputs within their acceptable intervals for reasons of safety, corrosion, optimality, etc., the focus of an initial operability study should be on a few critical process variables. If one lumps the entire chemical process plant as a single processing unit, process variables related to the feed and product streams of the process can be called exogenous (external) input and output variables, respectively (refer to Figure 1). These exogenous input and output variables are connected by the main process path.29 To keep the process within the allowed operating region, several internal manipulators at intermediate locations in the process, such as steam pressure/flow, cooling water, etc., are available; here, we call these as

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Figure 2. Schematic of achievable production output space (APOS).

endogenous (internal) inputs. The state of the plant is generally measured using several variables, such as pressure, temperature, and liquid level; these variables are grouped as endogenous output variables. Though these endogenous output variables have to be held in a range for satisfactory operation of a chemical plant using both the exogenous and endogenous inputs, the overall objective of the process is normally set in terms of the exogenous output variables which are related to the production rates and product qualities. Keeping this point in mind in the operability analysis of large scale chemical processes, we limit our focus to these exogenous output variables. The set-point-controlled outputs discussed by Subramanian et al.13 can be highly related to the exogenous outputs discussed here. On the basis of these exogenous output variables, we introduce an operating space called achievable production output space (APOS). It can be defined as the entire feasible operating region in the exogenous output variables related to production (rates and qualities) that is achievable at steady-state with the specified available input space (AIS) and without violating the process constraints. Clearly, APOS is a lower-dimensional subset of the previously known achievable output space (AOS). This is because we are limiting our attention to a few output variables while constraining some, if not all, of the endogenous output variables to be within their acceptable limits. The dimensionality of APOS, as defined here, will depend on the number of products a plant makes and the number of independent quality variables associated with each product. In general, the dimensionality of the APOS will be much smaller than the total number of process output variables. A schematic of a two-dimensional APOS is shown in Figure 2. APOS is typically bounded by an upper and lower limit on the production rate, and possibly on the product quality; the latter constraint is not shown in the figure. Because higher-dimensional spaces are difficult to graphically illustrate, we limit our analysis in this work to two-dimensional APOS. This would imply that we consider plants with one product stream, characterized by one quality variable. The case of higher-dimensional APOS may require several 2-D projections for its accurate depiction and full examination. Concepts similar to APOS can be found in the literature. Wu and Yu32 used the possible range of production rates to discriminate different control structures. It was evaluated only at the nominal value of the product quality variable. They also did not treat the process constraints explicitly. The feasibility region discussed by Swaney and Grossmann3 has certain similarities and substantial differences with the APOS. The main difference is in the selection of coordinates. Their feasibility region is constructed in the uncertain

parameter space, but APOS focuses only on the exogenous outputs. Moreover, they limit their attention to calculating the size of the largest uncertainty box that would fit inside the feasibility region, which is their recommended measure of flexibility in a given design. Here, in addition to the operability index, we are interested in the entire operability region itself. Let us now turn to formulating an algorithm for the calculation of a two-dimensional APOS. For a generic nonsquare system discussed by Subramanian et al.,13 the calculation of a point on the boundary of the APOS can be formulated as the following optimization problem:

P1: Maximize production of a given product such that f(x,u,d) ) 0 h1(x,y,u,d) ) 0

(1)

h2(x,y,u,d) e 0 (quality and d given) Here, x, u, y, and d represent the state, input, output, and disturbance vectors, respectively. The nonlinear function f represents the steady-state model equations, while the functions h1 and h2 represent the equality and inequality constraints on the process, respectively. By solving P1 for the different values of the quality variable in the range of interest, one can establish the upper bound of the APOS. Similarly, the lower bound of the APOS can be found by repeating the calculations with the minimization operator in the place of the maximization in P1. In each boundary point of the APOS, the identification of the constraints that become active helps to establish the design limitations of a given plant. This will become clear in the examination of the specific example processes. Note that the above formulation could also be used to obtain two-dimensional projections/slices of a higherdimensional APOS. The key assumption for the algorithm to work is that the APOS is 1-D convex3 in the fixed-variable direction, which is product quality. Furthermore, if one is interested in obtaining a quantitative operability index, an equation similar to those defined by Vinson and Georgakis8,9 can be used. Here, the DOS in the production-related output variables is compared to the APOS to obtain a quantitative operability index as

OI )

µ[APOS∩DOS] µ[DOS]

