EUROPEAN SYMPOSIUM ON COMPUTER AIDED PROCESS E N G I N E E R I N G - 11
COMPUTER-AIDED CHEMICAL ENGINEERING Advisory Editor: R. Gani Volume Volume Volume Volume
1: 2: 3: 4:
Volume 5: Volume 6: Volume 7: Volume 8: Volume 9:
Distillation Design in Practice (L.M. Rose) The Art of Chemical Process Design (G. L. Wells and L.M. Rose) Computer Programming Examples for Chemical Engineers (G. Ross) Analysis and Synthesis of Chemical Process Systems (K. Hartmann and K. Kaplick) Studies in Computer-Aided Modelling. Design and Operation Part A: Unit Operations (1. Pallai and Z. Fony6, Editors) Part B: Systems (1. Pallai and G.E. Veress, Editors) Neural Networks for Chemical Engineers (A.B. Bulsari, Editor) Material and Energy Balancing in the Process Industries - From Microscopic Balances to Large Plants (V.V. Veverka and F. Madron) European Symposium on Computer Aided Process Engineering-10 (S. Pierucci, Editor) European Symposium on Computer Aided Process Engineering-11 (R. Gani and S.B. J~rgensen, Editors)
COMPUTER-AIDED CHEMICAL ENGINEERING, 9
EUROPEAN SYMPOSIUM ON COMPUTER AIDED PROCESS E N G I N E E R I N G - 11 34 * European Symposium of the Working Party on Computer Aided Process Engineering ESCAPE- 11, 27-30 May, 2001, Kolding, Denmark
Edited by
Rafiqul Gani Sten Bay Jorgensen CAPEC, Technical University of Denmark, Department of Chemical Engineering, Building 229, DK- 2800 KGS, Lyngby, Denmark
2001 Elsevier Amsterdam
- London
- New York-
Oxford
- Paris - Shannon
- Tokyo
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9 2001 Elsevier Science B.V. All rights reserved.
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Preface This book contains papers presented at the 11th European Symposium of Computer Aided Process Engineering (ESCAPE-11), held in Kolding, Denmark, from May 27-30, 2001. The ECSAPE series started in 1992 at Elsinore, Denmark, on a strong foundation of 23 events of the European Federation of Chemical Engineers (EFCE) Working Party on Computer Aided Process Engineering (CAPE). The first event on computer applications was organized by the CAPE working party in Tutzing, Germany in 1968. ESCAPE-11 is the 34 th event of the CAPE Working Party. The most recent symposia were organised in Florence, Italy 2000 (ESCAPE10), Budapest, Hungary 1999 (ESCAPE-9) and Brugge, Belgium 1998 (ESCAPE-8). The ESCAPE series serves as a forum for bringing together scientists, researchers, managers, engineers, and students from academia and industry, who are interested in CAPE. The scientific aim of the symposium is to present and review the latest developments in CAPE or Systems (Process/Product) Engineering. This research area bridges fundamental chemical and biological sciences with various aspects of process and product engineering. The objective of ESCAPE-11 is to highlight the use of computers and information technology tools, that is, the traditional CAPE topics as well as the new CAPE topics of current and future interest. The theme for ESCAPE-11 is Process and Tools Integration with emphasis on hybrid processing, cleaner and efficient technologies (process integration), computer aided systems for modelling, design, synthesis, control (tools integration) and industrial case studies (application of integrated strategies). The papers at ESCAPE-11 are arranged in terms of the following themes: computer aided modelling, computer aided design/synthesis, computer aided control/operations, computer aided manufacturing, process & tools integration, and, new frontiers in CAPE. A total of 188 papers, consisting of 5 keynote and 183 contributed papers are included in this book. All the papers have been reviewed and we thank the members of the international scientific committee for their evaluations, comments and recommendations. It was a very difficult task since we started with more than 450 submitted abstracts. The selection process involved review of abstracts, review of manuscripts and final selection of the revised manuscript. We hope that this book will serve as a valuable reference document to the scientific and industrial community and that it will contribute to the progress in computer aided process and product engineering. Rafiqul Gani Sten Bay JCrgensen
vi
International Scientific Committee
R. Gani (Denmark, Co-chairman) S. B. JCrgensen (Denmark, Co-chairman) J. Aittamaa D. Bogle B. Braunschweig D. Cameron M. Doherty Z. Fonyo U. Gren X. Joulia B. Kalitventzeff Z. Kravanja D. R. Lewin T. Malik Y. Naka
(Finland) (United Kingdom) (France) (Norway) (USA) (Hungary) (Sweden) (France) (Belgium) (Slovenia) (Israel) (United Kingdom) (Japan)
Y. Natori C. SCrlie S. Pierucci P. Pilavachi E. N. Pistikopoulos M. Pons L. Puigjaner H. Schmidt-Traub S. Skogestad E. L. S~rensen J. Van Schijndel J. Vinson
(Japan) (Norway) (Italy) (Belgium) (United Kingdom) (France) (Spain) (Germany) (Norway) (Denmark) (The Netherlands) (USA)
National Organizing Committee
R. Gani (CAPEC-DTU, Co-chairman) S. B. JCrgensen (CAPEC-DTU, Co-chairman) B. Christensen N.-J. Friis E. Hansen S. W. Jensen G. Jonsson L. B. J Crgensen C. Koch
(Kemira Agro Oy) (Cheminova Agro) (I~vens Kemiske Fabrik) (Novo Nordisk) (KT-DTU) (Danisco Sugar) (Statoil Raffinaderiet)
H. K. Nielsen M. B. Sommer E. L. SCrensen L. Wiebe M. R. Eden P. M. Harper
(Novo Nordisk) (H. Lundbeck) (Haldor TopsCe) (Danisco Ingredients) (CAPEC-DTU) (CAPEC-DTU)
Conference Secretariat
Computer Aided Process Engineering Center (CAPEC) Department of Chemical Engineering Building 229, Technical University of Denmark DK-2800 Kongens Lyngby, Denmark Phone: +45 4525 2800, Fax: +45 4588 2258 E-mail: capec @kt.dtu.dk URL: http://www.capec.kt.dtu.dk
vii
Contents Keynote Papers Samad, T., Cofer, D. Autonomy in automation: trends, technologies, tools Pantelides, C.C. New Challenges and Opportunities for Process Modelling Cordiner, J.L. Use of Prediction and Modeling in Early Evaluation of Process Options Ng, K.M. A Multiscale-Multifaceted Approach to Process Synthesis and Development Stephanopoulos, G., Schmitt, W.A. System Biology: an Emerging Theme in Biological Research
15 27 41 55
Contributed Papers Computer Aided Systems for Modeling Alexandridis, A.P., Siettos, C.I., Sarimveis, H.K., Boudouvis, A.G., Bafas, G.V. Modelling of Nonlinear Process Dynamics using Kohonen's Neural Networks, Fuzzy Systems and Chebyshev Series Barnard, J.P., Aldrich, C. A Systematic Methodology for Empirical Modeling of Non-linear State Space Systems Barnard, J.P., Aldrich, C. Modelling of Air Pollution in an Environmental System by use of Non-linear Independent Component Analysis Batres, R., Aoyama, A., Naka, Y. A Life-cycle Approach for Model Reuse and Exchange Baur, R., Taylor, R., Krishna, R. Dynamics of a Reactive Distillation Column for TAME Synthesis Described by a Non-equilibrium Stage Model Berezowski, M., Jacobsen, E.W., Grzywacz, R. Dynamics of Heat-integrated Heterogeneous Tubular Reactors with Axial Heat Conductivity in Reactor Wall Bj6rn, I.N., Gren, U., Svensson, F. Simulation and Experimental Study of Intermediate Heat Exchange in a Sieve Tray Distillation Column Cameron, D.B., Odegaard, R.J., Glende, E. On-line Modelling in the Petroleum Industry: Successful applications and future perspectives Charles, A.S., Azzaro-Pantel, C., Domenech, S., Pibouleau, L., Floquet, P., Jaume, D., Tilhac, F. Implementation of a Failure Model Validation Technique using a Discrete-event Batch Simulator: Application to semiconductor manufacturing
69 75 81
87 93 99 105
111
1t 7
viii Costa Jr., E. F., Vieira, R. C., Secchi, A. R., Biscaia Jr., E. C. Automatic Structural Characterization of DAE Systems Eliceche, A.M., Corvalfin, S.M., Ortiz, I. Steady State Analysis of Membrane Processes for the Treatment of Industrial Effluents F araoni, V., Mancusi, E., Russo, L., Continillo, G. Bifurcation Analysis of Periodically Forced Systems via Continuation of a Discrete Map Galen, O., Palazoglu, A., Romagnoli, J.A. Modelling and Optimisation of a High Density Fermentation Process using Multilinear Models: An Application to a Bench Scale Bioreactor Garea, A., Marqu6s, J.A., Hechavarria, T.L., Irabien, A. Simulation of the FGD In-duct Injection Technology using Complex Kinetic Models Harding, S.T., Floudas, C.A. EQUISTAR: Reliable Software for Design of Nonideal and Reactive Systems Hjertager, L.K., Hjertager, B.H., Solberg, T. CFD Modeling of Fast Chemical Reactions in Turbulent Liquid Flows K6hler, R., Rieber, J., Zeitz, M. Symbolic Discretization of Distributed Parameter Process Models on Self-adaptive Moving Grids Kohout, M., Schreiber, I., Kubicek, M. Computational Tools for Nonlinear Dynamical and Bifurcation Analysis of Chemical Engineering Problems Kosek, J., Stepanek, F., Novak, A., Grof, Z., Marek, M. Multi-scale Modeling of Growing Polymer Particles in Heterogeneous Catalytic Reactors Kr~xner, S., Gesthuisen, R. Semi-Batch Emulsion Copolymerisation: A Gereral Model for a Copolymer Formed from n Monomer Units Kristensen, N.R., Madsen, H., Jorgensen, S.B. Computer Aided Continuous Time Stochastic Process Modelling Lakner, R., Hangos, K.M., Cameron, I.T. Assumption Retrieval from Process Models Lim, Y.I., Le Lann, J.M., Meyer, X.M., Joulia, X. Dynamic Simulation of Batch Crystallization Process by using Moving Finite Difference Method Liu, Y., Jacobsen, E.W. Effective Model Reduction for Analysis of Distributed Parameter Systems Lucia, A., Yang, F. Global Terrain Methods for Chemical Process Simulation Lovik, I., R~nnekleiv, M., Olsvik, O., Hertzberg, T. Estimation of a Deactivation Model for the Methanol Synthesis Catalyst from Historic Process Data Mancusi, E., Maffettone, P.L., Gioia, F., Creseitelli, S. Nonlinear Analysis of an Industrial Ammonia Reactor with Heterogeneous Model Marchetti, M., Rao, A., Vickery, D. Mixed Mode Simulation - Adding Equation Oriented Convergence to a Sequential Modular Simulation Tool
123
129
135
141
147 153 159
165
171
177 183 189 195 201 207 213
219 225
231
ix Martinez, E.C., Lopez, G.D. Adaptive Optimal Operation of the Fenton's Batch Process for Industrial Wastewater Treatment Moharir, A.S., Shah, S.S., Gudi, R.D., Devereux, B.M., Vanden Bussche, K., Venimadhavan, G. Generalized Reactor Model: An Object Oriented Approach to Reactor Modelling Morton, W., Kozel, L., Lim, P.P.S., Douglas, D. Step Restriction for a Bounded Newton's Method Paloschi, J.R. Improving Robustness using Homotopy as an Open Solver in a Dynamic Simulation Package Preisig, H.A. Using Wavelets in Process Identification: A New Link to the State Space Rouzineau, D., Meyer, M., Prevost, M. Evaluation of Coupled Reactive Distillation Performances by Means of a Rigorous Simulation Procedure Sakizlis, V., Bansal, V., Ross, R., Perkins, J.D., Pistikopoulos, E.N. An Adjoint-Based Algorithm for Mixed Integer Dynamic Optimization Santana, P.L., Vasco de Toledo, E.C., Meleiro, L.A.C., Scheffer, R., Freitas Jr., B.B., Maciel, M.R.W., Maciel Filho. R. A Hybrid Mathematical Model for a Three-phase Industrial Hydrogenation Reactor Schneider, R., Kenig, E.Y., G6rak, A. Complex Reactive Absorption Processes: Model Optimisation and Dynamic Column Simulation Shacham, M., Brauner, N., Cutlip, M.B. A Web-based Library for Testing Performance of Numerical Solvers for Solving Nonlinear Algebraic Equations Siepmann, V., Haug-Warberg, T., Mathisen, K.W. Analysis and Consistency of Process Models with Application to Ammonia Production Teoh, H.K., Sorensen, E., Tumer, M., Titchener-Hooker, N. Dynamic Modelling of Chromatographic Processes: A Systematic Procedure for Isotherms Determination Tolsma, J.E., Barton, P.I. Process Simulation and Analysis with Heterogenenous Models Vale Lima, P., Saraiva, P.M. A Structured and Selective Framework for Hybrid Mechanistic-Empirical Model Building Wolf-Maciel, M.R., Soares, C., Barros, A.A.C. Validations of the Nonequilibrium Stage Model and of a New Efficiency Correlation for Nonideal Distillation Process through Simulated and Experimental Data Yiagopoulos, A., Yiannoulakis, H., Morris, J., Kiparissides, C. Simulation of an Industrial Olefin Polymerization FBR Operating under Condensed Mode Zhuang, H., Chiu, M.-S. Extended Self-Organizing Map with Application to the Modelling of Pulse Jet Fabric Filters
23 7 243 249
255 261 267 273 279 285 291 297 303 309 315
321 327
333
Computer Aided Systems for Synthesis and Design Agrawal, R., Herron, D.M. Feed Pretreatment for Binary Distillation Efficiency Improvement Bayer, B., Weidenhaupt, K., Jarke, M., Marquardt, W. A Flowsheet-Centered Architecture for Conceptual Design Bertok, B., Friedler, F., Feng, G., Fan, L.T. Systematic Generation of the Optimal and Alternative Flowsheets for Azeotropic Distillation Systems Bfihner, C., Schembecker, G. Reactor Selection and Design for Heterogeneous Reaction Systems Caballero, J.A., Grossmann, I.E. Generalized Disjunctive Programming Model for the Synthesis of Thermally Linked Distillation Systems Camarda, K.V., Sunderesan, P., Siddhaye, S., Suppes, G.J., Heppert, J. An Optimization Approach to the Design of Value-Added Soybean Oil Products Cismondi, M., Brignole, E.A. ECOFAC- Computer Aided Solvent Design and Evaluation in Environmental Problems, Based on Group Contribution Methods with Association Csukas, B., Balogh, S. Evolutionary Synthesis of Almost Closed Conservational Processes Eliceche, A.M., Hoch, P.M., Ortiz, I. Analysis of Azeotropic Distillation Columns combined with Pervaporation Membranes G~imt~s, Z.H., Floudas, C.A Nonlinear Bilevel Programming: A Deterministic Global Optimization Framework Hostrup, M., Balakrishna, S. Systematic Methodologies for Chemical Reaction Analysis Ierapetritou, M.G. An Efficient Approach to Quantify Process Feasibility based on Convex Hull Jim6nez, L., Costa, J. Design, Sizing and Modeling of a Reactive Extractive Distillation Unit and Solvent Recovery System Kahn, D., Plapp, R., Modi, A. Modeling a Multi-Step Protein Synthesis and Purification Process: A Case Study of a CAPE Application in the Pharmaceutical Industry Ko, D., Kim, M., Moon, I., Choi, D.-K. New designed TSA bed with cooling jacket for purification and regeneration of benzene and toluene Kr6ner, A., Kronseder, Th., Engl, G., v. Stryk, O. Dynamic Optimization for Air Separation Plants Li, X.-N.L., Rong, B.-G., Kraslawski, A. TRIZ-Based Creative Retrofitting of Complex Distillation Processes- An Industrial Case Study Liu, F., Hallale, N. Retrofit of Refinery Hydrogen Systems
339 345 351 357
363 369 375 381 387 393 401 407 413 419
427 433 439 445
xi Marcoulaki, E.C., Kokossis, A.C., Batzias F.A. Computer- Aided Synthesis of Molecular Mixtures and Process Streams Marechal, F., Kalitventzeff, B. A Tool for Optimal Synthesis of Industrial Refrigeration Systems Omota, F., Dimian, A.C., Bliek, A. Design of Reactive Distillation Process for Fatty Acid Esterification Pajula, E., Seuranen, T., Hurme, M. Selection of Separation Sequences by using Case-based Reasoning Patsiatzis, D.I., Papageorgiou, L.G. Optimal Multi-floor Process Plant Layout Reneaume, J.-M., Niclout, N. Plate Fin Heat Exchanger Design using Simulated Annealing Reyes-Labarta, J.A., Grossmann, I.E. Optimal Synthesis of Liquid-Liquid Multistage Extractors Rigopoulos, S., Linke, P., Kokossis, A. Development of Novel Process Designs for Simultaneous Oxidation and Denitrification of Wastewaters Rodriguez-Donis, I., Gerbaud, V., Joulia, X. Middle Vessel Heterogeneous Batch Distillation of an Azeotropic Mixture Samanta, A., Jobson, M. Optimisation of Heat Integrated Distillation Sequences in the Context of Background Process Samanta, A., Jobson, M. A New Heat Integration Model for Streams of Variable Temperature and Constrained Matches Sanchez Daza, O., Perez-Cisneros, E., Bek-Pedersen, E., Hostrup, M. Tools for Reactive Distillation Column Design: Graphical and Stage-to-Stage Computation Methods Schroer, J.W., Wibowo, C., Ng, K.M., O'Young, L. Development of Software Tools for Crystallization System Synthesis Sobocan, G., Glavic, P. Optimization of Ethylene Process Design Steinbach, W., Friedl, A., Hofbauer, H. Optimization of an Acidic Chlorine Scrubber with a Rate-based Simulation Engine Strouvalis, A.M., Heckl, I., Friedler, F., Kokossis, A.C. An Accelerated Branch-and-Bound Algorithm for Assignment Problems of Utility Systems Suh, M.-h., Friedler, F., Park, S., Lee, T.-y. Retrofit Design of Chemical Processing Networks under Uncertainties: Application to Petrochemical Industry Szitkai, Z., Lelkes, Z., Rev, E., Fonyo, Z. Optimisation of an Industrial Scale Ethanol Dehydration Plant. A Case Study Takano, K., Gani, R., Kolar, P., Ishikawa, T. Multi-Level Computer Aided System for the Design and Analysis of Processes with Electrolyte Systems Torres Alvarez, M.E., Martini, R.F., Wolf-Maciel, M.R. Characterization and Simulation of the Pervaporation Process for Separating Azeotropic Mixtures
451 457 463 469 475 481 487
493 499
505
511 517 523 529 535 541
547 553
559
567
xdi
Uerdingen, E., Fischer, U., Hungerbtihler, K. A Screening Method for Identifying Economic Improvement Potentials in Retrofit Design Vasquez-Alvarez, E., Pinto, J.M. MILP Models for the Synthesis of Protein Purification Processes Wang, Y.P., Achenie, L.E.K. A CAPD Approach for Reaction Solvent Design Wasylkiewicz, S.K., Castillo, F.J.L. Automatic Synthesis of Complex Separation Sequences with Recycles
573 579 585 591
Computer Aided Systems for Control and Operation
Aartovaara, M. Model-based Temperature Control of an Exothermic Semi-batch Reactor Akay, B., Ertunc, S., Bursali, N., Hapoglu, H., Alpbaz, M. Adaptive General Predictive Controller for a Nonlinear Bioreactor Aziz, N., Mujtaba, I.M. Optimal Control of Semi-batch Reactors Blanco, A.M., Figueroa, J.L., Bandoni, J.A. Feedback Control Design by Lyapunov's Direct Method Bonn6, D., Jorgensen, S.B. Batch to Batch Improving Control of Yeast Fermentation Coffey, D.P., Ydstie, B.E., Andersen, T.R., Jorgensen, S.B. Distillation Control Using Passivity Dechechi, E.C., Meleiro, L.A.C., Maciel Filho, R. A Novel Adaptive Multivariable DMC Controller: Application to an Industrial Reactor Ender, L., Scheffer, R., Maciel Filho, R. Computer Design of a New Predictive Adaptive Controller Coupling Neural Networks and Kalman Filter Eo, S.Y., Chang, T.S., Lee, B., Shin, D., Yoon, E.S. Function-Behavior Modeling and Multi-Agent Approach for Fault Diagnosis of Chemical Processes Gehan, O., Farza, M., M'Saad, M., Binet, G. Robust Predictive Control Combined with Nonlinear Observation for Monitoring (Bio)chemical Processes Govatsmark, M.S., Skogestad, S. Control Structure Selection for an Evaporation Process Grosman, B., Lewin, D.R. MPC using Nonlinear Models Generated by Genetic Programming Huang, Y., Reklaitis, G.V., Venkatasubramanian, V. Wavelet Shrinkage Based Coariance Matrix Estimation from Process Measurements Jordache, C., Temet, D., Brown, S. Efficient Gross Error Elimination Methods for Rigorous On-line Optimization
597 603 609 615
621 627 633 639 645 651 657 663
669 675
xiii Kint, E., Samyudia, Y., de Jong, P. A Combined Data and Gap Metric Approach to Nonlinear Process Control
Macias, J.J., Feliu, J.A. Dynamic Study of Inferential Sensors (Neural Nets) in Quality Prediction of Crude Oil Distillation Tower Side Streams M6ndez, C.A., Cerd~i, J. An Efficient MILP Continuous-Time Formulation for the Optimal Operation of General Multipurpose Facilities Michiel Meeuse, F., de Deugd, R.M., Kapteijn, F., Verheijen, P.J.T., Ypma, S.M. Increasing the Selectivity of the Fischer Tropsch Process by Periodic Operation Mourikas, G., Seferlis, P., Morris, A.J., Kiparissides, C. On-line Optimal Operating Policy and Control of a Batch Free Radical Polymerization Process Nagy, Z., Agachi, S., Allgower, F., Findeisen, R., Diehl, M., Book, H.G., Schloder, J.P. Using Genetic Algorithm in Robust Nonlinear Model Predictive Control Preuss, K., Le Lann, M.-V. Inferential Control of Microbial Batch Culture Rovaglio, M., Manta, D., Cortese, F., Mussone, P. Multistability and Robust Control of the Ammonia Synthesis Loop Ruiz, C., Basualdo, M.S., Molina, A., Jim6nez, L., Parisse, B., Richalet, J. Predictive Functional Control (PFC) Applied to an Unstable System- An Industrial Application Schmidt, H., Jacobsen, E.W. Selecting Control Configurations for Performance Shah, S.S., Madvadhan, K.P. Design of Controllable Batch Processes Silva, D.C.M., Oliveira, N.M.C. Optimization and Nonlinear Model Predictive Control of Batch Polymerization Systems Silva-Beard, A., Flores-Tlacuahuac, A., Fernandez-Anaya, G. Interval Matrix Robust Control of a MMA Polymerization Reactor Simon, L., Karim, N.M. Model Predictive Control of Apoptotis in Mammalian Cell Cultures Singer, A.B., Bok, J.-K., Barton, P.I. Convex Underestimators for Variational and Optimal Control Problems Skotte, R., An, W., Lenz, D.H., Baptiste, D.R., Lapham, D.S., Lymburner, C.J., Kaylor, J.M., Pinsky, M., Gani, R., Jorgensen, S.B. Modeling, Simulation and Control of an Industrial Electrochemical Process Smets, I.Y.M., Van Impe, J.F.M. Generic Properties of Time and Space Dependent Optimal Control of (Bio-) Chemical Processes Szederk6nyi, G., Kristensen, N.R., Hangos, K.M., Jorgensen, S.B. Nonlinear Analysis and Control of a Continuous Fermentation Process Torgashov, A.Yu. Nonlinear Process Model-based Self-optimizing Control of Complex Crude Distillation Column
681 687 693 699 705 711 717
723 731 737 743 749 755 761 767 773 781 787 793
~dv
Tousain, R.L., Michiel Meeuse, F. Closed Loop Controllability Analysis of Process Designs: Application to Distillation Column Design Tresmondi, A., Domingues, A., Maciel Filho, R. Online Optimization Integrated with Online Analyzers and Multivariable Predictive Controller in Industrial Airlift Reactors Verdijck, G.J.C., Lukasse, L.J.S., Preisig, H.A. A Control Methodology for Product Quality Control in Climate Controlled Operations involving Agro-materials Xaumier, F., Ettedgui, E., Le Lann, M.-V., Cabassud, M., Casamatta, G. A Model-Based Supervisory Control Routine for Temperature Control of Batch Reactors: Experimental Results Zeaiter, J., Gomes, V.G., Barton, G.W., Romagnoli, J.A., Gilbert, R.G. Strategies for Optimisation and Control of Molecular Weight and Particle Size Distributions in Emulsion Polymerization
799 805 811
817
823
Computer Aided Systems for Manufacturing Aoyama, A., Batres, R., Naka, Y. Process Safety Management for Batch Process Operation Badell, M., Ruiz, D., Puigjaner, L. Dynamic Cross-functional Factory-to-Business Links in the Batch Industry Canton, J., Afonso, A., Graells, M., Espuna, A., Puigjaner, L. Ad-hoc Scheduling~Planning Strategies in Distributed Computing Systems: An Application to Pipeless Batch Plants Castro, P., Barbosa-P6voa, A.P.F.D., Matos, H., Duarte, B. Dynamic Modelling and Scheduling of an Industrial Batch Digester Cooking System Dash, S., Kantharao, S., Rengaswamy, R., Venkatasubramanian, V. Application and Evaluation of Linear~Restricted Nonlinear Observers to a Nonlinear CSTR Gabbar, H.A., Suzuki, K., Shimada, Y. Design Considerations of Computer-Aided RCM-based Plant Maintenance Management System Gatica, G., Shah, N., Papageorgiou, L.G. Capacity Planning under Clinical Trials Uncertainty for the Pharmaceutical Industry Gupta, A., Maranas, C.D. Multiperiod Planning of Multisite Supply Chains Under Demand Uncertainty Harjunkoski, I., Grossmann, I.E. Combined MILP-Constraint Programming Approach for the Optimal Scheduling of Multistage Batch Processes Henning, G.P. Development of Interactive Facilities in a Knowledge-based Scheduling Framework H6treux, G., Perret, J., Pingaud, H. Computer Aided System for Short-term Scheduling of Batch Processes based on Hybrid Simulation Kim, D., Lee, Y., Moon, I., Lee, Y., Yoon, D. Automatic Accident Scenario Generation for Petrochemical Processes
829 835 841 847 853 859
865 871 877 883 889 895
XV
Mockus, L., Vinson, J.M., Luo, K. The Integration of Production Plan and Operating Schedule in a Pharmaceutical Pilot Plant Ortiz-G6mez, A., Rico-Ramirez, V., V/tzquez-Rom/m, R. Mixed-Integer Multiperiod Model for the Planning of Oilfield Production Sequeira, S.E., Graells, M., Puigianer, L. Decision-making Framework for the Scheduling of Cleaning~Maintenance Tasks in Continuous Parallel Lines with Time-decreasing Performance Strouvalis, A.M., Kokossis, A.C. A Conceptual Optimisation Approach for the Multiperiod Planning of Utility Networks Venkatasubramanian, V., Zhao, J., Viswanathan, S., Zhao, C., Mu, F., Harper, P., Russel, B.M. An Integrated Environment for Batch Process Development- From Recipe to Manufacture Zhao, J., Viswanathan, S., Zhao, C., Mu, F., Venkatasubramanian, V. Knowledge-based Management of Change in Chemical Process Industry Zhu, X.X., Majozi, T. A Novel Continuous Time MILP Formulation for Multipurpose Batch Plants Integrated Planning, Design and Scheduling
901 907 913
919
925 931 937
Process and Tools Integration
Alfadala, H.E., Sunol, A.K., E1-Halwagi, M.M. Retrofitting of Mass Exchange Networks with Temperature Effects Azzaro-Pantel, C., Davin, A., Pibouleau, L., Floquet, P., Domenech, S. Implementation of Multiobjective Optimisation for Multipurpose Batch Plant Planning Banares-Alcantara, R., Fraga, E.S., Perris, T. Concurrent Process Engineering & The Implications for CAPE Bansal, V., Perkins, J.D., Pistikopoulos, E.N. A Unified Framework for Flexibility Analysis and Design of Non-Linear Systems via Parametric Programming Belaud, J.-P., Alloula, K., Le Lann, J.-M., Joulia, X. Open Software Architecture for Numerical Solvers." Design, Implementation and Validation Bildea, C.S., Dimian, A.C., Iedema, P.D. Multiplicity and Stability of CSTR-Reactor-Separator-Recycle Systems Dua, V., Bozinis, A., Pistikopoulos, E.N. A New Multiparametric Mixed-Integer Quadratic Programming Algorithm Dunn, R.F., Wenzel, H. A Process Integration Design Method for Water Conservation and Wastewater Reduction in Industry Fraser, D.M., Harding, N., Matthews, C. Retrofit of Mass Exchange Networks Georgiadis, M.C., Schenk, M., Gani, R., Pistikopoulos, E.N. The Interactions of Design, Control and Operability in Reactive Distillation Systems
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Grancharova, A. General Strategy for Decision Support in Integrated Process Synthesis, Design and Control Henriksen, J.P., Russel, B.M. Static and Dynamic Optimisation of CAPE Problems using a Model Testbed Hertwig, T.A., Xu, A., Nagy, A.B., Pike, R.W., Hopper, J.R., Yaws, C.L. A Prototype System for Economic, Environmental and Sustainable Optimization of a Chemical Complex Kheawhom, S., Hirao, M. Decision Support Tools for Process Design and Selection Kwok, Y.-Y., Hui, C.-W. Site-wide Energy Optimization with Steam Pressure Changes Li, H.-Q., Chen, H.-Z., Li, Z.-H., Li, B.-H., Yao, P.-J. A Combined Approach for the Overall Energy Integration and Optimal Synthesis of Low-Temperature Process Systems Lid, T., Skogestad, S. Implementation Issues for Real Time Optimization of a Crude Unit Heat Exchanger Network Ma, K., Bogle, I.D.L. An Approach to Controllability and Economic Design Analysis of Nonlinear Systems with Multiplicity Minet, F., Heyen, G., Kalitventzeff, B., Di Puma, J., Malmendier, M. Dynamic Data Reconciliation of Regenerative Heat Exchangers coupled to a Blast Furnace Okada, H., Shirao, T. New Chemical Process Economic Analysis Methods Pingen, J. A Vision of Future Needs and Capabilities in Process Modelling, Simulation & Control Schlegel, M., Binder, T., Cruse, A., Oldenburg, J., Marquardt, W. Component-based implementation of a dynamic optimization algorithm using adaptive parameterization. Sequeira, S.E., Graells, M., Puigjaner, L. Integration of available CAPE Tools for Real Time Optimization Systems Shang, Z., Kokossis, A.C Design and Synthesis of Process Plant Utility Systems under Operational Variations Shethna, H.K., Jezowksi, J., Castillo, F.J.L. Near-independent Subsystems in Heat Exchanger Networks Design Sorsak, A., Kravanja, Z. Simultaneous MINLP Synthesis of Heat Exchanger Networks Comprising Different Exchanger Types Subramanian, D., Pekny, J.F., Reklaitis, G.V. SIM-OPT: A Computational Architechture to Address Valuable Business Aspects of Research & Development Pipeline Management Szitkai, Z., Lelkes, Z., Rev, E., Fonyo, Z. Solution of MEN Synthesis Problems using MINLP: Formulations of the Kremser Equation
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xvii Wang, K., Clark, G.A., Chung, P.W.H., Rossiter, D. Modelling Interface for Chemical Plantwide Process Design Yang, A.-D., von Wedel, L., Marquardt, W. An Open Thermodynamics Server for Integrated Process Engineering Environments
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Elgue, S., Cabassud, M., Prat, L., Le Lann, J.M., Casamatta, G., Cezerac, J. Optimisation of Global Pharmaceutical Syntheses Integrating Environmental Aspects Garg, S., Achenie, L.E.K. Genome Wide Functional Annotation using Mathematical Programming Georgiadis, M.C., Kostoglou, M. On the Optimisation of Drug Delivery Devices Halim, I., Palaniappan, C., Srinivasan, R. An Integrated Framework for Developing Inherently Safer and Environmentally Benign Processes Hill, P.J., Ng, K.M. Particle Size Distribution by Design Maranas, C.D. Optimization in Molecular Design and Bioinformatics Smith, R.L., Mata, T.M., Young, D.M., Cabezas, H., Costa, C.A.V. Designing Efficient, Economic and Environmentally Friendly Chemical Processes Xie, X., Hua, B., Chen, Q., Liang, R., Zeng, M. Study on Lifecycle and Agility of Process Industry Yang, T.C.-K., Lin, H.-S., Wang, S.-F., Cheng, M.-C. Dynamic Assessment of Induced Thermal Stresses on the Semiconductor Packaging Substrates in a Radiation Belt Furnace by Computational Fluid Dynamics AUTHOR INDEX
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European Symposiumon ComputerAided ProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsewerScience B.V. All rights reserved.
Autonomy in automation: trends, technologies, tools Tariq Samad and Darren Cofer Honeywell Laboratories, 3660 Technology Drive Minneapolis, MN 55418, U.S.A. tariq, samadldarren.cofer @ honeywell.com We focus on the topic of autonomy in automation and control systems. The trend toward increasing autonomy is discussed and illustrated with examples from multiple domains. Economics, performance, and human safety are highlighted as key considerations driving research into autonomous systems. We note that autonomy implies an ability to react appropriately to unforeseen situations, and identify two technical concepts that are instrumental for realizing this ability: multimodels and dynamic resource management. The need for new tools is discussed in this context with particular emphasis on integrating diverse aspects of models and systems. Finally, we speculate that some degree of "consciousness" will be required of automation systems if they are to become truly autonomous. 1. I N T R O D U C T I O N There are many perspectives we can take in examining the progress of automation in complex engineering systems such as process plants. In this paper we focus on the topic of autonomy. This is perhaps not an obvious principal concern today in the process industries, but we maintain that it has always been at least an implicit consideration in the development of automation and control technology in general and one that is increasingly capturing the interest of the process engineering research community. By autonomy here we mean the substitution by automated tools of functions that are or were performed by people. Complete automation of any significant system is not feasible now and will not be feasible in the foreseeable future. Increased autonomy thus implies essentially that the role of the human is shifted from lower to higher level tasks. What used to be accomplished with a large team may now be done with a smaller team, or with one person. Our discussion will largely be focused on the autonomous operation of systems. (Automated system design is another related topic and one that is well advanced in terms of tools available in many domains.) However, we do not limit our discussion to process engineering, although we frequently refer to it. The trend toward autonomy spans many industries and there are numerous points of similarity between parallel developments in different areas. The following section illustrates the trend toward autonomy with some examples, drawn from different domains. In Section 3, we briefly note three reasons why autonomy is being sought: performance improvement, economics, and human safety. Next, we note that autonomy implies an ability to respond appropriately to unforeseen situations, and we highlight two technical concepts that are critical for this ability: multimodels and dynamic
resource management. We discuss the types of new tools that are needed to facilitate the development of autonomous control systems in Section 5. Before concluding with a summary we reflect on the topic of consciousness vis-~.-vis autonomy. Some of the ideas discussed here are further elaborated in (Samad and Weyrauch, 2000). 2. T O W A R D A U T O N O M Y m E X A M P L E S
A couple of examples may help underscore the trend toward autonomous operation. In the process industries, metrics such as "loops per operator" are being used and there is continuing emphasis on increasing this quantity. In fact, operator employment has been shrinking. From 1980 to 1998, the number of production workers involved in petroleum refining in the United States shrank from 93,000 to 60,000--even as total refinery production increased from 5.3 to 6.2 billion barrels (U.S. Bureau of the Census, 1999). Analogously, a little over fifty years ago Lockheed introduced what was then a revolutionary new aircraft, the Constellation. It required a cockpit crew of five: a pilot, a copilot, a radio operator, a flight engineer, and a navigator. Because of improvements in navigation tools and avionics, the newest airliners today operate with just a pilot and copilot. The trend toward the replacement of human operation by automated systems can also be discerned through the levels of automation in engineering systems. Our engineered systems are multilevel, complex entities. From the lowest levelqsingle-loop control--to the highestqenterprise-wide optimization--concepts of feedback, dynamics, and adaptation are relevant. The degree of autonomy can be related to the levels through which operations are largely automated. Requirements for knowledge of the system's dynamics, regulation through feedback control, and adaptation to changing conditions can, in principle, be fulfilled through manual or automatic means. In all industries and application domains, we can see steady advances in levels of automation. In the early days of flight, control automation as we know it today was nonexistent. The pilot received sensory inputs directly (through visual and vestibular channels) and operated the actuators. The first step toward automatic control was the development of what we now refer to as inner-loop flight controllers. These allowed pilots to specify higher level commands to the aircraft, such as the desired pitch, roll, and yaw, with the controller responsible for the subsecond-scale feedback loop for sensing and actuation. The next step was the development of controllers that could take as input a heading command and automatically produce a sequence of desired states based on predefined "handling qualities." Today, all commercial aircraft have flight management systems (FMSs) on board that remove even this level of responsibility from the pilot (under normal conditions). A sequence of flight _ route h..I
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Figure 1. Control loops in commercial aviation today.
waypoints can be entered through the console of the FMS and the aircraft can automatically fly the route with no further manual assistance (Figure 1). Similar advances in the level of complexity under investigation have also occurred in the process industries. In a meeting to identify future needs for process automation with some aluminum processing industry representatives a few years ago, the discussion centered not on regulatory or multivariable control of individual processes, but on the management of the enterprise as a whole (Figure 2). Each of the processes shown in the figure will have several control loops associated with it, but it is automated solutions for the end-to-end system, from the bauxite ore as input to refined alumina as product, that are now desired. Increased autonomy, and the technological infrastructure that has enabled it, also implies that larger-scale systems are now falling under the purview of automation. Sensors, actuators, processors, and displays for entire facilities are now integrated through one distributed computing system. The control room in an oil refinery can provide access to and/or manipulate 20,000 or more "points" or variables. One of the largest integrated control system implementations, the Munich II international airport building management system, can be used to control everything from heating and cooling to baggage transport. The system controls more than 100,000 points and integrates 13 major subsystems from nine different vendors, all distributed over a site that includes more than 120 buildings (Ancevic, 1997).
