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Lahiere, R. J. Mass Transfer and Hydraulic Characteristics of a Supercritical Fluid Sieve Tray Extractor. PbD. Dissertation, Department of Chemical Engineering, The University of Texas at Austin, 1986. Lahiere, R. J.; Fair, J. R. Mass Transfer Efficiencies of Column Contractors in Supercritical Extraction Service. Ind. Eng. Chem. Res. 1987,26, 2086-2092. Larson, K. A.; King, M. L. Evaluation of Supercritical Extraction in the Pharmaceutical Industry. Biotechnol. Prog. 1986,2, 73. Martin, C. L.;Seibert, A. F. Supercritical Fluid Extraction-Process Simulation and Design. Separations Research Program Publication No. E-87-2;The University of Texas at Austin, TX,1987. McHugh, M. A,; Krukonis, V. J. Supercritical Fluid Extraction: Principles and Practice; Butterworth Stoneham, MA, 1986. Moses, J. M.; de Fillippi, R. P. Critical Fluid Extraction of Organics from Water. Contract No. DE-ACOl-79CS40258,U.S. Department of Commerce, 1984. Moses, J. M.; Goklen, K. E.; de Fillippi, R. P. Pilot Plant Critical Fluid Extraction of Organics from Water. Presented at the 1982 AIChE Annual Meeting, Loe Angeles, CA, Nov. 1982;paper 127c. Panagiotopoulos, A. Z.;Reid, R. C. Multiphaae High Pressure Equilibria in Ternary Aqueous Systems. Fluid Phase Equilib. 1986,29,525-534. Paulaitis, M. E.; Krukonis, V. J.; Kurnik, R. T.; Reid, R. C. Supercritical Fluid Extraction. Rev. Chem. Eng. 1982, 1, 179. Paulaitis, M. E.; McHugh, M. A.; Chai, C. P. Solid Solubilities in Supercritical Fluids at Elevated Pressures. In Chemical Engineering at Supercritical Fluid Conditions; Paulaitis, M. E., Penninger, J. M. L., Gray, R. D., Davidson, P., E&.; Ann Arbor Science: Ann Arbor, MI, 1983;Chapter 6,pp 139-158. Peng, D.-Y.; Robinson, D. B. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976,15, 59. Rachford, H. H.,Jr.; Rice, J. D. Procedure for Use of Electronic
Digital Computers in Calculating Flash Vaporization Hydrocarbon Equilibrium. J. Pet. Technol. 1952,4 (10,Section l), 19. Rathkamp, D. F. Mass Transfer Efficiency of a Supercritical Fluid Extraction Column. M.S. Thesis, Department of Chemical Engineering, The University of Texas at Austin, 1986. Riggs, J. B.An Introduction to Numerical Methods for Chemical Engineers; Texas Tech University Press: Lubbock, TX, 1988. Rizvi, S. S.H.; Benado, A. L.; Zollweg, J. A.; Daniels, J. A. Supercritical Fluid Extraction: Fundamentel Principles and Modeling Methods. Food Technol. 1986,40 (6),55. Scott, R. L.; van Konynenburg, P. B. Static Properties of Solutions-Van der Waals and Related Models for Hydrocarbon Mixtures. Diecws. Faraday SOC.1970,49,87. Seibert, A. F.; Mooebeg, D. G.Performance of Spray, Sieve-tray and Packed Contactors for High Pressure Extraction. Sep. Sci. Technol. 1988,23,2049-2063. Seibert, A. F.; Mooaberg, D. G.; Bravo, J. L.; Johnston, K. P. Spray, Sieve-Tray, and Packed High-pressure Extraction ColumnaDesign and Analysis. In Proceedings of the International Symposium on Supercritical Fluids; Perrut, M., Eds.; Societe Francaise de Chemie: Nice, France, 1988; pp 561-570. Stahl,T. G.; Quirin, K. W.; Gerard, D. Dense Gases for Extraction and Refining; Springer-Verlag: Berlin, 1988. Tzouanae, V. K.; Luyben, W. L.; Georgakis, C.; Ungar, L. H. Expert Multivariable Control. 1. Structure and Design Methodology. Ind. Eng. Chem. Res. 1990,29, 382. Williams,D. F. Extraction with Supercritical Gaeee. Chem. Eng. Sci. 1981,36, 1769.