(2)

where µ is a measure function calculating the size of the corresponding space. For example, in two dimensions, it represents area, and in three dimensions, it represents volume. Another important question that would be pertinent to ask here is how sensitive the APOS is to changes in the disturbance variables and the uncertain parameters. In other words, for disturbances described in the expected disturbance space (EDS), how much of the previously calculated APOS under the no-disturbance assumption will still be realizable. A conservative estimate can be obtained by calculating the APOS, which will be achievable when any of the disturbances described in the EDS enter the process. Drawing parallel to the overall DIS of the operability methodology,


which was defined as the union of different servo or regulatory DIS, this conservative APOS notated by APOS can be defined as the intersection of the APOS obtained for all the disturbances in the EDS. Mathematically stated,

APOS )

∩ APOS(d) d∈EDS

(3)

In the case of two-dimensional APOS, for a given product quality, the upper bound of APOS can be obtained by solving a nested min-max problem, where the min operator searches over the disturbance space and the max operator maximizes the production as in P1. Similarly, the lower bound of this APOS can be found by solving a set of max-min problems, where the max operator looks for the worst disturbance case and the min operator minimizes the production. A less conservative approach might be possible if the disturbances/uncertain parameters are expressed stochastically; then robust optimization techniques33 can be used to establish a probabilistic maximum or minimum bound on production. 2.1. Effect of Control Structures. In addition to characterizing different plant designs, the APOS methodology can also be useful in assessing the steady-state performance of different control structures (CSs). Though most of the processes are inherently multiple-input multiple-output (MIMO) systems, it is well-known that many chemical process plants are controlled using a set of single-input single-output (SISO) controllers. Often one could come up with many different CSs for the same process. Thus, a tool to characterize the steady-state limitations of these CSs would be useful in screening them. The main steps in the design of a CS with SISO controllers are as follows: (1) selecting the set of input/ maipulated variables and the set of output variables to be controlled, (2) pairing the input and output variables, (3) choosing set points for these controllers, and (4) tuning the controllers. In some cases, it might be beneficial to cascade some loops. Designing a set of SISO controllers to regulate a MIMO chemical process has been a subject of many articles and books.28-30 Different choices of CSs may limit the APOS calculated using P1 to varying degrees. This could be computed by including the control structure information in the model equations. In many of the problems, it just amounts to letting only the manipulated variables of the SISO controllers be the decision variables and including the necessary set-point constraints. An implicit and important assumption in this analysis is that the SISO controller gains do not change signs over the entire operating range. Stated differently, the process does not exhibit input multiplicity behavior. It could be a restrictive assumption, depending on the nonlinearity of the process. We also assume that each controller is an offset-free SISO controller, which need not be the case. For example, controllers which give offsets, such as P-only controllers, can be explicitly included in the model equations. It might be worth distinguishing the present approach to the steady-state disturbance sensitivity analysis of Yi and Luyben.34 They used the changes in the manipulated variables that would be required to recover from persistent disturbances as a measure to screen different control structures. Obviously, control structures which have a lower sensitivity to disturbances are preferred. Their approach is “local”

in nature because they perform sensitivity analysis locally; in contrast, the APOS approach is “global” in perspective. It should be borne in mind that this approach, which is quite effective in distinguishing CSs that differ in their list of input or output variables, would not distinguish them if the CSs differ only in the pairing of these variables. In the same way, tuning of the controllers would not be an issue in our steady-state analysis. These factors could influence the dynamic operability of a process quite significantly. The effect of changing the set points to the SISO controllers can be studied in this method. It should also be kept in mind that the existence of a steady-state predicted value using these calculations does not guaranty its reachability in the dynamic sense. 2.2. Illustrative Example. Let us illustrate the APOS methodology with a simple problem. A jacketcooled CSTR system with a liquid-phase, first-order exothermic reaction of type A f B is considered.10,35 We further make a simplifying assumption that the coolant flow rate is not restricted. This allows us to specify the desired reactor temperature and to assume that there exists a coolant flow rate to achieve this temperature. With the assumptions of complete mixing and constant physical properties, the steady-state molar balance for component A can be written as

F)

k(TR)VRx k(TR)VR ) x0 - x x0 -1 x

(4)