3. W H Y A U T O N O M Y ? What is motivating the development of increasingly autonomous systems and the enabling technology for them? We note three key drivers. Performance. Automation can provide superior performance compared to manual operation. This can be due to the reduced sensing and actuation latencies of automated systems, their greater calculation prowess, greater memory fidelity in some cases, etc.
Ore [ GrindingI Caustic ~ [ ~ Steam Water~ ~ [ Digestion,[~Wash~ Water ~,........ I ,I Europe, I HeatExchl
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Figure 2. Processes involved in alumina refining. Performance means different things in different applications. Response times, settling times,
disturbance rejection, and setpoint tracking are some common measures for low-level control. At higher levels, yield, throughput, environmental impact, and energy efficiency are important parameters for many processes. Economics. We can also find many examples of a preference for an autonomous solution even where the manual alternative is no worsemor is even better~in terms of performance. Typically, in these cases there is an additional cost associated with manual operation that the associated performance improvement cannot compensate for. Our favorite example of this is the now obsolete traffic policeperson whose job consisted solely in standing in the middle of an intersection, generally on a raised and somewhat protected platform, and through hand signals directing roadway traffic. He or she could respond in an intelligent, context-sensitive manner to a wide variety of situations and incidents. From the point of view of performance a traffic light is a poor substitute, but it is considerably cheaper. H u m a n Safety. In some cases autonomy is sought because the task to be performed is dangerous for people. Examples include toxic material handling, bomb disposal, and combat. Considerable research is underway focusing on the development of uninhabited vehiclesm aerial, terrestrial, and undersea~for such applications. We note that both economics and performance are often considered supercriteria, in the sense that any other factor (including economics for performance and vice versa) can be subsumed by them. We are using these terms here in more specific senses. 4. T E C H N O L O G I E S FOR AUTONOMOUS SYSTEMS The discussion above has presented, as evidence of the trend toward autonomy, several examples of the advances in automation. Autonomy, however, is much more than automation. Today's engineered systems may be highly automated, but they are brittle and capable of "hands-off.' operation only under more-or-less nominal conditions. As long as the system only encounters situations that were explicitly considered during the design of its operational logic, the human element is dispensable. As soon as any abnormal situation arises, control reverts to the human. A pump fails in an oil refinery, or there is excessive turbulence on the route being flown by a commercial airliner, or the paper being manufactured in a paper machine consistently exceeds its caliper tolerancemin all these cases the human is immediately put back in the loop. Automation is part of the solution currently, but its role is in support of the operatormusually by supplying some relevant information about the state of the system. Ultimately the intelligence required to fix the problem resides in the operator. Autonomy, as distinct from automation, is not characterized by an ability to handle a complex system without human assistance in normal operating regimes. An autonomous agent must be capable of responding appropriately to unforeseen situationsmthat is, situations unforeseen by its designers. Some degree of circumscription of a system's operating space will always exist, since survival under every environmental extreme is inconceivable, but "precompiled" behaviors and strategies are not sufficient for effective autonomy. What does it mean to be able to react appropriately to unforeseen situations? To be capable of exhibiting behaviors that are not precompiled? We highlight two technical concepts in this section, multimodels and dynamic resource management.
Table 1 Types of models ......Characterizing dimension ....
. . . . . . . . . . . . . . . .
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,,
Examples First-principles, heuristic, and statistical models Temporal and algebraic models Operationally and componentially localized models
Knowledge source Complexity Domain of competence H
Multimodels
Domain knowledge is a central requirement for control and automation. Current control solutions fulfil this requirement in various specific ways, but generally they incorporate their designers', implementers', and users' understanding about the application domain. Automation is a multifarious proposition, and its different aspectsmfeedback and supervisory control, control system design, prognostic and diagnostic procedures, operator display, etc.m demand different types of knowledge. Even within one area, the knowledge required often needs to be differentiated; thus system dynamics often vary drastically with the operating regime of a physical system. Increasing automation requires increasing articulation and representation of domain knowledge within the automation system. Today, when aircraft can fly from point A to point B with no human interaction, it is in large part because we now have available explicit representations (models) of the dynamics of the aircraft under different conditions: the rate of fuel burn at different altitudes, speeds, and payloads; terrain topography; locations of navigation aids; special use airspace regions; etc.
4.1.1 Multimodel Types and Mechanisms In the process industries, model-predictive control technology, which integrates a predictive model of the system to be controlled (e.g., a distillation column in an oil refinery), is now well established. These models are still limited in various waysmthey are generally linear and valid only over a small operating regime. Higher fidelity models are also sometimes available, but not for online control. Instead, their use is generally limited to offline activities, such as operator training and process design. Just as linear, dynamic, highdimensional multivariable models used to be beyond the capability of process control platforms a decade or two ago, we similarly foresee that more complex models will be widely used in real-time. The trend is clear: increasing amounts of knowledge, in the form of models, will be incorporated within automation systems and tools. This should not evince much surprise given the objectives of increasing autonomy. If detailed and accurate models are not accessible online, the generation of appropriate responses to unforeseen situations is impossible. Control and automation of complex systems cannot be based on any single, unitary concept of model. Thus, a fundamental research need in modeling for autonomous systems is active multimodeling: the online use of a variety of model types. Some examples of different types of models, characterized by different dimensions, are listed in Table 1. The knowledge embedded in a model can result from a first-principles understanding of the physics and chemistry involved in the operation of the system, from the heuristic understanding of expert personnel, and from empirical data. Models can be more or less complex both algebraically
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(c) Figure 3. Selected multimodeling mechanisms. (linear to increasingly nonlinear) and temporally (static versus dynamic). Finally, a model can be localized to a particular operating regime and/or it can describe the behavior of a particular component or subsystem. Approaches for integrating multiple models have become an increasingly popular topic. Figure 3 shows some common examples. Figure 3a shows a superposition of a first principles model and an empirical model (in this case a neural network); the neural network is fit to predict the error between the actual data and the first principles model prediction (Su et al., 1992). In Figure 3b the outputs of different individual models are blended through a supervisory function (Murray-Smith and Johansen, 1997). A simple hybrid dynamical system model (Lemmon, 2000) is shown in Figure 3c. The ellipses represent different operational modes with customized models. The arcs represent mode transitions triggered by conditions. A diagram such as this one could represent the operation of a simple batch chemical reactor, distinguishing between reactor charging, heating, and product draw, with potential for reversion to recharging or reheating during product draw. A final interesting example of a multimodeling approach is multiscale and multiresolution models. For example, the wavelet transform of a signal can be seen as providing a multiplicity of separate yet consistent representations of the signal.
4.2
Dynamic Resource Management
Until recently, only highly constrained algorithms could be implemented and executed on real-time control system platforms 9 PID controller calculations and PLC ladder logic were about the limit of online computational complexity. Multivariable control, optimization, estimation, and process monitoring were implemented on nonreal-time computers in the plant, such as VAXes, with limited interfaces to DCSs and other real-time platforms. A PID algorithm executes periodically, deterministically, and in constant time: When installing the code realizing the algorithm on the control platform, engineers know how frequently the code needs to be executed and a good upper bound on its execution time, assumed constant. With online processing consisting entirely of such tasks, the execution profile or schedule for the real-time system can be determined in advance and can be assumed to hold indefinitely 9 Capabilities for different operational modes may be available, but even these provide essentially just a small finite set of processing options 9 While today's control
systems are more capable than yesterday's, and can allow, for example, iterative MPC calculations, they still require static, offline-determined task scheduling. The control system infrastructure, as it currently exists, does not permit dynamic adaptive resource management and thereby severely limits the "intelligence" that can be incorporated within automation. With human operators available to handle abnormal conditions, this limitation has by and large not caused much concern. As we attempt to endow automation systems with the algorithmic capabilities to autonomously manage complex enterprises, the lack of dynamic resource management technology is becoming a serious bottleneck.
4.2.1 Algorithmic Complications The need for dynamic resource management becomes evident when we consider the computational support needed for anytime algorithms. Anytime algorithms are fexible, scalable methods to solve particular problems, whether related to regulation, optimization, system health management, or other functions. They are characterized by their ability to use profitably as much time as is available. That is, the longer an anytime algorithm is run, the better will be its computed result. Depending on the current operational objective, the state of the system, and other computational requirements, an anytime algorithm can be given more or less CPU cycles. In addition, anytime algorithms are not limited to periodic execution; they can be event-driven and aperiodic as well. All of this stands in sharp contrast to the algorithms that are currently employed on real-time platforms, where rates of execution, priorities, and computational requirements can be fully characterized offline. Examples of anytime algorithms include: Data-centric forecasting. Online-accessible databases provide a new approach for modeling and prediction (Kulhav3), Lu, and Samad, 2001). Instead of relying on a first principles or empirical model generated offline as a computationally static entity, relevant operational data can be dynamically accessed and used for "just-in-time" modeling. In such a scenario, the number of samples accessed can vary dramatically, depending on the frequency with which the particular operational regime of interest has occurred in the past as well as on accuracy and response time requirements. Computational complexity will vary accordingly. Evolutionary Computing. Randomized optimization algorithms inspired, however loosely, by biological evolution are increasingly being explored for high-dimensional, analytically untractable problems. The quality of solution obtained with these algorithms is related to the number of iterations performed. Genetic algorithms are the most familiar example, but several other methods also exist (Fogel, 1995). Multiresolution Modeling. For high-performance applications and for failure tolerance, online modeling and identification will often be necessary. Depending on circumstances and resource availability, it may be appropriate to undertake model development or model execution at different levels of detail.
4.2.2
Optimizing Performance by Control Task Adaptation
The challenge to resource scheduling arises when we contemplate the need to run a large number of processing tasks in a finite amount of time where these tasks can include anytime and fixed-time; periodic, aperiodic, and event-driven; and deterministic and nondeterministic tasks. In addition to any control optimizations being performed, the allocation of available computing resources must be actively adapted to optimize the performance of the overall system. The relationships between the overall application goals, computing resource models,
and control models can be thought of as follows (Figure 4). Based on the computed or observed system state, task criticalities and computing requirements are assigned. Computing
system& ,~ ~'P~~iti~ce,~ . environment/ ~ "~taskcriticality
Figure resources are made available to tasks based on criticality, pending requests, and a schedulability analysis. Control tasks then execute within their allotted time. These tasks must adapt to meet the application constraints (deadlines, accuracy, etc.). Active models will be configured as schedulable tasks with a variety of service requirements that must be satisfied by the execution environment. Models will specify their requirements as ranges of values within which the execution environment may optimize overall vehicle performance. For example: 9 A model may be able to run at a range of rates, within a given minimum and maximum, and adapt its execution time correspondingly. Example: servicing a message queue. 9 A model may be able to run at several discrete rates. Example: a control loop or state estimator that can use different tables of values for gains and time constants. 9 A collection of models may provide the same data but by using algorithms of varying fidelity and execution time. Example: process models for fault detection and identification may include a simple static gain model, an approximate empirical mode, and a full dynamic model. The set of currently executing models and tasks will have to adapt to both internal and external triggers to make optimal use of available computational resources. Mode changes will cause tasks to start and stop, and external influences (new targets, feedstock changes, etc.) will cause changes in task criticality and computing loads.
5. THE NEED FOR TOOLS This paper has generally been concerned with articulating concepts for the autonomous operation of complex engineering systems. As such, we have not specifically discussed the need for new tools. However, there are some important implications of our discussion for process engineering tools.
5.1
Multimodei Tools
Most modeling and identification tools available today assume a unitary notion of model. Even if a tool permits more than one type of model to be developed, provisions for
compositions of models into effective multimodel architectures are not likely to be available. Multimodeling tools, not just modeling tools, are needed. In this context it is important to note that the model-centric nature of complex automation solutions requires that tools be developed that are informed by a suitably rich notion of model. Thus we should distinguish between, and integrate within tools, three "facets" of models (Figure 5): 9 A model is a set of mathematical formulae that capture some important characteristics of the behavior of a physical system. This is the typical control science and engineering perspective, and most tools limit themselves to it. 9 Models are computational entities with processing, memory, and communication requirements. For online execution of multiple models, these requirements must be satisfiedmgenerally a nontrivial undertaking given resource constraints of control system platforms. This computational facet needs to be provided by tools addressing real-time applications. 9 Models can be viewed as objects, with inputs, outputs, and functional capabilities abstractly specified. This view is especially useful for tools that purport to facilitate model composition.
5.2
Tools Integrating Performance and Computation
A related gap in the tools available today concerns design, simulation, and evaluation of automation and control systems that integrate performance and resource usage aspects. The incorporation of anytime algorithms within real-time platforms complicates design and assessment. Designers need to be concerned not only with the performance of a particular algorithm under a fixed resource availability, but they must consider and effect tradeoffs between performance and computing requirements--given multiple complex algorithms, bounds on computational resources, and diverse scenarios. We have recently developed a prototype simulation and evaluation tool as a first attempt to address this gap. This development is an outcome of an ongoing project focusing on the autonomous operation of uninhabited aerial vehicles (UAVs) (Godbole, Samad, and Gopal, 2000). This tool combines real-time scheduling and execution of control algorithms with aircraft and environmental models running in simulated time (see Figure 6). This framework can be populated with a variety of control and computational models and algorithms. The system permits control tasks (active models) to be run as real executable code (not simulations) that have real-time deadlines. The aircraft and environmental simulations can be based on high fidelity models since they are not required to execute in real time. Each aircraft is created as a separate process that can later be transferred to separate
Figure 5. Facets of a model. hardware to more closely approximate a real multi-aircraft application.
Real time
10 performance data for the control algorithms and scheduling infrastructure can be tracked independently of the (nonreal-time) simulation. Execution time can be scaled to simulate the improved hardware performance expected for future platforms. More details are available in (Agrawal et al., 2000). Simulation and evaluation tools such as the UAV-specific one discussed above are also needed for the process industry domain. While the details will differ considerably, the overall framework of Figure 6 is, we believe, still relevant.
Figure 6. Multimodel simulation tool for a UAV application. The separation of the nonreal-time execution of system models with the simulated and scaled real-time execution of control models and algorithms, the allowance for dynamically managed computational loads among different tasks, and possibly the componentwise integration for assessing system-level automation solutions (e.g., multiunit integration for plantwide automation) should all be of interest in this case. 6. A PHILOSOPHICAL SIDEBAR: AUTONOMY AND CONSCIOUSNESS The notion of developing engineered sensors or actuators, or even low-level models of computation, that are based on biologically gleaned principles is uncontroversial. Embodying higher-level cognitive capabilities in computational systems, however, is another matter. Some researchers argue that such capabilities cannot even in principle be realized by the sorts of machines we are contemplating. The levels of autonomy, intelligence, and adaptability exhibited by humans are thereby excluded (the argument goes) from realization in engineered systems. At the center of this controversy lies the indeterminate notion of consciousness. There is no accepted precise definition of the term, but it is generally held that it is a key to human (and possibly other animal) behavior and to the subjective sense of being human. Consequently, any attempt to design automation systems with humanlike autonomous characteristics requires designing in some elements of consciousness: in particular, the
11 property of being aware of one's multiple tasks and goals within a dynamic environment and of adapting behavior accordingly. The nexus between consciousness and computation is a favorite topic of some philosophers and neuroscientists. There are two theoretical limitations of formal systems that are driving much of the controversymthe issue under debate is whether humans, and perhaps other animals, are not subject to these limitations. First, we know that all digital computing machines are "Turing-equivalent"--they differ in processing speeds, implementation technology, input/output media, etc., but they are all (given unlimited memory and computing time) capable of exactly the same calculations. More importantly, there are some problems that no digital computer can solve. The best known example is the halting problem--we know that it is impossible to realize a computer program that will take as input another, arbitrary, computer program and determine whether or not the program is guaranteed to always terminate. Second, by G6del's proof, we know that in any mathematical system of at least a minimal power there are truths that cannot be proven. The fact that we humans can demonstrate the incompleteness of a mathematical system has led to claims that G6del's proof does not apply to humans. In analyzing the ongoing debate on this topic, it is clear that a number of different critiques are being made of what we can call the "computational consciousness" research program. In order of increasing "difficulty," these include the following: 9 Biological information processing is entirely analog, and analog processing is qualitatively different from digital. Thus sufficiently powerful analog computers might be able to realize autonomous systems, but digitally based computation cannot. Most researchers do not believe that analog processing overcomes the limitations of digital systems; the matter has not been proven, but the Church-Turing hypothesis (roughly, that anything computable is Turing-Machine [i.e., digitally/algorithmically] computable) is generally taken as fact. A variation of this argument, directed principally at elements of the artificial intelligence and cognitive science communities, asserts that primarily symbolic, rule-based processing cannot explain human intelligent behavior. [] Analog computers can of course be made from non-biological material, so the above argument does not rule out the possibility of engineered consciousness. Assertions that the biological substrate itself is special have also been proposed. Being constructed out of this material, neural cells can undertake some form of processing that, for example, silicon-based systems cannot. Beyond an ability to implement a level of self-reflection that, per G6del, is ruled out for Turing machines, specifics of this "form of processing" are seldom proposed, although Penrose's hypothesis that the brain exploits quantum gravitational effects is a notable exception (Penrose, 1989). (It is worth noting that no accepted model of biological processing relies on quantum-level phenomena.) [] It has also been argued that intelligence, as exhibited by animals, is essentially tied to embodiment. Disembodied computer programs running on immobile platforms and relying on keyboards, screens, and files for their inputs and outputs, are inherently incapable of robustly managing the real world. According to this view, a necessary (not necessarily sufficient) requirement for an autonomous system is that it undertakes a formative process where it is allowed to interact with the real world. 9 Finally, the ultimate argument is a variation of the vitalist one, that consciousness is something extra-material. For current purposes this can be considered a refrain of the Cartesian mind/body dualist position. Contemporary explorations of this theme include
12 Chalmers (1995)--an article that also includes a rebuttal by Christof Koch and Francis Crick. Consciousness is a multifaceted phenomenon. Reflective, deliberative decision making is an important element, although admittedly not the only one. Thus the technical concepts discussed earlier~multimodels, anytime algorithms, dynamic resource allocation--which, we have argued, are essential for high-performance autonomous behavior, are by the same token necessary correlates of consciousness. (Our observation of) our own conscious processing supports this contention--we dynamically allocate cognitive resources as appropriate for an unforeseen situation, scale the precision and resolution of our processing accordingly, and rely on our knowledge of the various systems and phenomena that constitute our environment. 7. SUMMARY Any but the most indomitable technophile today would, we expect, refuse to be a passenger in a pilotless airliner or have an uninhabited yet operational refinery in her neighborhood. But, regardless, the trend toward reducing human involvement in the operation of complex engineering systems, driven as it is by considerations of economics, performance, and safety, appears inexorable. Further substantial improvement in process automation, however, will require more than evolutionary technological advances. Our focus in this paper has been on autonomy, a property absent in today's automation systems. In order to satisfy the demands of industry and society, we will need to make our automation solutions autonomous--they will need to be able to respond appropriately to unforeseen situations, not just limited to precompiled behaviors. We have noted two research directions that are central for engineering autonomy. First, diverse representations of knowledgemof process dynamics, failure modes, the environment, and other factorsmwill be needed and will need to be integrated so that unanticipated situations can be explicitly reasoned about. Models are widely recognized as principal determiners of control system performance; we must broaden our attention now to multimodels. Second, dynamic resource management technology must be developed to allow multiple, competing, heterogeneous computational tasks to execute on real-time platforms under hard and soft deadlines and resource constraints. The full space of relative priorities that could be faced by an autonomous system cannot be predicted; tradeoffs among resource allocations for different tasks will need to be made online and adaptively, not through static scheduling. Engineering autonomy will require maturing concepts into tools. Tools that can truly enable multimodeling and dynamic resource management are unavailable today. A key, general differentiating attribute that distinguishes the tools that are needed from those that are currently in use is the level of integration. Whether it is models of different subsystems, or knowledge from disparate sources, or performance and resource usage aspects of computational tasks, tools for autonomous systems will require integration of diverse phenomena, features, and representations. Finally, we suspect that our pursuit of autonomy in automation and control will bring the topic of consciousness~at least in the limited sense of an agent's awareness of its
13 environment and context and its deliberative adaptation of its information processing priorities--into the foreground of discussion.
Acknowledgement: This research is supported in part by the U.S. Defense Advanced Research Projects Agency (DARPA) under contract number F33615-98-C-1340. REFERENCES 1. 2. 3. .
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Agrawal, M. et al. (2000). Real-time adaptive resource management for multi-model control. Submitted for publication. Ancevic, M. (1997) Intelligent building system for airport. ASHRAE Journal, November, pp. 31-35. Chalmers, D. (1995). The puzzle of conscious experience. Scientific American, pp. 8086, December. Fogel, D.B. (1995). Evolutionary Computing: Toward a New Philosophy of Machine Intelligence, IEEE Press, Piscataway, N.J., U.S.A. Godbole, D., T. Samad, and V. Gopal (2000). Active multi-model control for dynamic maneuver optimization of unmanned air vehicles. Proc. IEEE Int. Conf. on Robotics and Automation, San Francisco. Kulhav3~, R., J. Lu, and T. Samad (2001). Emerging technologies for process optimization. To appear in Proceedings of CPC-VI, Tucson, Ariz., U.S, January. Lemmon, M. (2000). Supervisory hybrid control systems. In Perspectives in Control Engineering: Technologies, Applications, and New Directions, T. Samad (ed.), IEEE Press, Piscataway, N.J., U.S.A. Murray-Smith, R. and T.A. Johansen (eds.) (1997). Multiple Model Approaches to Modelling and Control Taylor & Francis Ltd., London. Penrose, R. (1989). The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics, Oxford Univ. Press. Samad, T. and J. Weyrauch (eds.) (2000). Automation, Control and Complexity: An Integrated View, John Wiley & Sons, Chichester, U.K. Su, H.-T., et al. (1992). Integrating neural networks with first principles models for dynamic modeling. Preprints of the IFAC Symp. on DYCORD+, College Park, Md. U.S. Bureau of the Census (1999). Statistical Abstract of the United States: 1999, 118th edition, Washington, D.C.
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European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rightsreserved.
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New challenges and opportunities for process modelling Constantinos C. Pantelides abt aCentre for Process Systems Engineering, Imperial College of Kingdom bprocess Systems Enterprise Ltd., 107a Hammersmith Bridge Road, London W6 9DA, United Kingdom Over the past decade, process modelling has made substantial progress with respect to both methodological advances and applications in engineering practice. This paper discusses some of the major issues that need to be addressed for this kind of progress to be sustained over the next decade. Particular emphasis is focused on three areas, namely the modelling of complex distributed systems, the construction of validated models, and multiscale modelling. For each of these areas, recent progress is reviewed and some key outstanding problems are identified. 1. I N T R O D U C T I O N Process modelling has made substantial progress over the past decade. We are now capable of building mathematical models with a degree of detail and predictive accuracy that would be almost unimaginable in the early 1990s. In some cases, this reflects our improved understanding of process physics. In other cases, the main driver for recent progress has been the advent of generic modelling concepts and software that permit this knowledge to be transformed into practically usable descriptions. Perhaps more impressively, we are actually able to solve many of these models and to perform various other manipulations based on them. For some types of problem (e.g. steady-state simulation), the underlying algorithms and codes existed well before the 1990s; although some improvements have undeniably taken place since then, much of the credit for progress has to be given to improved computer speed and memory availability. However, in other areas (such as the simulation of hybrid discrete/continuous processes and dynamic optimization), the past decade has indeed witnessed much advancement in both mathematical understanding and numerical algorithms. The above progress has provided the driving force behind the increasingly central role played by process modelling within most process engineering activities. However, this increased usage has also exposed certain obstacles and bottlenecks to the wider deployment of this technology. This paper identifies three of the major issues in process modelling that need to be addressed if further significant progress is to be achieved over the next decade. These are the modelling of complex distributed systems (section 2), the construction of validated models (section 3) and multiscale modelling (section 4). In all cases, the paper is primarily concerned with process modelling methodology. Naturally, substantial improvements in physical understanding contFAX: +44-20-7594 6606, email: c. pantelides@ ic. ac. uk
16 tinue to be a major requirement in several areas of importance to process engineering. However, this topic is beyond the scope of the present paper.
2. MODELLING OF COMPLEX DISTRIBUTED SYSTEMS As a significant part of the process industries moves away from commodity chemicals towards higher value-added products, there is an increasing requirement for the accurate prediction and control of the quality of these products. This had led to a shift in emphasis from traditional measures of process performance, such as yield and conversion, to other measures that directly affect product quality, such as selectivity. As far as process modelling is concerned, the above developments have often required the abandonment of "perfect mixing" simplifications and the direct consideration of both spatial and temporal variations within process equipment. This is necessary as many of the properties of interest are affected by mixing inhomogeneities. From the mathematical point of view, this leads to distributed (as opposed to lumped) systems described by mixed sets of integral, partial and ordinary differential, and algebraic equations (IPDAEs). Many of the higher value-added products mentioned above involve materials, such as polymers and crystalline solids, that are more complex than the relatively simple liquids and gases that have been the mainstay of the process industries in the past. The design and operation of such processes requires the prediction of product quality in terms of properties that are of interest to the end-user. These properties are often determined by complex characteristics of the product such as the distributions of crystal sizes and shapes (for crystalline solids) or of chain lengths and degree of branching (for polymeric materials). Thus, we have distributed processes, the mathematical modelling of which again requires the introduction of IPDAE systems. In view of the above, it is not surprising that a decisive factor in the wider use of generalpurpose process modelling tools during the past decade has been their ability to describe and solve some classes of distributed systems [1, 2, 3]. Interestingly, this feature has made such tools useful not only to engineers concerned with describing entire process flowsheets but also to specialists who are concerned with the detailed modelling and design of a single item of processing equipment (e.g. chemical reactor or separation device). The use of common tools for modelling both individual equipment items and the entire process has been a highly desirable recent development. It has to be recognised, however, that the distributed modelling capabilities of general-purpose process modelling tools are still limited in some important ways. One restriction is that the domains that can be described by current technology must be expressed as the cartesian product of one or more line segments. This limits the applicability of these tools to relatively simple geometries (e.g. rectangles, cylinders, spheres) and presents a serious obstacle to the accurate modelling of processing equipment of irregular shapes. A second limitation concerns the reliability and efficiency of numerical solution. For example, although it is possible to describe 3-dimensional models of fluid flow within the latest generation of process modelling tools, the resulting models cannot easily be solved using generic numerical codes of the type typically incorporated in these tools. The two limitations mentioned above are already addressed quite well by a different type of technology which has been evolving along quite separate lines to general process modelling tools. More specifically, Computational Fluid Dynamics (CFD) tools (see, for example, [4])
17 can describe equipment of arbitrarily complex geometry. They also employ specialised solution techniques that exploit the nature of the underlying equations to deal with numerical complexity. These characteristics have allowed CFD tools to make important contributions to process modelling in recent years [5]. On the other hand, they are limited to a specific set of equations and, consequently, lack the wide scope of general-purpose process modelling tools (e.g. with respect to describing complex multicomponent mass transfer phenomena) and the capabilities they offer in terms of making a common process model available to diverse types of application. Moreover, the specialised numerical methods that they employ cannot always deal effectively with highly nonlinear and/or stiff systems, such as those involving very fast reactions. Recent work has sought to combine the capabilities of CFD tools with those of more conventional modelling technology. Work along these lines has been reported for the modelling of industrial crystallisation [6], gas/liquid bubble column reactors [7] and exothermic batch reactors [8]. Despite the diversity of the systems studied in these papers, a number of common features can be discerned: 9 The combined model encompasses a "simple" model (e.g. a network of well-mixed regions in [6], a set of 1-dimensional vertical zones of gas or liquid in [7], and a perfectly mixed tank in [8]) coupled with a CFD model. 9 Both the CFD model and the simple model describe the same spatial domain. However, the CFD model incorporates only phenomena that are directly related to fluid dynamics while all other phenomena (e.g. crystal nucleation and growth in [6], mass transfer and reaction in [7], reaction in [8]) are incorporated in the simple model. 9 The CFD model is used to compute parameters that are necessary for the simple model, being treated as fixed constants by it (e.g. the mass fluxes between adjacent regions and the turbulent energy dissipation rate (which affects crystal nucleation) in [6], the lateral flows between the gas and liquid zones in [7] and the heat transfer coefficient between the process fluid and the cooling jacket in [8]). 9 The solution of the CFD model requires knowledge of properties that depend on the solution of the simple model (e.g. the effective density and viscosity of a pseudo-homogeneous fluid in [6], the bubble size and gas-liquid mass trasfer rate in [7] and the liquid density and viscosity in [8]). 9 In view of the last two points, an iteration between the simple and the CFD models is necessary to obtain a converged and consistent solution. In all cases, a successive substitution approach has been used. The publications mentioned above, as well as other unpublished material, indicate that this approach can lead to striking improvements in predictive accuracy for models of industrialscale equipment. However, much remains to be done to transform it into a tool that can be used routinely for process modelling. First, we need a set of genetic concepts for deciding the partitioning of the process physics between the CFD and the simple model, and for mapping the physical domain into each of the two models. Secondly, any procedure that involves repeated solutions of CFD models (some of which may take several hours or days) may lead to serious efficiency concerns. Finally, the successive substitution procedure may not be the most reliable
18 way of combining the two models - especially in cases where there is strong two-way coupling between the fluid mechanical and the other (e.g. reaction) phenomena. More fundamentally, for strongly coupled systems, the loss of accuracy that is inherent in the construction of any simplified model may be unacceptable in some applications. In these cases, a simultaneous solution approach that treats all phenomena together appears unavoidable. Such an approach would require a certain degree of convergence between general-purpose process modelling technology on one hand and CFD technology on the other. Some of the issues that have to be addressed in this context (e.g. the description of irregular geometries) are primarily of a software nature, and satisfactory solutions already exist and can be borrowed from the CFD area. Extending the numerical solution techniques so that they can deal with general IPDAE systems will be more challenging. On past experience, however, the ever increasing computer power may well diminish the importance of using solution techniques specifically designed to solve particular classes of problems, and widen the range of applicability of more general methods ~. However, the real difficulties lie at the fundamental mathematical level. Our current understanding of the mathematical properties of IPDAEs (including such basic questions as wellposedness and validity of initial and boundary conditions) is arguably at approximately the same level as our understanding of DAEs was in the mid-1980s. Even for specific well-studied PDE systems such as the Navier-Stokes equations of fluid mechanics, the formulation of a correct set of boundary conditions has been the subject of intense study and debate (see, for instance, [9-12]). For more general systems, it is only very recently that ao index classification of PDAE systems has been introduced [13] and that this has been related to the specifications (e.g. initial or boundary conditions) that may independently be imposed on the system along a particular co-ordinate direction [14, 15]. Moreover, our understanding of the relation between the mathematical properties of IPDAE system and the methods that may be applied for its numerical solution is also at a preliminary stage (see, for instance, [16]). !n summary, much theoretical progress is still required before we can have practical tools that are able to guide the modeller towards the correct formulation and solution of complex distributed models. 3. CONSTRUCTION OF VALIDATED PROCESS MODELS The previous section identified some outstanding issues that relate to the modelling of complex distributed systems. However, there are other important problems that affect almost all process models, even some relatively simple ones. Almost all non-trivial models involve parameters that describe various thermodynamic and kinetic phenomena. In fact, the unavailability of values for these parameters is one of the most frequently cited obstacles to the wider use of process modelling in industrial practice. The problem has been exacerbated by the move of the process industry towards processes involving newer and/or more complex materials and reactions that are not yet well characterised, and towards higher fidelity modelling that often requires information relating to different types of phenomena (e.g. parameters for modelling finite rates of mass transfer) beyond those that were routinely modelled in the past (e.g. phase equilibria). The use of rigorous optimisation techniques for the estimation of model parameters from ~Analogous developments took place in numerical methods for nonlinear algebraic equations and differentialalgebraic equations (DAEs) in the 1970s and 1980s.
19 measured data is now well established. The objective functions primarily used are based on maximum likelihood or nonlinear least squares criteria for both steady-state and dynamic models. Progress in this area has benefited from general developments in numerical optimisation algorithms and codes, often adapted to exploit the special form of the objective function in the parameter estimation problem. For example, dynamic optimisation techniques have formed the basis for fitting parameters to data from dynamic experiments [ 17] while, more recently, global optimisation techniques have been used to ensure that the correct (globally optimal) solution of the parameter estimation problem is obtained [18, 19]. Beyond point estimates of the parameter values, the computation of measures of statistical significance of these estimates has also received attention, especially for the case of nonlinear models for which standard linear measures may be misleading [20]. In principle, the parameter estimation techniques mentioned above apply equally well irrespective of whether the measured data used come from laboratory-scale experimentation or from measurements carried out in pilot or, indeed, industrial-scale plants. In particular, fitting models to match their predictions to observed plant behaviour has been, and still is, common practice. However, an interesting consequence of our increasing ability to handle complexity in process modelling, especially where this arises because of inhomogeneities in mixing, is that we can now often afford to adopt a more systematic approach towards the development of high fidelity models. As always in process modelling, we have to start with the identification of all important phenomena that may occur in the process. Ideally, this is followed by the quantitative characterisation of each such phenomenon in isolation. This is achieved using laboratory experiments specifically designed for this purpose; for example, homogeneous reaction kinetics can be measured under conditions of high mixing intensity, thereby removing any mixing effects. Such a complete decoupling may not always be possible, especially in heterogeneous systems where mass transfer and reaction phenomena may be difficult to separate. Nevertheless, there is a definite advantage in minimising the number of phenomena that are involved in any individual experiment so as to achieve a better statistical characterisation. Once all phenomena of interest are characterised in this manner, they can be combined within a single plant model. The latter can often adequately describe the plant behaviour with little, if any, need for additional adjustment. Where successful, the application of the above approach can lead to some important benefits. Since diverse unit operations within the same or different processes involve combinations of the same phenomena, the physical knowledge regarding these phenomena may be re-used, thus reducing the overall cost of process modelling. Moreover, models that have been built from the combination of phenomena characterised in isolation are likely to predict plant behaviour accurately even when changes in operating conditions affect the relative importance of these different phenomena (e.g. when significant changes in flow and mixing shift the balance between mass transfer and chemical reaction). Of course, laboratory experimentation is often an expensive and time consuming activity itself. The need to carry out such activities in the most effective and efficient manner possible has led to increased interest in systematic techniques for the design of experiments during the last decade. Asprey and Macchietto [21] have recently outlined a comprehensive scheme for constructing validated models via a combination of laboratory experimentation and model-based optimisation techniques. The scheme starts with one or more candidate models, each involving
20 one or more unknown parameters. The models are first tested for parametric identifiability, i.e. the ability to determine unique values for the parameters appearing in them given a set of quantities that can be measured experimentally. Models that are deemed to be identifiable in principle are then tested against the results from experiments specifically designed to discriminate between competing models. Once the best model is selected, further experiments are designed to allow the accurate estimation of the parameters in it. The above scheme comprises a number of computational steps, each involving the solution of an appropriate optimisation problem. The step that is perhaps best understood at present is the final one, namely that of designing experiments for optimal parameter estimation. Dynamic experiments are particularly important in this context as they have the potential of generating a large amount of information in a single run. The design of a dynamic experiment typically involves several decisions, including its initial conditions and duration, the variation of any available control variables over the latter, and the times at which various measurements are to be taken. Several alternative measures for the information content of an experiment have been proposed in the literature [22, 23, 24]. The determination of an optimal experiment design involves the solution of a dynamic optimisation problem. Thus, like their counterparts for parameter estimation, experiment design techniques have also benefited from the significant progress in dynamic optimisation algorithms and codes over the past decade. A more fundamental problem is that the optimisation is carried out using a model that involves parameters, the values of which are still subject to significant uncertainty. Consequently, experiment design is an iterative process: once an experiment is designed, it is executed in the laboratory, and the data collected from it are then used to re-estimate the model parameters. A new experiment design can then be determined on the basis of this improved model, and the process is repeated. An attempt to accelerate this iteration by designing "robust" experiments that take direct account of the parametric uncertainty in the underlying models has recently been reported [25]. In summary, the last decade has witnessed some methodological progress for the construction of validated models, with emphasis shifting away from plant-level data fitting and towards laboratory-scale experiments targetted at the characterisation of individual physical phenomena. There may still be scope for further decomposition in certain applications such as those involving complex reaction networks. For example, it would be beneficial to determine systematically a set of distinct experiments, each of which involves only a (hopefully small) subset of the species and reactions in the network. We now also have a clearer idea of an integrated approach to this problem, and the various computational and experimental components that it may comprise. However, much remains to be done in terms of both the mathematical formulation and the numerical solution of some of these computational components. On the other hand, both hardware and software for automated laboratory experimentation have achieved significant progress in recent years (see, for instance, [26, 27]), partly driven by the demanding needs of the pharmaceutical and biotechnological industries. This ultimately opens the way for the seamless integration of the experimental components with their computational counterparts, with substantial potential savings in both the time and the cost required for the development of validated process models.