Received for review December 11, 1990 Revised manuscript received August 5, 1991 Accepted August 15, 1991
Nonlinear Plant-Wide Control: Application to a Supercritical Fluid Extraction Process Balshekar Ramchandran, James B. Riggs,* and Hubert
R. Heichelheim
Department of Chemical Engineering, Texas Tech University, Lubbock, Texas 79409-3121
A framework has been developed to control multiunit proceases using phenomenological models based on material and energy balances and phase equilibrium. The nonlinear control strategy permits the controller to “look” at the entire process a t one time while considering multivariable decoupling and process nonlinearities when selecting a control action. The models are only approximate and use adjustable parameters to make them true to the process behavior. The models account for the characteristic nonlinearities of the process, and hence can also be used for on-line unit optimization. Supercritical fluid extraction (SFE) is a highly coupled, multiunit process, and its interactive nature makes processing efficiency and product quality very sensitive to the process conditions. Approximate models for the key units of the SFE process were developed and two different nonlinear process model-based control strategies were tested for set-point tracking and disturbance rejection using a dynamic SFE process simulator.
Introduction Any modern chemical process industry is made up of many multiunit processes in an effort to increase energy efficiency and improve the yield and product quality. Multiunit processes are coupled and interactive, and their interactive nature presents an extraordinary challenge to the control structure. The interactions between the units may occur over a time frame larger than that for the response of any single unit, making them significant enough to be accounted for to obtain satisfadory control over the entire process. Conventional control is based on the assumption of linear behavior of the process. Moreover, specific control problems associated with multivariable control, like dead
*Towhom a l l correspondence should be sent.
time, multivariable interactions, nonstationary behavior, and unmeasurable process states, severely limit conventional control performance. Due to the limitations of conventional control, advanced control strategies have become an important part of the control structure for improving performance in such a class of systems. Model-based control forms one class of advanced control strategies. Model-based control strategies use mathematical models of the process to infer a control action. While there are many different formulations available in model-based control, the techniques are similar in the sense that they make use of models to predict the behavior of the process over some future time interval and base control calculations on the model predictions. Most of the models used for these predictions are linear approximations of the process or experimentally obtained step-response data.
0888-5885/92/2631-0290$03.00/00 1992 American Chemical Society
Ind. Eng. Chem. Res., Vol. 31, No. 1,1992 291 Real chemical processes generally are quite nonlinear and often show changing behavior over a period of time. Linear models (Garcia and Morari, 1982; Cutler et al., 1983), therefore, are not characteristicof chemical processes and, at best, are locally valid and require substantial upsets for statistically confident parametrization. They may not be valid for either economic optimization or control as the process states or operating conditions change. Nonlinear models are based on the process phenomena and are derived from conservativeand constitutive relationships, such as the fundamental mass and energy balances, thermodynamics, and kinetics. The control strategy that uses nonlinear phenomenological models is termed nonlinear process model-based control (nonlinear PMBC). The nonlinear process models that are used for control purposes are called approximate models. As a result, they may not always be completely true in their predictions. For effective control, the model must be true to the procesa and must, therefore, adapt to the changes in the process. Consequently, a few quantities are chosen to be the adjustable parameters, and they are adjusted using measured process data. The model-adaptation step is called parametrization and is very important, particularly when the process characteristics (e.g., catalyst activity, heat-exchanger fouling rate, or mass-transfer efficiencies) change with time. Typically, parameters of the approximate model are chosen such that process quantities about which the least information is available are included in a single adjustable variable. The approximate models have an additional advantage in that they can also be used for steady-state optimization, apart from serving as control models. One type of nonlinear PMBC method is the generic model control (GMC) technique (Lee and Sullivan, 19881, which was used for control and optimization of a binary distillation column (Cott et al., 1986; Cott and Sullivan, 1987). The method has been applied successfullyto many individually simulated processes, such as coal gasification (Pandit and Rhinehart, 19891, binary distillation with side-stream draw (Riggs, 1989, 1990), wastewater neutralization (Choi and Rhinehart, 1986; Williams et al., 1990), single-effectevaporation (Leeet aL,19891, and batch polymerization (Zheng, 1990),and to experimental systems, such as a nonideal binary distillation (Pandit et al., 1990). The control strategy has also been implemented on industrial equipment (Riggs et al., 1990) and is still operational and in use. To date, nonlinear PMBC has not been applied to any multiunit process. Literature in the area of multivariable control abounds with references to the application of a number of different strategies, especially applications to distillation columna (Tzownaa et al., 1990). But most techniques employ conventional linear multiloop singleinput4ngleoutput (SISO) control structures (Ding and Luyben, 1990). Cygnarowicz and Seider (1990,1991) performed control studies on a SFE process to recover 8-carotene using conventional proportional-integral (PI) controllers. They observed that the PI-controller performance for disturbance rejection was poor and suggest that advanced control strategies be applied to yield a more reliable performance. Other nonlinear model-based control techniques that have been investigated on various nonlinear systems include differential geometric approach (Isidori et al., 1981; Nijmeijer and Schumacher, 1986; Ha and Gilbert, 1986), globally linearizing control (GLC) (Kravaris and Chung, 1987), internal decoupling (Balchen et al., 19881, and unified differential geometric approach (Henson and Seborg, 1989,1990). Even though the model-based control
SolVcnI co2
coz
Msl;c-up
-.
