1218
Ind. Eng. Chem. Res. 1990,29, 1218-1226 Trommsdorff, E.; Kohle, H.; Lagally, P. Polymerization of methyl methacrylates. Makramol. Chem. 1948, 1, 169. Tulig, T. J.; Tirrell, M. On the onset of the Trommsdorff effect. Macromolecules 1982, 15, 459.
Sourour, S.; Kamal, M. R. Differential scanning calorimetry of epoxy cure: Isothermal cure kinetics. Thermochem. Acta 1976,14,41. Stauffer, D. Introduction to Percolation Theory; Taylor and Francis: London, 1985. Stauffer, D.; Coniglio, A.; Adam, M. Gelation and critical phenomena. Adu. Polym. Sci. 1982, 44, 103. Stockmayer, W. H. Theory of molecular size distribution and gel formation in branched polymers. J. Chem. Phys. 1944, 12, 125.
Received for review September 12, 1989 Revised manuscript received March 12, 1990 Accepted March 26, 1990
PROCESS ENGINEERING AND DESIGN Design of Resilient Controllable Chemical Processes: An Autothermal Reactor Case Study Richard W. Chylla, Jr.,t and Ali Char* Department of Chemical Engineering, Illinois Institute of Technology, Chicago, Illinois 60616
A technique for the analysis of statespace linear systems is applied to the problem of selection of resilient chemical process designs. Structural Dominance Analysis affords the evaluation of many process design and control configurations and assessment of the effects of potential manipulated variables and disturbances. After a brief presentation of the analysis method, a complex multibed tubular autothermal reactor system is examined. Resilient process configurations, ease of control, and effects of various inputs on reactor state variables and outputs are considered, and effective control configurations are selected, 1. Introduction An ever-changing world marketplace and stiff foreign competition have placed severe burdens on the chemical process engineer. Companies are more than ever being forced to squeeze profitability out of existing processing facilities. With the role of specialty chemicals and polymers coming of age, the problems have intensified. One manufacturing plant may be used to produce many different products, each requiring different process schedules and operating conditions. The control engineer must design control strategies that are flexible enough to regulate the process over a wide variety of conditions subject to both economic and physical constraints. One approach toward achieving resilient process operation is to couple the control system design with the chemical process design. Indeed, the most elegant process design is virtually worthless if the plant cannot be controlled about the nominal operating conditions with acceptable performance. Major contributions to control system performance and, hence, process operability often derive from perceptive and clever modifications of the process itself (Foss, 1973; MacGregor, 1985). The process design stage or process retrofit stage is the time to consider and resolve many important control issues along with the design decisions. Design and control decisions for a tubular packed-bed autothermal reactor system can illustrate the benefits of such an approach. Assuming that the plant capacity and chemical reaction characteristics dictate use of fixed-bed tubular reactors and energy management assessment Current address: Innochem Process Development, S. C. Johnson & Son, Inc., Racine, WI 53403. 0888-5885/90/2629-1218$02.50/0
dictates autothermal reactor operation, a major design decision that remains is the mechanism of heat exchange. In the tubular autothermal reactor shown in Figure 1, the exothermic heat of reaction is removed by using the cold feed gas as a heat-exchange medium, thus bringing the feed gas to reaction temperature, and the process becomes self-sustaining. This heat exchange may be done by internal heat exchange by using an annulus (Figure 1)and/or by a feed-effluent heat exchanger following an adiabatic reactor bed. When wall cooling is chosen, the coolant may be passed cocurrently or countercurrently. These reactor design decisions must be made when designing fixed-bed catalytic reactors for reactions that are highly exothermic such as ammonia synthesis (Baddour et al., 1965; Eschenbrenner and Wagner, 1972; Handman and Leblanc, 1982), methanol synthesis (Stephens, 1975), and others (Froment, 1980). The large variety of designs found in industry is due at least in part to the huge task of evaluating and optimizing all the possible reactor configurations. As will be presented later, these selections have a large influence on the stability of the reactor system and on the difficulty of controlling it. Once the reactor design is determined, the selection of controlled and manipulated variables remains. If on-line measurements of the controlled variables are not available, alternate measurements should be selected and the need for state estimation should be assessed. These are all design/control synthesis problems that must be addressed when designing an autothermal reactor system. Many of these control and design issues can be tackled simultaneously through a systematic analysis of detailed mathematical models of the process. Detailed process representations of large collections of algebraic and differential equations based on first principles avoid the issue 0 1990 American Chemical Society
Ind. Eng. Chem. Res., Vol. 29, No. 7,1990 1219
t
Feed-Effluent Heat Exchanger
I
I
rl
1
h
Figure 1. Autothermal tubular reactor with internal heat exchange and feed-effluent heat exchanger.