Here, F is the volumetric feed rate, which is also product flow rate; k(TR) is the reaction rate constant as a function of temperature (Arrhenius dependency); VR is the reactor holdup; x is the molar concentration of component A in the reactor; and x0 is the feed concentration. At the nominal operating point, the reactor temperature TR is 140 °F, the reactor volume VN R is 3800 ft3, and the reactor concentration is x ) 0.05 lbmol/ft3. We desire to operate the rector in the quality interval, defined in terms of the concentration of component A in the reactor as follows:

Q ) {x | x∈[0.01, 0.1] lbmol/ft3}

(5)

Further, the reactor temperature and reactor volume holdup are constrained as follows:

132 °F e TR e 158 °F

(6)

N 0.3VN R e VR e VR

(7)

Now we want to calculate the APOS for this system. We intuitively know that the reactor will reach its maximum production capacity when it is operating at full volume (maximum residence time) at the maximum allowed reactor temperature (highest kinetic rate). Conversely, it will reach the minimum production levels when the reactor volume and the reactor temperature are at their lower limits. These bounds of this APOS are calculated for the conversion range of interest and are shown in Figure 3 as a continuous line (marked as CS0). Note that both axes are in the logarithmic scale. This performance is the best any control structure could deliver at steady-state, because it fully exploits the flexibility available in changing both the reactor volume

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Figure 3. Achievable production output space (APOS) of a single CSTR with a first-order, irreversible exothermic reaction of type A f B (labeled as CS0). The APOS of two simple SISO control structures CS1 (fixed reactor volume) and CS2 (fixed reactor temperature) are compared.

and the reactor temperature. It will require a constrained MIMO controller to realize the full range of the APOS. On the other hand, if one decides to use a simpler decentralized SISO control structure, this choice will limit the size of the APOS. To illustrate the point, we consider two control structures, denoted as CS1 and CS2. Both the control structures treat the feed flow and, thus, production changes as disturbances. In CS1, the reactor volume is controlled at its nominal value of VN R by manipulating the reactor product flow rate, easily achieving the desired production changes when the feed flow changes. The other SISO loop controls the reactor composition by changing the reactor temperature, which is in turn achieved by manipulating the jacket flow rate. It is easy to see that the system controlled by CS1 would reach its maximum and minimum production capacities when the reactor temperature reaches the higher and lower limits, respectively. The lower limit of the APOS for this case (CS1) is shown in Figure 3 as a dashed line, while the upper limit of APOS for CS1 coincides with that for CS0. This is due to the fact that the APOS for CS1 was obtained for a nominal reacting volume, VN R , which is also the maximum allowed value. If the calculations were repeated after fixing the reactor volume at a lower value than this maximum, both the curves characterizing the maximum and minimum bounds of the APOS would shift to the left. In either case, the APOS for CS1 is a subset of the APOS for CS0. In CS2, the reactor temperature is controlled at its nominal set point of TR ) 140 °F, while the desired reactor volume is manipulated in order to control the reactor concentration at each desired level. The reactor volume is controlled, in a cascade, by manipulating the product flow rate. As before, we can deduce that CS2 will reach its limits when the reactor volume hits the lower and upper bounds. The APOS calculated for this case is shown as a dash-dotted line in Figure 3. It falls inside of both the higher and lower bounds of CS0. This is due to the fact that the selected reactor temperature, 140 °F, is not the maximum allowed. If we were to repeat this calculation for a reactor temperature of 158 °F, then the upper (right) boundary of APOS will coincide with that of CS0. One can see that different

Figure 4. Flowsheet and nomenclature of CSTR-separator processes: (a) CSTR-stripper process and (b) CSTR-column process.