21 4. MULTISCALE M O D E L L I N G The previous section considered the use of laboratory experimentation for the characterisation of the fundamental phenomena that need to be described in process models. An increasingly viable alternative is provided by techniques of computational chemistry that attempt to model matter at the molecular, atomic or sub-atomic levels based on classical Newtonian mechanics, quantum mechanics, or combinations of the two. At the simplest level, computational chemistry techniques may be used to generate "pseudoexperimental" points that can then replace or complement the results of real experiments. This type of approach is particularly feasible for simple thermodynamic and transport properties, for which the predictive accuracy of computational chemistry has improved significantly in recent years. Moreover, it has the advantage that it does not actually require any changes to standard methodologies and tools of either computational chemistry or process modelling. A more challenging, but also potentially much more powerful, mode of utilisation of computational chemistry techniques is in their direct incorporation within process models. This kind of approach is best understood within the wider context of the multiscale nature of phenomena and operations of interest to process engineering [28], ranging from the nanoscale of molecules, atoms and sub-atomic particles (involving distances of O(10 -1~ m and times under O(10 -12) s) to the megascale of global supply chains (with distances of O(107) m and times of O(108) s). Intermediate scales include the microscale of particles, eddies and bubbles, the mesoscale of process equipment items, and the macroscale of process plants. It should be stressed that, albeit commonly used, neither the terminology introduced above nor the delineation of the boundaries between different scales are unique. In fact, sometimes the same terms are used to describe different delineations- for example, material scientists allocate quantum mechanical and classical mechanical phenomena to different scales, with the high end of their classification reaching up to what was called "microscale" above (see, for example, [29]). What cannot be doubted is the widely different scales of both space and time involved in process engineering, and the challenges that these pose to our attempts to encode our knowledge in terms of mathematical models. The traditional way of addressing multiscale complexity can, perhaps, best be described as scale decoupling. This simply focuses scientific and engineering endeavour on each individual scale with the aim of building the best possible understanding and description of the phenomena taking place at that scale. For example, at the nanoscale, such descriptions may take the form of models of molecular and intra-molecular motion; at the microscale, we have descriptions of mixing and fluid flow of the type mentioned in section 2 of this paper; mesoscale models involve detailed descriptions of dynamic behaviour of processing equipment; at the macroscale, we have models of the type used to schedule the operation of complex multipurpose plants; and at the megascale, there are the models used to analyse the dynamics of supply chains. Of course, the different scales are not truly independent of each other. After all, our mesoscale models of dynamic unit operation models invariably require some description of the behaviour of both the materials involved and of fluid flow, which are precisely the objects of study at the nanoscale and microscale respectively. The traditional approach in dealing with such interactions has largely been based on scale aggregation, leading to simplified descriptions of behaviour at each scale in terms of quantities that are directly relevant to higher scales. For example,
22 9 the nanoscale's detailed descriptions of the behaviour of matter are aggregated into equations of state and relatively simple kinetic laws so that they can be used in higher-level models; 9 the complexities of fluid flow at the microscale are hidden behind the well-mixed region (or networks of well-mixed regions) approximations used for modelling process equipment; 9 the details of the dynamic behaviour of batch processing equipment studied at the mesoscale are replaced by the simple concept of a task with a finite duration and fixed demands on resources, of the type that can be used for plant scheduling; 9 the large networks of interacting resources and tasks used for modelling multipurpose plants at the macroscale are replaced by a few simple linear constraints describing overall production capacity for the purposes of modelling supply chains involving interacting manufacturing and distribution operations. The scale aggregation approach has proven very successful in handling the inherent complexity of process engineering, representing a pragmatic trade-off between the predictive accuracy of a model and its complexity at both the conceptual and the computational levels. It has to be recognised, however, that any aggregation operation involves an inherent approximation. As our demands for model accuracy at a given scale (e.g. the mesoscale of processing equipment) become more stringent while the predictive accuracy of models at lower scales improves, there often comes a point at which the loss of useful information involved in the aggregation of lowerlevel behaviour becomes unacceptable. At this point, we are forced to consider an alternative to scale decoupling, namely scale integration. Scale integration involves the use of descriptions at different scales within the same model. At present, the most common ways of achieving this are the so-called serial and parallel integration strategies [29]. A serial integration strategy is one in which the finer scale model is simply used to generate some of the parameters or data required by the higher-scale one. This is not very different from the scale decoupling approach except that the aggregate description (e.g. the equation of state) is formally derived from the lower-scale model (rather than, for instance, being determined empirically by fitting to experimental data). On the other hand, a parallel integration strategy involves the simultaneous use of descriptions at different scales applied to the same computational domain. The results of one description form inputs to the other, and vice versa. Thus, an iteration between the two models is normally required to achieve a consistent overall description. The combination of CFD with conventional process models described in section 2 of this paper is a good example of such a parallel strategy. In addition to the serial and parallel strategies mentioned above, we could also envisage a third hierarchical integration strategy in which the finer-scale model is formally embedded within the higher-scale model to represent a set of relations among macroscopic quantities occurring in the latter. Finally, there is a simultaneous strategy in which the higher-scale model is formed completely from finer-scale descriptions. This is currently possible only in certain applications, mostly towards the higher scales of interest to process engineering. For example, advances in numerical methods for dynamic simulation and increased computing power now routinely allow us to build dynamic (macroscale) models of entire plants simply by assembling
23 detailed dynamic (mesoscale) models of individual equipment items without the need for any simplification at this stage. Also (megascale) models used for short-term scheduling of combined production/distribution operations can be constructed from detailed (macroscale) models of each individual manufacturing site; mathematical decomposition techniques can then be used for the solution of these very large problems [30]. Multiscale modelling is key to the effective utilisation of computational chemistry techniques for process modelling applications. A recent example of such an approach is the work by Rodgers and Jensen [31] on modelling of chemical vapour deposition devices. Although the bulk of the fluid can be described by standard continuum equations, the rate of reaction at the solid surface is determined by the latter's inhomogeneities, the size of which is small compared with the mean free path of molecules in the gas phase. Consequently, the continuum hypothesis does not apply to this part of the system; instead, a molecular simulation approach is used to estimate the nett flux from the gas to the solid phase. This flux then forms the boundary condition for the continuum model describing the gas phase. Of course, the results of the molecular model depend on the gas phase composition and temperature. Consequently, the two models are coupled and an iteration between them is necessary. Another important example of multiscale modelling is in studying the rheology of complex fluids. The standard equations of fluid dynamics are an asymptotic approximation based on the assumption that the time scales of molecular and intramolecular motion are much shorter than those of the fluid flow [32]. While this is a reasonable assumption for most simple fluids, it is certainly not true for complex molecules such as polymers. The conventional way of dealing with this effect is via the use of constitutive equations for stress that attempt to account for the history of deformation of the fluid. In the general context of multiscale modelling outlined above, these can be thought of as aggregate descriptions of molecular behaviour. However, the predictive accuracy of such models is fundamentally limited since the state of the system at any particular time is determined by a much larger number of variables (i.e. those describing the distribution of molecular conformations) than is encompassed in these constitutive equations. Moreover, the additional computational complexity involved in the introduction of these constitutive equations within the standard Navier-Stokes equations is far from negligible. In view of these difficulties, an increasingly attractive alternative is provided by the direct combination of the continuum conservation equations with a stochastic simulation of molecular motion [33]. Ultimately, detailed atomistic models may be necessary to describe certain areas of the flow such as those adjacent to solid walls. A review of these and other important issues in computational rheology has recently been presented by Keunings [34]. As evidenced by the references mentioned above, much of the work in multiscale modelling to date has been carried out by pioneers who are interested in solving particular types of problems. Looking at it from the viewpoint of process systems engineering, it is clear that a major aim should be to use these examples as the basis for the development of generic methodologies for multiscale modelling, and for identifying and solving the underlying mathematical problems. These essential pre-requisites for the wide and reliable use of these approaches become increasingly important as we move towards strategies in which the models at different scales are more tightly coupled. Consider, for instance, a hierarchical integration strategy in which a fine-scale model is embedded within a coarse-scale one. An important question is whether the results of the solution of the former model correspond to a well-defined function of its inputs. Moreover, depending
24 on the mathematical techniques used for solving the coarse-scale model, the continuity and differentiability of this function become important issues. It must be stressed that it should not be taken for granted that techniques (e.g. for CFD or molecular modelling) that have evolved over long periods of time for "stand-alone" use automatically possess the properties that are necessary for their integration within wider computational schemes. Indeed, our experience has been that the contrary is often true and that significant theoretical and algorithmic effort may have to be invested in ensuring that these additional requirements are met. An illustration of these issues for the case of molecular dynamics is provided in [35-37]. 5. CONCLUDING REMARKS
This paper has focused on issues that are directly related to process modelling methodology and which, in the author's opinion, represent significant, but hopefully attainable, challenges for the next decade. These issues must be seen in the context of several other important problems in the related areas of mathematical solution methods (e.g. for obtaining globally optimal solutions of optimisation problems [38, 39] and for the optimisation of hybrid processes [40]) and of software developments (e.g. the move towards open system architectures for process modelling software [41 ]). Regarding the first of the three issues considered in this paper, namely that of modelling complex distributed processes, it is likely that significant progress can be made in the relatively short term by the combination of CFD with conventional process modelling technology. Longer-term progress is likely to involve a certain degree of convergence between the two technologies. However, a better mathematical understanding of the properties of general PDAE systems will be required before this can be achieved. Addressing the second issue considered, that of the construction of validated process models, in a satisfactory manner will require the solution of several novel types of optimisation problem beyond those that have already been widely recognised in the literature. Some of these new problems will require significant developments in global optimisation techniques, especially for the solution of time-dependent problems (cf. [18]). Finally, multiscale modelling is an essential pre-requisite for making full use of advances in scientific understanding within engineering applications of practical interest. Albeit sometimes regarded as primarily a problem of software integration, multiscale modelling in fact involves fundamental questions of at least three different kinds: 9 conceptual issues (e.g. the optimal way of partitioning the descriptions of the relevant physical phenomena among the different scales considered); 9 mathematical issues (e.g. regarding the precise definition and well-posedness of the mathematical problem that is actually posed by the multi-scale model); 9 numerical issues (e.g. regarding the reliability and efficiency of the overall solution algorithm). Multiscale modelling is still in its infancy as far as general systematic methodologies are concerned. It is the author's belief that process systems engineering can make valuable contributions to this field, providing some of the concepts and techniques that are needed to bind the different scales together.
25
Acknowledgements The author's work on process modelling is partially funded under grant GR/N08636 of the United Kingdom's Engineering and Physical Sciences Research Council (EPSRC). The author is indebted to Dr Steven Asprey and Professor Sandro Macchietto for many useful discussions on some of the material appearing in this paper.
REFERENCES 1. M. Oh, Modelling and Simulation of Combined Lumped and Distributed Processes, PhD thesis, University of London (1995). 2. M. Oh and C.C. Pantelides, Comput. chem. Engng., 20 (1995) 611. 3. S.E. Zitney and D.M. Laing, paper 259g, AIChE Annual Meeting, Los Angeles, CA, November 2000. 4. J.D. Anderson Jr., Computational Fluid Dynamics, McGraw-Hill, New York, 1995. 5. A.D. Gosman, Chem. Engng. Res. Des., 76 (1998) 153. 6. Z. Urban and L. Liberis, in Proc. Chemputers'99 Conference, Dtisseldorf, Germany, October 1999. 7. M. Bauer and G. Eigenberger, Chem. Eng. Sci., 54 (1999) 5109. 8. E Bezzo, S. Macchietto and C.C. Pantelides, Comput. chem. Engng., 24 (2000) 653. 9. J.C. Strikwerda, Comm. Pure Appl. Math., 30 (1977) 797. 10. E Dutt, SIAM J. Numer. Anal., 25 (1988) 245. 11. T. Miyauchi, M. Tanahashi and M. Suzuki, JSME Int. J., Series B, 39 (1996) 305. 12. E Moin and K. Mahesh, Ann. Rev. Fluid Mech., 30 (1998) 539. 13. S.L. Campbell and W. Marszalek, Math. Comp. Model. Dynam. Systems, 5 (1999) 18. 14. W.S. Martinson, Index and Characteristic Analysis of Partial Differential Equations, PhD thesis, Massachusetts Institute of Technology (2000). 15. W.S. Martinson and EI. Barton, SIAM J. Sci. Comput., 21 (2000) 2295. 16. B.-M. Pfeiffer and W. Marquardt, Math. Comput. Simul., 42 (1996) 617. 17. I.B. Tjoa and L.T. Biegler, Ind. Eng. Chem. Res., 30 (1991) 376. 18. W.R. Esposito and C.A. Floudas, Ind. Eng. Chem. Res., 39 (2000) 1291. 19. C.Y. Gau and M.A. Stadtherr, Comput. chem. Engng., 24 (2000) 631. 20. J.S. Albuquerque and L.T. Biegler, AIChE J., 43 (1997) 986. 21. S.E Asprey and S. Macchietto, Comput. chem. Engng., 24 (2000) 1261. 22. D.M. Espie and S. Macchietto, AIChE J., 35 (1989) 223. 23. L.C. Zullo, Computer-Aided Design of Experiments - An Engineering Approach, PhD thesis, University of London (1991). 24. I. Bauer, H.G. Bock, S. K6rkel and J.E Schl6der, J. Comput. Appl. Math., 120 (2000) 1. 25. S.E Asprey, S. Macchietto and C.C. Pantelides, in Proc. of ADCHEM'2000 Conference, L.T. Biegler, A. Brambilla and C. Scali (eds.) Vol. II, IFAC (2000) 869. 26. H. Du, L.A. Corkan, K. Yang, EY. Kuo and J.S. Lindsey, Chemometr. Intell. Lab. Syst., 48 (1999) 181. 27. D.G. Cork, T. Sugawara, J.S. Lindsey, L.A. Corkan and H. Du, Lab. Robot. Autom., 11 (1999) 217. 28. J. Villermaux, Chem. Engng. Res. Dev., 73 (1995) 105. 29. D. Maroudas, AIChE J., 46 (2000) 878.
26 30. A.D. Dimitriadis, Algorithms for the Solution of Large-Scale Scheduling Problems, PhD thesis, University of London (2000). 31. S.T. Rodgers and K.E Jensen, J. Appl. Phys., 83 (1998) 524. 32. G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1967. 33. K. Laso and H.C. Ottinger, J. Non-Newtonian Fluid Mech., 47 (1993) 1. 34. R. Keunings, in Proc. 13th Intl. Congress on Rheology, D.M. Binding et al. (Eds), British Society of Rheology, Glasgow, 1 (2000) 7. 35. J. Stefanovi6 and C.C. Pantelides, in Proc. 5th Intl. Conf. on Foundations of ComputerAided Process Design, M.E Malone and J.A. Trainham (eds.), CACHE Publications (2000) 236. 36. J. Stefanovi6, B. Fouchet and C.C. Pantelides, in Proc. FOMMS 2000: Foundations of Molecular Modeling and Simulation, P. Cummings (ed.), Keystone, Colorado, July 2000. 37. J. Stefanovi6 and C.C. Pantelides, "On the Mathematics of Molecular Dynamics", Parts I, II, and III, to appear in Molecular Simulation, (2001). 38. C.A. Floudas, J. Process Control, 10 (2000) 125. 39. P.M. Pardalos, H.E. Romeijn and H. Tuy, J. Comput. Appl. Math., 124 (2000) 209. 40. C.C. Pantelides, M.P. Avraam and N. Shah, in Scientific Computing in Chemical Engineering, E Keil, W. Mackens and H. Voss (eds.), Springer Verlag (1999) 62. 41. B.L. Braunschweig, C.C. Pantelides, H.I. Britt and S. Sama, in Proc. 5th Intl. Conf. on Foundations of Computer-Aided Process Design, M.E Malone and J.A. Trainham (eds.), CACHE Publications (2000) 220.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
USE OF PREDICTION AND MODELLING OF PROCESS OPTIONS
27
IN EARLY
EVALUATION
JL Cordiner Technology and Projects, Syngenta (Formally Zeneca Agrochemicals), Leeds Road, Huddersfield, England HD2 1FF Often in the chemical industry modelling tools are used to design unit items for processes that have already been fixed. Predictive techniques and modelling tools can be used at route and process selection stages to facilitate rapid evaluation of process options. The tools can be used to maximise the return on experimental time required in the development of alternative chemical routes and processes. In addition making the tools available to and useable by both chemical engineers and chemists gives a common language, facilitating joint working and enabling them to bring their different insights to select and design processes. This work has proven to be very successful in Syngenta where there are many examples where alternative processing options have been considered and evaluated. 1. I N T R O D U C T I O N Fine Chemicals Manufacturing is increasingly looking at reducing the time to market, this means that decisions about the process are pushed further and further back the decision train. These decisions are then required when less and less of the apparently required information is available. Conventional wisdom needs to be tested to consider what information is really needed and what level and quality of decision is required at each stage. In some cases, for example pharmaceuticals, the process route needs to be decided very early for registration reasons. The choice of the route can have large implications on the costs of production and capital requirement. It is then advantageous to have methods to challenge the normal route selection and development processes. This paper considers how early evaluation tools have been used and developed to address these issues within Syngenta. In addition the paper demonstrates benefits of wide spread use and the format of the tools best suited for the wide range of potential users working on development of processes.
2. THE D E V E L O P M E N T PROCESS Delivering a plant with an agrochemical ready for market involves a complex process as described in figure 1 which is a schematic of the overall development process from a research route.
28
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[1,21) One can see that major impact of early evaluation tools are at the key stage of route selection, ensuring the best overall route is chosen taking into account issues such as manufacturing safety, health and environment, meeting business requirement for initial and (and possibly longer term) business requirements. This is discussed further by Carpenter [ 1,2]. 3. G E N E R A T I N G A N D R E V I E W I N G A L T E R N A T I V E R O U T E S Clearly the synthetic routes from Research are usually not practical for a manufacturing setting. The chemist and engineer need to work together to consider how all the routes for consideration will be operated at the manufacturing scale desired by the business. At this stage it is vital the Early Evaluation Tools are able to aid this process in generating processes that can be radically different from conventional wisdom. Each chemical route can be operated at a manufacturing scale in a number of different ways and these need considered in any route evaluation. In addition the Early Evaluation tools are required to enable comparison of routes and processes to enable the most practical options to be chosen. Clearly the level of information on each route will be sparse at this stage and therefore the tools must allow quality decision to be taken on the limited data. It is therefore important to remember that comparison requires the data to be consistent but not necessarily accurate at this stage. As it is important to consider the whole supply chain in route selection one should use the tools alongside experience from different professionals rather than expecting the tools to do the whole job. A trap one can fall into is trying to develop a tool to perform a large complex remit. This becomes unworkable particularly by the non-frequent or non expert user. A tool with a smaller remit could perform a useful task and considerably speed up the total process and enable novel ideas.
29 4. C U S T O M E R S / U S E R S O F E A R L Y E V A L U A T I O N T O O L S Very many synthetic routes will be considered at the early stages and reduced down to a small number for further investigation until a single route is chosen and moves into development (process, formulation and packing). Plant (3) has reviewed the number of active molecules on average coming from discovery to reach the end of development, in a typical agrochemical business (table 1). Phase
Activity
Time Number per year ..... (years) .required for 1.produc ~Phase 1-Invention ...........investigates interesting activity 5 - 1 0 12 .... Phase 2 - Evaluation Clarifies the preferred candidate 2 4 Phase 3 - Development Answers critical questions 4 1 Table 1" A typical agrochemical development chain (after Plant [4]) '" This means that often a diverse and large number of people are involved in many different processes. Most of these people are non-specialists and in-frequent users of CAPE tools. It is imperative therefore that these people have access to the tools to enable them to benefit from early evaluation. This brings the benefit of the expertise of the specialist to the largest numbers of processes. Such people need training to use the early evaluation tools to help them make informed decisions for route selection or design of experiments etc. The in-house software SMSWIN was developed to provide early evaluation tools for all chemists and engineers working in process and product development. The large number of processes, speed required and typically batch operation means that large simulations of a process are often not appropriate and certainly not at the early stages where route selection takes place. Much consideration was given to the format and training the tool would have as described below. The in-house training delivered along with the launch of SMSWIN included fundamental thermodynamics, phase equilibria, solvent selection, short cut separation selection, crystallisation etc. This training used case studies to show the use and power of the tools. The training was also developed into Intranet Expert Systems. This allows rapid access for the personnel, where they can get assistance in following the evaluation process, selecting a separating method or making use of the tools. Training has been given to chemists and engineers together from all facets of the product process. This enables common language and mutual understanding of the issues that can be considered using the tools. These training sessions also give feedback on the content using the tools to facilitate improvements and usability. The tools and training allow the chemists and engineers to work more closely together to look at the routes. This helps focus their efforts and time on the key issues by rapidly evaluating each, looking for potential issues. The teams are them able to plan their experiments, make more detailed models and work programmes to concentrate on these issues. This leads to more cocurrent development, again shortening the development time.
30 It can be beneficial to have relatively trivial calculations (to a person experienced in thermodynamics or CAPE) available in very visual tools or tables. This can speed the work of the developers and allows other disciplines to carry out the calculations or work alongside the chemical engineer. Examples of this can be seen later in the paper. Giving the user the choice of discrete calculations rather than just an expert system that is set up for the whole problem aids understanding of the process options. In addition the discrete calculations can be used for process improvements of parts of the process either at route selection or at any time of the process life.
5. CHALLENGES FOR THE EARLY EVALUATION TOOLS. The traditional simulation tools are extremely comprehensive. The tools are very daunting and difficult for the in-frequent or non-expert user who would have to go through many forms and make a number of selections. The tools therefore need to be user friendly, robust and easy to use. In particular the tools need to be as intuitive as possible for the infrequent user, minimising the number of forms to be filled in or clicks required. This can be seen in setting the most commonly used information at very easy and fast reach as shown in figure 2.
Figure 2 Property selection From SMSWIN Where ever possible an expert system to select items or calculation methods needs to be employed in such a way that it is easy for the non specialist, whilst still giving enough information and guidance to give some training/understanding whilst the tool is being used. For example the physical property method for early evaluation and setting up of this method needs to be made very easy. This can be demonstrated by pre-setting the groups for UNIFAC (and it's
31 variants) for as many molecules and frequently used building blocks for molecules as possible as is done typically for UNIQUAC for molecules. Many of the developers will need help in selecting a property method, where an expert system would be beneficial. Such a tool has been developed by Gani and O'Connell [4]. A decision tree (as shown in figure 3), written for typically encountered processes, is also a useful tool for the non expert. This allows rapid selection and points to when further advice from the expert system is required.
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Figure 3 Decision Tree The tools should also be as visual as possible. Many visualisation tools have been provided in SMSWIN (in-house programme) to help developers rapidly access route options. For example, residue maps, eutectic diagrams, solubility plots, and scatter diagrams for evaluating crystallisation's feasibility - some of these are discussed later. Having diagrammatic ways of presenting the same data can aid understanding of a process. For example, looking at ternary VLE data in a triangular diagram shows two liquid phase regions well. A pseudo 2 component diagram shows what and how much entrainer or solvent breaks or sufficiently moves azeotropes. A residue diagram shows feasible separation regions.
6. FACETS OF ROUTES NEEDING CONSIDERATION There are many issues that need to be considered from materials handling, where solids as well as fluids need to be handled with issues relating to toxicity, environmental impact etc. through to the issues of packing and formulation. A variety of tools, for example, databases and
32 expert systems, aid the professional in decision points by having the commonly used information easily at hand. It is useful to have a tool that allows a rapid means of reviewing each research process assessing the manufacturing process options. The chemist or chemical engineer can carry out a rapid assessment using diagrams and tables (some of which are shown later in the paper) seeing the potential issues easily and quickly. The reaction chemistry can have a large impact on the choice of route depending on the yield and selectivity obtained. Typically yield is considered most important given that materials are often not recycled due to complexity of separation. Where recycling is possible the best route for this needs to be considered and yield is therefore not such an issue. Often the reaction chemistry can be highly altered both in products and in rate via the choice of solvent. This is graphically demonstrated in the aromatic nucleophilic substitution of the azide ion with 4fluoronitrobenzene (Cox (5,6)), where the rate changes by 6 orders of magnitude depending upon the solvent as shown in figure 5. Clearly choice of solvent is very important. F
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The solvent chosen potentially can be used for the separations required or indeed may need to be separated. In most fine chemical processes there tends to be more than one stage of chemistry required and often different solvent may be chosen for each stage with initial manufactures requiring purification and separation at each stage. This can mean a large number of different solvents are used throughout the process which involves much separation leading to yield losses. Selecting the solvents for each reaction stage carefully to minimise solvent swaps can be very cost effective and can also increased yield. Clearly any tool which aids solvent selection can radically reduce the capital and operating costs of any route. The tools can lower the experimentation time required by reducing the number of solvents to be tested in the laboratory. One can look at using the tool as doing the experiments quicker. Reducing the experimentation time and hence aid faster development enables more radical processes to be tried. The techniques can then also be used to look at selecting a stage wide or process wide solvent rather than having a number of solvents and solvent swaps through the process. The tools used, to aid solvent selection for reaction, range from using taxonomies of solvent properties and effects, the use of principle component analysis to develop further taxonomies for example by Chastrette [7] as shown in figure 5, through to using optimisation techniques to evaluate potential solvents for a given set of constraints.
33
Figure 2. Solvent Taxonomy These classifications solvents are however restricted to known or commonly used solvents. This needs to be challenged with increasing environmental issues and legislation driving use away from the problematic traditional solvents as discussed by Odele and Macchietto [8], Pistikopoulos and Stefanis [9], Hostrup et al. [10] and Harper [ 11 ]). Therefore, tools to generate novel solvent options are particularly valuable. The tools being developed take a number of formats all making use of optimisation MINLP or Global Optimisation. These allow the generation of molecules which fit selection criteria for the solvent. The generation of the molecules allows novel solvents to be considered. The criteria to set the objective function and constraints being based on a number of requirements for a successful solvent. The tool developed by Gani et al at CAPEC Consortium of the Danish Technical University (ProCamd) makes use of UNIFAC for solvent property prediction have been used successfully and integrated with in-house software tools. In order to test out these methods a process wide solvent selection for the nitric acid oxidation of anthracene to anthraquinone was carried out using solvent selection tool Pro-Cared, an inhouse programme SMSWIN and Global Optimisation in collaboration with Dr C Adjiman and P Bavishi [12] of Imperial College. The techniques looked at solvent effect on reaction, solubilizing the starting material, solubility of nitric acid, recovery of product, separation scheme for recovery of solvent, recovery of nitric acid, boiling point and melting point, vapour pressure, price, safety and toxicity factors etc.
34 Some of the techniques can be used independently in SMSWIN being discrete calculations and visualisations as shown later, however the optimisation selection tools 'Pro-CAMD' and Global optimisation were set covering as wide a range of the selection criteria as possible. The solubility of the starting material and the product can be visualised with SMSWIN as shown in the example in figure 6.
Figure. 6. Solid Solubility This allows selection of solvent with highest solubility of the starting material, it can also be used to select a solvent with the highest recovery of product. The scatter graph shows how the solubility of solute varies with solvents that SMSWIN has selected. The Productivity Index is the product of solubility at the higher temperature and % Recovery hence the best solvents for recovery of product are in the top right hand corner, the names appearing as required. The crystallisation of the product can then be considered using a eutectic diagram as shown in figure7. SMSWIN allows the selection of crystallisation temperature and calculates the recovery of Product into the solid phase.
35
Figure 7: Crystallisation calculations. Alternatively if a drown out crystallisation is used, the solvent swap or drown out can be visualised in SMSWIN very quickly allowing a graph of composition versus recovery at different temperatures or a table allowing for multiple easy recalculations as shown in figure 8. In selecting the solvent recovery system, due to the presence of water, solvents were required to be immiscible in water as one alternative. A means of looking at miscibility ranges over a range of solvents is therefore useful or a calculated table which allows sorting of the various solvents data to find the immiscibility range required. This could also be seen by use of Ternary diagrams for the non experienced personnel. Where extractive distillation is considered pseudo binary or ternary data can also be used to look at minimum entrainer usage possible (Knapp and Doherty [ 13]). Further separations can be evaluated by use of residue and distillation curve diagrams shown in figures 9 and 10. as proposed by Doherty and Perkins [14,15,16,17], Dongen and Doherty [18] and Stichlmair [19,20]. Where the early evaluation tool allows rapid access to such diagrams, a number of distillation can be considered for each process and compared, looking for potentially difficult or prohibitive separations.
36
_Figure 8 Solvent Swapping and Drownout Calculations._
Figure 9 Example of Residue Curve Map
37
Figure 10. Example of Distillate Curve Map
Figure 11. Example of Short cut method taken from training programme.
38 This can be further developed by rapid, or short cut tools ,as shown in figure 11, to allow the developer to look at the scale of equipment that might be required, although trivial calculations, the rapid access can be very useful for the developer whether they be a chemist or an engineer.
7. FUTURE CHALLENGES AND REQUIREMENTS FOR EARLY EVALUATION TOOLS Many of the typical processes contain very complex molecules of which there is little information. These complex molecules have many functional groups and be in the presence of similar molecules which are produced as by products or as pre or post stage products. Indeed many final molecules are required in a particular enantiomer. Some typical molecules are shown in figure 12 (from Carpenter [2]). The selection of the separation task therefore becomes complicated. It is important therefore to have good predictive tools for the important physical properties and the ability to improve these predictions with as much known information as possible. This sort of tool has been developed by the CAPEC Group of the Technical University of Denmark. The tools however are limited in their range of applicability. The large complex molecules which are common in the fine chemicals industry cannot be reliably modelled using the existing group contribution techniques. There are however ways forward by using as much information as available from the molecule and similar molecules to give some guidance. This is where using the tools along side experience and experiment can work very well. BF
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o
A green azo dyestuff for dying polyester
Figure 12. Typical Molecules. A joint project with the CAPEC group looked at a Gap Analysis of the physical properties required (Cordiner and Nielsen [21] and what Gaps existed in the available technology. This showed that predictive ability in complex multi-functional molecules, along with electrolytes, salt forms of complex molecules, surfactants and surface properties were key areas. In many of
39 these areas it is common to use molecular modelling and dynamics to obtain properties of interest. The physical property "gaps" were prioritised by demand, difficulty to resolve and ease of measurement. The simple to measure properties are best measured where possible. It is common in many processes to have by-products and intermediates that are very similar in structure to the product, indeed it is also common to have enantiomers where one is the active compound and all other enantiomers inactive. This makes the separation selection and also the prediction of the properties more difficult. Measurement of the required physical properties can also be problematic due to the difficulty of producing a pure sample of any by-product. There is therefore a substantial gap in the currently available property prediction methods to be filled. The optimisation techniques mentioned for solvent selection need to be further developed to take account of wider solvent issues and could also be widened to route selection including formulation of active products i.e. surfactant selection etc. In addition visualisation tools along with optimisation that allow selection of separation scheme taking into account efficiency of separation are being developed by the CAPEC group (Bek-Pedersen et al.[22])and others and will prove very useful. Solvent selection tools will also be greatly improved when reaction effects are better predicted. 8. CONCLUSION Early Evaluation tools are proving very useful in improving route selection practise, bringing chemical engineers and chemist together and facilitating co-current development that is focussed much earlier reducing the necessary experimentation and development time-scales. The paper considered the benefits of making the tools available to the numerous and wide range of people involved in the many development processes that are required to facilitate the manufacture of a new successful product.
REFERENCES 1. Carpenter, K.J., 2000, Chemical engineering in Product Development- the application of engineering science, Entropic 223,4 2. Carpenter ,K.J. 16th International Symposium on Reaction Engineering (ISCRE 16), to be published in Chem.Eng.Sci., 2001 3. Plant. P.,July 1999, Internal Zeneca Agrochemicals, The development chain. 4. Gani, R. and O'Connell, J. P., Computers Chem Engng, 13(4/5), pp397-404, 1989. 5. Cox, B. G., 1994, Modern liquid phase kinetics, Oxford Chemistry Primer Series 21, Oxford University Press 6. Cox B.G. and Parker, A.J,1973, J. Am. Chem. Soc., 95,408 7. Chastrette JACS Vo1107 No 1 1-11 1985 8. Odele O. and Macchietto S. Fluid Phase Equilibria. Vol 82, pp 57-54, 1993 9. E.N. Pistikopoulos ,S. K. Stefanis Computers Chem. Engng 1998 Vol 22, pp 717-733, 10. Hostrup, M., Harper, P.M., Gani, R., Comput Chem Eng, 23, pp 1394-1405, 1999. 11. Harper, P.M., 'A Multi-Phsae, Multi-Level Framework for Computer Aided Molecular Design, Ph.D.-thesis, 2000 CAPEC Danish Technical University 12. Bavishi P, MEng Thesis-2000, Imperial College supervised by Adjiman C. 13. Knapp and Doherty AIChE Journal Vol.40 No2 p243-268 1994
40 14. 15. 16. 17. 18. 19. 20. 21. 22.
Doherty M. and Perkins J. Chemical Engineering Science1978 Vol.33 281-301 Doherty M. and Perkins J. Chemical Engineering Science1978 Vol.33 569-578 Doherty M. and Perkins J. Chemical Engineering Science 1979 Vol.34 1401-1414 Doherty M. and Perkins J. Chemical Engineering Science 1982 Vol.37 381-392 Dongen and Doherty M. Chemical Engineering Science 1984 Vol.39 No.5,883-892 Stichlmair Chem.Eng.Prog. 1989 63-69 Stichlmair AIChE Journal 1992 Vol.38 No.10, 1523-1535 Cordiner JL, Nielsen TL, "Gap analysis" Syngenta Internal Report 1999 Bek-Pedersen, E., Gani, R., Levaux, O., Comput Chem Eng. 24 (2-7) pp253-259, 2000
ACKNOWLEDGEMENTS Permission to publish from Syngenta is gratefully acknowledged. Thanks to a great many friends and colleagues for advice and information, especially: Dr Keith Carpenter and Dr. Alan Hall, Dr Will Wood of Syngenta Technology and Projects and James Morrison Consultant.
European Symposiumon ComputerAided ProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rights reserved.
41
A Muitiscale-Multifaceted Approach to Process Synthesis and Development K a M . Ng Department of Chemical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong This paper presents a strategy for the synthesis and development of chemical manufacturing processes. By viewing an undertaking from various length scales and its different facets, this approach is expected to lead to an improved process with reduced time and effort. Some of the issues, methods and tools for this approach are illustrated with processes involving crystallization and solids processing steps. 1. I N T R O D U C T I O N The global chemical processing industries (CPI) are undergoing profound changes. With the increasingly free flow of capital, technology, and human resources, the CPI in developing countries are rapidly expanding. For example, Asian countries with a large internal/regional market and low labor costs are witnessing an annual growth rate of over 10%. While a large fraction of production is destined for import substitution for countries such as China, other countries such as Singapore and Thailand are poised to become significant net exporters. In response to the increased competition, the companies in the developed countries have been locating some of their chemical plants overseas as well as repositioning themselves through divestitures, mergers and alliances. They are also intensifying their effort to innovate and to refocus on high-value-added (HVA) chemicals in search of a higher return. The allure is clear when one compares the 8% profit margin in a typical chemical firm to the 20% figure of a pharmaceutical company in the U.S. [ 1]. All of this has led to new challenges to the CPI (Table 1). One is the accelerated pace of process design and development [2]. For commodity chemicals, it used to take eight years Table 1. Challenges to Process Synthesis and Development Accelerated pace of process design and development Increasing product complexity Increasing process complexity Cost reduction from a product life viewpoint Environmental considerations Technology transfer among people, laboratories and sites Expanding project scope to entire business chain
42 or so to build a large-scale grassroots plant starting from basic chemistry [3]. Now the desirable time is 3 to 4 years. For pharmaceuticals, rapid process development is essential in order to produce enough materials for clinical trials and to prolong effective patent lives. Due to the shift to HVA chemicals, product specifications are becoming more complex. Rather than just purity, particle size distribution, particle shape, morphology and polymorphism are specified product attributes. This is accompanied by an increase in process complexity, involving multiple processing steps and complex chemistry. The economic pressure has also greatly intensified. Whether the objective is to reduce cost or improve profit, it is critical that process development be viewed over the entire product life. For example, the regeneration of monomers from PET is becoming increasingly common for environmental reasons. An accelerated pace of development demands an even closer collaboration among traditionally different functional units such as chemistry, chemical engineering and business. If one thinks it is challenging to coordinate the efforts of a chemist and a chemical engineer in close proximity, it is now necessary to coordinate efforts of people with different cultures in laboratories all over the world. All these technology transfer issues need to be resolved effectively. To take advantage of any possible synergism and leverage, the project scope is also expanding. It is desirable for a company to capture a coherent set of technologies and supply chain for the entire enterprise or product line in order to gain a competitive advantage. 2. M U L T I S C A L E - M U L T I F A C E T E D APPROACH These multiple demands necessitate a multifaceted approach to process synthesis and development (Table 2). To demonstrate how some of these facets come together as a coherent whole, we consider a specific area-the synthesis and development of processes where crystallization and solids processing play an important role. This is particularly relevant to HVA chemical and pharmaceutical manufacturing processes, many of which have solids as an intermediate or the final product.
43 Table 2. A Multiscale-Multifaceted Approach Facet
Conventional Approach
New Strategies
Length scale
Individual scale one at a time
Multiscale
Scope
Single plant
Enterprise
Financial goal
Cost reduction / Improved profit
Product life economic analysis
Knowledge base
Scattered
IT-based integration
Process vs. Product
Process engineering
Product-based processing
Technical focus
Science and Engineering
Science/Engineering aided by design rules and workfiow
Experiments and Design
Separated
Integrated
Chemists / Chemical Engineers
Limited collaboration
An integrated team
Scaleup
Rule-based
Fundamental models
Timeline
Sequential
Concurrent
Procedure
Ad-hoc
Systematic, hierarchical
The foremost element of this approach is a multiscale perspective. The time and length scales for a solids plant is shown in Figure 1. Similar concepts have been advanced by Villermaux [4], Sapre and Katzer [5], Lerou and Ng [6], and Grossmann and Westerberg [7].
44 10-16
10-10
_//J//I
10-6
10 -4
10-2
10 0
10 2
I
,
,
, . . . . . . . . . . .