of prebiasing the solution of the optimization and control problem caused by the use of so-called simple models (Garcia and Prett, 1986). Any control system synthesis must begin with a thorough understanding of the steadystate and dynamic operation of the prwess to be regulated. Highly complex integrated processes create the greatest modeling challenge, and many times the inclination is to avoid the difficulty by overconservative designs and operation. It is precisely because of the complexity that the importance of modeling should be emphasized. The power and availability of low-cost digitial computing provides the means to tackle problems that could not be attacked as recently as a decade ago. This study focuses on the design of an autothermal reactor and its control system. A detailed reactor model that is based on fundamental equations, the reaction rate expression for CO oxidation with the catalyst used, transport relations obtained from the literature, and physical properties pertaining to the reactor system (Chylla, 1987; Adomaitis, 1988) constitutes the information available. A detailed model based on this information and use of sound numerical techniques resulted in a fairly accurate representation of the process. Model improvement through experiments resulted in the adjustment of one parameter by about 7 % . Obviously, a systematic analysis tool is required to thoroughly evaluate the many reactor design and control alternatives. Various generations of synthesis tools for addressing these issues have been discussed by Morari (1982), and the need for new techniques to design resilient controllable processes has been underlined. Linear system theory has provided various mathematical tools that can be used for the development of such techniques. Singular value decomposition (SVD) has been used for assessment of dynamic resilience and robustness of chemical plants (Arkun et al., 1984; Morari, 1982, 1983; Palazoglu et al.,
19851, for interaction analysis and controller pairing (Lau et al., 1985; Moore, 1986),and for evaluation of changeover control policies (Lau and Jensen, 1985). SVD based techniques have also been developed for integral equation models of distributed parameter systems (DPS) for identification and model based control of linear systems (Gay and Ray, 1986) and have been extended to nonlinear DPS by means of local linearization techniques (Gay and Ray, 1988). Another useful tool is the structural dominance analysis (SDA) based on the model reduction technique proposed by Litz (1980) and extended by Bonvin and Mellichamp (1982a). The SDA approach has been applied successfully for model reduction and sensor location selection for modal control (Wong et al., 1983). This study presents not only another successful application of the SDA approach for reduced-order model development and sensor positioning but, furthermore, it illustrates the use of SDA for the design of a resilient chemical reactor system. Both SVD and SDA results are affected by scaling of variables. Both techniques provide a means for assessing the difficulty of control for a specific system configuration, the sensitivity of a system at a given set of operating conditions, the sensor location, the selection of manipulated variables, and the controller configuration selection. SVD results for interaction analysis and controllability assessment (using the largest and smallest singular values along with their ratio, the condition number) are more straightforward, while SDA enables development of accurate reduced-order models and gives detailed information on which output is affected by which input through which mode. The systematic analysis of the reactor system in this study is conveniently done by using the generalized SDA (Bonvin and Mellichamp, 1982a). The choice to use linearized process models is a pragmatic one. Chemical processes generally behave in a nonlinear manner; however, few if any systematic analysis methods have been developed to handle nonlinear systems. Furthermore, Shinnar (1986) observed recently that, for a large class of nonlinear systems including many chemical reactors, the frequency properties of damped higher harmonics permit the use of linear control and analysis methods in highly nonlinear systems. In examining a linearized process representation, the acknowledgment is made that the conclusions formed me valid within a region about which the linearized model is valid. When there is a need to relinearize the process model, the SDA technique can be applied again. In fact, automation of the linearization and SDA implementation can be combined in a single program easily. SVD is used to assess further the manipulated variables and sensor locations selected for the reactor system. While both methods were used, this study is not intended to compare and rate the techniques available for process/controller design but to illustrate the use of these recent recent powerful techniques. Considering the extent of “art” in such design efforts, it is natural that many “favorite” techniques would coexist. The results of the case study will be reported in three parts. The first part (this paper) is devoted to the design of resilient controllable processes. The second part focuses on synthesis and experimental validation of model based controllers for the single-bed autothermal reactor (Chylla and Cinar, 1990). The third part will treat the multibed autothermal reactor control. The structure of this paper may be summarized as follows: Section 2 reviews previous applications of SDA and SVD with some discussion of the various outputs from these techniques. The reactor and associated control system design for the autothermal CO