control structures limit the maximal APOS to varying extents. This approach clearly quantifies the operability losses that would occur if one chooses to limit the control structures to the SISO one. One would be able to design some additional overrides or other logic on top of these control structures to fully realize the APOS characterizing CS0 for this process. The purpose of this simple example is to illustrate the idea without the need for significant computation. In the following two sections, we will demonstrate the APOS methodology with a more realistic CSTR-stripper process and the TE process. A quantitative measure of operability can be obtained by using the intersection calculations as prescribed in eq 2. Here, we assume a DOS as shown in Figure 3. Its intersection with the full APOS is shown as a shaded area; the corresponding operability index is 0.84. Control structures CS1 and CS2 yield operability indices of 0.73 and 0.39, respectively. More details on the intersection calculations are available in Subramanian and Georgakis.10 Intersection calculations were not performed for the other systems studied in the article, mainly to avoid overcrowding of figures that already compare several designs/control structures at once. 3. CSTR-Separator Processes The process consists of a CSTR and a separation unit. A first-order, liquid-phase exothermic reaction of type A f B takes place in the CSTR. The fresh feed, rich in component A, enters the reactor. The reactor output is fed to the separation unit. The product, rich in B, is withdrawn from the bottom of the separator, and the distillate is recycled back to the reactor. For the separation unit, we consider two choices, namely, (1) a simple stripper column where the feed enters the top tray and the stripper operates without any reflux and (2) a full distillation column with reflux. The flowsheets and nomenclature of these two configurations are shown in Figure 4. Many researchers have studied different aspects of similar flowsheets.17,32,36 The CSTR-stripper system parameters taken from Jaisathaporn37 are listed in Table 1. In modeling the system, we assume that the reactor is completely mixed and the physical properties remain constant. The columns are modeled with the following assumptions: constant relative volatility, equimolal overflow, theoretical trays, and molar heat of vaporization independent

Ind. Eng. Chem. Res., Vol. 44, No. 20, 2005 7775 Table 1. System Parameters for CSTR-Separator Processes37 parameter

notation

value

feed flow feed composition feed temperature reactor temperature distillate temperature jacket feed temperature Arrhenius Factor activation energy relative volatility heat of reaction fluid specific heat capacity coolant specific heat capacity fluid density coolant density molecular weight of A heat transfer coefficient

F0 z0 T0 TR TD TJ0 k0 EA R ∆H Cp CpJ F FJ MWA U

108.8 0.9 294.26 333.15 333.15 294.26 2.789 × 1010 6.978 × 107 2 6.978 × 107 2.0934 × 103 4.186 × 103 800.92 997.95 50 6.1325 × 106

units kmol/h m.f.A °K °K °K °K 1/h J/kmol J/kmol J/(kg K) J/(kg K) kg/m3 kg/m3 kg/kmol J/(m2 K)

of the binary composition. Other simplifying assumptions we made are that the separator feed is a saturated liquid and the recycle from the separator is returned at the nominal reactor temperature. The mathematical model of this system is adopted from Jaisathaporn37 and simplified for steady-state calculations. 3.1. Variable Selection and Specifications. The input variables which are available for manipulation are the following: the vapor flow rate in the separator (V), the volume holdup in the reactor (VR), and the reactor coolant flow rate (Fc). The desired reactor holdup is achieved by controlling the feed flow rates to the reactor or the product flow rate from the reactor. In this steadystate analysis, the reactor holdup is taken as an independent manipulated variable. A process with a 3 nominal reactor volume VN R ) 50 m and the number of trays in the stripper NT ) 15 is considered as the nominal design. At the nominal operating point, the system processes FN 0 ) 108.8 kmol/h of the feed, and the product flow obtained is of the same value with a composition of xB ) 0.05 mole fraction of A (m.f.A). Different operating spaces are defined around the nominal operating point as follows:

Q ) {xB | xB∈[0.01, 0.1] m.f.A}

(8)

N AIS ) {(V, VR, Fc) | V∈[FN 0 , 2.5F0 ] kmol/h; 3 3 N N VR∈[0.3VN R , VR ] m ; Fc∈[0, 4Fc ] m /h} (9)

The superscript N in the above equation stands for the corresponding nominal value. Further, the reactor temperature and the column feed are constrained as follows:

50 °C e TR e 70 °C

(10)

F g FN 0

(11)

The lower constraint on the column feed is set to be equal to the nominal feed rate so that one avoids drying up the column. For the CSTR-stripper process, this constraint is redundant because the lower limit of vapor flow is activated first. This is because the vapor flow, V, also has the same lower limit (refer to eq 9), but the column feed is the combination of vapor flow and the fresh feed; thus, the vapor flow reaches this constraint first. In the CSTR-column process with reflux, this lower limit constraint on the column feed became active when the production is minimized. The upper constraint