A I
Solvent Condenser
L
t I
I
Expansion Valve Aqgueous
Feed
i-Ropsnol-wwr
c
I-
C
Extractor
Shipper
Trim Cooler
/
Reboier COZ - i-Ropanol
for final flash se+ulation
Figure 1. Process flow diagram of the SFE process.
techniques have different approaches, they are some modification of a general differential geometric framework (Brockett, 1976; Kokotovic, 1985; Kantor, 1987) which is based on input-output linearization and decoupling. While GMC, internal decoupling, and the unified differential geometric approach are applicable to models in which the control variables appear nonlinearly, the other techniques in comparison are restricted to linear models. Several multiunit processes are candidates for implementing PMBC strategy, such as a train of distillation columns, heat-integrated multiple-effect evaporators, or a synthesis reactor followed by a separation sequence. A supercritical fluid extraction (SFE) process was chosen as the example multiunit process for the following reasons: (i) The SFE process is an extremely nonlinear process characterized by a high degree of multivariable coupling and exhibits varying response times in different units of the process. (ii) SFE is a developing technology in the field of separations that has the potential for substantial energy savings and product quality enhancement, if properly utilized. (iii) The U.S.Department of Energy (DOE) has identified that one of the reasons for industry’s reluctance in applying SFE on a commercial basis is the lack of demonstrated control for maintaining product quality (Energy Conservation, 1989).
The SFE Process and Process Simulator Figure 1shows the schematic of the process flows in a typical SFE process. The process consists of an extractor in which an aqueous feed containing a solute (isopropyl alcohol, P A ) is contacted with a supercritical fluid solvent (carbon dioxide, COz). Supercritical C02extracts the solute. The high-pressure extract is flashed to a lower pressure and the solvent is recovered by distillation. The recovered solvent is compressed and reused in the extractor. The heat required for the distillation is provided by the recompressed solvent. A dynamic process simulator
292 Ind. Eng. Chem. Res., Vol. 31, No. 1,1992
for the SFE process has been developed (Ramchandran et al., 1990) by simulating individually the key units of the SFE process. The individual simulator units were finally tied together to form the dynamic process simulator. The dynamic process simulator accounts for instrument or measurement noise, process drifts, analyzer dead times, and transport delays in order to enhance the realism of the simulated results. The SFE process simulator was compared with operating data obtained from a pilobscale SFE process at the Separations Research Program (SRP) at The University of Texas at Austin and was found to be in agreement with the pilot-scale SFE process. The dynamic simulator was used to test the two PMBC control configurations and compare the performances of the controllers. Approximate Models The nonlinear PMBC strategy uses nonlinear approximate models directly to calculate the control action. Even though the models are only approximate, they must account for the dominant phenomenological process characteristics. The SFE process simulator consists of five major units: the extractor, the flash process, the stripper, the reboiler, and the trim cooler. An approximate model is required for each of these units to implement the nonlinear PMBC strategy. Approximate Model for Extraction. The approximate model for the extractor was based on the development by Smith and Brinkley (1960a). They proposed a general short-cut steady-state design equation for equilibrium stage processes and presented a generalized solution which is applicable to all equilibrium stage processes containing any number of stages and handling any number of components. Under the assumptions of (i) constant partition coefficients and (ii) constant flow rates of the liquid and the fluid phases within the column, a simple analytical solution of the difference equations (Smith and Brinkley, 1960b) can be obtained. The generalized Smith-Brinkley model was adapted for the extraction process and resulted in the following equation: (xs
- X F ) ( S n - 1)+ xs(1 - SnN+') -xR = 0 1 - SnN+l
(1)