108m
10 4
,
r //
109S
4
iEnterprise Plant i.............,..................z ~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Reaction Chemistry
1
:................
!
............... r........... :ji Crystallizer
J
--
10 4
--
10 2
--
100
_
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--
10-4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fluid Dynamics and Transport
I ii
1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
~ . . . . . . .
Particle Nucleation i and Growth ........ ~................................... ~........... i'
i
i
i i i i
/ / /
-
10-14
Molecular / Electronic e i
10-16
Figure 1. The length and time scales covered in the multiscale-multifaceted approach for a crystallization-based plant. The boundaries for each region are not meant to be exact. Contiguous regions signify potential interactions and synergism between them. The more the regions overlap, the more the scales interact. For example, there is a significant overlap between the enterprise and the plant. Indeed, a chemical plant is never designed in isolation. It is absolutely critical to consider the entire global enterprise, that is, the product line both upstream and downstream of the business chain. Make-buy decisions, toll manufacturing, and alliances can have an enormous impact on profitability. The interface between the plant and crystallizer (equipment scale) is the traditional area of process synthesis of a single plant [8]. The effect of reaction, and transport on equipment design has been a rich and fruitful research domain. Molecular considerations are critical for reaction chemistry, etc. but there is limited benefit in linking them to enterprise level issues. As stressed in Table 2, the scope of work in process systems engineering (PSE) has expanded tremendously. Driven by competitive pressures, in addition to our traditional tasks in process research and pilot plants (Figure 2), we have to redouble our efforts on technology transfer, and manufacturing and operations [9, 10]. Building on its strengths in manufacturing, the Huntsman Corp. has risen to the fourth place in terms of sales among all U.S. chemical producers in 1999 [ 11 ].
45
Process Research & Development
Pilot Plants
Technology Transfer
~] Manufacturing & Operations
9 Processchemists, Process synthesis engineers,
9 Processengineers
9 Processengineers, technicians and 9 Testingand validation operators
9 Processengineers, technicians and operators
Financial analysts 9 Laboratory experiments
9 Redesignand adaptation
9 Adaptationat manufacturing sites
9 Benchmarkingand best practices
9 Trainingand documentation
9 Process improvements
9 Full-scalevalidation Figure 2. Synthesis and development of manufacturing processes. In considering the business issues, it is imperative to conduct product life economic analysis to obtain maximum return. It is also important to tie together product and process design [12]. Product attributes are defined based on customer demands. The materials with the proper properties are identified and the process to transform the materials into the final desired product is then synthesized. As we proceed to the equipment scale and below, we begin to take advantage of the relevant basic sciences. However, there is a limit as to where basic sciences can take us in process development. The use of design rules can significantly reduce the search space, thus shortening the time required for process synthesis and optimization. It is equally important to closely integrate process synthesis and experiments. One has to identify the critical experiments to perform, and the regions of parameter and composition space where the data can have the highest impact [13]. For example, while reaction data focusing on high yield are useful, selectivity data at a low conversion can be more important because of the recycle of unconverted reactants. In this regard, the integration of chemistry and chemical engineering is essential to create a high quality process in the shortest possible time. This sentiment was echoed in a discussion of the experiences, barriers, benefits and the path forward in building an integrated team of chemists and chemical engineers in process development [ 14]. Because of time pressure or technological limitations, scaleup tests are performed by necessity. While some scaleup rules are based on experimental observations or simply experience, we need to fathom the fundamental reasoning behind the scaleup rules whenever possible. For example, it has been shown that the feed addition time should be considered in scaling up a semibatch liquid phase reactor in the micromixing regime although this is not commonly practiced. Also, the use of constant agitator tip speed is not a good scaleup rule [ 15]. For a typical process development project, the performance of a process in terms of economics, safety, operability, etc. generally increases while the uncertainty of the expected performance decreases with the total amount of effort (Figure 3). However, one has to recognize that there is a technological limit beyond which our additional effort has a rapidly diminishing return. For example, in the design of a new large-scale multiphase reactor with complex reactions, there is always a degree of uncertainty until the commercial reactor is up and running for a period of time. To minimize time, it is essential that development tasks be performed concurrently instead of sequentially. Obviously, there is an upper limit to effective
46 effort/time (i.e., concurrent effort) in that, depending on the nature of the problem and the available resources, some tasks have to be performed sequentially.
Total Effort = Time x Concurrent Effort i.e.
Effective Effort T-~me )
Figure 3. The dependence of process performance and uncertainty on the total amount of effort invested in the process development project. This is exactly why systematic, hierarchical design procedures can play an important role in process synthesis and development. Before delving into the details of a development project, a preliminary plan can be drawn (Table 3):
47
Table 3. Preliminary Planning in Process Development 1. Identify and prioritize the project objectives and decision points. 2. Identify the tasks to meet those objectives. 3. Identify the technical expertise, time and resources required for achieving those objectives. 4. Identify the relevant in-house functional units and the possibility of outsourcing R&D. 5. Organize the workflow among the functional units. 6. Identify the achievables and non-achievables based on the available time, technological limits, and both human and monetary resources. 7. Identify the time horizon for the entire project (Be Realistic). Such design procedures can help us anticipate the key decision points and the data and tools to achieve the project objectives. It also helps us to devise the workflow to integrate the contributions from disparate functional units. Admittedly, some of the stumbling blocks are hard to foresee even for experts in systematic design, particularly for completely new processes. Yet this preliminary exercise, if properly executed, is expected to provide a roadmap to expedite the entire project. 3. SYNTHESIS AND DEVELOPMENT OF CRYSTALLIZATION AND SOLIDS PROCESSES To be more concrete, let us examine a few selected issues of the multiscale-multifaceted approach in the context of crystallization and solids processing [16]. Enterprise and molecular scale issues will not be considered. Technical Focus
Table 4 shows a taxonomy of the issues in the design of solids processes as viewed from four length scales-plant, unit operations (or equipment), continuum and particle. Tools from business, management and basic sciences are required for a full consideration of a process development project.
48 Table 4. Subjects Examined at each Length Scale, and the Typical Problems and Tools Scale Plant
Unit Operation
I
Subjects Process synthesis Process simulation Process optimization Process control
Typical Problems Multicomponent crystallization Solid-liquid separation downstream of crystallizers Granulation / tableting operations
Equipment performance Equipment sizing Equipment costing
Mixing-Demixing Blender, hydrocyclone, filters Size Change Processes Crystallization Comminution, granulation
Heat & mass transfer Kinetics Discretized population equations CFD
SLE phase diagrams Flow in filter cakes Fluidization Pneumatic conveying
Thermodynamics Continuum mechanics
Solvent effect on crystal shape Nucleation and growth Particle breakage Interparticle forces
Solid-state physics Quantum mechanics Solid mechanics Statistical mechanics Colloid and interface science
Continuum SLE, Solubility Flow of powders and slurries Particle
Particle Attributes Composition, PSD, density, strength, shape, chirality, polymorphism
i
Tools Hierarchical design Optimization tools Simulation tools Process economics
At the plant level, the subjects to be considered are those of the entire chemical plant. Hierarchical design procedures and optimization techniques are some of the tools for addressing these problems. Process synthesis is impossible without the necessary knowledge about the units. At the unit operation scale, we are concerned with equipment selection, sizing, operations, control and innovations. There are two classes of solids processing equipment, depending on whether the particle size changes or not. In addition to the fundamentals of transport, population balance equations are essential for the problems at this scale. Computational fluid dynamics (CFD) tools are now routinely used in process development. Description of the continuum behavior necessitates both thermodynamic considerations such as solid-liquid equilibrium (SLE) and dynamic considerations such as flow of powders and slurries. At the particle scale, we focus on how to obtain the various desirable solids attributes such as size, shape, chirality, polymorphism and density for which tools such as solid-state physics, and solid mechanics are needed. Recently, separation of enantiomers by means of crystallization [17] and chromatography [18], and prediction of crystal shape [ 19] have received much attention.
49
Preliminary Planning Workflow is devised based on the problems and tools thus identified. Specifically, let us consider a process development project where an active pharmaceutical ingredient (API) is recovered from a multicomponent mixture by crystallization. After filtration, washing, dewatering and drying, we obtain the cut with the acceptable particle size distribution (PSD) through bulk solids processing. Then, the API is blended with various excipients before tableting and coating. Figure 4 shows the overall workflow for this project. The central column shows the tasks or problems to be considered. The rectangles on the left indicate the data required for the corresponding problems. Experiments are necessary because the state of the art simply does not allow reliable predictions of these quantities [20]. While it is possible to predict SLE for ideal simple eutectic systems, it is difficult for complex molecules. And it is impossible for systems with compound formation. Washing efficiency is simple to measure but it can have considerable impact on cost because the wash solvent is recycled through the distillation system for purification. Similarly, breakage depends not only on the material properties, but also on the processing history of the particles. Obviously, some of these measurements can be and should be carried out concurrently. SLE data, and nucleation and growth kinetics were often not measured in a project because of time constraints. However, in the past several years, a number of industrial laboratories have been set up to measure the relevant quantities on a routine basis. A similar situation existed for reaction kinetics although systematization of those measurements are much farther along. In comparison, except for a few companies, measurements for bulk solids processing are less systematic, with the work relegated to specialists in individual unit operations, who may overlook some of the systems problems. The rounded rectangles on the right indicate the techniques, design procedures and basic sciences required for the corresponding tasks. Let us examine some of the methods and procedures although, within the scope of this article, it is not possible to get into the myriad workflow details required for coordinating all the functional units.
50
SLE Experimental Data
\ ,~/ Crystallization ~ ] Separation "~ Scheme /
Filter Cake Resistance; Washing Efficiency
~/,-\,, DownstreamCrystaliZerproces~sing
(Visualizationof High-D'~ PhaseDiagrams; | Movementsin | ~.. Composition Space )
Solid-Liquidsynthesis ProcedurePr~ 1
Nucleation and Growth Kinetics
~/
Crystallizer "~ Design and ~ "~ Operating Policy ,/
(Heat & Mass Transfer;'~ Crystallizer Types & ~ ~. OperationModes /
Specific Rate of Breakage; Work Index
~/ "~
BulkSolids ~_~Synthesis Procedure to~ Processing / " ~MeetPSD Requiremen~
r
Hamaker Constant, Dielectric Constant, Unconfined Yield Stress
(\ Operating Issues ~ - ~ ~ in Solids Processing
V~ and Effects 1 Acting on Particles
Figure 4. Overall workflow for the synthesis and development of a process with crystallization and solids processing steps. Systematic Design Procedures Plant Scale
There exist a wide variety of crystallization techniques, ranging from fractional crystallization [21-25], extractive crystallization [26, 27], reactive crystallization [28, 29], and drowning-out crystallization [30]. It can be shown that these crystallization processes can be represented by four basic crystallization-relatedmovements in composition space, namely, cooling/heating, stream combination/split, solvent addition/removal and MSA (mass separating agent) addition/removal [31]. Depending on the relevant features on the phase diagram, this synthesis procedure generates the feasible schemes without being restricted to a specific separation technique. It is applicable to multicomponent mixtures by using cuts and projections to visualize high-dimensional systems (i.e., 4 or higher) [32, 33]. This approach is also applicable to other unit operations such as distillation, and membrane processes [34]. However, the crystallization system does not exist in isolation. It is closely linked to the downstream processing system in which dry crystals are produced. A systematic procedure is available for the synthesis of flowsheet alternatives for treating the crystals as well as the crystallization, wash, and recrystallization solvents [35]. The PSD of the dried crystals can be one of the specified product requirements. For example, in a pharmaceutical tableting
51 process, the PSD of the API and that of the excipients have to be controlled for a number of reasons. The relevant synthesis procedure as well as the simulation method for the entire plant based on discretized population equations has been proposed [36-37].
Equipment Scale We have developed a hierarchy of solids processing models. The basic ones are heat and mass balances. The intermediate ones consider kinetics and mass transfer while the detailed ones account for changes in PSD. Particle size can change due to breakage, agglomeration, nucleation, growth and dissolution (Figure 5). Each mechanism has been modeled separately [38-41]. An equipment unit with any combination of these mechanisms can be modeled by selecting the corresponding mechanisms in the population balance equation, as shown below:
dNi-'dNi (L_~_i_)Agglomerafion+ dNi (L_~)Growth& (--dT-)Breakage +dNi
+dNi Qout (L'~)Dissolution+Qin V, miin +--(-, m i
Equipment Unit
Ts
I Mechanisms
Breakage Agglomeration ~.~ Nucleation and Growth "Dissolution
Figure 5. Modeling by functionality.
Particle Scale Solids plants are plagued by operational problems. To identify the potential problems and resolve them effectively, we have to consider both the continuum and particle scales. As Figure 6 indicates, the combination of material characteristics, particle attributes, equipment design and operations determines the bulk mechanical properties of the powder as well as the forces acting on the particles. Whether an operational problem such as particle breakage, adhesion, segregation, arching (as in a hopper) occurs or not depends on the causal effect and the effect that opposes it. For example, encrustation on equipment walls can be a problem if the adhesion force is greater than the forces that entrain the particle. A systematic procedure has been proposed to examine these problems [42]. 4. S O F T W A R E TOOLS To facilitate the execution of this approach for process synthesis and development, we have been developing a number of software tools such as those for the visualization of phase diagrams, for tracking PSD around a chemical plant, etc. While related, these codes are modular in nature. They are based on Visual Basic or Java, and can be linked to other codes (Figure 7). In addition to computations, each program includes design rules to identify decision points and shows how to make those decisions, parameter maps for typical model parameter values, and cause-effect tables for showing the likely consequence of an action.
52
Material Characteristics
~
es~ article Attribut
Hamakerconstant Dielectricconstant Young'smodulus
[ ] L
PSD 1 Shape 1 Composition/
~
gl quipment Desi
(' /
[ Geometry l [ Constituentparts 1 L Materialproperties/
Operating ~ 1 Conditions /
| Speedofmovingparts I 1 Temperature [ L Humidity )
Performance of Individual and Interconnected Units Figure 6. Interrelationships of factors affecting the performance of a solids processing unit (after Wibowo and Ng, 2001).
Visual Basic / Java User supplied codes in Fortran, C++
LL
9 Calculates and plots high-D phase diagrams. ~ Tracks PSI) around the plant. o Back-calculates model parameters.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Public codes such as Excel, GAMS
OLE
9 Things-to-consider tables. ~ Parameter maps. ~ Cause-effect tables. ~
I
I
J
"
Figure 7. Modular computer codes with built-in knowledge-base and graphical output for process synthesis and development. 5. CONCLUSIONS Process systems engineering is an evolving subject [43]. As the conventional part of chemical engineering as a discipline matures, PSE is destined to expand in both scope and
53 depth. This trend is captured in this multiscale-multifaceted approach for process synthesis and development. The enterprise is viewed scale by scale in a hierarchical manner. In conjunction with other facets of this approach, it is expected to help exploit more fully the synergism within the enterprise, thus leading to an improved process with reduced time, effort and money. To put this approach in practice, the perspective is merely a starting point. It serves as a framework to which methods, procedures, techniques, etc. need to be added so that the user can tackle a product-based process synthesis and development project with an efficient workflow. Some of the design procedures and methods, in conformity with this approach, for the synthesis of processes with crystallization and solids processing steps are discussed. Similar methods have been developed for reaction systems [44-49]. It should be emphasized that this approach does not replace the need for experts in various sub-fields yet. For example, segregation of solids is a complex subject and demands careful treatment if it is encountered in process development. Similarly, mixing and reaction, particularly those in multiphase systems, can still be a challenging problem. However, technology integration relentlessly marches forward, providing new computer tools for both traditional tasks and innovations. While not illustrated in this article, business issues, detailed workflow, technology transfer, and molecular considerations are an integral part of this approach. Contributions from researchers in these areas and industrial sectors other than solids processing are needed to realize the full potential of this approach. We anticipate an expanding role of PSE in an increasingly integrated global CPI in the coming years. 6. ACKNOWLEGMENT The financial support from the National Science Foundation (Grant No. CTS-9908667) is gratefully acknowledged. The author also likes to acknowledge the invaluable impact of the industrial collaborations on my thinking. Specifically, I would like to thank George Stephanopoulos, Yukikazu Natori, and Lionel O'Young of Mitsubishi, Jan Lerou formerly of DuPont, Prabir Basu of Pharmacia, and Alan Leviton of Rohm and Haas for their advice and guidance. REFERENCES
1. Arora, A., Landau, R., and Rosenberg, N., Chemicals and Long-Term Economic Growth: Insights from the Chemical Industry, Wiley, New York, NY, 1998. 2. Pisano, G. P., The Development Factory: Unlocking the Potential of process Innovation, Harvard Business School Press, Boston, MA, 1997. 3. Vogel, H., "Process Development," Ullman's Ency. Ind. Chem., 5th ed. Vol. B4, B. Elvers, S. Hawkins, and G. Schulz (eds.), VCH Verlagsgesselschaft, Weinheim, p. 438, 1992. 4. Villermaux, J., Trans. IChemE, Part A, 73, 105 (1995). 5. Sapre, A. V., and Katzer, J. R., Ind. Eng. Chem. Res., 34, 2202 (1995). 6. Lerou, J. J., and Ng, K. M., Chem. Eng. Sci., 51, 1595 (1996). 7. Grossmann, I. E., and Westerberg, A. W., AIChE J., 46, 1700 (2000). 8. Douglas, J. M., Conceptual Design of Chemical Processes, McGraw-Hill, New York, 1988.
54
9. Technology Vision 2020. The U.S. Chemical Industry, ACS, Washington, D.C., 1996. 10. The Process Development Division of the American Institute of Chemical Engineers, www.pd-aiche.com. 11. Chem. Eng. News, June 26, 2000. 12. Westerberg, A. W., and Subrahmanian, E., Comput. Chem. Eng., 24, 959 (2000). 13. O'Young, L., Natori, L., Pressly, T. G., and Ng, K. M., Comp. Chem. Eng., 21, $223 (1997). 14. Chemists and Chemical Engineers: An Integrated Team for Process Development, www.pd-aiche.com. 15. Samant, K. D., and Ng, K. M., AIChE J., 45, 2371 (1999). 16. Rajagopal, S., Ng, K. M., and Douglas, J. M., Comput. Chem. Eng., 16, 675 (1992). 17. Schroer, J. W., Wibowo, C., and Ng, K. M., AIChE J., in print (2001). 18. Migliorini, C., Mazzotti, M., Zenoni, G., Pedeferri, M., and Morbidelli, M., AIChE J., 46, 1530 (2000) 19. Winn, D., and Doherty, M. F., AIChE J., 46, 1348 (2000). 20. Basu, P. K., Mack, R., and Vinson, J. M., Chem. Eng. Prog., 95(8), 82(1999). 21. Dye, S. R., and Ng, K. M.,AIChE J., 41, 2427 (1995). 22. Cistemas, L. A., and Rudd, D. F., Ind. Eng. Chem. Res., 32, 1993 (1993). 23. Berry, D. A., and Ng, K. M., AIChE J., 42, 2162 (1996). 24. Ng, K. M., Separations Tech., 1, 108 (1991). 25. Cesar, M. A. B., and Ng, K. M. 1rid. Eng. Chem. Res., 38, 823 (1999). 26. Rajagopal, S., Ng, K. M., and Douglas, J. M., AIChE J., 37, 437 (1991). 27. Dye, S. R., and Ng, K. M.,AIChEJ., 41, 1456 (1995). 28. Berry, D. A., and Ng, K. M.,AIChEJ., 43, 1737 (1997). 29. Kelkar, V. V., and Ng, K. M., AIChE J., 45, 69 (1999). 30. Berry, D. A., Dye, S. R., and Ng, K. M., AIChE J., 43, 91 (1997). 31. Wibowo, C., and Ng, K. M., AIChE J., 46, 1400 (2000). 32. Samant, K. D., Berry, D. A., and Ng, K. M.,AIChEJ., 46, 2435 (2000). 33. Samant, K. D., and Ng, K. M., AIChE J., in print (2001). 34. Pressly, T. G., and Ng, K. M.,AIChEJ., 45, 1939 (1999). 35. Chang, W. C., and Ng, K. M., AIChE J., 44, 2240 (1998). 36. Wibowo, C., and Ng, K. M., AIChE J., 45, 1629 (1999). 37. Hill, P. J., and Ng, K. M.,AIChEJ., 43, 715 (1997). 38. Hill, P. J., and Ng, K. M.,AIChEJ., 41, 1204 (1995). 39. Hill, P. J., and Ng, K. M.,AIChEJ., 42, 727 (1996) 40. Kumar, S., and Ramkrishna, D., Chem. Eng. Sci., 52, 4659 (1997). 41. Hill, P. J., and Ng, K. M., AIChE J., 42, 1600 (1996). 42. Wibowo, C., and Ng, K. M., AIChE J., in print (2001). 43. Perkins, J., Comput. Chem. Eng., 24, 1367 (2000). 44. Kelkar, V. V., and Ng, K. M., AIChE J., 44, 1563 (1998). 45. Kelkar, V. V., and Ng, K. M., AIChE J., 46, 389 (2000). 46. Samant, K. D., and Ng, K. M., AIChE J., 44, 1363 (1998). 47. Samant, K. D., and Ng, K. M.,AIChEJ., 44, 2212 (1998). 48. Samant, K. D., and Ng, K. M., AIChE J., 44, 2689 (1998). 49. Samant, K. D., and Ng, K. M., AIChE J., 45, 1808 (1999).
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rightsreserved.
55
Systems Biology: an Emerging Theme in Biological Research Gregory Stephanopoulos and William A. Schmitt Chemical Engineering Department, Room 56-469 Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307
The sequencing of genomes is rapidly altering the landscape of biotechnological and biomedical research. This is due to the direct access to gene-coding and intergenic sequence information, as well as genomics-based and other technologies that allow high throuput measurement of important classes of biological molecules. This information has the potential of elucidating gene regulation mechanisms and identifying genes implicated in disease, or perfecting industrially important high performance strains. To fully utilize this wealth of information, raw data must be appropriately upgraded through a variety of computational methods. A review of the computational issues associated with the utilization of genomic and physiological data is provided in this paper. These methods aim to integrate such data in a physiologically coherent framework and also provide the basis for a quantitative description of cell function. The introduction of computational methods to integrate and quantify biological data is an important step in the development of the emerging field of systems biology.
1. INTRODUCTION We have entered a period of rapid change in biological research. These changes are transforming traditional approaches in biological studies and are also generating new opportunities in biotechnology and biomedical research. At the core of these changes are recent developments in genomics and the associated need for integrating in a coherent framework diverse measurements of biological function. This integration underlies the emergence of systems biology as a field that aims to provide a global and quantitative understanding of cellular function. In order to materialize this vision a number of problems need to be solved, both experimental and computational. The objective of this review is to identify and describe in some detail the computational challenges before us. There are four driving forces that provide the impetus for these changes. The first is genomics: specifically the sequencing of genomes of many species. At least 50 species had been sequenced by the end of 2000 and it is expected that the genome of all industrially important organisms and many biomedically relevant species will be sequenced within the next 1-2 years (http://www.ncbi.nlm.nih.gov/). The second is the extensive monetary investment in the life sciences from both the public and private sectors. It is estimated that during the past 20 years more than 200 billion dollars have been invested in the life sciences in the US alone. Although most of the research activity has been aimed towards health-
56 related problems, the resulting fundamental advances in the life sciences are equally applicable to other fields. There are presently molecular biological methods available that can be applied on a routine basis to introduce controls at the genetic level and to construct optimal genetic backgrounds for potential medical and industrial applications. The enormous opportunities created by these fundamental advances are the third driver for change. Besides obvious applications in the field of medicine, other areas include production of chemicals (cell factories), pharmaceuticals and specialty chemicals (in particular, molecules of specific chirality), materials with special properties, and environmental applications [1]. The final driver of change is the development of technologies that probe the expression, proteomic, and metabolic phenotypes. The development of DNA microarrays allowed the genome-wide measurement of gene expression in cells [2, 3, 4]. This advance has sparked an interest in the development of other high-throughput technologies for the measurement of other important classes of cellular variables. Ongoing research is investigating the possibility of measuring on a cell-wide basis proteins (proteome) [5, 6, 7] and metabolites (metabolome) [8, 9]. The integration of such measurements could provide a rigorous understanding of cell physiology in its entirety (physiome).
Fig. 1. Probing cellular function. A schematic of cellular processes initiated by a receptorligand binding event that leads, through signal transduction to transcription (mRNA), translation, and ultimately metabolic and other cellular reactions, mRNA transcript populations and fluxes can be presently assessed through developing tools such as DNA microarrays and metabolic flux analysis techniques. Other methods aiming at the high throuput measurement of proteins and metabolites are under development (from M. Klapa). Figure 1 demonstrates that, as important as such measurements are, they do not individually provide a complete picture of cellular function. Instead, all available information must be considered simultaneously to understand the complex interactions of cellular function. To achieve this goal, these measurements must be integrated in a coherent framework of cell physiology and quantified to the extent that this is possible. A number of computational problems arise from this integration effort. These problems can be classified
57 as those associated with the utilization of sequence information, the measurements of cellular parameters, the mechanistic analysis of cellular black-box analysis of cell-wide measurements. All of these problems deal of biological information that defines, in essence, the field of bioinformatics
upgrade of raw data, and finally with the upgrade [ 10].
2. SEQUENCE-DRIVEN COMPUTATIONAL PROBLEMS The raw data generated by sequencing projects consist of base-pair sequence lists of individual DNA fragments obtained from restriction digests of the entire genome. Before any systematic sequence analysis is undertaken, these individual DNA sequences must be organized in a contiguous and coherent sequence that is also complete and unique. Additionally, absent in these data are the location and function of the various genes and other genomic factors. Therefore, key computational challenges in the use of raw sequence data include [11, 12, 13, 14]: a) Integrating sub-sequences together to form the entire genome sequence by uniquely matching the prefixes and suffixes of a large number of individual sub-sequences. As this sequence reconstruction may yield non-unique answers, multiple genome coverage is often desired and determining the exact extent of coverage is an interesting problem in its own fight b) Identifying open-reading frames (genes) once the entire genome sequence has been reconstructed c) Identifying gene splicing sites and intron location in eukaryotic cells d) Determining gene function (gene annotation) e) Determining sequence patterns of regulatory sites that are important in understanding gene regulation and expression in normal and diseased tissues f) Hypothesizing evolutionary relationships between organisms by construction and analysis of sequence-based phylogenetic trees Although quite diverse in nature, many of the above questions can be answered by solving a few generic types of problems including [11, 12]: a) Given two sequences align them optimally and determine the extent of homology between them (sequence alignment) b) Solve the above problem (a) for many sequences (multiple sequence alignment) c) Discover patterns characteristic of sequences that belong to the same family by virtue of the fact that they code for the same type of gene in different genomes (pattern discovery in sequences) d) For the case of protein sequences (see below), discover small characteristic sequences (motifs) that are common in proteins of the same family or by virtue of appearance in a protein database at significantly higher than expected frequency Similar problems need to be solved in the analysis of protein sequences. Such problems include a) identification of functional motifs in protein sequences, b) correlation of protein sequence to protein structure and function, and c) extensive determination of protein sequence homologies among proteins of similar function [ 15, 16, 17]. These problems have been the subject of intense investigation in recent years. Methods such as BLAST, FASTA, ORFinder (http://www.ncbi.nlm.nih.gov/gorf/gorf.html), TIGR's Glimmer (http://www.cs.jhu.edu/labs/compbio/glimmer.html), and IBM Research's Teriesias (http://www.research.ibm.com/bioinformatics/) have been developed and are extensively
58 used for sequence comparisons, homology analyses, gene identification, and the development of function discovery techniques of varying success. It is important to note that concepts familiar in the systems theoretic and optimization literature, such as dynamic programming [11, 18], Markov chains [19], and other statistical methods [20, 21] provide the basis of these sequence analysis approaches. These methodologies and algorithms make use of extensive genomic and proteomic databases, many of which are in the public domain and easily accessible through the web (http://www.ncbi.nlm.nih.gov/, http://www.expasy.ch/). The continued efforts of researchers in this area are certain to lead to both improved and novel tools for upgrading the content of the immense sequence libraries being generated.
3. COMPUTATIONAL ISSUES IN UPGRADING RAW MEASUREMENTS At the present time, there are three classes of biological measurements which can be performed at a relatively high-throughput rates: mRNA transcript levels using DNA microarrays, protein levels using 2-D gel electrophoresis and HPLC-MS, and metabolic flux measurements using NMR spectroscopy and/or mass isotopomer measurements by GS-MS. We describe below problems associated with the derivation of the relevant biological parameters from the raw measurements. 3.1. DNA microarrays The acquisition, cleaning, adjustment, and analysis of micro-array data are all important steps in the determination of reliable mRNA transcript measurements requiring customized designs for each individual system. Filtering the raw fluorescence intensities is important in removing spurious signals. Replicate spots on the same array provide measurement multiplicity that can be used for error analysis, background noise and signal to noise ratio calculations, and determination of confidence levels. Ratio adjustments (normalization) have a large effect on the true and false positives but a very small effect on discovered classes from clustering techniques [22, 23] typically used for microarray expression analysis [24]. Typically, gene specific data are normalized by the average or total signal for each fluorophore. This procedure is based on the assumption that differences among microarrays in brightness and total RNA added for each fluorophore will be corrected by such normalization. This normalization strategy, however, alters the data such that values are now reported as fractions of the overall RNA pool. While this is inconsequential when the overall RNA population per cell is constant, under conditions such that total RNA population is undergoing dramatic changes on a per cell basis a more robust normalization basis is required. This is particularly true in the increasingly popular cases of genome-subset (or partial) arrays, where the specific genes probed by the array are only those genes expected to change under the experimental conditions evaluated. Furthermore, by definition total mRNA levels are expected to differ at the different conditions studied raising questions about a normalization that is based on the total RNA. Depending on the goal of the study, care should be taken to normalize and adjust data in a manner consistent with the experimental questions under consideration. Specifically, overall conclusions about cell transcriptional profiles gene induction or repression and associated cell physiology are strongly influenced by the technique of normalization and can lead to inconsistent conclusions about cellular states.
59 3.2. Proteomic data Although 2-D gel electrophoresis effectively separates most proteins, the analysis of individual spots to determine the identity and amount of the protein represented in each spot poses experimental and computational challenges. Mass spectroscopy (MS) analysis typically is applied to peptide protein fragments resulting from the trypsinization of the protein(s) in each spot of the 2-D gel. Such peptides are identified by their molecular weights and then the original protein is determined through an elaborate reconstruction process that makes use of extensive databases of protein sequence and molecular weights. MS is applied routinely for this purpose, however, MS readings can be complicated by the presence of bound sugars (in glycosylated proteins), phosphate groups (phosphorylated proteins), or other posttranslational modification steps. Additionally, the possibility that more than one proteins may be present in a single spot adds an additional dimension to the reconstruction problem. This problem is similar to the problem of reconstructing the entire genome sequence from the sequences of individual DNA fragments discussed in a previous section. Multiple coverage of a protein, resulting from replicate trypsinizations and analyses of the resulting protein fragments enhances the accuracy of reconstruction and reliability of protein identification. [25].
3.3.
Isotopic tracer analysis
The determination of metabolite fluxes through metabolic networks is a critical step in the description of cellular function and physiology. Fluxes are the actual rates at which metabolites and proteins are processed through pathways or signaling networks and, as such, they represent the actual cellular physiological state. Macroscopic fluxes can be calculated from overall metabolite balances and extracellular metabolite measurements [26, 27], however, more detailed flux determination relies on the ability of the researcher to observe individual reactions within the network. The existence of reversible pathways, splitting pathways and metabolic loops complicates this process, requiring additional information beyond the rates of cellular inputs and outputs. Isotopic tracers allow us to mark specific carbon atoms of the input metabolites that are subsequently distributed throughout the cell's metabolic network. In general, the distribution of tracer depends on the fluxes that one wishes to determine. Measurements of these tracer distributions can be obtained from the degree of label enrichment and mass isotopomer measurement of secreted or intracellular metabolites, made possible by 13C NMR and (LC) GC-MS analysis [1, 26, 27, 28]. Metabolic network fluxes are then determined by solving the inverse problem. This process of tracer data upgrade for flux determination is depicted in Figure 2.
60
Labeled Tracer
r-
luxe
~ Isotopomer~ ~ A , istributi ,,,......_.....~ ~ D o~
~
Isotopomer Measurements (NMR, GC-MS)
""............................. Informationupgrade ............................"'" Fig. 2. Schematic of metabolic flux determination. A labeled tracer is introduced and distributed among intracellular metabolites by metabolic reactions. The latter are mapped onto tracer dependent measurements, such as label enrichment or mass isotopomer fractions that can be obtained by NMR spectroscopy or GC-MS. Fluxes are determined by inverting this mapping thus upgrading the isotopic data information content.
4. M E C H A N I S T I C LEARNING FROM DATA Learning from experimental data can be accomplished in two separate forms: mechanistic and black-box learning. Black-box learning, discussed in Section 5, refers to those methods that do not seek to directly associate measurements with known cellular mechanisms, but rather use all available information to uncover data structure. Mechanistic learning, on the other hand, generates insight into the mechanics of cellular function and usually involves the calculation of important variables from primary measurements. Current challenges in mechanistic learning are discussed below. 4.1. Reconstruction of regulatory networks One generic problem that arises in connection with differential gene expression data (obtained from DNA microarrays) is to use such data to uncover the correlational structure of their expression patterns and then elucidate the gene regulatory mechanisms. We illustrate this problem by means of an example from the transcriptional control of a well-understood system, the lac operon. As depicted in Figure 3-A, transcription of the structural genes of the operon lacZ, lacY, and lacA (coding, respectively, for ~-galactosidase, permease and galactoside transacetylase), is controlled by a repressor protein coded by lacI. In the absence of lactose, this repressor protein (indicated by a star in Fig. 3-A) binds the operator region, thus physically hindering the binding of RNA polymerase at the promoter region. When lactose is present, it binds on specific sites of the repressor molecule. This binding allosterically diminishes the repressor's DNA binding affinity with the lac operator and allows RNA polymerase to carry out its function of transcribing the polycistronic mRNA. Besides the above negative control by lactose, the lac operon also has a positive control by glucose mediated by CAP (catabolite activator protein). When the energy level (ATP) is low or cAMP is high, CAP binds to cAMP and the resulting CAP-cAMP complex binds the CAP binding site of the lac promoter, an event that promotes helix destabilization and RNA polymerase binding. For illustrative purposes we have also added in Figure 3-A a positive control of cap and lacl expression by sigma factor c~3 under growth conditions, and negative controls of the same genes by sigma factors ch and ~2 in the presence of a rich medium.
61
Fig. 3. Illustrative example of the probing of gene regulation through differential expression analysis. Dark spots represent upregulated genes, light spots downregulated genes. Figure 3-B shows a schematic of simulated gene expression patterns that would have been obtained for the genes listed on the left column under the growth conditions indicated at the top row. For simplicity only three levels of gene expression are shown: induced, basal and repressed levels indicated by black, gray, and white dots, respectively. For illustrative purposes a set of maltose inducible genes (mal) has also been added with an assumed regulation similar to that by lactose. The data of Figure 3-B are representative of the type of data that will be generated from differential gene expression experiments aiming at the elucidation of transcriptional control mechanisms similar to that of Fig. 3-A. This means that one is called to solve the inverse problem of determining the regulatory structure of gene expression from differential gene expression data obtained under a variety of experimental conditions. An important difference between the simulated example of Figs. 3-A and 3-B and a real situation is that, in the latter case, one is dealing not with a well defined and small set of genes such as those of
62 Fig. 3-B but with the expression profile of the entire genome. As the latter typically encompasses a few thousand genes, identifying those that contribute in a meaningful way to the observed cellular physiological state presents a challenge that must be addressed with further research. As more and more expression data accumulate, a characteristic pattern of gene expression is bound to emerge. Determining such a pattern will allow for the identification of the key players in a particular genetic regulatory scheme. The most straightforward approach to finding these mechanistic structures is to search for correlation [20, 22, 29, 30] between gene expression patterns, but often issues of observability hinder this approach. For example, if all intermediate players are not included in the study, it is impossible to determine whether a correlation between an experimental condition and a particular gene's expression is a direct effect or a secondary relationship [31 ]. Thus, we are currently limited in many cases to understanding the overall connectivity of genetic regulatory networks and not all of the intermediate players [7, 32]. Furthermore, while the illustrative example shows a static system, many interactions will be transient [33] and therefore the frequency of sampling becomes critical to ensure such interactions are observed [29]. As costs decrease, however, we can expect access to time-scales at finer and finer intervals. The potential of this quantity of data, combined with well-designed experiments to ensure quality of observations, will allow maximum insight into the cell's regulatory structure.
4.2. Flux determination from isotopomer data Intracellular fluxes cannot be measured directly, but they are rather estimated indirectly from measurements of extracellular metabolite consumption and production rates, as well as those of the isotopic distribution of extracellular and/or intracellular metabolites after the introduction of labeled substrate(s). This indirect estimation is possible because unknowns (fluxes) and measurements are related through mass (extracellular rates) and isotopomer (isotopic distribution) balances [26, 27, 35, 36]. Positional isotopomers are the different labeling patterns of a molecule. A molecule of n carbon atoms has 2 n isotopomers. Figure 4 shows the balance equations around the isotopomers of metabolite D in the depicted artificial pathway. The latter converts a four carbon molecule A to B and then to C, D, E, and F through metabolic reactions v l, v2, v3, v4 and v5. Metabolite carbon atoms are shaded to indicate how atoms are distributed from a substrate to a product. In reaction 2, for example, the top two carbon atoms of molecule B partition into F while the bottom two carbon atoms form product molecule D. Similar balances can be written for all isotopomers in the metabolic network. The important point is that all these isotopomer balances are intrinsic functions of the individual metabolic fluxes, vj, that we wish to determine, and the distribution of the input and measured isotopomers.