on this flow was not set directly, but it is indirectly limited by the vapor flow constraints. 3.2. APOS of CSTR-Stripper Process. The APOS for different designs is calculated by solving the optimization problem P1. We compare the APOS of the nominal plant design (VR ) 50 m3 and NT ) 15) with two other competing designs in Figure 5a. The first one uses a larger reactor to achieve a higher conversion with fewer trays in the stripper (VR ) 75 m3 and NT ) 10). The other design uses a smaller reactor but employs more trays in the stripper (VR ) 40 m3, NT ) 20). Their normal operating points are given in the first three rows of Table 2. To make the comparison more interesting, let us say that these designs have similar steady-state economics. A broad range of product purities is examined for better understanding the possible differences in the operability characteristics of the systems. Figure 5a clearly brings out the strengths and weaknesses of each of the three designs considered. The plant with the smallest reactor has a narrower range of feasible production rates for lower-purity products and a wider range for higher-purity products. The design with the largest reactor shows the opposite behavior. The nominal design falls between these two designs in both regions. It can be observed that the major differences in APOS occurs in the maximum possible production rates rather than in the minimum limits. Aided with the above operability characteristics of different designs, the designer would be able to make the critical decision as to which of these designs is desirable for a particular application. This application illustrates the utility of this method in ranking competing designs in a direct fashion from the steady-state operability perspective. If a quantitative operability measure is desired, then eq 2 can be used. The optimization results revealed that the processes reached their maximum capacity, when the reactor temperature, reactor volume, and vapor flow rate (recycle from the stripper) were at their maximum limit, which is in agreement with our intuition. Note that, in this analysis, we did not have an upper constraint on the liquid flow rate in the column. It was indirectly restricted by the vapor flow rate, which had an upper limit. It should also be pointed out that the constraints on vapor flow expressed in the AIS are common to all the designs considered. It can be justified by assuming that the column diameter in all the designs is the same. The minimum productions were achieved when the reactor temperature, reactor volume, and vapor flow rate reached their lower limits. The profiles of certain key variables, corresponding to the maximum production in APOS for different designs, are compared in Figure 6. The following overall component A balance around the process is helpful in understanding some of these profiles:

F0 )

kVRz (z0 - xB)

(12)

For a given reactor volume, reactor temperature, and product purity, any operation which increases the reactor composition of A and z would also increase the production, since the reaction rate is directly proportional to the reactant concentration. To examine the effect of the number of trays in the stripper, in Figure 5b we compare three designs with different number of trays in the stripper but with the

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Figure 5. Comparison of different reactor-stripper processes based on their achievable production output space: (a) three base-case designs, (b) effect of number of stages in the stripper, and (c) effect of reactor volume. Table 2. Nominal Operating Points for Different CSTR-Stripper Designsa VR (m3)

NT

rxr conc., z (m.f.A)

yNT (m.f.A)

recycle flow, V (kmol/h)

coolant, Fc (m3/h)

40 50 75 75 75 40

20 15 10 15 20 15

0.428 0.342 0.228 0.228 0.228 0.428

0.599 0.508 0.365 0.370 0.371 0.597

239.70 191.45 141.43 136.26 135.24 241.81

62.74 57.21 50.63 50.63 50.63 62.74

a

F0 ) 108.8 kmol/h and xB ) 0.05 m.f.A.

same reactor volume of 75 m3. It is obvious that increasing the number of trays improves the maximum possible production in the high-purity region, but the low-purity region is relatively unaffected. In other words, the high-purity production is limited more by the achievable separation. Figure 6 shows a consistent observation that flowsheets with a larger number of trays are able to maintain a higher concentration in the reactor (for a fixed reactor volume) and, thus, provide higher production in the high-purity region. In Figure 5c, we try to elucidate the effect of varying the reactor volume for designs with the same number of trays in the stripper. It can be seen that increasing the reactor volume improves the maximum possible production in the entire range considered, but this increase is more so in the low-purity product region. We can, thus, conclude from the three parts of Figure 5 that increasing the reactor volume will enhance process operability in the lower-purity region significantly, while increasing the number of trays in the stripper will improve the operability in the high-purity region. 3.3. Effect of Control Structure on CSTR-Stripper Process. The APOS presented in the previous section assumed the presence of a MIMO controller capable of handling constraints on process variables, along with the ability to optimally use the available extra degrees of freedom in the process. Extra degrees of freedom arise here because product purity, xB, is the