xFis the composition of iPA in the feed entering the extractor, and x s = y s / K , where ys is the composition of iPA in the fresh solvent entering the extractor and K is the average value of the partition coefficient. xR is the composition of iPA in the raffinate leaving the extractor. S , is called the stripping factor and is expressed as S, = K E / R = K G / F (2) where F = R and G = E are the flow rates of the liquid and the fluid phases, respectively. Equation 1 is a nonlinear equation that can be solved for s, to determine the solvent flow rate, G. The total number of theoretical stages, N, is the model parameter which is adjusted when the process is identified to be operating at steady-state conditions. Equation 1can be solved analytically to determine the total number of theoretical stages, N, and is given as
rate, F,are taken into account in the approximate model ( F appears indirectly in the stripping factor). Approximate Model for the Flash Process. Even though the composition of the feed to the stripper is expressed as an overall composition in mole fraction of Cot, the approximate model for distillation requires knowledge of the vapor-liquid split in the feed. The Peng-Robinson equation of state (PR-EOS) (Peng and Robinson, 1976) is used to solve an adiabatic flash (Ramchandran et al., 1990)and predict the vapor fraction of the feed, q, after the flash. Approximate Model for Distillation. The approximate model for distillation is a tray-to-tray formulation based on the material balance on each stage and a single adjustable parameter. The following assumptions are made in the development of the model: (i) equimolal overflow and (ii) negligible heat effects. The phase equilibria for a binary mixture can be expressed in terms of the relative volatility of the two components. The dynamic process simulation for distillation (Ramchandran et al., 1990) uses a Murphree stage efficiency to define the approach to equilibrium on each stage. The relative volatility and the Murphree stage efficiency depend on the process states and the flow rates of the vapor and liquid phases (Van Winkle, 1967). They affect the overall performance of the proceas, and therefore, fairly accurate knowledge of the relative volatility and stage efficiency is required to obtain good predictions from an approximate model. We have combined the relative volatility and the Murphree stage efficiency into a single adjustable parameter, a, which we call "effective Separability", defined by
- Y- - a- X l-y l-x
y and x are the compositions of the vapor and the liquid
phases leaving each stage, not necessarily in equilibrium. The approximate model is called the effective separability model for distillation. The effective separability accounts for the thermodynamic equilibrium, as well as the stage efficiency. The effective separability defines a "pseudoequilibrium" which specifies the composition of the vapor and the liquid phases leaving any stage under the actual operating conditions. Figure 2 is a tray-to-tray schematic of the distillation process of N stages. Stage 1 is the reboiler, and the overhead condenser is a total condenser. Under steadystate operating conditions, F and z are the flow rate and the composition of the feed to the stripper, respectively, and q is the vapor fraction in the feed. L and L' are the liquid flow rates above and below the feed tray, and V and V'are the vapor flow rates above and below the feed tray, respectively. B and xB are the flow rate and the composition of the bottom product, respectively. D and xD are the flow rate and the Composition of the overhead product, respectively. The feed stage is denoted by NF. The component balance on any loop i is given as
L'Xi+l if i
BXB+ V'yj
It should be noted that the two major disturbances that
(5)
< NF and L x ; ++~ FzF = B x B
can affect the process, feed composition, xF, and feed flow
(4)
+ Vyi
(6)
if i > NF. The tray-to-tray model can be used to determine the boil-up rate, V'. An iterative procedure uses the component balance and the definition of effective separability (eq 4) to predict the composition of the liquid and vapor phases leaving each stage. The difference between the
Ind. Eng. Chem. Res., Vol. 31, No. 1, 1992 293 Table I. Average Relative Percentage Error ( W E ) for the Approximate Models approximate model average RPE extractor model