63
Fig. 4. A bioreaction network schematic depicting carbon atom distribution among network metabolites. The fluxes of the metabolic reactions dictate the fate of each carbon atom (ovals) in the substrate A. Balances can be drawn around the species within the network such as the one shown for metabolite D. Even though the isotopomer fractions cannot be measured directly, they are linearly connected with quantities measurable by Nuclear Magnetic Resonance (NMR) Spectroscopy or Mass Spectrometry (MS). Therefore isotopomer balances provide a framework for the estimation of the unknown fluxes using measurements from isotopic tracer techniques. Compared with carbon enrichment analysis, which allows the use of label enrichment measurements only for the estimation of metabolic fluxes, isotopomer analysis provides a more extended framework for flux determination. It allows the use of all possible measurements from isotopic tracer techniques, label enrichments, fine structure of NMR spectra, and mass isotopomer fractions for the estimation of metabolic fluxes. This integration can be accomplished because all of the above mentioned measurable quantities are related to the isotopomer fractions. The ability to integrate such varied types of biological measurements maximizes the insight that can be drawn from an experiment.
4.3.
Signal transduction
A final application of mechanistic learning from high throughput biological data is the analysis of signal transduction pathways. Signal transduction pathways are the means by which extra-cellular conditions are communicated to the interior of the cell. Signaling occurs
64 via consecutive phosphorylation-dephosphorylation steps whereby the phosphorylated (active) form of an intermediate protein acts as a catalyst (kinase) for the phosphorylation of the subsequent step. The final outcome of a signaling pathway is often the activation of a transcription factor that, in turn, initiates gene [37]. To date, signal transduction pathways have been investigated in isolation from one another. It has become abundantly clear, however, that there is a great degree of interaction (cross-talk) of signal transduction pathways for the simple reason that they share common protein intermediates [38]. This introduces the possibility that one ligand may effect the expression of more than one gene or that the expression of a single gene may be effected by more than one ligand. Again, the network features of signaling provide a fertile ground for the application of concepts from network analysis in conjunction with expression and, in particular, proteomic data. The key here is to recognize that the main function of signaling pathways is propagation of information instead of molecular inter-conversions. As such, conservation equations, like the ones used in metabolic network analysis, are not available. This fact that complicates quantitative network analysis and the correct formulations and applicable principles that take this fundamental difference into consideration are yet to be developed.
5. BLACK-BOX LEARNING Black-box learning refers to those techniques which are based purely on statistics and not on fundamental biological mechanisms about the system. These techniques are advantageous because they are not limited by lack of knowledge about the underlying system. Because mechanistic knowledge is especially incomplete in many biological situations, these tools can be used to solve problems such as characterization and classification based purely on the experimental data on hand. If this data is sufficient in its coverage of the system in question, then differentiating characteristics and structure will be uncovered with very low probability of detecting accidental or insignificant relationships.
5.1. Data visualization through dimensional reduction Because of the high dimensional nature of high-throughput data, visualization of the results is impossible using traditional plots. This is particularly true of DNA microarray data, where thousands of genes may be represented in dozens of experiments, creating a tangle of information requiring innumerable 2-dimensional plots to untangle. However, the genes in these experiments are not expressed independently, but rather are correlated to each other through the underlying gene regulatory network. The method of principal component analysis (PCA) [18, 21, 39] allows for a simplification of the data to eliminate these redundancies. Essentially, PCA uses eigenvalue (or singular value) decomposition to express experiments as linear combinations of the variables (genes) present. Those linear combinations that describe the largest portion of the data are retained, while redundant information is easily eliminated. In this way an experiment may be reduced to a few principle components (PCs) which then may be plotted more easily on a typical 2- or 3-D plot. This allows the researcher to quickly see if there is underlying structure in the data and use this information to proceed in the analysis. A related technique is that of canonical discriminant analysis (CDA) [18, 21] which uses the same concept to instead identify combinations of genes that are most suited to the distinction of classes under investigation. Consider Fig. 5, which shows the distinction
65
between 3 sub-types of leukemia with CDA (40, data from 30). CDA identifies two canonical variables (CVs) which are of primary importance to this discrimination. CV1 separates T-ALL from the other two disease states, while CV2 separates B-ALL from AML samples. Use of this visualization technique shows that the data easily supports the three classes, and that further analysis into the nature of their distinction is warranted. If this visualization fails to show any underlying structure, then the experiment should be reconsidered to determine how better data can be obtained.
Fig. 5. Canonical discriminant analysis of leukemia using gene expression data
5.2.
Sample characterization
Samples from different physiological conditions ranging from normal to diseased cells or wild type versus super producing industrial strains in optimized media, will have a characteristic "fingerprint" of genes that are induced or repressed. Quick identification of those genes whose expression are characteristic of a certain physiological difference is possible through careful selection and use of appropriate statistical tools. A large number of samples (> 30 for each class) for the types under consideration must be taken to accurately establish an expression distribution, and frequently such data is not available. Therefore techniques which make justifiable assumptions and are reasonably robust to small sample size are a key consideration. Statistical tests, such as 2-tailed t-tests [20] and mean hypothesis tests [41 ], adjust distributions according to sample size and are therefore generally reasonable for such purposes, although they are limited to the consideration of two classes at a time. Other techniques such as the Wilk's lambda criteria [21] and misclassification rate can be used to identify discriminatory genes for experiments involving 3 or more sample types, such as the situation shown in Fig. 5. However, when large numbers of samples are present, simpler correlation techniques [30] will provide robust results, so it is always advisable to ensure many replicates of any important experiment.
66
5.3. Classification The complimentary problem of how to use these "fingerprints" of gene expression in different samples has also been explored. The direct application lies in the field of classification or diagnosis, where the expressions of all available genes define a "fingerprint", and thus offer a broad (but incomplete) identifier for the physiological state of a sample. Traditionally, simple clustering has been applied on samples to identify which samples from known classes have a characteristic expression pattern closest to samples from unknown classes [42]. A more statistically-oriented perspective can also be obtained through the generation of a discriminant function which will classify new samples. This approach has been successfully applied to distinguish disease samples for diagnostic purposes [30]. Because DNA microarrays provide such a wealth of data in only a few experiments, wellconstructed classifiers can make very fine distinctions between disease states that would otherwise be difficult to distinguish, with obvious clinical applications. There is still a great body of modeling and classification knowledge in engineering that can be applied to improve the accuracy of these models, including the use of adaptive neural nets and other more advanced techniques. 5.4. Pattern discovery When the distinction of two or more classes of samples is known a priori, the problem of finding discriminatory features can be approached directly. However, in the case of exploratory research, it is not necessarily known which states are most similar and why. For this reason work in pattern recognition should be developed to tackle the problem of uncovering recurring themes or structure in biological data. Even experiments designed with a specific hypothesis in mind are bound to contain biological patterns that will remain hidden in the data unless data mining tools are employed for their discovery. The discovery of patterns and underlying structure in data will assist researchers in developing a new set of hypotheses which will guide further research.
6. CONCLUDING REMARKS We've arrived at a turning point in the study of biological systems, as data generation becomes less of a problem and the thorough analysis of data comes into the limelight. There are several goals of such analyses, as pointed out in this review. One goal of particular importance is to organize the data such as to uncover and highlight the relationships that exist between measures of cellular phenotype (such as gene expression, metabolite concentrations, protein levels) and measures of cellular function such as fluxes, resistance to antibiotics, production capacity, etc. This linkage, depicted in Figure 6, will help elucidate the role of individual genes and proteins in bringing about a specific biological result. Additionally, the coordination required among different cellular players will become better understood through the suggested linkage between phenotypic markers and functional metrics. As such relationships are unveiled they will undoubtedly generate specific hypotheses that can be tested experimentally. The important difference will be that such hypotheses are generated by data instead of critical analysis of the state of the knowledge about a particular system. We believe that such data driven hypotheses will shape to a significant extent future biological research with far reaching implications about the future organization of the research enterprise. This underlines the importance of developing and deploying rigorous
67 computational methods in mining as much knowledge as possible from the ever expanding data fields.
Fig. 6. Linking phenotypic markers such as expression data with metrics of cellular function such as metabolic fluxes. REFERENCES
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68 16. Branden, C.-I., Tooze, J. Introduction to Protein Structure (Garland Publishing Inc., 1999). 17. Durbin, R., Ed., Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids (Cambridge University Press, 1999). 18. Rao, S. S. Engineering Optimization: Theory and Practice (John Wiley & Sons, Inc., New York, 1996). 19. Rabiner, L. R. Proceedings of the IEEE 77, 257-286 (1989). 20. Dillon, W. R., Goldstein, M. Multivariate Analysis, Wiley Series in Probability and Mathematical Statistics (Wiley, New York, 1984). 21. Johnson, R. A., Wichern, D. W. Applied Multivariate Statistical Analysis (Prentice Hall, Englewood Cliffs, New Jersey, 1992). 22. Eisen, M. B., Spellman, P. T., Brown, P. O., Botstein, D. Proceedings Of the National Academy Of Sciences Of the United States Of America 95, 14863-14868 (1998). 23. Tamayo, P. et al., Proceedings Of the National Academy Of Sciences Of the United States Of America 96, 2907-2912 (1999). 24. Personal communication, Gill, Ryan T. 2001. 25. Pandey, A., Mann, M. Nature 405, 837-46 (2000). 26. Vallino, J. J., Stephanopoulos, G. Biotechnology and Bioengineering 41,633-646 (1993). 27. Klapa, M. I., Stephanopoulos, G. in Bioreaction Engineering Schugerl, K., Bellgardt, K.H., Eds. (Springer-Verlag, Heidelberg, 2000). 28. Stephanopoulos, G. Metab Eng 1, 1-11 (1999). 29. Arkin, A., Ross, J. Journal Of Physical Chemistry 99, 970-979 (1995). 30. Golub, T. R. et al., Science 286, 531-537 (1999). 31. Ren, B. et al., Science 290, 2306-+ (2000). 32. Thieffry, D. Bioessays 21, 895-899 (1999). 33. Gardner, T. S., Cantor, C. R., Collins, J. J. Nature 403, 339-342 (2000). 34. McAdams, H. H., Shapiro, L. Science 269, 650-656 (1995). 35. Klapa, M. I., Park, S. M., Sinskey, A. J., Stephanopoulos, G. Biotechnol Bioeng 62, 375391 (1999). 36. Park, S. M., Klapa, M. I., Sinskey, A. J., Stephanopoulos, G. Biotechnol Bioeng 62, 392401 (1999). 37. Lauffenburger, D. A., Linderman, J. J. Receptors: Models for Binding, Trafficking, and Signaling (Oxford University Press, New York, 1993). 38. Roberts, C. J. et al., Science 287, 873-880 (2000). 39. Alter, O., Brown, P. O., Botstein, D. Proceedings Of the National Academy Of Sciences Of the United States Of America 97, 10101-10106 (2000). 40. Work performed at corresponding author's laboratory, currently in submission under the title Defining Physiological States from Microarray Expression Measurements. 41. Kamimura, R. T. Ph.D. Thesis, Massachussetts Institute of Technology (1997). 42. Hughes, T. R. et al., Cell 102, 109-126 (2000).
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
Modelling of nonlinear process dynamics using Kohonen's Networks, Fuzzy Systems and Chebyshev Series
69
Neural
A.P. Alexandridis, C.I. Siettos, H.K. Sarimveis, A.G. Boudouvis and G.V. Bafas* Department of Chemical Engineering, National Technical University of Athens, Zografou Campus, Athens 15780, Greece This paper introduces a new approach to the problem of nonlinear system identification with the aid of neural networks, fuzzy systems and truncated Chebyshev series. The proposed methodology is of general use and results in both a linguistic and an analytical model of the system under study. The method was successfully used for identifying certain operating regions of a Continuous Stirred Tank Reactor (CSTR) where highly nonlinear phenomena, such as limit cycles and multiple steady states appear. 1. INTRODUCTION Mathematical models, which can describe efficiently the dynamics of the system under study, play an essential role in process analysis and control. However, most of the real-world processes are complicated and nonlinear in nature, making the derivation of mathematical models and/or subsequent analysis formidable tasks. So far, many approaches based on nonlinear time series [1], Poincare maps [2] and Lyapunov exponents [3] have been applied in nonlinear system modelling and analysis. During the last decade, a considerable amount of work has been published on the dynamic modelling of nonlinear systems using neural networks [1, 4] and/or fuzzy logic methodologies [5, 6]. Neural networks (NN) have proven to be very powerful tools, still providing only a black-box representation of the system dynamics. On the other hand, fuzzy logic can incorporate expertise and a-priori qualitative knowledge of the system, but due to the complexity of nonlinear processes, it is rather difficult to construct a proper fuzzy rule base. Moreover, the lack of analytical models remains the major drawback for both methodologies. This paper presents a new systematic methodology that facilitates the development of both linguistic and nonlinear analytical models with the aid of Kohonen's self-organizing neural networks, fuzzy logic and truncated Chebyshev series. The result is a model structure, which contains only few essential polynomial terms that are suitable to capture both the qualitative and the quantitative characteristics of the system dynamics. The methodology is applied to the identification of a CSTR, which, depending on the operating conditions may exhibit multiple steady states and limit cycles. The results demonstrate that the produced model captures the qualitative dynamic behaviour of the process and, furthermore, offers a satisfactory quantitative approximation. For comparison purposes, the performance of the proposed approach is compared with two other identification methods: one based on feedforward neural networks and one based on normal form theory. *Corresponding author
70 2.
O V E R V I E W OF THE I D E N T I F I C A T I O N A L G O R I T H M
The proposed identification algorithm can be split into the following basic steps: 9 The output space is clustered using a Kohonen's Self Organizing Map (SOM) network [4]. Self-organizing maps form a special class of neural networks that use an unsupervised learning approach to partition a finite set of data into a certain number of natural subsets based on similarity measures. 9 Based on the clustering of the output space, a fuzzy dynamical model is constructed to describe qualitatively the process dynamics. 9 The fuzzy system is then approximated by truncated Chebyshev series [7], resulting in low-order model structures suitable for vital tasks such as stability analysis and model predictive control applications. Assuming the availability of N training examples [xi Yi], i=l,...,N, where xi, are vectors containing the values of the input variables and yi are the values of the output variable at time point i, the proposed algorithm can be summarized as follows: Step 1. Implement a Kohonen's self-organizing map to cluster the output data: a) Select the number n of fuzzy sets, which will be used to describe the output variable. b) Set the number of neurons of the self-organizing map equal to n. Initialize the synaptic weights wj-,j = 1,...,n of the self-organizing map. c) Select randomly a value yi and find the best matching neuron e i.e., the one that minimizes the distance between yi and wj:
w =m}nly,- al,
j = l , 2 .....
n
(1)
d) Update the synaptic weights of each neuron according to wj (t + 1) = w~ (t) + rl(t)ha. (t)(y i - wj (t)) , t = 0,1,2.....
(2)
where ~/(t) is the learning parameter which decays exponentially with time and h(t) is the neighborhood function, which decreases exponentially with time and with distance between We and wj. Go back to step 1c until the synaptic weights converge. Step 2. Construct a fuzzy dynamical model as follows: a) Define fuzzy sets for each input variable. b) Set the center values of the output fuzzy sets equal to the synaptic weights that are determined by the SOM algorithm. c) For every Yi compute the membership function/z(wj,yi) of each fuzzy set j:
:IE
d) For each pair [xi, yi] find the fuzzy sets with the greatest membership in the fuzzy input and fuzzy output space, respectively. e) Construct a fuzzy rule using as the rule-antecedent the Cartesian cross product of the fuzzy sets in the input space (from step 2d) and as the rule-consequent the fuzzy set in the output space (from step 2d). f) The rules that are most frequently activated enter the rule base.
71
Step 3. Derive analytical models based on truncated Chebyshev series: a) Set a maximum order m for the Chebyshev polynomials. Use the roots of a m-th order Chebyshev polynomial as input values to the fuzzy system derived in step 2 to numerically calculate the input-output mapping in the normalized interval [-1, 1], using common fuzzification, inference and defuzzification techniques [6]. b) Use the least squares method (LS) to calculate the Chebyshev polynomial coefficients that best fit the fuzzy input-output mapping. c) Perform an analysis of variance to select the polynomial terms that mostly contribute to the variation in the data. d) Rearrange the model using only the selected terms. Fit the reduced model to the process trajectories within the operating region of interest by the LS method. If the approximation is unsatisfactory go back to step 3a and choose a larger m. 3. CASE STUDY: IDENTIFICATION OF THE NONLINEAR DYNAMICS OF A CSTR The adequacy of the methodology described above will be demonstrated by the identification of a CSTR that exhibits rich nonlinear dynamics. The dynamic behavior of the reactor is described by the following set of dimensionless nonlinear differential equations [8]: 2) exp(x 2) -
YcI = - x I + D a ( 1 - x 1 ) e x p ( x ic 2 = - x 2 + B D a ( 1 - x l )
fix, 2
(4)
where xlandx2represent the dimensionless conversion and temperature inside the reactor, respectively, D a is the Damk6hler number, B is the dimensionless heat of reaction and fl is the dimensionless heat transfer coefficient. The identification of the reactor dynamics is very challenging in two operating regions: one containing two stable steady states, say region I (figure 1a) and another with a periodic state (limit cycle), i.e. Hopf bifurcation, say region II (figure l b). The aim is to approximate the process dynamics in the operating region of interest, by extracting analytical models based on input-output data observations. Using the state variables and x~, x 2 as inputs, and their time derivatives ~l and 2 2 as outputs the fuzzy dynamical model will be formulated as follows:
Ri: If x~ is
FA, '
and is x 2 FA,' Then is J?l Fc,, and is 22 Fc2' .
3.1 Identification of the operating region with multiple steady states. A set of 1300 input-output data is used for the identification. For demonstration, figure 2a shows the derived fuzzy rule base, which describes the behaviour of kl with respect to the input variables. Seven fuzzy sets have been used for each variable assignment : Very Small (VS), Small (S), Medium Small (MS), Medium (M), Medium Big (MB), Big (B), and Very Big (VB). The fuzzy model was utilized to generate 1600 input-output data. That is, 40 points for xs and 40 points for x2 have been used to construct the output surface, applying the maxmin fuzzy inference operator and the centroid defuzzification method [6]. The truncated Chebyshev series approximation of the derived fuzzy model produces the following analytical model:
72 12 10
8 x
J
12
10
8
6
6
4
4
2
J
2
$1
0
0
0.0
0.2
0.4
0.6
0.8
Xl (a) Da=0.072, B=8.0, 13=0.3
1.0
0.0
0.2
0.4
0.6
0.8
Xl (b) Da= 0.32, B =11.0,13=3.0
Figure 1. Process phase portraits: (a) Two steady states, Sl and
S2.
(b) A limit cycle, LC.
Xl = 3.04539 + 0.23041 x 2 + 0.00278 (4x3-3x2 2) - 5.61076 x 1 - 0.58220 x~ xz + 0.02868x, (2x 2 _ 1) + 2.97509 (2x I2 - 1 ) -0.74373 (4x 3 - 3 x 1) - 0.00357(4x 3l - 3 x l ) (4X32 - 3 X 2) + 0"26440(16 Xl5 _ 20x 31 + 5Xl) + 0"00031(16 Xl5 _ 20x 3~+5X~ )(4x 32 _ 3x 2 ) k2 = 5.693x2 1.589(2x22 _ 1) + 0.255(4x32 - 3x 2) - 1.314x I - 9.508XlX 2 + 1.868x 1(2x 2 - 1 ) 0.264 x I (4x32-3x2) + 2.892 (2x 1= - 1 ) + 0.984 (2x 12 _ 1 ) x 2 - 1.814 ( 4 x ~ - 3 x ) 1 -
-
The selected polynomial terms cover the 95% of the total variance in the data. The phase portrait of the model given by the above equations is shown in figure 2b. Comparing the resulting phase portrait with the original shown in figure l a, makes clear that the produced model captures qualitatively the dynamic behavior of the process but also offers a very good quantitative approximation.
3.2 Identification of the operating region containing a Hopf bifurcation. For the operating region containing the Hopf bifurcation, a total number of 700 inputoutput data have been used for identification. Figure 2c shows the derived fuzzy rule base for describing the behavior of :~2 with respect to the input variables. The application o f the Truncated Chebyshev series methodology results in the following analytical model: k 1 = 1.2753 - 2.2542x 2 + 0.7002 (2x~ - 1) - 0.0005(4x32 - 3x2) - 1.46764 x I + 2.07313XlX 2 0.6522x, (2x22 - 1 ) - 0.9777(2x~ - 1 ) - 0.0307(2x~ -1)(2x 2 _ 1) + 0.2608(4x~ - 3x,)x 2 0.2602(16Xls _ 20x 31 + 5 X l ) - 0.0027(16x] - 2 0 x ~ + 5x,)(Zx~ - 1 ) 5:2 =9.390 -14.401x 2 +9.210 (2x22-1) -0.011(4x32-3x2)+2.556 x 1 -9.456 x I (2x22-1)8.961(2x~ - 1) + 8.062(2x~ - 1)x 2 + 0.652(2x~ - 1)(2x~ - 1) + 0.030(4x~ - 3x,)x z The resulting phase portrait of the model is shown in figure 2d. A comparison with the phase portrait of figure 1b, shows that the two-phase portraits are almost identical.
3.3 Comparison with other identification techniques: neural networks and normal form The proposed algorithm was compared with two other identification schemes, one based on feedforward neural networks and another based on normal form theory [8]. Neural networks are black-box models where no qualitative a-priori information can be incorporated. The normal form is the representation of a specific nonlinear phenomenon within the operating region of interest, using the simplest possible class of equations.
73
(c) Fuzzy rule base for :~2 for region II
(d) Model phase portrait for region II
Figure 2. Resulting fuzzy rule bases and model phase portraits. A four-layer neural network with five nodes in each of the two hidden layers was used for the prediction of each output variable. The neural network structure was selected so that the system dynamics are adequately approximated, while avoiding undesirable phenomena such as data overfitting. In the case of the two stable steady states region, training was based on 1130 input-output data, while 457 data were used for the identification of the Hopf bifurcation. Training was achieved using the Levenberg-Marquardt algorithm [1 ]. The phase portraits for the two stable steady states and the Hopf bifurcation region are shown in figures 3a and 3c, respectively. As it is clearly shown neural networks approximate well enough the process dynamics within both operating regions. The phase portraits produced for the two stable steady states region and the Hopf bifurcation region, using normal forms that qualitatively describe the dynamics of a system in the operating region of interest [8] are given in figures 3b and 3d, respectively. As it is shown, both normal forms describe the process dynamics only in a qualitative way at best. In fact the phase portrait given in figure 3b depicts clearly that the corresponding model fails to approximate the process dynamics in the left part of the region. 4. CONCLUSIONS In this work a new systematic methodology for the identification of nonlinear systems was proposed, by integrating neural and fuzzy techniques in a common framework. The proposed method results to two models: a fuzzy model, which gives a linguistic description of the process behaviour and a truncated Chebyshev series model suitable enough to represent accurately the process dynamics within the operating region of interest. The applicability of the methodology was demonstrated by means of approximating the nonlinear dynamics of a
74
(c) NN phase portrait for region II
(d) Normal form phase portrait for region II
Figure 3. Phase portraits of the neural networks and normal forms. CSTR within two operating regions: one containing multiple steady states and one containing a Hopf Bifurcation. A comparison with two other nonlinear identification methods, one based on neural networks and one based on normal forms revealed the effectiveness of the proposed approach. REFERENCES
1. J. Sjoberg, Q. Zhang, L. Ljung, A. Benvensiste, B. Deylon, P. Glorennec, H. Hjalmarsson and A. Juditsky, Nonlinear Black-box Modeling in System Identification: a unified overview, Automatica 31 (12) (1995) 1691-1724. 2. M. Henon, On the numerical computation of Poincare maps, Physica D 5 (1982) 412-414. 3. A. Wolf, J. B. Swift, Swinney, H. L. and J. A. Vastano, Determining Lyapunov exponents from a time series, Physica D 16 (1985) 285-317. 4. Haykin S., Neural Networks, 2nd Ed.,Prentice Hall, 1999. 5. R. Babuska and H. B. Verbruggen, An overview of fuzzy modeling for control, Control Eng. Practice 4(11) (1996) 1593-1606. 6. H. J. Zimmermann, Fuzzy set theory and its applications, 3rd Ed., Kluwer, 1996. 7. Rivlin, T. J., An introduction to the approximation of functions, Dover Publications, Inc., 1969. 8. N. K. Read and W. H. Ray, Application of nonlinear dynamic analysis in the identification and control of nonlinear systems-I. Simple dynamics, Journal of Process Control 1 (1998) 115.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
75
A systematic methodology for empirical modeling of non-linear state space systems J.P. Barnard and C. Aldrich Department of Chemical Engineering, University of Stellenbosch, Private Bag X1, Matieland, Stellenbosch, South Africa, 7602. Email:/
[email protected] In this paper the authors formulate a theoretical framework for the empirical modelling of non-linear state space systems. The classification of non-linear system data, selection of model structure and order, system parameterisation, stationarity of the data, handling of outliers and noise in the data, parameter estimation and model validation can all be addressed with established, though loosely associated numerical techniques, often referred to as nonlinear process modelling. Relatively few researchers in system identification are comfortable with the application of these numerical techniques, such as time series embedding, surrogate data methods, non-linear stationarity, Lyapunov exponents for chaotic processes and nonlinear predictability. The authors reinterpret some of the above non-linear empirical concepts against the established background for linear state space system identification. Hereby we lay a basis for a systematic methodology to address empirical modelling of non-linear process dynamics, which can be implemented in a non-linear system identification toolbox. In particular, we apply surrogate data methods for the classification of data as stochastic or deterministic. For deterministic data, we embed the individual observations of the process and separate the embedding variables by non-linear factor analysis to arrive at a state space parameterisation of the system. The separation function makes no prior assumptions about the probability distributions of the observations and is robust against dynamic and measurement noise. An ensemble learning technique is used to estimate the parameters of the separation function. After parameterisation of the system a multiple-layer perceptron neural network maps the time evolution of the state vector onto the observations, one sample step ahead. In this manner, the dynamics of the process are captured. Model order is established against the Schwarz information criterion, formulated for multidimensional observations as a function of the model order and modelling error. Model validation is performed against the R 2 statistic, as well as in terms of free-run prediction performance. 1.
INTRODUCTION This paper presents a formal methodological framework for empirical modeling of nonlinear multivariate dynamic systems that can be parameterised as state space systems. Identification is based on multiple time series observations. The methodology addresses classification of observations, using surrogate data techniques, parameterisation of the system by way of multiple time series embedding and prediction of the time series by using multiplelayer perceptron neural network or other suitable models.
76 I D E N T I F I C A T I O N OF N O N - L I N E A R S T A T E SPACE SYSTEMS System identification is well defined for linear dynamic systems and described in severa comprehensive publications (Ljung, 1987, Norton, 1986, Eykhoff, 1974). Amongst parametrit system identification methods, state space methods are generally regarded as superior to othe methods and therefore forms the basis of our methodology. In this section we treat mode selection, data classification and model validation.
2.
2.1. Model selection No single comprehensive mathematical model sufficiently represents all classes o dynamic systems. Thus we are interested in the class of deterministic, non-linear dynamica systems that can be represented mathematically by a state equation in a number of stat~ variables. The dynamics of a non-linear state space system are interpreted as follows. Starting fron some initial conditions, the system's state vector follows a trajectory with time, that i: confined to some bounded subspace of the total available state space. The dynamic attractor to which the trajectory thus converges, is a smooth, non-linear manifold in this state space am defines the true dynamics of the system (Thompson et al., 1995). In mathematical terms fo: discrete-time non-linear systems, the state equation is: xt+l - f[xt, ut]
(1
where x is the state vector, u the input vector of independent variables and f the stat{ transition function that maps the temporal evolution of xt to xt+l. The output vector o dependent variables of the system is defined as yt = g[xt, ut] (21 where g(.) is a non-linear function that projects xt and ut onto the output vector yr. In the first part of system identification, the evolution of xt is reconstructed from the observec system outputs yt. The remaining steps of system identification focus on approximating gof at g(.)" [xt, ut]--~yt+l as well as validating the model. 2.1.1
Parameterisation
Parameterisation plays a critical role in the ability of any model structure to estimate non. linear dynamics. For the class of non-linear state space models, parameterisation introduce,' the concept of state space reconstruction by embedding. We have previously proposed ar extension to Takens' embedding theory in which we approximate the unknown original stat~ space by a phase space, constructed by time-series embedding of each observation componen and combining these subspaces (Barnard and Aldrich, 2000). In this paper we adapt thi,, methodology by applying singular spectrum analysis and non-linear factor analysis to th~ initial combined embedding space. According to Takens (1981), one can reconstruct an equivalent representation of th~ system state space from a one-dimensional time series observation, y~ 9~n, under the conditior that the observation function h(.) is smooth. Such a reconstruction is called an embedding o: the observed time series by way of delay co-ordinates (equivalent to phase variables). Th~ number of these co-ordinates is the embedding dimension, m and the time delay, k (ir multiples of the sample period) is the delay between each co-ordinate. Optimal delay ensure,, linear, statistical independence among delay co-ordinates - a fundamental requisite of phas~ variables and thus also of delay co-ordinates. The optimal time delay between the delay co-ordinates is usually determined by th~ average mutual information (AMI) criterion of Frazer and Swinney. (1986), while the optima number of co-ordinates is typically calculated using the method of false nearest neighbour,
77 (Kennel et al., 1992). However, since inferring k from AMI is confounded by noise, we have chosen to apply singular spectrum analysis instead to determine the initial phase space. In other words, for the embedding of each observation component, yi use a default delay of k = 1 and determine the embedding dimension, mi as the linear decorrelation point in the autocorrelation sequence ofyi: mi = min[first min(y.yt_k), first(y.yt_k= 0)]i (3) The individual embedding spaces are then concatenated columnwise and separated initially by singular value decomposition. The first q significant eigenvectors are selected to span the resultant phase space. A = embed(Y, m, k), Y ~ 9~ n x p , A ~ ~l](n'J 0) x Xmi (4) where j0 = max(m/- 1) + 1, 1< i
Model class
Several non-linear empirical model classes have been proposed as functional approximations for fg(.). Multiple layer perceptron (MLP) and radial-basis function neural networks are strong functional approximators (Judd and Mees, 1995), because the optimal approximation error grows more slowly with dimension than for weak functional approximators. Since the class of dynamic systems under discussion requires strong functional approximations, we shall focus on MLP networks. These model structures have the added advantage that they are relatively easy to apply and are well supported by large commercial software systems, such as Matlab, G2 with NeurOn-line and Process Insights, used in the process industries. The set of MLP model structures, A4*FF, are often implemented as non-linear regressors and classifiers (Rumelhart et al., 1994). Provided that the input space is carefully selected and the topology correctly specified, an MLP model, MFF(O), can successfully simulate or predict multidimensional non-linear data (Funahashi, 1989). A MLP model structure, MFF, is specified in terms of the model class, the topology and the nodal activation function of each layer. In our methodology, we have implemented the bipolar sigmoidal or hyperbolic tangent function, of the form ~ . ) - [1-exp(.)]/[ 1+exp(.)]. Further, we specify a linear input layer, one non-linear hidden layer, and a linear output layer. Weights are assigned to the connections between layers, which carry the output of nodes forward from one layer to the next. In addition, a layer can also incorporate a bias constant. The weights and biases constitute the model parameters and are estimated by so-called training algorithms. The number of nodes in
78 the input layer is equal to the dimension of the input space, while the size of the output laye: is equal to the dimension of the output space. 2.1.3 Model order Model order is defined as the number of model parameters of a model structur,
(Ljung, 1987). For M*FF, the model order depends on the dimensions of the input and outpu spaces, as well as the numbers of nodes in the hidden layer(s). Determining the order off non-linear model can be approached in two ways (Judd and Mees, 1995). According to the first approach, model order is usually determined iteratively by testin~ several models of increasing order for generalisation against the sum-of-squared-errors (SSE or other suitable norm. In the presence of noise, it is possible to overfit by implementing model of too high an order. Such a model will fit the training data well, but will not generalis~ well, because the model also partially represents features of the noise component. The second approach, which is followed in this paper, starts with a subclass of mode structures and then optimises model order by calculating, for example, Rissanen's minimtm description length (MDL) for each model structure (Judd and Mees, 1995). According to thi: approach, the model parameters and model error are encoded as a bit stream of information The more complex the model and the larger the modelling error, the more bits will be requirec in the encoding The model structure corresponding with the lowest MDL is therefore optimal We use an approximation of MDL, the Schwartz Information Criterion (SIC), which i: simpler to calculate than MDL and is formulated as: SIC = N Z [log(mSE)] + r*log(N) (81 r = [dim(x) + dim(y) + 1] S + dim(y) (91 where r = model order, N = number of samples, MSE = mean square error of prediction, S number of hidden nodes in MLP model structure and dim(.) indicates the number o components in an array. 2.2. Data classification A major problem with empirical systems is to determine a priori whether deterministi( dynamics are inherent in the data. The method of surrogate data is a statistical approach tc data classification (Takens, 1993; Theiler and Pritchard, 1996; Theiler and Rapp, 1996). Thi: method involves a null hypothesis against which the data are tested, as well as discriminating statistic. In this paper, the null hypothesis states that the data belong to th~ class of amplitude adjusted Fourier transform (AAFT) stochastic processes (Small and Judd 1998). Surrogate data are subsequently generated, with the same probability distribution am Fourier spectra as the data, based on the assumed stochastic process. An appropriat~ discriminating statistic, correlation dimension as formulated below, is calculated for both th~ surrogate and the original data (Theiler et al., 1992). The correlation dimension, dc, is defined as follows:
d e = lim lim l~ c~0 U~oo 1ogc where CN is the correlation function, defined by: cu(~)
=
(10
21110,-o.,11<)
O
I(-) is a Heavyside function that returns 1 if the distance between point i and j is within c and 0 otherwise, while N is the number of observations in the data set.
79 Judd (Judd and Mees, 1995) proposed that the correlation dimension be estimated as a function of attractor scale c0 using the following equation, valid for c < c o :
CN(C ) ~ c dcq(c) where q(.) is a polynomial of order of the topological dimension. 2.3.
(12)
Model validation
Model validation is often based on one-step prediction, which is not necessarily a good indicator of the ability of the model to generalise the underlying (dynamic) process represented by the data (Zhu and Rohwer, 1996). In contrast, a free-run prediction is a considerably more rigorous test of the validity of the model (Small and Judd, 1998). To achieve this, one has to generate a free-run time series with the model, reconstruct the dynamic attractor from these predicted values and characterise the attractor by use of the correlation dimension. Likewise, the dynamic attractor for the actual system is reconstructed from the observed time series and characterised. The reliability of the model can thus be assessed systematically by comparing the correlation dimension of the model output with that of the experimental data at different scales of the respective attractors. However, since nonlinear factor analysis is an iterative procedure, free-run predictions, which implement repeated embedding and separation, are computationally prohibitive and therefore have not been pursued in this paper. 3.
CONCLUSIONS In this paper we have presented a formal methodology for empirical state space modelling of non-linear dynamic systems, based on simultaneous multiple observations. In a companion paper, we demonstrate the application of this methodology in a case study, involving prediction of NO2 in an environmental system. Our method has expanded the fundamental embedding theory of Takens to multidimensional observations. By using singular spectrum analysis (Vautard et al., 1992) and non-linear factor analysis, the method also avoids the problems encountered with optimal embedding in the presence of noise. In addition, we classify the observations as deterministic or stochastic, prior to estimating models. Finally, validation can be performed more conclusively by free-run prediction than is the case with using linear statistics and one-step ahead predictions. Since the computational complexity around free-run prediction when combined with nonlinear factor analysis has prohibited demonstration of this method of validation, we recommend an adaptation to our methodology to reduce computational complexity. As a compromise, one can increase the signal to noise ratio of the observations by embedding in the fashion of singular spectrum analysis, followed by separation, using non-linear factor analysis. The noise-reduced initial embedding can then be projected to an optimal phase space by singular spectrum analysis. Since the parameterisation now requires only linear separation, the computational complexity decreases markedly and encourages the use of free-run validation of the models. 4. REFERENCES Barnard, J. P. and Aldrich C. 2000. Multivariate dynamic process systems by use of independent component analysis, Proceedings of the Second International Workshop on Independent Component Analysis and Blind Signal Separation, June 19-22, 2000, Helsinki, Finland. Eykhoff, P. 1974. System identification: Parameter and state estimation, Wiley, Chichester.