only set-point-controlled variable, but we have three input variables, namely, vapor flow rate V, reactor volume VR, and coolant flow rate Fc, available for control. The controller should also meet the constraints on the availability of the inputs, reactor temperature and column feed. As the APOS was calculated by solving the optimization problem P1, exploiting the extra degrees of freedom, these solutions represent the best possible control action from the steady-state point of view. In the following discussion, we refer to this optimal control scheme as CS0. As discussed in Section 2, many processes are controlled using a set of SISO controllers. Performance obtained under such control structures are generally more restrictive than the optimal controller discussed before. This loss could be negligible in certain cases but quite severe in some others. The impact of a particular CS on the steady-state performance of a process can be assessed by comparing its APOS obtained under the control scheme with the optimal controller (CS0). It is worth repeating that this steady-state approach would not be able to distinguish control structures differing only in the pairing of variables because they all would reach the same steady state. In this work, we shall compare two SISO control structures. The first one, called CS1 here, was referred to as conventional by Luyben36 and Wu and Yu.32 While both of these researchers considered a CSTR-distillation column process, it is adapted here for the CSTRstripper process following Jaisathaporn.37 In this structure, the reactor volume and reactor temperature are held at a constant value by manipulating the reactor product flow rate and jacket flow rate, respectively. The bottom composition is controlled at set point by the vapor boil-up. Changes in the fresh feed flow are treated as load disturbances to the process. Luyben and coworkers28,36 reported that this process with CS1 exhibited a higher sensitivity to load changes, such as feed rate and feed composition. They observed that slight


Figure 6. Comparison of key process variables for different CSTR-stripper designs corresponding to the maximum production conditions shown in Figure 5. Flow rates are in kmol/h, and concentrations are in m.f.A.

changes in the feed rate or the feed composition caused large changes in the reactor product flow rate. This was referred by them as the “snowball effect”. Luyben et al.28 provide a lucid discussion on this subject. This was attributed to the fact that this control structure does not let the reactor level or reactor temperature change, and thus, any change in the reaction rate can only be brought about by changing the concentration of component A in the reactor. Thus, the separation unit has to absorb a larger proportion of any disturbance leading to a change in the production rate. On the basis of this analysis, Luyben36 proposed that, in liquid recycle systems, the flow rate of a liquid stream somewhere in the recycle loop should be controlled to a fixed value to avoid such a snowball effect. The idea behind this heuristic is to maintain the load to the separation units at a reasonable level. For this particular system, he recommended fixing the reactor product flow rate at its nominal value. In this control structure, referred to here as CS2, the reactor product flow rate is held constant at the nominal value and the bottom composition in the stripper is, as before, controlled to the set point using the vapor flow rate. The desired production rate changes are obtained by changing the set point for the reactor level, which is in turn controlled by the fresh feed rate. Wu and Yu32 presented an interesting analysis of CS2 and pointed out that this control configuration makes the reactor overwork. In fact, they showed that a snowball effect in the reactor volume occurs. Wu and Yu32 also presented an alternate control structure which they called the balanced control structure because it distributed the load to both the units evenly. We will analyze the characteristics of the control structure proposed by Wu and Yu after we discuss CS1 and CS2. To make the comparisons among these control schemes fair, we restricted the maximum allowable reactor volume in CS2 to be the nominal reactor volume, as in the AIS. For each control structure, two situations are analyzed separately: (1) the reactor temperature is fixed at the nominal operating point and (2) the reactor temperature is allowed to change as per the limits in eq 10. When the temperature in the reactor is allowed to change, an additional supervisory controller will be

required to change the set point of the reactor temperature controller in a cascade. This supervisory controller can be activated by appropriate override loops. In Figure 7, we compare the APOS obtained with CS0, CS1, and CS2 for the nominal CSTR-stripper design. The top half of this figure presents the cases with constant temperature, while the bottom half is obtained when the temperature is allowed to change in the range. Obviously, flexibility in temperature reflects very favorably on the APOS for all the control structures. Interestingly, CS1 is performing as well as the best controller when the production is maximized. As CS1 fixes the reactor volume, it does not allow as much flexibility when the production rate needs to be minimized. Control structure CS2 is limiting the APOS substantially, which is the price one pays for keeping the reactor product rate constant. In the minimum production region, CS2 does better than CS1 but not as well as CS0, because it lets the reactor volume change to the lower limit. Another interesting observation is that, for the fixed temperature case in the high-purity region, CS1 and CS2 are very close to each other. This seems to be due to the limiting effects of the constraints of holding the reactor volume constant (CS1) or the column feed constant (CS2). A modification of CS1 and CS2, following the balanced control structure of Wu and Yu32 and the tiered plantwide control framework of Price and Georgakis,29 can be suggested as follows. Instead of holding the reactor product flow constant as in CS2, it can be used as the production rate manipulator. The reactor level and the stripper bottom composition can be controlled using the fresh feed rate and the vapor flow, respectively. If the reactor level is held constant at the nominal value, this control structure will perform the same way as CS1 when the production is maximized or minimized. If the reactor level set point is changed using a supervisory controller, the operability performance of this control structure can be improved, even while minimizing the production. Such a scheme would fully realize the APOS of isothermal CS0. Similar results can also be obtained by adding a volume set-point loop to CS1. It will be interesting to compare the dynamic performance of