1220 Ind. Eng. Chem. Res., Vol. 29, No. 7 , 1990
oxidation process are presented in section 3. Both model and experimental results for the reactor designs are presented in section 4. Discussion of results and conclusions follow in section 5. 2. Structural Dominance Analysis and Singular Value Decomposition The SDA technique proposed by Litz (1980) and extended by Bonvin and Mellichamp (1982a) is useful in the evaluation of large-scale systems and reduction of highorder models. Bonvin and Mellichamp (1982b) compared other earlier modal approaches to model reduction (such as Davison's original and modified methods, Marshall's technique, and Fossard's method) with the Litz method and indicated the strengths of the SDA approach with respect to these earlier techniques. The SDA approach has been generalized (Bonvin and Mellichamp, 1982a) to aid in determining not only the dominant modes, as in the method proposed by Litz, but also the most sensitive state variables and the most effective inputs of a linear system. The existence of notable dominant poles improves the effectiveness of modal techniques. The existence of such poles in the autothermal reactor models makes SDA a good choice for analysis and model reduction. Another model reduction technique, balanced realization (Moore, 1981), has been used successfully in recent years. Balanced realization was initially proposed for model reduction of open-loop stable systems only. Recently the method has been extended to unstable systems with small dimensions as well. Unfortunately, for the autothermal reactor system study, some of the intermediate matrices computed during the execution of balanced realization algorithms did not satisfy the requirements needed for continuing with the computations of the proposed algorithm (a matrix failed to be positive definite due to negative eigenvalues having a magnitude on the order of 10-l8),preventing a comparison of reduced-order models developed by two different techniques. During the last decade, SVD has become a versatile tool for analyzing large-scale linear systems (Klema and Laub, 1980; Moore, 1981). When used to assess a process and/or its control configuration, SVD provides two useful measures; the condition number, y (the ratio of maximum to minimum singular values, uM/ om), indicates the sensitivity of the system, and the magnitude of the minimum singular value, nm, discloses the magnitude of disturbances that can be handled without saturation of the manipulated variable (Morari, 1982). A good control system would have large om and small y. Furthermore, the right and left singular vectors of the process transfer function matrix are useful for controller pairing, selection of sensor location, and manipulated variables (Moore, 1986). Both SDA and SVD are sensitive to unit scaling; direct numerical comparison of values requires some normalization of results. Bonvin and Mellichamp (1987) have successfully addressed the scaling problem, resulting in scaling variables that are extensive; they indicate the amount of a chosen reference quantity that is present in each system state variable or brought into the system by each input variable. Their paper clearly outlines the scaling procedure, so it will not be discussed further. It should be noted that, in some cases, direct numerical comparison of values is not always desired. When a good "physical feel" of the various system variables is known, the authors have found that unity scale factors are preferred. Scaling based on both extensive variables (Bonvin and Mellichamp 1987) and unity scale factors has been used in this study. Analysis of Dominance Measures. SDA provides two useful measures that, characterize a state-space system.
The normalized coupling coefficient, q,,,, is a quantitative measure of the effectiveness of the ith input on the sth state variable acting through the mth mode. This can be used to evaluate the effectiveness of various inputs on the chemical process without detailed simulation studies. The generalized dominance measure, d,,,, indicates the dominance of the ith input on the sth state variable acting through the mth mode. The difference between the two dominance measures is that d,, is time weighted to account for the fact that slower modes tend to dominate the system dynamics. For each input, therefore, the important modes (eigenvalues) of the reactor system can be immediately identified by determining for which modes the d,,,'s are the largest, and when desired, the selection of a reduced order model of appropriate order is facilitated. By comparing the actual numerical value of corresponding ysmrfor different process configurations, decisions can be made about the effect of various inputs on the key state variables of the process. The speed at which a particular input affects a process can be gauged by which modes are excited by that input. The eigenvalues closest to the origin will be the slowest, and hence, inputs that effect state variables principally through these modes will take the most time to influence the system behavior. The net effect of a particular input on the system can be determined by examining the static gain of the input on that state variable; this is done by simply summing ,,y, over all modes: h' Qs,
=
qsmi m=l
(1)