80 Frazer, A.M. and Swinney, H.L. 1986. Independent co-ordinates for strange attractors. Physical Review Letters A, 33, 1134-1140. Funahashi, K. 1989. On the approximate realisation of continuous mappings by neural networks. Neural Networks, 2, 183-192. Judd, K. and Mees, A. 1995. On selecting models for non-linear time series. Physica D, 82, 426-444. Kennel, M. B., Brown, R. and Abarbanel, H. D. I. 1992. Determining minimum embedding dimension using a geometrical construction. Physica Review A, 45, 3403-3411. Ljung, L. 1987. System Identification, Prentice-Hall (New Jersey). Norton, J. P. 1986. An Introduction to Identification, Academic Press (London). Rumelhart, D. E., Widrow, B., and Lehr, M. 1994. The basic ideas in neural networks. Communications of the ACM, 37, (3), 87-91. Small, M. and Judd, K. 1998a. Comparison of new non-linear modelling techniques with applications to infant respiration. Physica D, 117, 283-298. Takens, F. 1981. Detecting strange attractors in turbulence. Lecture Notes in Mathematics, 898, Springer (Berlin), 366-381. Takens, F. 1993. Detecting non-linearities in stationary time series. International Journal of Bifurcations and Chaos, 3, 241-256. Theiler, J. and Pritchard, D. 1996. Constrained realisation Monte Carlo method for hypothesis testing, Physica A, 94, 221-235. Theiler, J. and Rapp, P.E. 1996. Re-examination of the evidence for low-dimensional nonlinear structure in the human electroencephalogram. Encephalography and Clinical Neurophysiology, 98, 213-222. Theiler, J., Eubank, E., Longtin, A. and Galdrikian, B. 1992. Testing for non-linearity in time series: The method of surrogate data. Physica, 58D, 77-94. Thompson, J. M. T. and Bishop, S. R. 1995. Nonlinearity and Chaos in Engineering Dynamics, Wiley and Sons Ltd. (Chichester), 1-10. Valpola, H. 2000. Non-linear independent component analysis using ensemble learning: Theory. Proceedings of the Second International Workshop on Independent Component Analysis and Blind Signal Separation, June 19-22, 2000, Helsinki, Finland. Vautard, R., Yiou, P. and Ghil, M. 1992. Singular spectrum analysis: A toolkit for short noisy and chaotic time series. Physica D, 58, 95-126. Zhu, H. and Rohwer, R. 1996. No free lunch for cross-validation. Neural Computation, 8, 1421-1426.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
81
Modelling of Air Pollution in an Environmental System by use of Nonlinear Independent Component analysis J.P. Bamard and C. Aldrich Department of Chemical Engineering, University of Stellenbosch, Private Bag X1, Matieland, Stellenbosch, South Africa, 7602. Email: jsteyl@ing, sun.ac.za In this paper an empirical, non-linear state space model of a metropolitan environmental system is constructed by use of singular spectrum analysis and non-linear independent component analysis. Environmental systems are complex, high-dimensional and non-linear. Conventional modelling techniques demand expensive fundamental models, as well as costly supercomputers to effectively simulate and predict these systems. On the other hand, numerical methods such as empirical state space parameterisation and multiple-layer perceptron neural networks promise simpler models that can be accommodated on affordable desktop computers. The state space model presented in this paper is constructed by embedding and separation of the individual observations of the polluting agents. The observations are regarded as a nonlinear mixture of the underlying process state variables and are classified as deterministic by using a surrogate data technique. It is shown that non-linear separation enhances the ability of the non-linear model to predict the dependent observations, especially in the presence of unknown levels of dynamic and measurement noise. No pre-assumptions are made on the statistical distributions of the original state variables or the noise content. Instead, these distributions are estimated as mixtures of Gaussian distributions. An ensemble learning technique is implemented in the parameter estimation algorithm for the separation model. The results show a reduction in complexity in the attractor and satisfactory one step ahead predictions. 1. INTRODUCTION This paper presents a practical application of a methodology for the identification of nonlinear dynamics from simultaneous multiple observations. The theory behind this methodology is covered in a companion paper at this conference. In principal parameterisation of one-dimensional observations is sufficient to estimate the dynamics of a source (Takens, 1981). However, in practice, owing to measurement and dynamic noise in observations, as well as nonstationarity of observations, the simultaneous observation of multiple sources is often necessary. This simultaneous observation of multiple system outputs or sources can be compared to oversampling in order to improve the signal to noise ratio, as well as to gain more information on the dynamics, which may be contained insufficiently in nonstationary one-dimensional observations. These concepts form the foundation of the proposed methodology.
82 We approximate the original, unknown state space of the observed system by constructing a suitable phase space. The reconstruction is based on a multidimensional generalisation of Takens' embedding of one-dimensional observations. According to this parameterisation method, each observation component is embedded individually with a default unary delay and dimension equal to the point of linear decorrelation. Columnwise concatenation of the resultant phase spaces forms an initial combined phase space. The optimal phase space dimension is determined iteratively over increasing embedding dimension, by embedding of the observations and estimating a multiple-layer perceptron model per iteration. The model predicts the relationship between output at time t+ 1 and phase space at time t. The optimal phase space is then reconstructed by non-linear factor analysis and the model is re-estimated. In addition to parameterisation, we also demonstrate data classification by using surrogate data and correlation dimension. In terms of the correlation dimension, this technique tests the hypothesis that the observations originate from a known stochastic process class. Finally we build and validate an optimal non-linear multiple-layer perceptron model that predicts the output at time t+l in terms of the phase vector at time t. The application of the methodology is demonstrated on a real environmental system. The system comprises the photochemical formation of NO2 from NO, under the influence of solar radiation and environmental air temperature. The effectiveness of the methodology is compared to a conventional model of [NO2], expressed directly in terms of the measured [NO], solar radiation and air temperature, as well as a model of NO2 based on an optimal parameterisation achieved by using singular spectrum analysis (Vautard et al., 1992). 2. DATA CLASSIFICATION Data for our case study were recorded in Cape Town in South Africa over 365 days during 1997. The hourly means of the original samples, recorded at minute intervals, provided us with 8664 records. The first step was to remove statistical outliers and nonsensical samples from the observations. Statistical outliers were identified by principal component analysis of the observed data. Hotelling's T 2 statistic was calculated and a 99.9% upper control limit was set. This indicated statistical outliers in less than 2% of the data. After pre-processing the data, classification of the dependent observation component was performed by surrogate analysis (refer to companion paper in this proceedings). The classification results for [NO2] are shown in Figure 1. A set of 10 surrogate time series was generated, using an amplitude adjusted Fourier transformed stochastic process so that the surrogate time series (broken lines) had the same distribution function, as well as Fourier spectrum as the [NO2] observations (solid line). Each surrogate time series was embedded in a 10-dimensional phase space, using singular spectrum analysis. The observations of [NO2] were likewise embedded. Next we calculated the correlation dimension vs. attractor scale for each of these phase spaces.These correlation dimension curves were then compared in order to classify the observations, as indicated in Figure 1. According to this figure, the data are not stochastic, albeit high-dimensional, as seen in the high correlation dimension at small attractor scales. If we take twice the mean correlation dimension as an indication of the optimal embedding dimension, a sensible embedding dimension for this system is approximately 10.
83
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3. PARAMETERISATION OF OBSERVATIONS Optimal parameterisation of observations is crucial to the success of estimating a sufficient prediction model of the dependent observations in terms of all observation components. We apply a generalisation of Takens' embedding theory to the multidimensional observations in our case study. The observation space is constituted by four observation components, viz. [NO2], [NO], solar radiation and environmental air temperature. All these observations were recorded simultaneously at the same physical location. Our aim was to optimally parameterise these observations to form a phase space to serve as an approximation of the original, unknown system state space. Since conventional embedding is sensitive to noise, we decided to implement singular spectrum analysis instead. This technique does not require the explicit calculation of embedding delay against the average mutual information statistic, as with the conventional method of delayed coordinate embedding. Also, we do not need to calculate false nearest neighbours to determine the embedding dimension. Instead we embed each observation in a phase space of dimension equal to the point of linear decorrelation for the particular component, using a delay of one. The individual phase spaces are concatenated columnwise to form a combined phase space. A singular value decomposition of the combined phase space linearly separates this space. Guided by the classification results, we decided to test two phase space dimensions, namely a five dimensional space and a ten-dimensional space. The individual embedding dimensions for each observation component, determined as mi = min[first m i n ( y t . Y t _ k ) , first(yt.yt_k = 0)]i, came to m i = { 13, 8, 7, 13 } for [NO2], [NO], solar radiation and temperature respectively. The dimension of the combined initial phase space was therefore 41. For the five-dimensional space we retained the first five eigenvectors of the correlation matrix of the
84 initial phase space, which corresponded to 83% of the total variance. The ten-dimensional phase space explained 93% of the total variance.
Figure 2
Free-run performance of MLP prediction models, based on a 5 dimensional phase space (top) and a 10-dimensional phase space (bottom).
For each phase space we estimated a prediction model as N: ~t--~Yt+l (1) where E indicates the final phase space after projection. The model with the best R 2 statistic and the longest coherent free-run prediction was selected. In addition we also estimated a model which related the [NO2] direct in terms of the [NO], solar radiation and temperature, not using parameterisation. The results are shown in Figure 2. As can be seen from Figure 2, none of the two parameterisations and models performed particularly well in free-run mode. However, from the free-run performance over the first 100 points, it is clear that the 10-dimensional parameterisation performed better. Also, the onestep performance was better for the 10-dimensional parameterisation (R2 = 0.70) than the 5dimensional parameterisation (R2 = 0.61). This result concurs with the indication from classification of the dependent variable, which has suggested a state space dimension of approximately 10. In comparison, the model using no parameterisation performed dismally (not shown here, R 2 < 0.1 on validation data). Our next step was to determine the system phase space by non-linear factor analysis. According to this technique, the required phase space is regarded as an unknown source that is mixed by a non-linear mixing function to produce the observed outputs. Using the algorithm by Valpola (2000) we simultaneously estimated the phase space and a Bayesian neural network model, which acted as non-linear mixing function. Valpola provided software code for Matlab 5 for this purpose (software location on the Intemet: http://www.cis.hut.fi/proj ects/ica/bayes/). We heuristically chose 20 tan-sigmoidal nodes for the hidden layer of the MLP model structure. The input space of dimension 8498 x 41 was the initial phase space, formed from
85 individual embedding of each observation component. The output space was the yet unknown phase space of dimension 8498 • 10. The calculations are numerically intensive and ran in Matlab 5.3.1 for about 5 hours over 1500 iterations on a dual Pentium| 400 with 384 MB RAM, under Microsoft| NT4 SP6 (cost function over iterations appears in Figure 3). ,-- 1.5
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4. MODEL ESTIMATION After parameterisation of the observations we proceeded to estimate a feed-forward multiple layer perceptron model to predict the [NO2] at time t+l from [NO], solar radiation and air temperature at time t. The Levenberg-Marquardt (Levenberg, 1944; Marquardt, 1964) algorithm was used to optimise the model parameters against the mean square error norm. We determined the optimal model order, r, in terms of the number of hidden nodes against the Schwartz information criterion (SIC). The optimal model order was indicated by the global minimum in SIC(r) over a range of monotonically increasing model orders, from the equivalent range of 8 through to 16 hidden nodes. According to Figure 4, fourteen hidden nodes turned out to be optimal. -1.3 _0
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5. MODEL VALIDATION Model validation ensures the proper generalisation of a model. For dynamic systems, we suggested in our companion paper at this conference that the model performance under freerun prediction is the most stringent test for model quality. However, when applying non-linear factor analysis as in the methodology presented here, it is computationally prohibitive to attempt free-run, since the separation function must be iteratively calculated for each step. Instead we present the R 2 statistic for one-step prediction of the validation set in Figure 5.
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Figure 5 One-step prediction of validation data by an MLP model based on parameterisation using non-linear factor analysis. R 2 = 0.62 6. CONCLUSIONS In this paper we demonstrated the practical application of a novel methodology for identification of non-linear dynamics based on simultaneous observation of multiple system outputs. A parameterisation method that implements singular spectrum analysis and nonlinear factor analysis produced a phase space based on the observations of four factors in an environmental system. From the prediction results it appears that the parameterisation by singular spectrum analysis produced better-correlated one step predictions of the dependent variable than non-linear factor analysis. However, the non-linear separation method removed noise from the reconstructed attractor in phase space which, smoothed prediction, thereby reducing the R E to 0.62 compared to 0.70. From a pragmatic viewpoint parameterisation by singular spectrum analysis is a practical compromise, with much lower computational cost and comparable prediction accuracy. REFERENCES 1. Levenberg, K. 1944. A method for the solution of certain non-linear problems in least squares. Quart. Applied Mathematics, 2, 164-168. 2. Marquardt, D. W. 1963. An algorithm for least squares estimation of non-linear parameters. Journal of SAIM, 11, 431-441. 3. Takens, F. 1981. Detecting strange attractors in turbulence. Lecture Notes in Mathematics, 898, Springer (Berlin), 366-381. 4. Valpola, H. 2000. Non-linear independent component analysis using ensemble learning: Experiments and discussions. Proceedings of the Second International Workshop on Independent Component Analysis and Blind Signal Separation, June 19-22, 2000, Helsinki, Finland. 5. Vautard, R., Yiou, P. and Ghil, M. 1992. Singular spectrum analysis: A toolkit for short noisy and chaotic time series. Physica D, 58, 95-126.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
87
A life-cycle a p p r o a c h for model reuse and e x c h a n g e Rafael Batres, Atsushi Aoyama and Yuji Naka Tokyo Institute of Technology, R1 Bldg., Midori-ku, Nagatsuta 4259, Yokohama 226-8503, Japan This paper discusses a formalism in which plant, processes and products are represented as structural, behavioral, and operational interrelated objects. A metamodel approach based on this paradigm is exploited to develop a simulation model representation that supports model exchange with varying level of detail and coarseness. This work evaluates Modelica as an interchange modeling language to represent equation-models in this context. 1. I N T R O D U C T I O N Recent research has shown that there is a need for a paradigm shift from tools based on unit operations to tools based on explicit phenomenological models [ 1 ][2]. Unit operations define functional views of equipment and operations but fail in restricting innovative solutions [3 ]. This paper suggests the use of metamodels to represent simulation models of physicochemical behaviors. This concept differs from unit operation in that metamodels define phenomena that are independent of the equipment and operation information. The term is borrowed from the bond-graph modeling technique as described by [4] in which metamodels abstract and describe the mental process behind developing models for specific physical systems. Metamodels have the properties of aggregation and abstraction (hierarchical decomposition). Metamodels in lower levels have associated equation models in such a way that information about the topology and the structure of metamodels can be used to generate a larger set of equations. This work evaluates the use of Modelica as a model representation language in the metamodel context. Modelica [5] is a modeling language that is intended to facilitate the exchange and reuse of metamodels, and the construction of model libraries. Conceptual information models or ontologies that define a common agreed view of the product, process, and plant fife-cycles play a very important role in the integration of engineering software tools [6][7]. Related ongoing work includes the development of conceptual information models for process design [8] in which functional description of the chemical process is realized by plant items and described by behavior models. Behavior models are assembled from plant control and phenomenological models. Ontologies are being developed that are intended to support the exchange of information and knowledge (such as simulation models) between CAD systems, dynamic simulators, equation solvers, thermodynamic-property estimators, control systems [9]. These ontologies specify the syntax and semantics of formal descriptions of knowledge about the plant, processes, and products. They are organized around a multi-dimensional formalism that arranges information, activities and tools about the plant, process, and product life-cycles into physical, behavioral, and operational dimensions or perspectives [10]. The taxonomy of two of these ontologies is shown in Figure 1, using the UML graphical notation [11 ]. Formal definitions that are understandable by both humans and computers are structured in KIF [12]. For example, the following axiom is used to reinforce the fact that instances of material-phase or process-material (material ontology) necessarily should have at least one component.
88 (=> (Phase-System ?X) (forall ?X (Exists (has-material-component ?X ?A))))
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Figure 1. Taxonomies of the process behavior and material ontologies. The ontologies in the behavior dimension comprise definitions for physicochemical behavior that comes out as a result of the interaction between the physical dimension and exogenous factors. Ontologies of the physical dimension specify concepts for the representation of the plant structure such as classes of equipment and plant entities or for the representation of the product structure (material). The operations and management ontology defines plans and actions needed for controlling, operating and manipulating objects of the physical dimension. This work emphasizes on the use of the process behavior ontology for model representation and reuse. Each dimension has multiple views with mappings between them and across dimensions. For example, plant structure objects are defined from abstract, mechanical, functional and operational perspectives. Facilities in the abstract view are mapped one to one to equipment item instances in the mechanical view. Metamodels in the process behavior ontology and equipment facilities in the structure dimension of the plant life-cycle are defined as separate but related objects, which allows software interoperability among plant and process life-cycles. In the design of the ontologies the following aspects about behavior models were considered: 1) easiness of the information sharing and exchange between intelligent CAD systems and process
89 simulation sotLvare tools; 2) simulation model exchange with varying level of detail and coarseness; 3) unnecessary structural modifications to the existing flowsheet and the re-entering of new instances of "unit operations"; 4) modeling of undesired behaviors (e.g. leakages) without affecting the flowsheet topology; 5) exploration of operation alternatives without modifications to the simulation models; 6) the ability to associate more than one kind of behavior associated to a particular equipment item; 7) the representation of matter, energy, and information flows that do not flow through defined pipes or channels, such as a vessel leakage or sunshine heating over the equipment. 2. T H E P R O C E S S B E H A V I O R O N T O L O G Y Behavior in this ontology refers to the physicochemical phenomena that are manifested through changes in the properties of the material that is processed in the plant. This definition implies the existence of physicochemical behavior models that can be defined independently of where the modeled phenomena occur. For a given piece of equipment, engineers can combine multiple metamodels to represent proposed, assumed or actual behaviors as well functional information (roles or expected behaviors). Consequently, instead of a single flowsheet diagram, a process topology diagram is worked out in parallel to the development of 2D or 3D plant representations (such as PFD, P&ID or 3D diagrams). The behavior that takes place in a piece of equipment is formulated using metamodel behavior entities. Metarnodels can be interconnected via ports for transfen~g energy, information and mass. In this context, connecting ports is comparable to combining sets of equations. Two important properties of metamodels are aggregation and abstraction (hierarchical decomposition). Hierarchical decomposition allows a metamodel to be decomposed into a number of other metamodels. Aggregation is the property of combining and linking instances of metamodels. Metamodels that are not composed of other metamodels are called protomodels. Protomodels represent material and energy balance equations as well as thermodynamic models that are independent of the geometry or other features of a particular equipment item. Definitions about the properties of the material are defined in an ontology that belongs to the physical dimension of the product. 3. A P P L I C A T I O N OF T H E P R O C E S S B E H A V I O R O N T O L O G Y In order to illustrate the use of metamodels, an example based on the boiler model presented in [13 ] is provided. Steam Instances of top-level metamodels describe the behavior that Feed takes place in one equipment item. Figure 2 shows a simplified diagram of a fired-tube boiler in which products of combustion pass through tubes surrounded by water. In this process, water is evaporated to produce steam. The fuel is bumt with a mixture of air to provide the energy for producing the steam, but this part is excluded for space reasons. The top level metamodel associated to the boiler is shown in Figure 3. At this level an instance of balance- Figure 2. Boiler diagram (facility) volume is used to define the region of interest that is to be modeled. The energy flow between the fired section of the boiler and the tank is represented as a thermal-energy-connection. From an implementation perspective, each facility can be assigned a window upon which metamodels reside. In this modeling scenario, the modeler (a modeling person or a program) decides to construct models for each physical phase, assuming the presence of vapor and liquid. These are represented by the balance volumes fiquid_balanee_volume and vapor_balance_volume.
90 Parameters such as volume of the vessel, cross-section of the tank, etc. are available from a provision-model that is associated to the tank. In general, a provision-model supplies information about the geometry and function of the equipment item. As a model it can incorporate assumptions that facilitate the description of the physical aspects of the equipment. For example, while the cross-section of a vertical tank can be considered as a parameter for the calculation of the liquid level, an expression for the cross-section of a spherical-tank may require the value of the level. In this example, the provision model tank_geometry_model is used to represents the constraint that the volume of the tank imposes over the volume of the material of each phase. In a lower level, the modeler (probably with the assistance of a modeling wizard) selects from a model library a model for the accumulation of material (liquid_inventory_model) and a model that describes evaporation phenomena (evaporation_model). Figure 3. Metamodels associated to a boiler Ports are automatically created when connections are drawn from the ports of liquid_balance_volume to the metamodel boxes. Alternatively, ports can be interactively assigned to the metamodels. At this level, metamodels are not decomposed further. These metamodels are referred to as protomodels. Protomodels have associated equation models based on physicochemical principles such as conservation laws and chemistry. At this point, the model is incomplete, as no equations have been defined so far. 3.1 Use of Modelica as a model representation language Modelica is a model representation language that is being developed in an intemational effort that aims at providing an approach for model reuse and exchange between simulation systems. Modelica code is non-causal, which means that equations are declared in a mathematical sense and not as a computational procedure. Thus models can be formulated independently of how they are to be solved. The Dymola modeling and simulation environment was used to edit and validate the models in Modelica. Ontology definfions such as process material, metamodel parent class, and ports were defined as records, classes, and connectors. Classes were also used to define the metamodels of the example. Modelica code for liquid_inventory and evaporation metamodels is shown in Figure 4 and Figure 5 respectively. Once metamodels are instituted, Dymola checks the syntax and attempts to generate a combined set of equations from the lower level equation models and connectivity information. Subsequently, degrees of freedom are verified, the Modelica code is translated, compiled and simulated. In a combined design and simulation environment, once metamodels for each equipment item are defined, equation-models for the entire plant can be generated using the topological information of the interconnected facilities. class liquid_inventory e x t e n d s metamodel;
SIunits. Pressure P;
91 SIunits.Temperature T; SIunits.Mass M; SIunits.Volume VL; S I u n i t s . D e n s i t y D; SIunits.Length L "Level"; SIunits. SpecificEnthalpy H "Acumulated Heat"; parameter S I u n i t s . S p e c i f i c H e a t C a p a c i t y Cpliq "Specific Heat Capacity"; parameter S I u n i t s . T e m p e r a t u r e TR=273 "Reference Temperature"; parameter S I u n i t s . M a s s F l o w R a t e R R = I "Evaporation reflux parameter"; annotation ...
equation
P = material_input.material info.mdoom_pressure; material_output.material_info~mdoom_massflowrate = material_input. massflowrate; material _ info.mdoom der(M) = m a t e r i a l _ r e c y c l e _ i n p u t . m a t e r i a l i n f o . m d o o m _ m a s s f l o w r a t e ; H = Cpliq*(T - TR)*M; der(H) = material_input.material_info.mdoom_enthalpyflow material_recycleinput.material_info.mdoom_enthalpy_flow material_output.material_info.mdoom_enthalpy_flow; material_output.material_info.mdoom_temperature = T; material_output.material_info.mdoom_specific_enthalpy = Cpliq*(T - TR); material_output.material_info.mdoom_enthalpy_flow = Cpliq*(T - TR)* material_output.material_info.mdoom_massflowrate; material_output.material_info.mdoom_pressure = P; material_output.material_info.mdoom_density = material_input.material _ i n f o . m d o o m _ density; D = material_input.materialinfo.mdoom_density; VL = M/D; geometry_port.volume = VL; L = VL/geometry_port.area; end l i q u i d _ i n v e n t o r y ;
Figure 4. Modelica code for the liquid-inventory metamodel of Figure 2.
4. C O N C L U S I O N S Metamodels are specified for each piece of equipment without necessarily defining a network of metamodels for the whole plant. Metamodels are also useful in the creation of libraries of reusable models that can be exchanged between simulation tools. The property of abstraction together with the separation of behavior from the equipment information, enable a metarnodel to be replaced by another metamodel of different fidelity or coarseness without modifying the plant structure and operational information. This contributes to the consistency of the information that is exchanged among simulation tools and CAD systems. The development up to date of the metamodel concepts shows that there is a promising potential for developing modeling techniques and software tools adequate to support engineering activities along the life-cycles of the plant, processes and products. An example has been shown to illustrate the use of the metamodel concepts and to identify functional requirements of modeling tools. The example explores the use of Modelica as a model representation language as seen from the metamodel perspective. It shows how aggregation and abstraction of behavior models are both supported by Modelica. Four different categories of Modelica code were identified: 1) Modelica code for data structures; 2) automatically generated code; 3) building-blocks created by the modeler for immediate use; 4) building blocks created by the modeler to be stored in a model library. Additions to the Modelica language to explicitly distinguish among the categories would be beneficial. A number of issues are to be addressed, including the management of the level of detail in the models, determining whether the attributes of provision-models are to be fixed in the ontologies or defined by the modeler, and more importantly the development of a modeling methodology in detail based on physical principles. One aspect that the authors are exploring is the use of the same behavior models for both batch and continuous processes.
92 class e v a p o r a t i o n extends m e t a m o d e l ;
S I u n i t s . T e m p e r a t u r e T; SIunits. P r e s s u r e P; parameter S I u n i t s . S p e c i f i c E n e r g y l a m b d a "Heat of v a p o r i z a t i o n " ; parameter SIunits. S p e c i f i c H e a t C a p a c i t y C p l i q " L i q u i d S p e c i f i c Heat C a p a c i t y " ; parameter S I u n i t s . T e m p e r a t u r e TR " R e f e r e n c e T e m p e r a t u r e " ; parameter SIunits. D e n s i t y D " L i q u i d D e n s i t y " ; annotation (..);
equation
material_input.material_info.mdoom_massflowrate = material_recycle_port.material_info.mdoom_massflowrate + vapor_output.material _ info.mdoom_ massflowrate; material recycle_port.material_info.mdoom_enthalpy_flow = material_recycle_port.material_info.mdoom_massflowrate*Cpliq*(T
- TR);
material_input.material_info.mdoom_enthalpy_flow = -energy_input.mdoom_heat_flow vapor_output.material_info.mdoom_enthalpy_flow; vapor_output.material_info.mdoom_enthalpy_flow = vapor_output.material_info.mdoom_massflowrate*(lambda + C p l i q * ( T - TR)); vapor_output.material_info.mdoom_specific_enthalpy = vapor_output.material_info.mdoom_enthalpy_flow / vapor_output.material_info.mdoom_massflowrate; material_recycle_port.material_info.mdoom_specific_enthalpy = C p l i q * ( T - TR); vapor_output.material_info.mdoom_temperature = T; material_recycle_port.material_info.mdoom_temperature = T; material_recycle_port.material_info.mdoom_pressure = vapor_output.material_info.mdoom_pressure; material_recycle_port.material_info.mdoom_density = D; // The A n t o i n e c o r r e l a t i o n is u s e d for the v a p o r p r e s s u r e c a l c u l a t i o n P = exp(16.5362 - 3985.44/(T - 38.997))'1000; vapor_output.material_info.mdoom_pressure = P; end e v a p o r a t i o n ;
+
Figure 5. Modelica code for the evaporation metamodel.
REFERENCES 1. J. Bieszczad, A. Koulouris and G. Stephanopoulos, Proc. of the FOCAPD'99 Conference, Breckenridge, Colorado, USA. (1999) 2. A. A. Linninger, Proc. of the FOCAPD'99 Conference, Colorado, USA (1999) 3. N. Shah, N. J. Samsatli, M. Sharif, J. N. Borland, and L. G. Papageorgiou L. G., Proc. of the FOCAPD'99 Conference, Breckenridge, Colorado, USA. (1999) 4. P. Gawthrop and L. Smith, Metamodelling: For bond-graphs and dynamic systems, Prentice-Hall International, UK (1996) 5. H. Elmqvist, S. E. Mattsson and M. Otter, IEEE Symposium on Computer-Aided Control System Design, CACSD'99, Hawaii, August 22-27(1999) 6. W. Marquardt, Proc. of the FOCAPD'99 Conference, Breckenridge, USA. (1999) 7. R. Batres and Y. Naka, Proc. of the FOCAPD'99 Conference, Colorado, USA. 8.13. Bayer, R. Schneider, W. Marquardt, 7th International Symposium on Process Systems Engineering, PSE 2000, Keystone, Colorado USA, July 16-21 (2000) 9. R. 13atres and Y. Naka, AIChE Symposium Series No. 323, Vol. 96 (2000) 10. R. Batres, M. L. Lu and Y. Naka, Joumal of Concurrent Engineering, Vol. 7, No. 1 (1999) 11. G. Booch, I. Jacobson, J. Rumbaugh, The Unified Modeling Language User Guide. Addison-Wesley (1999). 12. M. R. Genesereth, R. E. Fikes, KIF Version 3.0 Reference Manual, [Online] http://logic.stanford.edu/papers/kif.ps (1992) 13. W. F. Ramirez, Computational Methods For Process Simulation, 2nd ed., 13utterworthHeinemann (1997)
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
93
Dynamics of a reactive distillation column for TAME synthesis described by a nonequilibrium stage model R. Baur a, R. Taylorb and R. Krishna a aDepartment of Chemical Engineering, University of Amsterdam, Nieuwe Acthergracht 166, 1018 WV Amsterdam, The Netherlands bDepartment of Chemical Engineering, Clarkson University, Potsdam, USA
The dynamics of a reactive distillation column for synthesis of tertiary amyl ether (TAME) has been analysed using a rigorous nonequilibrium (NEQ) stage model, employing the Maxwell-Stefan equations for mass transfer between fluid phases. Both pseudo-homogeneous and heterogeneous reaction models show the possibility of multiple steady states. Starting at the low-conversion steady state, a feed composition perturbation is shown to lead to a transition to the high conversion steady state, in conformity with the experiments of Mohl et al. (Chem. Eng. Sci. 54, 1029-1043, 1999). The liquid holdup within the catalytic reaction zone is seen to have a significant influence on column dynamics. 1. INTRODUCTION The design and operation issues for reactive distillation (RD) systems are considerably more complex than those involved for either conventional reactors or conventional distillation columns. The introduction of an in-situ separation function within the reaction zone leads to complex interactions between vapour-liquid equilibrium, vapour-liquid mass transfer, intracatalyst diffusion (for heterogeneously catalysed processes) and chemical kinetics. Such interactions have been shown to lead the phenomenon of multiple steady states (MSS) and complex dynamics, which have been verified in experimental laboratory and pilot plant units; for a comprehensive literature survey, see Taylor and Krishna (2000). In earlier publications we had developed rigorous nonequilibrium (NEQ) models to describe the steady-state and dynamics of RD columns (Baur et al., 2000a; Baur et al., 2000b; Higler et al., 2000). The major objective of this paper is to use our developed NEQ models to rationalise the recently published experimental results of Mohl et al. (1999) for TAME synthesis. 2. MULTIPLE STEADY STATES FOR TAME SYNTHESIS We first examine the steady-state multiplicity characteristics of the TAME process using both the pseudo-homogeneous and heterogeneous descriptions of the catalytic reactions. TAME is formed by an acid-catalysed equilibrium reaction from 2-methyl-l-butene, 2methyl-2-butene and methanol. The reaction kinetics is described by a LangmuirHinshelwood rate expression in terms of the liquid phase activities (Oost et al., 1995). We did not incorporate the isomerization reaction of the C5-olefins in the model. Furthermore, we assumed the acid group concentration per Raschig ring volume to be 0.9 eq(H+)/L and the
94 zomplete catalyst volume to be 1.2 L. Fig. 1 (a) schematically depicts the column zonfiguration of the TAME process presented in Mohl et. al. (1999). The column has an inner diameter of 76 mm. The top section of the column is packed with catalytic and the bottom section with inert (glass) Raschig rings. Both sections are 0.5 m high. The feed is located in the middle of the column between the reactive and inert packing. As specified by Mohl et. al. (1999) in their experiments the feed rate was chosen to be 0.96 kg/h. The column operates at a pressure of 0.25 MPa. In order to evaluate the vapour-liquid equilibrium at the interface the Wilson equation was used for calculating the liquid activity coefficients. Furthermore, a total condenser and total reboiler is employed. The reflux ratio was specified at 15, whereas the reboiler heat duty was used as a homotopy parameter. Each section has been divided in 20 slices. Simulations with the pseudo-homogenous model show that the overall performance correspond to an efficiency model with 10 stages and an efficiency of 0.7 proposed and experimentally validated by Mohl et. al. (1999). The column decomposition was subsequently maintained for both pseudo-homogenous and heterogeneous model. The pseudo homogeneous model replicates the steady state multiplicity experimentally observed by Mohl et. aL (1999) for a reboiler heat duty of 340 W; see Fig 1 (b) and (c). Mohl et. al. (1999) explained the steady state multiplicity with kinetic instabilities of the TAME reaction. The two steady states differ in their temperature profile and TAME purity in bottoms flow rate. As high conversion steady state is referred the steady state with higher temperatures and higher TAME purity in contrast to the low conversion steady. The TAME mole fraction in the bottoms flow rate is about 0.22 for the high conversion and 0.18 for the low conversion steady state, respectively; see Fig 1 (c). Fig 1 (b) also shows that the column is predicted not to operate in thermal equilibrium. Liquid temperature profiles, denoted by bold lines in Fig 1 (b), and vapour temperatures are not equal, except in the reactive section when operating at the low conversion steady state. This behaviour is not uncommon for small laboratory columns such as the one under consideration. It is caused by the differences in dew and bubble point temperatures when the liquid bottom mixture is vaporised in the total reboiler. For comparison purposes we also carried out simulations of the steady-state behaviour using a more detailed model additionally accounting for mass and heat transfer resistances at the liquid/solid interface and intra-catalyst diffusion (Dusty Fluid Model) and reaction; see our earlier paper by Higler et al. (2000) for details. The catalyst parameters used are: a)
f
-r ~"
0.24
~
0.22
',-'g 1 00
i
0.20
~3 0.75
~
o.18
~ m
016
- ~ n-pentane
c)
b) + methanol
Feed:
o~
15o '~
o".
='~_
E" 1.25
iso-olefins methanol n-pentane
g
E
0.50 O 25 325
t-
~ 330
n-pentane + TAM~E 9 o
Exp data (high con.) Exp.data (low con.)
-
~
-
~ o o 335 340
, 345
~ ~ ~' 350 355
' 360
250
200
Liq. temperature (high con ) Liq. temperature (low con.)
- - - Vap. - -
Vap.
temperature (high con.) temperature (low con.)
300
350
400
450
500
Reboiler heat duty / [W]
Temperature / [K] --
Pseudo homogenous model
Fig. 1 (a) Schematic depiction of the column configuration, (b) temperature profiles of the high and low conversion steady state at a reboiler load of 340 W, (c) bifurcation diagram for the pseudo homogenous model.
95
w
a)
....
" 0.24
Pseudo homogenous model
Base Case
b) 0.230
/ ~. 0.22 o
~'~ ~"
//
0.225
~= 1.5 (base case)
/
T=I.0
/ /
E
g
0.20
/.
-/ 7 /
E
/
/
10000m'~/m 3
interfacialarea
o.22o
o
uJ
:~
0.1800,
"T'
0.24
~' o~
,-,~, S 350
............. 400 450
500
c)
0.22
" "~ - - ~ ' - ' - " .
.
.
.
....~
~
~
J 500
9 o22.
, u , s e n 0..us.v..y (= 5 x average diffusivity)//
0,222
' '~'x, ~
-,-' '6
m
o
._ ~
', , - ' = , , , I , , , , 400 450
Knudsen diffusivity ignored (base case)
d)
Catalyst thickness: . lmm ~ '
=~
:~
0.215 . . . . . . . . 300 350
0.220
...
0.20
~o
~ .
( b2 e 2 m ~
,
,
,
,
:/,2//
350
i
400
i
i
i
I
I
450
Reboiler load I [W]
i
i
~
~. -"~
0.216
/.
E
i
I
500
i
300
i
i
i
i
i
350
,
i
i
i
400
i~Ik i
KKnudsen diffusivity UadvSen gdit
diffusivity)
l
I
I
450
I
I
I
I
I
500
Reboiler load / [W]
Fig. 2 (a) Bifurcation diagrams of the Dusty Fluid model for parameter variation of (a) interfacial area, (b) tortuosity, (c) catalyst thickness and (d) Knudsen diffusivity.
porosity= 0.5, tortuosity = 1.5, catalyst surface area = 567 m2/m3, mean pore diameter = 160 nm, catalyst thickness = 2 mm, thermal conductivity = 1.0x 10-6 W K ~ m -]. The parameter values are based on specifications given in Mohl et. al. (1999), Rapmund et al. (1998) and Sundmacher (1995). In order to solve the partial differential equations along the film and catalyst thickness numerically, we applied a finite difference scheme. Detailed description of the model and discretization scheme is given in Higler et. al. (2000). For the discretization of the catalyst thickness we applied 50 grid points per column slice and for liquid, vapour and solid/liquid films 5 grid points each. Finer grid spacing does not alter the results. Fig. 2 shows the bifurcation diagram of the TAME purity in the bottoms flow rate when the reboiler load is used as a continuity parameter. The pseudo homogenous model and the Dusty Fluid model exhibit similar steady state multiplicity characteristics; intra-particle mass and heat transfer resistances cause the overall TAME production of the Dusty Fluid model to be lower, as can be expected. On the other hand, the Dusty Fluid model coincides with the pseudo homogenous model as intra-catalyst resistances vanish. In this case the pressure and composition gradients disappear and the reaction rate becomes constant within the catalyst. Vanishing intra-catalyst resistances are achieved by increasing the interfacial area of the catalyst, as shown in Fig 2 (a). The influence of increased or decreased intra-catalyst resistances is exemplified in the Fig. 2 (b) and (c); decreased tortuosity or catalyst thickness reduces intra-catalyst resistances and so results in higher TAME production. Our Dusty Fluid
96 model also incorporates the Knudsen diffusivity, describing a diffusional resistance between the liquid and walls of the porous media. There is no correlation available to estimate the Knudsen diffusivity. Sundmacher and Hoffmann, (1994) proposed to ignore it since wall effects are negligible for macroporous media. Fig 2 (d) shows the bifurcation diagram for different values scaled on the basis of the average diffusivity. For the system under consideration the computations also appear to be insensitive to changes of the Knudsen diffusivity for a very large range. Therefore, the assumption of Sundmacher and Hoffmann (1994) seems not only to be appropriate, but also does not affect predictions for the TAME synthesis. The major conclusion to be drawn from the results shown in Fig. 2 is that the pseudo-homogenous model and the detailed heterogeneous model display essentially the same bifurcation features. This gives us confidence is proceeding further with the dynamic simulations using the, computationally simpler, pseudo-homogeneous model. 3. DYNAMICS SIMULATIONS OF STEADY STATE TRANSITIONS Dynamic simulations require additional information about the liquid holdup in the column, reboiler and condenser. The storage capacities of the condenser and reboiler are assumed constant and have been estimated to be 1 L each. Since the pseudo homogeneous model does not account for intra-particle mass storage in the catalyst we assumed a constant liquid hold-up in the catalyst with composition equal to the bulk phase. We estimated the time-independent hold-up to be 0.23 L in the entire catalytic section and 0.33 L in the entire inert section. This is a rather crude way of modelling the liquid hold-up in porous Raschig rings. As we showed before the compositions in the catalyst vary significantly. Further one can assume that the liquid hold-up in the catalyst too. Reason for this is the swelling or wetting of the catalyst depending on temperature and mixture composition. Our objective is to demonstrate that intra-catalyst liquid hold-up plays an important role for dynamics and one should keep in mind possible limitations of the underlying assumptions of a pseudo homogenous model. Consider steady-state operation at the low-conversion branch of the bifurcation diagram shown in Fig 1 (c) with the reboiler load fixed at 340 W. We adopted the perturbation scheme and the experimental data from Mohl et. al. (1999); see Fig. 3. At the beginning of the perturbation (t = 0 min) the feed is switched to pure TAME while the flow rate is maintained. a) Temperature at the bottom of the reactive section
b) Temperatureat the bottom
of the inert section 390
350 345
o
,,~ ] " I ~
/ /
,~
3 t..