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Figure 7. Effect of control structure on the APOS of nominal CSTR-stripper process: (a) reactor temperature fixed at 60 °C and (b) reactor temperature allowed to change between 50 and 70 °C. In both parts, the maximum production lines of CS0 and CS1 overlap each other.

Figure 8. Profile of different process variables when the production is maximized under different control structures in the nominal CSTR-stripper process. For the legend, refer to Figure 7. Flow rates are in kmol/h and concentrations are in m.f.A.

these different control structures after characterizing their minimum-time optimal control response.12 Profiles of different process variables corresponding to the maximum production locus when different control structures are in effect is shown in Figure 8. As expected, CS2 is holding the stripper feed at its nominal value, while the other two structures let the feed increase with increasing product purity requirements. It is also clear that the vapor flow rate constraint is active for CS0 and CS1. These discussions demonstrate the utility of APOS-based methods to characterize the expected performance of a process under different decentralized control structures. 3.4. APOS of CSTR-Column Process. It is only natural to question how the above operability calcu-

lations would change if we had a full distillation column with reflux instead of the stripper. In fact, many of the articles referred to in the previous discussion have worked with processes with a full distillation column. One might want to question whether this CSTR-column process has a broader APOS than the CSTR-stripper process. In the following, we pursue this issue. For comparison, we take a CSTR-column process 3 with the same reactor volume (VN R ) 50 m ) and number of trays in the distillation column (NT ) 15) as the nominal CSTR-stripper design. It is assumed that the column feed enters the tray numbered 10 (counted from the bottom) and that a portion of the condensed vapor leaving the column is refluxed back. The nominal reflux ratio is taken to be 2. We impose the same


Figure 9. Comparison of the APOS of a system with a stripper and of a system with a column.

Figure 10. Profiles of key process variables for a system with a stripper and for a system with a column. Results corresponding to both the minimum and maximum limits are plotted. Flow rates are in kmol/h and concentrations are in m.f.A.

set of constraints as given in eqs 8 and 9. Additionally, we restrict the reflux ratio to be in the interval of 0.5-20. In Figure 9, the APOS of the process with the stripper and the process with a full column are presented. Different process variables obtained for these two processes are compared in Figure 10. Interestingly, the process with stripper has a broader APOS than the process with the full distillation column. Although we have better control over the purity of the recycled stream in the process with a full column, it seems that, for the examined simple reaction network, the purity of the recycle stream does not have a significant impact. In fact, the stripper sends more material back to the reactor, as is evident from Figure 10. From one point of view, this should reduce the residence time of the material per pass in the reactor. In fact, recycling more material to the reactor results in an increased concentration of A, thus leading to higher reaction rate and

greater production for a given product purity specification (refer to eq 12). It will be interesting to find out how the dynamics of these processes compare. The process with the column, having an extra manipulated variable (reflux flow), might be able to reject disturbances quicker and/or track set-point transitions faster. Moreover, one should be careful in extending this result to reaction schemes of different types. For example, systems involving consecutive reactions, such as A f B f C, with the intermediate product B being desirable, would have different characteristics. In this system, recycling the desired product back to the reactor would mean losing out on the selectivity; therefore, we anticipate that the process with a column might do better than a stripper process. It should be borne in mind that we arbitrarily decided to introduce the column feed in tray 10 (counting from the bottom). From the results presented here, it is clear that, for this system, one would benefit by

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Figure 11. Simplified TE process flowsheet. Table 3. Operating Modes of the TE Process mode