If unity scale factors are used (i.e., w, = w, = l),the static gain represents the steady-state gain of J , to a unit step change in input u,. 3. Design of a Tubular Autothermal Reactor System and Its Control Configuration Depending on the exothermicity of a reaction, some chemical reactions may be carried out in adiabatic reactors while others necessitate removal of the heat generated. If the heat generated is very large, a multitubular reactor configuration is needed. Usually such reactors are modeled by assuming that all tubes behave identically (Froment, 19801, which implies that at any axial position the temperature and concentrations of all tubes are similar. If the heat generation is moderately high, energy exchange between reacting materials and cold feed through the reactor walls and/or heat exchangers become feasible. Ammonia and methanol syntheses are among such reactions of industrial importance. This study focuses on the design of a tubular autothermal reactor system and its control configuration for this class of reactions. The analysis in this study was performed on the model for a laboratorysized reactor since the various reactor design and control configurations were to be verified experimentally. Comparison of the results with published studies of industrial reactor behavior have shown that the conclusions are sound when extended to the industrial scale. The detailed reactor model of the packed-bed tubular autothermal CO oxidation system (Adomaitis,1988; Chylla, 1987) is given in the Appendix of Adomaitis and Char (1988). For various reactor configurations, the boundary conditions are modified appropriately. The analysis of various reactor configurations consisted of constructing the discretized models for these configurations and linearizing them about the steady state of interest. Structural dominance analysis was performed on each of these cases to construct the qsmrand d,, arrays, which were then analyzed for a variety of information. Selected elements for
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1221 Table V. Eigenvalue Dominance Studya
Table 1. Largest Coupling Elements qtmiof the Autothermal Reactor' state var T,
row
Tw TP Tc
x, Xb
reactant inlet concn inlet temp -5.0 X lo-' 2.7 X 10' -2.1 x 100 1.0 x 10' -1.0 x io-' 1.0x io-' 1.0x 10-1 -1.0 x 10-3 -1.0 x 10-5 1.0 x 10-3 4.6 X lo' -8.9 X lo2 2.1 x 10' -1.0 x 10-2
total flow rate -1.0 X -1.0 x 10-1 -1.0 x 10-3 -2.0 x 10-1 -1.0 x 10-2 1.3 X lo2 1.0 x 10-2
inlet coolant temp 3.9 X 10' 2.0 x 102 1.9 x 101 9.5 x 101 2.1 x 101 -7.1 X lo2 -4.3 x 10'
'Cocurrent coolant a t 395 K inlet temperature, 0.6% CO. Values are listed for the largest element in each state variable vector. Table 11. Dominance Analysis of the Autothermal Reactor' correcorresponding largest sponding largest d.,; state var a.m; state var inlet concn 5.0 X loo Xi0 1.5 X lo2 T:w T: 5.3 X lo5 T b reactant inlet temp 7.3 X lo3 total flow rate 1.7 X lo3 T; 1.7 X 10' T$ T: 9.3 X lo3 T:w inlet coolant temp 2.5 X
'Countercurrent coolant a t 407 K inlet temperature, 0.6% CO. 5": and X: denote the packed-bed temperature and the solid concentration at the ninth collocation point, respectively. Table 111. Static Gains for an Adiabatic Reactor inputs inlet concn reactant inlet temp total flow rate
T, Tg -4.4 X lo-* -1.8 X 2.5 1.9 X loo -9.1 1.8 X 10'
x, X X
loo
-4.8
X
lo4 -3.8
-4.2
X
'407 K inlet temperature, 0.6%
X
lo4
1.2
Xb
1.8 X loo lo-' -2.0 X lo-' X
CO.
Table IV. Dominance Analysis Using Unity Scale Factors state var mode largest lqsmil Qsi Ti0 59 1.048 0.896 TF 10 2.970 0.895 X :0 64 0.104 -0.700 X :0 59 0.729 -0.684 TkO 59 1.071 0.868 Ti0 29 31.0 0.854 T 63 0.436 0.839
a few cases are presented to illustrate key points in Tables I-V. The superscript on the state variables denotes the collocation point position in the reactor. The collocation points number from 1 to 12, reactor inlet to exit, respectively. The effect of four inputs on the reactor operation was examined: (1)inlet concentration, (2) reactant temperature at reactor inlet (reactant inlet temperature), (3) total flow rate, and (4) coolant inlet temperature. Adiabatic versus Wall-Cooled Reactor Beds. One must first choose between catalyst zones that are operated adiabatically or zones that exchange heat through the reactor walls. In a highly exothermic reaction, severe temperature gradients, which are detrimental to the selectivity and the catalyst itself, may develop in the reactor. In such cases, heat must be removed discretely or continuously. In discrete removal, several adiabatic beds are staged with intermediate cooling through either cold shot injection or feed-effluent heat exchange between the catalyst sections. This is in contrast to continuous heat removal by a coolant flowing outside the reactor walls. Both methods can be combined for a variety of reasons. For example, in order to use high-pressure material for outer reactor walls and high temperature material in inner reactor walls (with a
eigenvalue no. 1 2
3 4 5 6 7 8 9 10 11 12
13 14 15 16
eigenvalueb (-5.4243-4, 0.0) (-6.0173-3, 0.0) (-9.4353-2, 0.0) (-8.7193-2, 0.0) (-6.5643-2, 0.0) (-6.3823-2, 7.7873-3) (-6.3823-2, -7.7873-3) (-5.8113-2, 0.0) (-4.4663-2, 0.0) (-3.8843-2, 0.0) (-3.1663-2, 0.0) (-2.3453-2, 0.0) (-2.0913-2, 0.0) (-1.4003-2, 0.0) (-1.2283-2, 9.9753-5) (-1.2283-2, -9.9753-5)
max I L I 1.53-1 5.53-3 2.03-3 1.83-3 5.83-3 1.73-2 1.73-2 6.23-3 1.83-5 1.43-4 6.33-5 6.53-5 4.13-4 1.03-3 2.73-3 2.73-3
re1 dominance 1.0 0.037 0.013 0.012 0.039 0.113 0.113 0.041