335
., ~, | / J '~v I ~ / / / / / 350 ~
Experimental data Sim: Liq. temperature S i m : V a p . temperature
340 ~-
c)
t ~
330 .
"
.
o~
.
.
.
.
l,i
E
330
325
I
1
i
i
i
i
0
60
120
180
240
300
Time after perturbation / [min]
360
330
' 0
' 60
t 120
' 180
' 240
' 300
' 360
Time after perturbation / [min]
Fig. 3 Vapor and liquid temperature trajectories of low to high conversion steady state transition when switching the feed to pure T A M E for one hour. (a) Temperatures at the bottom of the catalytic section. (b) and (c) Temperatures at the bottom of the inert section.
97 One hour later the feed was reset to its original values. Fig. 3 shows the predicted vapour and liquid temperatures. We also see from Fig. 3 that the main dynamic features of the Mohl et al. experiments are captured by our model and the column undergoes a transition from a low to high conversion level. Fig 3 (a) shows the temperature at the bottom of the reactive section. The liquid and vapour temperature rises rapidly due to the vaporising pure TAME feed below the catalytic section. When the feed is reset at the end of the perturbation our model predicts a sharp drop in temperature and quickly recovers its final (high conversion) steady state. This trend does not correspond to the measurements of Mohl et. al. (1999), which show a large undershoot and it takes approximately another 4 h until the high conversion steady state is reached. Consider the bottom of the inert section. The magnitude of the temperature peak is matched quite well by our model. However the model predicts a much sharper rise than observed experimentally. The deviations between experiment and simulation are due to phenomena not incorporated in the pseudo homogeneous model, such as pressure fluctuations, not considered responses of the control and precise description of hydrodynamics in the laboratory column and packing. The hydrodynamics, in particular, seems to affect the dynamics significantly. The assumption of quasi-stationary liquid hold-up in the packing is not sufficient to capture changes in mass and energy within the packing. The steady state transition described above is mainly affected by the inverse reaction order of the TAME synthesis. Fig. 4 shows the composition profiles along the reactive section during perturbation. It can be seen that the feed disturbance causes the bottom of the reactive section to be purified from methanol. When the feed is reset to its original values high production rates are obtained at the bottom of the reactive zone due to the inverse reaction order of the TAME synthesis. As a consequence the system does not revert back to its original (low conversion) steady state. Fig. :i presents dynamic simulations when the quasi-stationary hold-up is doubled, which corresponds to the limiting case of completely wetted Raschig rings. In contrast to the base case discussed in the previous section no steady state transition is observed; the original steady state is recovered. We also note that the increase in temperature in the reactive section (see Fig. 5) is more gradual when the liquid hold-up is higher. As can be seen from Fig 5, the liquid temperature in the bottom of the catalytic section rises in two "steps". The simulations for the two cases show that the second temperature push takes about 20 min earlier for the 0.65
/~o.~
0.70,/ _~
o~ ,~E ,~
0.75 /
\ \,~
Temperature at the bottom of the reactive section
0 30
350
.\
~ = ~ , n ? ~ 0.25 %~.. 0.60/~'~'~/ ~'Y \ 02 0%. ~ initial /~, /~\(. 0 %& o~ / ~ c / ~ o r , vers,oo~\ -% ~
o o/
~',e~uy .... ?~
- - ~ 0.15
\/ \/ \
~ ~ \A 0.05 / \ / \ / \ of perturbation) "~ 0.00
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Mole fraction of TAME / [-] Fig. 4 Composition profile along the reactive section for low to high conversion steady state transition when switching the feed to pure TAME for one hour.
345
?
340
E
335
o/
A A A ~176
0"95/final~~-" ~' ~' steady l[ \ ~. \ / \ 1.00/ s t ~ / e n d
Liq. temperature (base case) Liq. temperature (double liq. cat. holdup)
330 325
0
60
" " ........... ' .... -~ 120 180 240 300 360
Time after perturbation / [min] Fig. 5 Liquid temperature trajectories for the base case amount of liquid hold-up in the catalyst and when it is doubled. The feed is switched to pure TAME for one hour at t = 0.
98 base case whereas the first push occurs almost simultaneously for the two cases. The first temperature push is caused by an increase of TAME in the bottom of the reactive section after pure TAME is fed and vaporised. Generally speaking, higher liquid hold-up increases the liquid residence time in the column and so resists a faster propagation of a disturbance. Consequently the composition profiles can be expected to change less pronounced in the reactive section. 4. CONCLUSIONS The RD column for TAME synthesis exhibits steady-state multiplicity; this phenomenon is captured by a pseudo-homogeneous description of the reaction kinetics. The use of a rigorous heterogeneous catalytic reaction model yields essentially the same bifurcation features. A dynamic nonequilibrium model, using the pseudo-homogenous reaction description, is able to predict the steady-state transition in the experiments of Mohl et al. (1999) starting at the low conversion branch of the bifurcation diagram. The precise transient behaviour is found to be very sensitive to the liquid holdup in the catalytic section. The need for a proper understanding of the hydrodynamics in order to describe the column dynamics is underlined. REFERENCES
1. Baur, R., Higler, A.P., Taylor, R., and Krishna, R. (2000a) Comparison of equilibrium stage and non-equilibrium stage models for reactive distillation, Chem. Eng. Journal, 76, 33 47. 2. Baur, R., Taylor, R., and Krishna, R. (2000b) Development of a dynamic nonequilibrium cell model for reactive distillation tray columns, Chem. Eng. Sci., 55, 6139 - 6154. 3. Higler, A., Krishna, R. and Taylor, R. (2000) Non-equilibrium modelling of reactive distillation: A dusty fluid model for heterogeneously catalysed processes, Ind. Eng. Chem. Research, 39, 1596 - 1607. 4. Kooijman, H.A. and Taylor, R. (1995) A Dynamic nonequilibrium model of tray distillation columns, A.I. Ch.E.J., 41, 1852 - 1863. 5. Mohl, K.D., Kienle, A., Gilles, E.D. Rapmund, P., Sundmacher, K. & Hoffmann, U., (1999) Steady state multiplicities in reactive distillation columns for the production of fuel ethers MTBE and TAME: theoretical analysis and experimental verification, Chem. Eng. Sci., 54, 1029 - 1043. 6. Oost, C. and Hoffmann, U. (1995) The synthesis of tertiary amyl methyl ether (TAME): microkentics of the reactions, Chem. Eng. Sci., 51, 329 - 340. 7. Rapmund, P., Sundmacher, K. and Hoffmann, U., (1998), Multiple steady states in a reactive distillation column for the production of the fuel ether TAME part II: Experimental validation, Chem. Eng. & Technology, 21, 136- 139. 8. Sundmacher, K. and Hoffmann, U. (1994) Multicomponent mass transfer and energy transport on different length scales in a packed reactive distillation column for heterogeneously catalyzed fuel ether production, Chem. Eng. Sci., 49, 4443 -4464. 9. Sundmacher, K. (1995) Reaktivdestillation mit katalytischen fuellkoerperpackungen - ein neuer Process zur Herstellung der Kraftstoffkomponente MTBE, Ph.D thesis, Universit~it Clausthal, Germany. 10. Taylor, R. and Krishna, R. (2000) Modelling reactive distillation, Chem. Eng. Sci., 55, 5183 - 5229.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
99
heat-integrated heterogeneous tubular reactors w i t h axial h e a t c o n d u c t i v i t y in r e a c t o r wall.
Dynamics
of
M.Berezowski a'c , E.W.Jacobsen h and R.Grzywacz c a Polish Academy of Sciences, Institute of Chemical Engineering, 44-100 Gliwice, ul.Baltycka 5, Poland, e-mail:
[email protected] hRoyal Institute of Technology, Stockholm, Sweden c Cracow University of Technology, Institute of Chemical Engineering and Physical Chemistry, Poland The study concerns a theoretical analysis of non-adiabatic heterogeneous tubular reactor with external heat integration and axial heat conductivity in the solid phase. Based on analysis and simulations it is demonstrated t h a t such a system can generate complex oscillatory profiles of temperature and concentration - periodic or chaotic. These profiles, especially those of aperiodic character, can seriously impair the performance of the system. The effect of one p a r a m e t e r on the reactor dynamics is studied, namely, the DamkShler number Da. All results presented in the paper are compared with those previously obtained using homogeneous [1] and pseudohomogeneous models [2]. 1. M O D E L L I N G The process considered in this paper is that of a single exothermic reaction taking place in a tubular reactor. The reactor is heat-integrated in the sense that the feed is pre-heated using the reactor effluent. The dynamic behavior of this process is a classic problem and has been extensively studied. See e.g., [3]-[5]. In particular, it is well known that the presence of heat-integration may introduce limit cycle behavior in the reactor. However, only recently was it shown t h a t this process is also capable of displaying highly complex behavior including chaotic oscillations [1]. The results presented in [1] were based on a homogeneous plug flow model, neglecting dispersion effects. In [2] a pseudohomogeneous model including mass and energy dispersion was employed, and it was shown that the results in [1] holds also for this case, but only for relatively small Lewis numbers. In [6] and [7] similar results for a loop reactor (heat and mass integration) are presented. In this paper we consider a heterogeneous model, i.e., a model including a solid phase. We assume the reactor wall to represent the solid phase, and take the axial heat conductivity of the wall into consideration. Including the heat capacity (or) of the wall makes the model more realistic from an industrial point of view.
100 The model we employ, based on dimensionless material and energy balances and including the above mentioned assumptions, is given below. See also Nomenclature. Mass balance of fluid phase: Oa Oa ~r ,34
(1)
heat balance of fluid phase:
-ar- + ar =~,(,~,~)+
st(o - ~)
(2)
heat balance of solid phase: Oz"
~a~~ Pe,. O~2
st(o
~)+~(o.
o)
(3)
with boundary conditions" dO for ~=0 a(0)- 0; ~b(0)-fib(l~, ~ - 0
(4)
dO ~=0
(5)
d~
where N(a,~) = Da(1-a)" exp
for ~=1
/ ) 1+ ,6'~
I
Here f is a p a r a m e t e r describing the heat-integration efficiency.
(6)
101
2. A N A L Y S I S OF T H E M O D E L The P D E model (1)-(6) is discretized in space using orthogonal collocation on finite e l e m e n t s . The n u m b e r of collocation points a n d e l e m e n t s w e r e chosen to e n s u r e acceptable consistency with static bifurcation points o b t a i n e d u s i n g the eigenvalue m e t h o d for the linearized PDE's. Bifurcation analysis is employed to trace out the d y n a m i c behavior for different p a r a m e t e r values, a n d we show t h a t the r e a c t o r m a y display highly complex d y n a m i c behavior, including chaos, for realistic p a r a m e t e r value. Figure 1 p r e s e n t s a bifurcation d i a g r a m of the s y s t e m for p a r a m e t e r values fl=1.4, f=0.3, Pq=150, St=50, 5=2, 0 H = - 0 . 0 7 , cr=0.03,
n=l. 0.4 0.35 0.3
r
i
!
LP~, ~ ....... C.,~B ............ :~. .............. ~ . . . . . : - : ~ : "... , ..............
........
i
0.25
" ....
................ ~ .............................................. --- ~ _ ~ _ ~ _ ,
................................
-: . . . . . . . . . .
::--:::;-
r-------
]"---
.............. "................ ::........... : .................. i ............... ~............... . ! .............
0.2 0.15 0.1
..............................................................
i-:::'~:~. ..... i ............... ~ .............. o
.............. -:.................................................
::............. :"i: .............. "-..............
0.05
-0.05 0.12
!l
i
0.13
0.14
D~t.
i
0.15
0.16
0.17
0.18
0.19
Fig.1. Bifurcation diagram. LP- limit point, HB- Hopf bifurcation, CB- chaos bifurcation. F r o m the above d i a g r a m we see t h a t periodic solutions of the s y s t e m exist b e t w e e n HB-CB a n d CB-HB points. Chaotic solutions a p p e a r b e t w e e n CB-CB points. The chaotic n a t u r e of the t e m p o r a l profile is reflected by its sensitivity to initial condition. Fig.2 illustrates two different t e m p o r a l p a t h w a y s . One p a t h w a y is denoted by a solid line, the other one by a b r o k e n line. In the initial s t a t e (r = 0 ) the difference b e t w e e n the two p a t h w a y s is infinitely small, a n d is practically invisible in the diagram. This state occurs, however, only up to a c e r t a i n m o m e n t (in our example until r ~ 17). F u r t h e r s i m u l a t i o n reveals a m a r k e d divergence of the two curves.
102 0.5
~(1)
O.45
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
- ........
0.4
li
0.35
0_25
9 0_2
i
i
I 5
0
10
I
15
20
"t;
25
30
Fig.2. Illustration of sensitivity to initial condition. Da=0.15. The type of oscillations can be identified also based on, among others, the analysis of a phase diagram and Poincar6 map. The phase trajectories in Fig.3 do not form any regular pattern and belong to a strange attractor. o_s
!
...............
.~ . . . . . . . . . . . . . . .
: ...............
~ ...............
~. . . . . . . . . . . . . . . .
J. . . . . . . . . . . . . . . .
...............
~.
-
, 0"2.65
0.7
111.75
0.8
0.85
0.9
i
, 0.95
cr I
Fig.3. Strange attractor. Da=0.15. In the present paper the Poincar~ section is defined as the plane constructed based on the extremes of the outlet temperature in the fluid phase r (1), as well as on the corresponding values of the conversion degree a(1). For every Da value in the interval CB-CB the picture obtained contains a cloud of points, indicative of chaos (Fig.4).
103
Fig.4. Poincar~ map. Da=0.15. Comparing the results in the present work, concerning heterogeneous reactor, with results from papers [1] and [ 2 ] , concerning homogeneous and pseudohomogeneous reactors, we see clearly that complex dynamic and chaos exist for the same p a r a m e t e r values as was found in the more idealized models in [1] and [2]. The results thus serve to indicate that the theoretically found complex behavior is possible to appear in a physical reactor system. For a detailed discussion of the source of the complex behavior we refer to the previous papers [1], [2]. From our results we find t h a t the high order of dynamics bifurcation and chaos are a consequence of the feedback effect imposed by the loop integration (heat or mass). F u n d a m e n t a l for this bifurcation is the pure logistic system [1]. All other models (pseudohomogeneous, heterogeneous, with dispersion etc.) can be seen as modifications of the logistic model only. 3. C O N C L U D I N G R E M A R K
The present study deals with theoretical analysis of the dynamics of the heterogeneous autothermal non-adiabatic tubular reactor with axial heat conductivity in the solid phase (wall of reactor). The significance of the results stem from the fact t h a t such systems are commonly employed in chemical engineering. Based on numerical simulations, various types of oscillations have been shown to occur in the reactor. Periodic, multiperiodic and chaotic oscillations have been found. The results are presented as so-called bifurcation diagram (Fig.l). Phase portrait, Poincar~ section and temporal trajectories are also shown. The oscillations of temperature and concentration, especially aperiodic oscillations (chaotic) can adversely affect the operation of the system.
104 4. N O M E N C L A T U R E
Da DamkShler number f thermal recycle coefficient n order of reaction Pe, thermal Peclet number for solid phase St Stanton number a degree of conversion fl dimensionless number related to adiabatic increase of temperature 5 dimensionless heat exchange coefficient dimensionless temperature of fluid phase y dimensionless activation energy O dimensionless temperature of solid phase dimensionless time dimensionless position coordinate along reactor Subscripts exextremum H cooling medium s solid
REFERENCES
1. E.W. Jacobsen, M.Berezowski, Dynamics of heat integrated homogeneous tubular reactor. 5th IFAC Symposium on Dynamics and Control of Process System. Corfu, Greece, June 8-10 (1998). 2. M.Berezowski, P.Ptaszek, E.W.Jacobsen, Dynamics of heat-integrated pseudohomogeneous tubular reactors with axial dispersion. Chem.Engng Process., 39,181-188 (2000). 3. S.Subramanian, V.Balakotaiah, Classification of steady state and dynamic behavior of distributed reactor models. Chem.Engng Sci., 51,401-421 (1996). 4. J.C.Morud, S.Skogestad, Analysis of instability in an industrial ammonia reactor. AIChE J., 44,888-895 (1998). 5. D.Luss, N.R.Amundson, Stability of loop reactors. AIChE J., 13,279-290 (1967). 6. E.W.Jacobsen, M.Berezowski, Chaotic dynamics in homogeneous tubular reactors with recycle. Chem.Engng Sci., 53,4023-4029 (1998). 7. M.Berezowski, Spatio-temporal chaos in tubular chemical reactors with the recycle of mass. Chaos,Solitons&Fractals, 11, 1197-1204 (2000).
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
105
Simulation and experimental study of intermediate heat exchange in a sieve tray distillation column Ingela Niklasson Bj6rn, Urban Gr6n and Fredrik Svensson Department of Chemical Engineering Design, Chalmers University of Technology, Sweden The aim with this study was to combine a simulation study with an experimental in the investigation of intermediate heat exchange as a method to change the distribution of the driving forces in the column. It was applied to a pilot plant sieve tray distillation column for a binary separation of ethanol from n-propanol. Intermediate heat exchange has been accomplished experimentally both in the stripping section and in the rectifying section. Optimal sidestream return in intermediate heat exchange is the focus of attention here. The experimental studies were performed on a pilot plant column with twelve sieve trays. When comparing experimental results from the pilot plant column, to simulation results the inclusion of tray efficiency is of interest. For this purpose a calculation routine was developed in MATHCAD based on a correlation by Chan and Fair. The resulting data was than transferred to the PROII simulation program. The results show that intermediate heat exchange can be used to improve the recovery, the separation capacity and to decrease the entropy production if the points of sidestream withdrawal and sidestream return are chosen properly. It is important that the heat exchanged liquid stream returned as vapour enters at a position in the column where the vapour phase has the same composition. Even if the composition coincides, the actual position for an added vapour stream is important for the flow profiles in the column. 1. INTRODUCTION Distillation is one of the most common separation operations in the process industries, but is also known to be an energy-consuming and sometimes thermodynamically ineffective operation. Interesting contributions regarding the different methods to improve the thermodynamic efficiency of the distillation operation have been presented (see e.g. 1, 3, 711). Special consideration should be devoted to the questions of what energetic improvements can be made in a given distillation column with the separation task maintained and in what way this will influence the performance of the column. Intermediate heat exchange may be regarded as a method to minimise the entropy production in distillation. It is equivalent to a minimisation of the net work consumption of the separation. When intermediate heat exchange is applied to a conventional column, the number of stages required for the separation is normally allowed to increase.
2. THEORY The strive for optimal sidestream return of phase changed streams due to heat exchange require that the position of sidestream withdrawal is matched with the position for retum of the stream. The extra energy added or withdrawn in the heat exchanger offer an opportunity to
106 either take full advantage of or to destroy the system separation performed depending on the combination of the outlet and inlet positions. However, to take full advantage of the optimal sidestream return method the amount of the intermediately heat exchanged and pumped around flow rate has to be taken into account. For a withdrawn and phase changed stream in the stripping section the vapour, Wihex, of the composition Yihexwill after return to the column be mixed with Vn from the tray below. To be able to judge different intermediate heat exchange couplings, not solely by comparing the composition of the vapour from the heat exchanger with the original vapour in the column, Dill*strip is introduced Yv.
(1)
Vmix *Ymix = Vn *Y v. + Vihex*Yihex
(2)
Diff;rip = Ymix --
A positive difference in Diff*strip will always for a fully phase changed incoming stream indicate that an internal separation increase has been achieved. The vapours have been mixed and the resulting vapour has a higher content of the light component then the original one on the tray, this means a gain in separation that is a combination of the amount and the composition of the vapour stream from the heat exchanger. Different pumparound flow rates in the heat exchangers will automatically affect the operating lines in the column. To make comparisons and to get a measure of the variations in driving forces represented by the distances between the equilibrium curve and the operating lines a variance describing the evenness has been expressed as: Variance
=
(~.,(y~q- Y o . l . ~ ) / N - ( E ( Y e q
- Yo.l.]]~2/IN 2
(3)
This can be a measure of the variations in driving forces and how they can be minimised. In order to express the tray efficiency the Murphree tray efficiency was used. To accommodate for varying operating conditions on the different trays the model of Chan and Fair (2, 3, 5 and 6) was used to estimate these variations. The basic concept in this model is that the concentration is not constant over the entire tray and in order to obtain a value at a single point on the tray a more complex model is needed describing the concentration gradient. By using two-film theory the following expression for calculation of the point efficiency can be derived:
Eo~ = 1 - exp(- Nov)
(4)
The derivation of the formula above can be found in many textbooks dealing with mass transfer. The formula above includes the overall transfer unit and is defined as:
Nov = 1/(1/g~ +,VN~)
(5)
107 There exists a large number of models for the prediction of transfer units. Chan and Fair have presented a model for the calculation of the liquid and vapour transfer units. The model is a modification of the AIChE model and the relationships for hydraulics and mass transfer were improved. This method was chosen for the evaluation of the transfer units since several authors recommended it for the system at hand. The number of transfer units in the vapour phase is calculated by: Nv = 0 0300 - 8670- F y ) - F j - ~ v . t v / X ~ c t
(6)
The equation for the calculation of the number of transfer units in the liquid phase is: N L = (0.40. F + 0.17). f f D L 91.97.
104.tL
(7)
An example of how the Murphree tray efficiency varies with vapour and liquid flows relevant for the pilot column is shown in figure 1.
Fig. 1. Murphree tray efficiency vs. vapour and liquid flow (50 mol-% ethanol/propanol). 3. EXPERIMENTAL The experimental studies were performed in a six-meter high pilot plant column with consists of a sieve tray distillation column with intermediate heat exchange and is shown in Figure 2. The intermediate heat exchange is performed in an external heat exchanger. The whole distillation column system has a rather flexible structure. This means that intermediate heat exchange can be studied for several sidestream connections in the stripping section and in the rectifying section. The column has got twelve plates with the diameter of 0.2 m. The distances between the trays are 0.35 m. The feed can be introduced at plate six, eight or ten. In the stripping section a liquid stream is vaporised and reintroduced to the stripping section. The vaporisation is carried out using admission steam as the heating medium. The liquid can be withdrawn from plate nine to plate twelve. And the vapour can be returned as appearing to come from plate nine to plate twelve.
108 To study intermediate heat exchange in the rectifying section a withdrawn vapour stream is condensed and returned as a liquid stream within the enriching section. The
Fig. 2. Pilot plant distillation column system intermediate heat exchange is then performed with cooling water as utility. Both the position for the outlet of the vapour stream and the position of the returned liquid stream can be varied to different locations. 4. RESULTS AND DISCUSSION As noted above the Murphree tray efficiency is mainly influenced by the vapour flow and liquid concentrations. Below is the Murphree tray efficiency presented for the simulated diabatic set-ups.
Fig. 3. The Murphree tray efficiency along the column at various intermediate flow rates.
Fig. 4. Entropy production as a function of intermediate flow rate.
The Murphree tray efficiency is rather constant through the column for the base case, the stripper part being slightly more efficient. The Murphree tray efficiency for the trays below the tray where the vapour is returned is seriously decreased. The higher the amount of pump-
109 around the lower the average Murphree tray efficiency gets since the Murphree tray efficiency is lowered considerable below the point of return. For 90 dm3/hr the Murphree tray efficiency drops to around 45 %. On the trays right above the point liquid extraction the Murphree tray efficiency is increased, the vapour flow in this section high and gives favourable flow conditions on the trays. The Murphree tray efficiency above the feed point is only marginally increased with the amount of pump-around. The entropy production of the column for different set-ups simulated is given above. The entropy production decreases as the amount of material being pumped around increases. The further up the liquid is extracted from the column the steeper the decrement is. The boiling point of the mixture is lowered as the concentration of ethanol increases and the sidedraw containing more ethanol thereby does not require as much heat to be vaporised as if the concentration of ethanol should have been higher. From this point of view the L9V 10, L9V 11 and L9V12 seem to be the most favourable set-ups. An interesting thing about the line for L9V12 is that the entropy production actually increases for high amounts of pump-around. Whether this is because the difference is so small so it lies within the margin of error or if it is because the flow conditions is being affected in an unfavourable fashion has been discussed. No answer with a hundred percent certainty has been concluded since the margin of error in PRO II could not be evaluated. It might be a combination of both.
0.85
Q04]
.................................
...................
]
0.75
~
0.65 I I
L9
0.6 !1 . . . . . 0.55
"
~
Lll
-
, V9
"'~\
,-,, LIO L12
, V10
Vapour retum
"'<%\ "~
, V11
i i
i i
V12
position
Fig. 5. Experimental column separation
411J . . . . . . . . . . . . . . . . . . . . . . . VamurM u . " b ~ m
J
Fig. 6. Calculated Diff
In Figure 5 experimental results of the column separation are shown. In Figure 6 calculated values of Diff support the experimental results, indicating that L9V10 is one of the more favourable set-ups.
REFERENCES 1. T.R. Andersen, G. Siragusa, B. Andresen, P. Salamon, S.B. Joergensen, Energy efficient distillation by optimal distribution of heating and cooling requirements, ESCAPE-10 (1999) 709 2. H. Chan, and J.R. Fair, Prediction of Point Efficiencies on Sieve Trays. 1. Binary Systems, Ind. Eng. Chem., Proc. Des. Dev. 23 (1984) 814 3. H. Chan, and J.R. Fair, Prediction of Point Efficiencies on Sieve Trays. 1. Multicomponent Systems, Ind. Eng. Chem., Proc. Des. Dev. 23 (1984) 820
110 4. V.R. Dhole, and B. Linnhoff, Distillation Column Targets, Computers Chem. Engng, Vol. 17, No. 5/6 (1993) 549 5. J.R. Fair, H. R. Null and W. L. Bolles, Scale-up ofplate efficiency from laboratory Oldershaw data, Ind. Eng. Chem., Proc. Des. Dev., 22 (1983) 53 6. J. Ilme, Estimating Plate Efficiencies in Simulation of Industrial Scale Distillation Columns, Doctoral Thesis, Lappeenranta University of Technology, 1997 7. L.R. Lynd, and H. E. Grethlein, Distillation with Intermediate Heat Pumps and Optimal Sidestream Return, AIChE J, Vol. 32, No. 8 (1986) 1347 8. O.C. Mullins and R. S. Berry, Minimisation of entropy production in distillation, J. Phys. Chem., 88 (1984) 723 9. S. Ratkje, Kjelstrup and A. J. De Swaan, Denbigh Revisited." Reducing Lost Work in Chemical Processes, Chem. Eng. Sci., Vol. 50, No. 10 (1995) 151 10. R. Rivero, T. Cachot, A. Ramadane and P. L. LeGoff, Diabatic or quasi reversible rectification, Int. Chem. Eng, Vol. 43, No. 2 (1994) 240 11. D. Tondeur and E. Kvaalen, Equipartition of entropy production. An optimality criterion for transfer and separation processes, Ind.Eng.Chem.Res., 26 (1987) 50 NOMENCLATURE
D DL
Dr, Diff*strip DSprod
Eo~ F
FI hct
NL Nov Nr,
Qc QR Qihex
tL tv
T8 T~ Tr,ihex XF XD Yihex
Yr,n
distillate, flow rate [kmol/s] liquid phase diffusivity [m2/s] vapour phase diffusivity [m2/s] concentration difference [-] entropy production [W/K] point efficiency [-] F-factor [(kg/m)~ feed flow rate [kmol/s] fractional approach to flooding [-] clear liquid height [m] number of transfer units for the liquid phase [-] overall number of transfer units [-] number of transfer units for the vapour phase [-] duty, condenser [W] duty, reboiler [W] duty, intermediate heat exchanger [W] average liquid residence time [s] average vapour residence time [s] temperature, bottoms [K] temperature, distillate [K] temperature, vapour, side stream [K] Concentration, feed [-] Concentration, distillate [-] Concentration, vapour, sidestream [-] Concentration, vapour, from tray n [-]
European Symposiumon ComputerAided ProcessEngineering - 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rights reserved.
111
On-line modelling in the petroleum industry: Successful applications and future perspectives David B. Cameron w R. Jorgen Odegaard*, and Erik Glende $ Fantoft Prosess AS, PO Box 306, N-1301 Sandvika, Norway The on-line use of a first-principles dynamic model of a hydrocarbon pipeline system is described. This system has been successfully implemented in an industrial plant and provides a wide range of practical information to the plant operators. Such systems, where the plant model is relatively simple, but still non-trivial, provide a valuable basis for developing more ambitious on-line systems for more complicated processing units. 1. I N T R O D U C T I O N : DYNAMIC SIMULATION IN T H E P E T R O L E U M INDUSTRY Dynamic process simulators are now standard tools for process design and operator training for petroleum production plants. Construction projects for new oil platforms in the North Sea include, as a matter of course, a sub-contract for an engineering simulator. This simulator is a high fidelity, dynamic model of the process, and provides valuable information about the operability and safety of the planned process to the design engineers. Later in the project, the engineering model can be used to check and validate the process control system. Finally, in the lead-up to commissioning, the same model is used to train the process operators. These applications of simulation are plotted on the well-known process lifecycle in Figure 1 on the next page. However, once the plant is started up, these models are largely discarded, and are seldom used to provide on-line information to operators. This is a shame, as a large amount of design knowledge is rendered inaccessible, just at the time when it can be used most profitably. Why is this so? We believe that the answer lies in the interdisciplinary challenges that confront the developer of an on-line modelling system. Many different systems, disciplines and processes need to function and collaborate properly in a successful on-line model: Such a model is a complex result of the application of process modelling, ergonomics, statistics, database technology, data communications, numerical methods, artificial intelligence and control theory. The resulting system is only as robust as its weakest link, and this has meant that many on-line modelling systems have not lived up to expectations. Such systems lie, neglected and unlamented, on a dusty workstation in the comer of the control room. However, successful implementations of on-line models are possible. Several such systems are described in the rest of this paper. Successful applications are characterised by (1) a high quality, accurate and tuneable model, based on process knowledge, (2) which runs
wTechnical Manager, Process Control and Optimisation. *Business Area Manager, Real-time Systems. Technical Director.
112 robustly, quickly and reliably. (3) Mismatch between the model and the plant needs to be analysed (using heuristics and statistics) so that process problems can be found and faulty instruments detected. Finally, (4) raw data and results need to be exchanged reliably and in a timely way with one or more third-party control systems and user interfaces over a network.
Figure 1. Dynamic simulation in the process lifecycle There are challenges to academic research and industrial development in each of these areas. The rest of the paper will outline these challenges, with a sketch of actual and possible solutions, using successful on-line modelling systems for hydrocarbon pipe systems as a framework for the discussion. 2.CASE STUDY: A PIPELINE MANAGEMENT SYSTEM (PMS) This system uses a high-fidelity model of a hydrocarbon transport network to provide operational information to the system's operators. The key requirement for a PMS is that it will raise an alarm when a leak occurs in the system. This is called leak detection. In addition, the system then uses the model to predict where the leak is most likely to have occurred. This leak location is important information, as the pipelines monitored are often several hundred kilometres long and run through inhospitable terrain. Scrapers, or pigs, are routinely sent through piping systems to clean out deposits and reduce pressure drop. Smart scrapers can also be used to collect information for preventative maintenance. The position of a scraper in the pipeline depends on the fluid flow, composition and topography. This can be readily calculated by a model, but is difficult to predict using rules-of-thumb. An on-line scraper-tracking tool is used to warn operators of an approaching scraper and to raise an alarm if a scraper is stuck in the pipe. Related, batch tracking calculations are relevant for multi-product pipelines, where it is important that the interface between different products (such as diesel, petrol and avgas) is tracked along the pipeline, so that operators can minimise the loss of off-spec product at the receiving station. Long pipelines contain appreciable amounts of fluid. They are therefore used as storage buffers to balance supply with demand. In gas pipelines, pressure can be increased in anticipation of demand peaks. The calculation of line pack, which is the amount of fluid in a pipeline, is therefore a vital tool for optimising profit and customer satisfaction. Finally, the line pack can be used to calculate and report a survival time for the pipeline. This is the time
113 during which normal supply can be maintained to customers after a stoppage in supply to the pipeline. A PMS uses a detailed model of a pipeline to calculate and report these control parameters, thereby providing a "window into the piping network" for the operator. Operators thus gain access to calculated results that are not available from instrumentation alone. Safer, more efficient operations result from this. Fantoft has successfully implemented systems for both single-phase and multiphase pipelines. Examples for both types of system are given in the following discussion. 3. PROCESS MODELLING The process model in the PMS is built using Fantoft's D-SPICE software for dynamic simulation. Single-phase pipelines are represented using a dynamic pipe module, called DPIPE. Multiphase pipelines are handled by interfacing to third party products, such as Olga (Shea et al., 1997). An interface has been written so that Olga, from a system-integration point-of-view, functions as a D-SPICE module. Figure 2 shows how the output from an OLGA model can be displayed an operator. The plot shows the predicted liquid fraction along the pipeline. The large "slug" of condensate ("kondensat" in Norwegian) 42 km from the platform is due to a scraper being sent along the pipe to clean out deposits. The operators can use this information to, for example, reduce set points for level in the shore processing plant so that this slug of liquid can be handled properly. Klemp et al. (1997) describe such an application.
Figure 2. Real-time results from a multiphase pipeline monitoring system. Our experience is that an on-line model usually consists of one or more complex unit operations in a network of "generic" unit operations (pumps, valves, branches). This critical unit is often modelled by specially written or third party code. Efficient software integration is thus critical for the effective rollout of novel on-line systems. We are therefore supportive of the efforts of Global CAPE-OPEN (Braunschweig et al., 2000). 4. ROBUST, EFFECTIVE MODEL EXECUTION An on-line model must be able to run for long periods with predictable and low maintenance requirements. It must be good-quality software. Errors, such as memory leaks, which are not usually critical in off-line models, are not permissible in an on-line application. Fantoft has chosen an architecture (originally developed for training simulators) in which the modelling calculations and user interfaces are supported by separate, independent programs. These programs communicate using TCP/IP. This structure is robust, as the modelling program (the kernel) is freed from the (error-prone) complexities of user interaction. System availability is thereby improved. This use of open interfaces also
114 supports mixed computing: an interface running under Windows can communicate with a kemel running on UNIX. The kemel supports a number of"real-time" facilities for managing and running models in specific ways. It is possible, for example, to maintain multiple instances of a model. One of the instances runs alongside the plant in real-time and provides the information used for leak detection. Other instances are created as required to run what-if calculations, provide lookahead trends or track scrapers or batches. These new instances can be initialised either with the current plant conditions, or a saved snap-shot of plant status. Robust on-line models require robust numerical methods. Pipeline models are complicated PDE modules with time-dependent two-point boundary conditions (Matko et al. 2000). Much work has been devoted to developing a stable, accurate interface between these models (in particular OLGA) and D-SPICE's algorithms for integration and solving flowpressure networks. 5. SENSOR VALIDATION AND MEASUREMENT CHECKING: DATA RECONCILIATION A successful pipeline monitoring system requires adequate instrumentation. Figure 3 shows the instrumentation that is needed for successful model-based pipeline monitoring. Flow and pressure measurements are needed at both the inlet and outlet of the pipeline. Pressure measurements at intermediate points are not strictly needed (and indeed are not present in underwater pipelines), but if they are available, the information they provide can help in leak location. High quality instrumentation is needed if small leaks are to be detected. Turbine meters are usually required for flow measurement. These have accuracies of around 0.5% of range. Given such accuracies, it is possible to detect leaks as small as 0.5% of nominal flow. However small degradations in measurement accuracy can seriously impair the performance of the system. Poor system performance is characterised by false alarms and missed leaks. In general, it is difficult to identify leaks of less than 1% of nominal flow. If additional accuracy is required, additional instrumentation is necessary. T
T ,l__
T
T
.... iIT
T ~
Figure 3. A pipe length in a model-based leakage detection system The system must be robust to measurement failure. We therefore use simple sensor validation techniques to ensure that bad measurements do not produce false alarms. These checks examine the measurements for bad status, freezing and invalid values. Leak detection is not stopped if measurements are bad: override values are used instead of the bad measurements. However, leakage alarms are suppressed, and only reported as warnings while status is bad. Leaks are detected by examining the mean residual of key measurements over a period. A non-zero mean indicates either a leak or a measurement bias. It is not possible, by examining the residual in measurement alone to determine whether the cause is a leak or a bias. More sophisticated methods are needed to discriminate between causes. A mixture of heuristics and logic has been found to work. It is important to examine pattems in both the temporal and spatial distribution of residuals. Thus, if a non-zero mean residual is detected
115 at a pressure sensor, but there is no discrepancy at the sensors immediately upstream or downstream, an instrument error can be suspected. If there is sufficient redundancy in the instrumentation, it is possible to perform formal data reconciliation calculations. However, it is seldom that a pipe system has sufficient redundancy for short-term data reconciliation. Data reconciliation is more applicable for checking the instrumentation for the network as a whole over longer time periods. 6. DATA COMMUNICATION, SOFTWARE ENGINEERING AND NETWORKING On-line applications require effective, reliable networks, data communications and software architectures. Open standards, such as TCP/IP, D-COM and OPC are valuable tools for speeding the implementation of such systems. Figure 4 gives a simplistic view of Fantoft's real time architecture. The real-time model communicates with the SCADA system or DCS using a link. This has historically been a proprietary protocol, which was different for each system vendor and system. Luckily, newer control systems now offer a relatively standard OPC interface. The model communicates with its maintenance interface and with other applications (such as third-party models) using TCP/IP and D-COM.