G/H mass ratio

production rate

1 2 3 4 5 6

50/50 10/90 90/10 50/50 10/90 90/10

7 038 kg h-1 G and 7 038 kg h-1 H 1 408 kg h-1 G and 12 669 kg h-1 H 10 000 kg h-1 G and 1 111 kg h-1 H maximum production rate maximum production rate maximum production rate

moving the feed tray toward the top tray and by reducing the reflux flow which would make the column operate like a stripper column. The optimal feed tray location for reaction schemes other than the one examined here may not be obvious. 4. Tennessee Eastman Process In this section, we characterize the steady-state operability of the well-known Tennessee Eastman (TE) challenge process31 using the APOS methodology. The TE process consists of five units, namely, a two-phase reactor, a partial condenser, a phase separator, a steamheated stripper, and a recycle compressor. A simplified flowsheet of the process is reproduced in Figure 11. The following gas-phase reactions catalyzed by a nonvolatile catalyst dissolved in the liquid phase are taking place:

A(g) + C(g) + D(g) f G(l), product 1 A(g) + C(g) + E(g) f H(l), product 2 A(g) + E(g) f F(l), byproduct 3D(g) f 2F(l), byproduct Products G and H both are of value and, at the base case, are made in equal mass proportion. The problem statement also included six different operating modes, as shown in Table 3. It can be seen that the plant operates over a wide range of operating conditions, especially in terms of the ratio of the products G to H. The product ratio is changed in response to the market demands. In this study, we investigate the G/H ratio starting from 0.1 to 10, encompassing the full range of interest. The process is open-loop unstable, mainly due to the recycle. Moreover, inventory of an inert component B entering in small amounts with the feed streams to the plant also has to be taken into consideration. Downs and Vogel31 had made FORTRAN subroutines available

Table 4. TE Process Constraints on Some Key Process Variables process variable

low limit

high limit

reactor pressure reactor level reactor temperature separator level stripper base level

none 50% none 30% 30%

2895 kPa 100% 150 °C 100% 100%

to simulate the plant, which was intentionally made obscure to discourage users from decoding the model equations. This package also checks for any violation of process constraints, in which case the simulation terminates with the shutdown flag. It is worth mentioning here that the TE subroutines were called with a time value of zero, one of the input arguments to the subroutine, so that no measurement noise was added. The time derivatives provided by the TE subroutine were constrained to zero in the APOS calculations, as we are dealing with the steady-state aspects of the problem. Many research groups in the process engineering community have successfully designed control schemes for this complex, large-scale industrial process. Some of them were simple decentralized control loops, with overrides to handle valve saturation, etc.38-40 Some others were model-based controllers.41-43 Later in this section, we will characterize the TE process with the decentralized control structure of Ricker.40 In analyzing the results discussed in the following sections, it might be worth recalling some of the properties of the TE process, both stated by Downs and Vogel31 and observed by other researchers. They are as follows: (1) The first reaction (producing G) and the byproduct-forming reactions have higher activation energies than that of the H producing reaction. (2) Reaction 1 (producing G) is faster than reaction 2 (producing H). (3) The reactions are approximately firstorder with respect to the reactant concentrations. (4) Components A, B, and C are effectively noncondensable. 4.1. APOS of TE Process. The APOS of the process was calculated by solving the problem P1 described in Section 2. The optimal points were obtained by maximizing the production of H (approximated by y17 × y41, where y17 is the volumetric product flow rate and y41 is the mole percent of H in the product), subject to the process constraints. The constrained outputs and their limits are shown in Table 4. When the reactor temperature was set free in the range given in the table, the


Figure 12. Comparison of APOS of the TE process with two different control structures, with and without the loss of feed A. The base case operating point, Mode 1 in Table 3, is plotted as a star. The results plotted as circle and square symbols are obtained from the closed-loop simulations with CS1. Legend: CS0 ) continuous; CS1 ) dashed; CS0 without makeup A ) dotted; and CS1 without makeup A ) dash-dot.

higher G/H operation employed a higher temperature. This favored the production of both G and F. Converting valuable reactants to the undesired byproduct F adversely affected the process economics. It was found that fixing the reactor temperature constant (∼124 °C) does not significantly affect the APOS, but this choice improved the steady-state economics. Therefore, in the results presented here, we fix the reactor temperature at 124 °C for all the cases. This value is in the temperature interval suggested by Ricker40 for optimal operation. The manipulated variables were also constrained to be within their limits of 0 and 100. In these optimization problems, the unreacted D and E in the product stream is also restricted to be

Methodology for the Steady-State Operability Analysis of Plantwide

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