Figure 4. Structure of a real-time system 7. ORGANISATIONAL CHALLENGES On-line modelling applications are unusually sensitive to problems related to project management and organisation. A successful application requires thorough planning, buy-in from the client's management, and active follow-up of the system. A technically excellent system can fail, and alas often does, because of a "trivial" problems. An overloaded router, an uncalibrated pressure measurement or a budget dispute between the IT department and the instrument department: these are typical problems that challenge on-line modellers. Adequate resource allocation is also important. A successful system requires process operators with skills, training and curiosity. Effort must be expended during commissioning and testing to win the operator's confidence and train them in the use of the system. Welldesigned contract and sub-contract arrangements are also necessary for successful applications. On-line applications are usually dependent upon optimal performance from a wide variety of systems (process, instrumentation, SCADA) from a number of vendors. Most of these vendors are large companies with large contracts. Successful commissioning
116 by a small systems vendor requires perseverance and tenacity in ensuring that all the supporting technologies are in place and working. 8. APPLICATION RESULTS Model-based pipeline management systems work. A Site Acceptance Test (SAT) was held in early 2000 for a 20 km long oil pipeline in the Middle East. Controlled leakage was induced by manually opening bleed valves at known locations at given times. The system detected leaks of 20-25% of normal flow within 2 minutes and leaks of 2% of total flow within an hour. The predicted locations were all within a few metres of the actual locations. Trials with batch and scraper tracking also gave good results. The arrival of the scraper at the end of the pipeline was predicted to within 11 seconds. The arrival of a new batch of product was predicted to within 10 seconds under continuous flow, and to within a couple of minutes when the pipeline was closed in for more than an hour during the test. Fantoft is currently working on extending this system to further pipelines in the same distribution system. The performance of a specific installation is acutely dependent on the quality and placement of instrumentation. For this reason, a leak detection demonstrator and prototype, built using D-SPICE, can be used to determine expected performance for specific instrumentation systems, measurement accuracies and noise patterns. 9.CONCLUSIONS On-line use of dynamic models is an essential to achieving the vision of lifecycle modelling in the process industries. This paper has presented industrial experiences and a novel approach to solving some of the problems associated with implementing on-line models in the petroleum industry. This experience can be directly applied to more complicated process systems, such as sub-sea production facilities and multi-phase pipeline networks. Technical challenges remain. Key areas for research and development are (1) software architectures, which speed-up the model building process and ease the burden of system maintenance, and (2) statistical methods for handling process data, which increase the power of the fault (leak and instrument failure) detection algorithms in the system. REFERENCES 1. KLEMP, S., MELAND, B., HUSTVEDT, E. AND OSTBY, O., (1997), "Operational experience from multi-phase transfer at Troll", 8th Int. Conf. On Multi-Phase Production, Cannes, BHR Group Conference Series No. 24, 477-496. 2. MATKO, D., GEIGER, G., GREGORITZA, W., (2000), "Pipeline simulation techniques", Mathematics and Computers in Simulation, 52, 211-230. 3. BRAUNSCHWEIG, B.L., PANTELIDES, C., BRITT, H.I., AND SAMA, S., (2000), "Process Modeling: The Promise of Open Software Architectures", Chem. Eng. Progress, 96(9), September, 65. 4. SHEA, R.H., RASMUSSEN, J., HEDNE, P., AND MALNES, D., (1997), "Holdup predictions for wet-gas pipelines compared", Oil and Gas Journal, May 19th.
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rights reserved.
117
Implementation of a failure model validation technique using a discreteevent batch simulator 9Application to semiconductor manufacturing A.S. Charles a, C. Azzaro-Pantel a, S. Domenech a, L. Pibouleau a, P. Floquet a, D. Jaume b, F. Tilhac b aLaboratoire de G6nie Chimique - UMR CNRS/INPAJPS 5503 ENSIACET - 18, Chemin de la Loge - 31078-Toulouse Cedex - France Email : Catherine.Azzaro @ensigct.fr bMOTOROLA Semiconducteurs SA Av. G6n6ral Eisenhower BP 1029F- F-31023-Toulouse Cedex - France This work emphasises proper techniques for data collection, failure model parameter estimation as well as equipment breakdown and scheduled maintenance modelling in a multipurpose batch plant. The proposed methodology is developed in the context of semiconductor manufacturing. 1. I N T R O D U C T I O N The process flow in a multipurpose plant is not only complex, but also altered by unpredictable situations such as operator unavailability, storage limitations, flow bottlenecks and equipment failures. In this context, maintenance and reliability play a crucial role in process operation. Consequently, plant managers and engineers are usually faced with important preventive maintenance decisions, which affect plant performance. The scope of this work presented is to consider the interaction effects of maintenance policies on batch plant scheduling. A simulation model previously developed in our laboratory, designed for equipment ideal behaviour, has been extended to incorporate equipment failures, set-up and repair times as well as maintenance operations. The objective of this work is to implement a failure model validation technique in an industrial context. 2. S E M I C O N D U C T O R M A N U F A C T U R I N G PROCESSES Semiconductor devices are built upon thin wafers of silicon. These wafers are typically between 4 and 10 inches in diameter. The devices vary in complexity from a few thousand to a few million transistors. The production of semiconductor devices is carried out in a socalled clean room, or wafer fab. It is achieved in a multistep process involving a variety of complex physico-chemical processes. A typical semiconductor factory contains hundreds of processing machines, exhibiting a wide variety of characteristics. Each equipment unit is associated with one of the unit operations involved in the global process. It is quite usual that an individual wafer batch visits one or more equipment units repeatedly during its course. In particular, each batch visits many times the same lithography or ion implantation workstation, giving the resulting process a cyclic character. On the contrary, deposition equipment and plasma etchers are often dedicated to a single operation to prevent contamination. In this
118 production environment, equipment breakdown or procedure drifting usually cause unscheduled production interruptions. Preventive interventions are also carried out in order to minimise the probability of unscheduled production interruptions. 3. MODELLING OF BREAKDOWN PHENOMENA AND MAINTENANCE 3.1 System definition An equipment item can be decomposed into several levels, i.e. functional groups, modules or pieces of equipment, each one representing a system. The components can be either dependent or independent, which interact or not with one another. In this paper, the equipment, viewed as a black box, is considered as a global system. The extension to multiunit systems is widely presented in Charles (2000). 3.2 Mathematical modelling and use of Weibull law Since in wafer labs, maintenance operations are usually preventive and corrective tasks, the model presented here will focus on these two types of maintenance operations. The model mathematically represents the time between a down/up status change to an up/down status change. Corrective maintenance interventions depend on the occurrence of breakdowns. Since breakdowns are random phenomena, they can only be estimated by modelling. Two laws apply, i.e. a reliability law and a maintainability law, describing respectively the breakdown phenomena (the lifetime distribution) and the corrective maintenance interventions. Lifetime distribution is a probability distribution frequently used in reliability to study the life cycle of components and systems. The most common lifetime distribution, chosen because of its flexibility, is the two parameters-Weibull law. The Weibull model is used to model failure frequency in any single of the so-called "bathtube" curve, which represents the three phases of
the equipment life (Pellegrin, 1997). The hazard function is given b y " ~ (t) = if_it]p_, , / . q \q7 where the parameter [3>0 is the shape factor which determines the time-dependent character ([3 < 1 for bum-in, [3 = 1 for stationary or [3 > 1 for wear-out) and the parameter I"1>0 is the characteristic time which represents the time scale of the effect. The expectation of the distribution represents the Mean Time Between Failure (MTBF). The time to repair also follows a statistical law (the lognormal distribution) in which the expectation of the distribution is the Mean Time To Repair (MTTR); the repair time includes the waiting time for a maintenance technician, the diagnosis time and the repair time. 4. INTEGRATION OF BREAKDOWN PHENOMENA M O D E L L I N G MAINTENANCE ACTIVITIES IN A DISCRETE EVENT SIMULATION
AND
4.1 Discrete Event Simulation principles To model the wafer fab in a high degree of detail, Discrete Event Simulation (DES) techniques have been implemented. The simulation tool called MELISSA C++ allows determining the exact chronology of discrete events occurring in the facility. In our formulation, time evolution occurs by "event jump", i.e. from an event to the following one. Typical events taken into account and managed in the simulation core have been widely presented in Navarre et al. (1998).
119
4.2 Downtime and maintenance assumptions Every equipment item has its own macroscopically failure model. Three types of maintenance procedures are considered: corrective and preventive operations performed by a technician as well as Total Productive Maintenance (TPM) actions (Pimor, 1991). These are usually performed by operators and represent simple maintenance actions of supervision and cleaning. Their frequency and duration are known. They are modelled independently of the other maintenance actions. - Corrective m a i n t e n a n c e actions can only be characterised by the MTBF and the MTTR since they occur randomly. Our model stipulates that every time a breakdown occurs, the corresponding corrective maintenance action is one of replacement that brings the system to its initial state, thus the intervention is based on an AGAN (As Good As New) assumption. - The two types of p r e v e n t i v e m a i n t e n a n c e actions (the classical ones and the TPM) are performed in accordance with a schedule. Although several types of scheduling for the preventive maintenance interventions can be used (Pellegrin, 1997), this work is focused on the classical periodic replacement model, in which preventive maintenance actions are of the AGAN type. Since the equipment set has been generally used for production for several years, a decrease in the probability of breakdown occurrence is practically observed, due to preventive replacement. Consequently, the collected failure frequencies (TBF, Time Between Failure) reflect directly the level and quality of preventive policy carried out in the wafer fab. This explains why a link between preventive and corrective actions exits intrinsically through the maintenance parameters. The data set necessary, on the one hand, to describe the workshop structure, operator/technician shifts and products and, on the other hand, the corresponding collecting mode are presented in table 1. The order of magnitude concerning the required number of parameters is also proposed, about 6000, for a workshop with 200 operating steps, 100 equipment units, 6 product families and shifts of 20 operators/5 technicians. Table 1 : Data collection mode and evaluation of parameter number for a wafer fab Data set Collecting mode Magnitude of parameter number Workshop description Available 1000 (number of operating steps, of zones, parallel units, capacity,...) 100 By production managers Production/maintenance teams (shifts, number, polyvalence...) Manufacturing recipes Operating time, Manufacturing specifications 3600 Load/unload time Measured by chronometers Production planning 72 By production managers Reliability data set (Breakdown time, By use of maintenance tuning preventive maintenance, repair time..) software 1200 At given times specified by the user during a simulation run, MELISSA-C++ records information about the workshop status. This information is particularly useful for case studies and concerns production level, inventory or WIP (Work In Process), cycle time, OEE (Overall
120 Equipment Effectiveness) which quantifies the different assignments of equipment (waiting for products, operators or technicians after a failure/for preventive maintenance, load/unload, stop for corrective/preventive/TPM actions, production). The OEE and its equivalent for operators or technicians is therefore an important indicator estimating the equipment and crew utilisation in production (Murphy et al., 1996). 5. I D E N T I F I C A T I O N M E T H O D S F O R R E L I A B I L I T Y AND M A I N T A I N A B I L I T Y LAWS: USE OF M A X I M U M L I K E H O O D E S T I M A T I O N F O R W E I B U L L L A W The available data set related to maintenance aspects includes the date and duration of both failures and predictive maintenance activities. A same methodology has been followed for carrying out a statistical data treatment in order to identify reliability and maintainability laws. The approach is illustrated for reliability law identification, for which Weibull law has been assumed. Two methods have been applied in this study : a practical one based on an empirical estimation of Weibull law hazard function using data graphical representation on the so-called Allan-Prait paper and a method based on maximum likehood estimation. For the sake of illustration, this latter is presented in what follows. As previously mentioned, the AGAN assumption has been put forward for all maintenance operations. Under this hypothesis, the TBF are easy to compute by difference between either an AGAN state (the last preventive or corrective operation) and a failed state or an AGAN state and a non-failed state (preventive operation). Let us note that the times are right censored data (Newton, 1991 ;Tan and Kramer, 1995; Pellegrin, 1997). The likehood function L is constructed as the product of the probabilities for the events k
that have occurred: L = 1-I
Pi 9Under an AGAN assumption, Pi for a failure event is the
i=l
probability density function f(t)dt normalised by the integral over the part of f(t) from which oo
~f(t)dt= 1-F(t, ), where tLi is the model time after the last ,L, maintenance (or beginning component time if no maintenance). F represents the cumulative distribution function for component failure time. Note that for an AGAN model tLi- tRi -- 0, thus cti = 1. (tRi is the reference time for event i). Likewise, for right censors (maintenance or survival without failure) Pi can be represented by R(t)/oq = (1 - F(t))/ctl In the same way, o~i = 1 under an AGAN hypothesis. Under these conditions, the likehood function is
the failure is sampled a i -
L - ~-I
flf,"tiq,, ,a-1"e-[-~) ( t,~a H m ~-~-qj
i=l
.1. 1 . i=l
1-e -I~] p
The search for the maximum value of the likehood function gives the model parameters with the highest probability. The maximisation of Ln L then gives relations satisfied by 13 and q. Using the formulation of (Newton, 1991), the following expressions are obtained : 1
~tiPlnti+s , , ~-[ tifl + s c,p 1 1
~lnts L , - fl- n
_ o
~
(1)
17=
~ c'P.l~ (2) t,,+ -n
121 The 13parameter is computed by the Newton iterative method, with 130as initialisation value. The 11 parameter is deduced from equation (2). Determination of Weibull law parameters involves reliability data collection (see table 1) so that the TBFs and censor times can be computed. The Weibull parameters are thus estimated using the aforementioned method. The MTBF is given by Frl, where F is the second species eulerian function of (1+1/13). Due to a relatively low number of data, the test of Little, Mc Clave et Often belonging to the Kolmogorov-Smirnov type) tests has been adopted for quality adjustment. 6. INDUSTRIAL VALIDATION 6.1 Validation on a manufacturing workshop
A reference state (an empty state) is considered for the workshop at simulation beginning and the corresponding simulation parameters are collected. For the following campaigns, the indicators reflecting the workshop behaviour are computed and also measured empirically to be compared. Figure 1 represents the evolution of the inventory state 1 (or WIP) vs. number of campaigns and shows that a good quantitative agreement is observed. The discrepancy (< 10% from 15th campaign) can be yet explained : first, the simulation model does not take into account the so-called "engineering" batches produced for investigations on new or improved processes as well as on new products; second, some batches may require rework actions. They both represent in average 2 to 5% of the total batches. Since WIP and product cycle time are strongly linked, it can be expected that predicted product cycle times will be slightly lower than those collected. Figure 2 shows that cycle times are 10% lower than those measured empirically, which can be explained as for WIP. Note that downtime equipment modelling involves a more important WIP level (multiplied by a factor 4) than in an ideal case with no production interruption by maintenance or breakdowns. A fluctuating evolution is also observed due to the random nature of occurring failures. Cycle time
Number of products
Averaoe cvde time (e.xpefimentall
ExperimentalWIP t
..........
20%
l
Product
1 (withbreakdown modelling) Product I (withoutbreakdownmodelling)
~WIP (withoutbreakdownmodelling) ......
!
:
0
5
Campaign,
number 10
, 15
Figure 1 WIP vs. campaign number
0
--
.....-.~
~ .........................................
5
10 Campaign
- ..................
number
15
20
Figure 2 Cycle time v s Campaign number
Another validation has been performed with the OEE indicator for technical distinct equipment (individual or collective wafer treatment, long or short operating time, with or without operator intervention ...) located at different manufacturing steps. Typical results obtained are presented in Table 2 and confirm that a good agreement is observed for experimental and computed values. 6.2 Validation on a "probe" workshop
The same operation was conducted in a "probe" workshop, dedicated to wafer test. Figure 3 represents the evolution of cycle time for a given product in function of campaign number. i For confidentiality reasons, the figures presented exhibit a "blind" scale in the Y-axis.
122 It can be seen that an equipment item has been stopped during the 10 th campaign, thus strongly affecting the product flow, as observed in the following campaigns. Table 2 9Comparison between experimental and computed O.E.E. Production Preventive Breakdown Values Waitingfor Operator
Technician
Product
Experimental
42%
4%
5%
25 %
3%
21%
Computed
39.7%
3.9%
5.3 %
25.1%
2.8%
23.2%
Relative discrepancy
6%
-2.5 %
5.5 %
1%
7%
10%
Figure 3 shows than the computed cycle times, although slightly inferior, are in good agreement with those measured in practice, which also means that some random phenomena have been neglected. The cycle times are much more fluctuating and dispersed than in a manufacturing workshop due to a limited number of equipment. Cycle time
~.~
0
i
2
"'---"pr%per~medntaluV:lue
i
4
t
6
/
i
8
Campaign number
i
10
12
Figure 3 9Evolution of product cycle time vs. campaign number (Probe workshop) 7. CONCLUSIONS This study has confirmed that reliability plays a pivotal role in production system manufacturing. Actually, plant modelling without taking into account equipment downtimes may actually misrepresent cycle times, throughput rates and inventory state. The use of reliability models, for which data collection must not be neglected, is essential for quality improvement of a production system behaviour. Reliability modelling, embedded in a discrete-event simulation tool for representing the workshop behaviour, provides a framework to investigate design, operating conditions and maintenance strategies in a batch plant. All these aspects have been largely studied in following works (Charles, 2000). REFERENCES
1. Charles A.S., 2000, INPT, PhD Thesis 2. Murphy, R., Saxena, P., Lewinson W., 1996, Semiconductor International, 125 3. Navarre, D., Pantel, M., B6rard, F., Pantel, C., Pibouleau, L, 1998, 17th IASTED Int. Conference, MIC'98, Grindelwald Switzerland 4. Newton D.W., 1991, Reliability Engng and System Safety, 34, 7 5. Pellegrin, C., 1997, Fondements de la ddcision de la maintenance, Economica 6. Pimor, 1991, TPM, La maintenance productive, Masson, Paris 7. Tan, J.S., Kramer, M.A., 1995, Reliability Engineering and System Safety, 49, 155-169
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
123
Automatic Structural Characterization of DAE Systems E.F.Costa Jr. b, R.C.Vieira b, A.R.Secchi a and E.C.Biscaia Jr. b a Departamento
de Engenharia Quimica- UFRGS - Porto Alegre - RS - Brazil
b Programa de Engenharia Quimica- COPPEAJFRJ - Caixa Postal 68.502 CEP: 21945 - 970 Rio de Janeiro, RJ - Brazil -
[email protected] The characterization of the DAE system is unavoidable for the construction of generalpurpose robust codes for numerical integration of DAEs. Such information is needed for index reduction and consistent initialization of DAE systems. In the present contribution, it is proposed the utilization of a graph theoretical approach associated with automatic differentiation tools for the automatic generation and resolution of the consistency system. All differentiation required is performed exactly and at an affordable computational cost, and the index reduction prior to numerical resolution is performed automatically. The resulting code from this project has been incorporated to the numerical integration code DASSLC. Several examples illustrate the potentiality of the approach. Encouraging results were achieved on both traditional benchmarks for initialization procedures and more realistic problems. 1. INTRODUCTION When solving DAEs one must concem about the index of the system and about the consistency of the initial conditions. The index of a DAE is the number of times that all or part of the system must be differentiated with respect to time in order to convert it into an explicit set of first order ODEs. Index 1 systems may be solved with modified ODE methods, while higher index systems (systems with index 2 or greater) require specific methods. Generally, higher index systems are rewritten in an index 1 formulation and solved with standard integration codes like DASSL [ 1] or DASSLC [2], written in FORTRAN and C. During the index reduction, some extra algebraic equations are obtained which generally correspond to derivatives of the original algebraic equations. Those hidden algebraic equations along with the original DAEs compose the extended system. Consistent initial values must satisfy not only the DAE system but also the underlying extended system [3]. Several rigorous methods for initialization have been proposed in the literature, and they all depend to some extent on the characterization of the DAE system. In other words, in order to perform the numerical integration of a DAE with standard integration codes, it is necessary to: (i)perform the characterization of the DAE, in order to determine its index and, if required, the set of equations to be differentiated; (ii)perform the differentiation; (iii)check for feasibility of the initial states arbitrarily set; (iv)solve the consistency system for a consistent initial state. In this contribution it is discussed an extension to the numerical integration code DASSLC. The code developed aims the preprocessing of the DAE in order to reduce its index and/or
124 determine consistent initial conditions. In w the DAE characterization issue is addressed. Existing code and algorithms are discussed and their potentiality and limitations are shown. In w the approach proposed in this contribution is detailed. Some numerical examples presented in w show the potentiality of the proposed approach and explain some of the main features of the code. In w some perspectives of the research in the area and the next developments of the code are discussed. 2. CHARACTERIZATION OF DAE SYSTEMS Consistent initial conditions for DAEs must satisfy not only the original equations in the system but also their differentials in respect to time up to some order. Whether or not this additional requirement actually imposes extra constraints on the initial values depends on the particular problem. Pantelides [4] derived a criterion for determining whether differentiation of a subset of the DAE system provides further constraints to be satisfied by initial conditions. A graphtheoretical algorithm was proposed to locate those subsets which need to be differentiated. Unger et al. [5] implemented two systematically different structural approaches as tools for the analysis of DAEs, in the codes PALG and ALGO. In structural computations, only hard zeros (0) and nonzeros (*) are distinguished, and a DAE system is merely represented by the pattern of its jacobian. Since the computation of the index and of the number of degrees of freedom require (repeated) differentiation, this operation has to be defined in the structural sense. Unger et al. [5] defined the structural differentiation to be of the type linear if the derivative of f(z) depends only on z'. On the contrary, if the derivative of f(z) is a fimction of z and z' the differentiation is said to be of the type highly nonlinear. Generally, neither way of structural differentiation yields the correct pattern of the Jacobian of the differentiated function. The linear and highly nonlinear definitions yield the patterns with minimum and maximum nonzeros entries, respectively. The structural computation as implemented in PALG and ALGO codes present two main drawbacks: (i) as the structural differentiation yields the wrong pattern for the derived equations, it impacts on the feasibility check of an arbitrary initial set; (ii) a purely structural gaussian elimination can not identify a set of equations that is structurally regular but actually singular. A pure numerical approach for the initialization of DAEs with known index v was proposed in [6]. The authors construct the consistency equations with all total differentials of the DAE with respect to time up to order v and extra user specifications. Then a family of forward finite difference approximations of these equations is constructed and the resulting system is solved in a least square sense. The solution of this rank-deficient algebraic problem poses a nontrivial numerical task, and existence and uniqueness of the solution can only be proven for linear constant coefficients DAEs. If only the necessary differentiation was performed in the previous algorithm, the resulting non-linear system would have a full rank Jacobian, and could be in principle solved exactly. Kr6ner et al. [7] proposed an algorithm that consists of three steps: (i) a structural analysis of the problem employing PALG/ALGO codes, in order to determine the index of the DAE, the minimum set of equations to be differentiated and the number of degrees of freedom; (ii) the numerical differentiation of these equations as in Leimkuhler's algorithm; (iii) the solution of the non-linear algebraic system. It must be stressed that steps (i) and (ii) are not exact, and
125 hence the resulting algebraic system may present severe numerical problems. The authors pointed out that the solution of the algebraic system was the most difficult step of the algorithm, and recommended restrictive damping strategies if there are steep initial gradients. Murata and Biscaia Jr. [8] developed the symbolical code INDEX for characterization of DAEs based on symbolical computation and MAPLE software. As output the code identifies general systems of index one or higher and Hessemberg forms of index two and three, besides checking for linearity and solvability of DAEs with certain properties. As an auxiliary task INDEX supplies the patterns of the jacobian matrices to ALGO/PALG structural analysis codes. In spite of the ever known tendency of computer algebra systems to use much memory space and computing time, the authors report success on the characterization of several examples. However, the code can not be coupled with a numerical integration software developed in a standard programming language (as FORTRAN or C), and most of all presents severe limitations on the size of the problems it can handle. 3. THE EXTENDED DASSLC CODE In this contribution, it is discussed the characterization/initialization module recently added to DASSLC. Based on Pantelides' algorithm, it uses the proposed graph-theoretical approach to identify the minimum set of equations to be differentiated in order to generate an index one formulation of the original DAE. Through AD code ADOL-C [9], the differentiation is performed automatically at an affordable computational cost, and, most important of all, the differentiation technique employed does not incur on truncation error. The structural pattern of the jacobian matrices required are computed via numerical perturbation of the original equations and of the exact extended system. The user must supply the DAE ..system F(z,z')=0 according to DASSLC standards. The code will apply Pantelides' algorithm, construct the extended system and inform the number r of degrees of freedom of the system. The user is prompted to give r arbitrary initial conditions, on which the code will perform a structural check for feasibility. It must be stressed that structural feasibility is a necessary but not sufficient condition for feasibility. That comes from the fact that structural computation can not determine if a structurally regular matrix is not in fact a singular matrix, as previously discussed. In the PALG/ALGO codes, structurally feasibility was neither necessary nor sufficient, as the pattern of the derived equations was computed in an approximated sense. In the present code, when an initial arbitrary set is found to be structurally feasible, the code will carry out the resolution of the extended system by a Newton type method. The extended system was constructed via automatic differentiation, and hence is exact. The initial conditions obtained from its resolution can be fed directly to the numerical integrator DASSLC. If there is a failure in any of these stages the user is warned and prompted to supply another arbitrary set. 4. NUMERICAL EXAMPLES 4.1. Condenser The model of a single component condenser presented by Pantelides et al. [10] is frequently adopted in literature for illustrative purposes. Although being a contrived example with non rigorous modeling, the widespread use of this model has transformed it into a kind of "benchmark" for initialization procedures [4,5,11 ].
126
iV = F - L PV=NRT
(la) (lb)
N C p l ~ = FCp (Tin- T)+ LAH + UAs(Tc- T) P = A exp(- B/(T + C))
(lc) (ld)
The index 2 formulation shown in Equations (1) was analyzed. The automatic index reduction was successfully performed, and the number of degrees of freedom (r = 1) was correctly determined. Equations (1 b) and (1 d) have been automatically differentiated in order to uncover the extended system. All variables are feasible arbitrary conditions, and the Newton code was capable to perform the numerical resolution of the extended system. 4.2. The Unavoidable Pendulum
The well known index 3 pendulum problem [4,5,8] is used to illustrate the superiority of the developed code over reported implementations of Pantelides' algorithm. This system presents two degrees of freedom (r=2). =G j, = Vy
(2a) (2b)
~x = ~,x
(2c)
~,y = ~ y - 1
(2d)
x 2 + y2 = 1
(2e)
PALG and ALGO codes, due to the inexact differentiation used, can not determine whether the set [Vx, Vy] is feasible or not. Depending on the type of the differentiation used, the codes reached to conflicting solutions. The code reported herein has been capable to correctly determine r and identify all feasible sets. Equations (2a) and (2b) have been differentiated once, and (2e) twice. Additionally, sets leading to singular consistency systems and/or impossible initial values have been eliminated by the resolution of the extended system with the Newton type algorithm implemented. 4.3. Batch Distillation Column
The following example illustrate the utilization of the developed code on a more realistic problem. The start-up of a batch distillation column [12] is typically a high index DAE system with highly nonlinear state equations and equilibrium relationships. Additionally, it can easily become a large scale system as mass and energy balances must be solved for every plate and there are several components involved. In this contribution a column with np plates and nc components is simulated, what leads to system with approximately (np+2).(2 nc+6) equations.
127
13/0 = V/+I Yi+l,i. + Li- ni-l'J
(3a)
Li -ni'j ~ -- V~ Yi,j
N, = ~--,~c ,,,j
(3b)
N i h i = Vi+1 Hi+ ~ + Li_ ~ hi_ 1 - Z i h i -I'~ H i + Qi
(3c)
yi,j = x i j K j ( T ~ ) ,
(3d)
Ni hi
tic = Zj=ll"li,j hj(Ti)
H i = Z~c=lYi,j Hj(T~) ~.c i=l Yi,j = 1 where:
(3e) (3f) i=0, 1..... np+l
j=l,2,...,nc
Hp,i(T)=~:=oe,,kZk~
(3g)
Ky(T):T~3k=oaj,kTk~
ej-Z~clYi,jep,i(rj) In the start-up of the column, the fbllowing relations between liquid and vapor fluxes hold: Vi+ 1 = L i
i: 0,1,..., np
(4)
In previous equations, x and y represents molar fractions on liquid and vapor phases, h and H are enthalpies on liquid and vapor phases, T is temperature, i represents plate and j represents component. All parameter used can be found in [12, pp.211-212]. With the extended DASSLC code discussed herein, it is possible to integrate the high index system directly. All index reductions and manipulation in order to achieve consistent initialization are performed automatically. The user should only give the original DAEs according to DASSLC standards, determine which are the arbitrarily set initial variables and provide initial values for those variables. Characterization results for an example with 5 components and 12 plates are shown in Table 1. It should be emphasized that other sets could be determined arbitrarily. Table 1: Characterization results for example 4.3. nc=5, np=12 Index 2 Equations (3b), (3d) (3e) and (3g) must be differentiated once Degrees of freedom: 70 [ nc (np+2) ] Feasible arbitrary initial set: moles of component i on plate j (nij) 5. CONCLUSIONS
It is presented the extension of the code DASSLC that permits its utilization for the numerical integration of high index problems. The characterization/initialization module added preprocesses the original DAE systems and converts it into an index one problem with consistent initialization. Based on Pantelides' criteria and on the Automatic Differentiation
128 tool ADOL-C, the exact and affordable computation of derivatives required is one of the key features. With the exact calculation of derivatives, structural feasibility became a necessary condition for feasibility, and this criteria is used to perform the first search for consistent arbitrary values. A Newton-type method is then used to eliminate singular but structurally regular subsets and to compute the consistent initial values. The condenser and the pendulum problems have been analyzed. These "benchmarks" have shown that the results achieved with the proposed approach are either equivalent to of better than previously reported characterization results. Additionally, the batch distillation column example has shown that the methodology proposed is specially suited to the highly nonlinear large scale models often faced in the simulation of chemical engineering processes. REFERENCES
1. 2.
3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
Petzold, L. DASSL code, Computing and Mathematics Research Division, Lawrence Livermore National Laboratory, L316, PO Box 808, Livermore, CA 94559 (1989) Secchi., A.R. "DASSLC: User's Manual, a Differential-Algebraic System Solver", Technical Report at Chemical Engineering Department, UFRGS, Porto Alegre, RS, pp.49 (1992), http://www.enq.ufrgs.br/enqlib/numeric.DASSLC Brenan, K., Campbell, S. and Petzold, L. Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, New York, Elsevier Sc. Publ. Co (1989) Pantelides, C.C. SIAMJ. Sci. Stat. Comp. 9 (1988), 213-231 Unger, J., Kr6ner, A. and Marquardt, W. Comp. Chem. Eng. 19 (1995), 867-882 B.Leimkuler, L.Petzold and C.Gear, SIAMJ. Num. Anal. 28 (1991), 205-226 Kr6ner, A., Marquardt, W. and Gilles, E. Comp. Chem. Eng. 16 (1992), S 131 Murata, V. and Biscala, E. JR., Comp. Chem. Engng. 21 (1997) $829-$834 Griewank, A., Juedes, D., Mitev, H., Utke, J., Vogel, O. and Walther, A., ACM TOMS 22 (1996) 131-167 - Algor. 755 Pantelides, C.C., Gritsis, D., Morison, K. and Sargent, R.W.S. Comp. Chem. Engng. 12(1988), 449-454 Kr6ner, A., Marquardt, W. and Gilles, E. Comp. Chem. Eng. 21 (1997) 145-158 Holland, C. and Liapis, A.I. Computer Methods for Solving Dynamic Separation Problems, McGraw Hill Publishers, 1983
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
129
Steady state analysis of membrane processes for the treatment of industrial effluents A. M. Eliceche a, S. M. Corval~n a and I.
Ortiz b
"PLAPIQUI-CONICET, Chem. Eng. Dept., Universidad Nacional del Sur, 8000 Bahia Blanca, ARGENTINA. bDpto.de Quimica, Universidad de Cantabria, 39005 Santander, SPAIN. e-mail: meliceche~0lapiqui.edu.ar
The feasibility of operating a new membrane technology for the treatment of industrial effluents in a continuous mode is analysed. The selection of the operating mode needs to be addressed at the conceptual design stage. The semicontinuous operation has been studied previously and the optimum operating conditions were reported by Eliceche et al. (2000). In this work a steady state configuration is proposed, the corresponding modelling equations are developed and a methodology for the selection of the operating conditions is presented. A given compound can be removed from the effluent in the extraction module and simultaneously it can be recovered in the stripping module, for recycling and re-use in the plant that generates the effluent. Environmental applications of this technology will reduce the amount of contaminant disposed off finally into the environment, leading to a cleaner technology. This fact motivates the need to promote its industrial application. The removal and recovery of Cr(VI) from surface treatment effluents is studied. 1. INTRODUCTION Membrane extraction processes using hollow fibres are of particular interest because of their versatility. In addition, the membrane process not only removes the required compound from the effluent but it can also recover and concentrate it simultaneously as a product. This product can be recycled and re-used in the same process that generated the effluents or in some other application. Extensive studies on dispersion-free solvent extraction with the use of microporous membranes have been carried out by D'Elia, Dahuron and Cussler (1986) and Prasad and Sirkar (1988) among other authors. Ho and Sirkar (1992) presented a general review of the NDSX technology and its applications. The simulation of a Non-Dispersive Solvent Extraction process, in batch and semicontinuos operation, was carried out by Alonso and Pantelides (1996) in gPROMS. An analysis to validate the models with the performance of a NDSX pilot plant is reported by Alonso et al. (1999). The optimisation of the semicontinuous mode of operation was implemented by Eliceche et al. (2000) in gPROMS and gOPT.
130 In this work, the steady state operation of NDSX membrane processes and the optimal operating conditions are found. As a case study, the removal and recovery of Cr(VI) from wastewater of a surface treatment plant is presented and the main numerical results reported. 2. NONDISPERSIVE SOLVENT E X T R A C T I O N PROCESS The main components of the process are two hollow fibre modules, the extraction and the stripping modules, as well as two tanks for the organic and stripping phases. A schematic diagram of the process, with both membrane modules operating in a continuous and co current mode, is shown in Fig. 1. In the semicontinuos mode of operation, the stripping stream runs in a batch mode. The stripping solution is concentrated in the stripping tank, until the required composition for recycling and re-use is reached.
Fig. 1 - Schematic diagram of the steady state operation In the membrane modules, the aqueous phases run through the lumen of the hollow fibre and the organic phase flows on the shell side. The organic solution wets the porous wall of the hydrophobic fibre. Organic phase dispersion into the aqueous phase is prevented by a slight overpressure of the aqueous stream. The organic phase, Fo, extracts the contaminant from the effluent in the extraction module. The contaminant is back- extracted from the organic phase and concentrated in the aqueous phase, in the stripping module. The organic stream is the carrier between the extraction and stripping modules. The organic stream leaving the extraction module is fed into the stripping module and the organic stream leaving the stripping module is recycled to the extraction module via a buffer tank. The stripping aqueous solution, Fs, takes the solute from the organic stream and is fed to the stripping tank for further recycling to the stripping module. An aqueous stream, Fp, free of contaminant enters the stripping tank and leaves it concentrated with the contaminant. The recovered contaminant, extracted from the effluent, leaves the plant in the purge stream, for recycling and re-use in the same plant that generated the effluent or other alternative industrial process that can re-use that stream.
131 3. STEADY STATE M O D E L OF A SELECTIVE M E M B R A N E S E P A R A T I O N PROCESS
The component mass balances in the organic and aqueous phases of the extraction, stripping modules and stirred tanks, including the mass transport through the membrane are described by the following set of ordinary differential and algebraic equations. For the development of the mathematical model it was assumed that: i) The main resistance to the solute transport lies in the micro porous membrane, therefore mass transfer resistances at both the aqueous and organic bulk phases were neglected. ii) The reactant species are in equilibrium in the whole interface. EXTRACTION MODULE Aqueous Stream 0
-
-
dCe A +__Km dz FeZ
E
(CoE - C o)
C e (Z=0)
--
Ce,in
(1)
Organic Stream __
dCE dz
+
A FoE
K m (foE
CEin ,
C E
-
o)
--
C S.... t
(2)
CT
(3)
STRIPPING MODULE Aqueous Stream 0
-
--
dCs A s + K m (CSo-Coi ) dz FsL
Cs,in
"--
Organic Stream. 0
-
--
dCS~ A --+ K dz FoL
s
m (CSo - C o i )
s
Co,in
--
E
(4)
C .... t
STRIPPING TANK 0
"--
Us ( C .... t - f T )
-
(5)
Up CT
Equations (1) to (5) represent the steady state behaviour of the extraction and stripping membrane modules. The solution of the ordinary differential equation (ODE) system allows the steady state simulation of the process that has been implemented in gPROMS (2000). 4. F O R M U L A T I O N OF T H E OPTIMIZATION P R O B L E M The objective function proposed is the maximisation of the effluent treated Fe. The algebraic
and
differential
equations
(f(:~,x,v)
=
0)
and
their
initial
conditions
( I(x, v) = 0 ), equations (1) to (5), that represent the steady state model of the NDSX process are formulated as equality constraints in the optimisation problem P1. Separation
132 specifications related to maximum composition allowed in the effluent for final disposal U < C .... t ) and minimum concentration in the stripping phase for further recycling and
(C .... t
re-use (CV > CV' c ) are formulated as inequality constraints g (x,v) in problem P 1. The vector v of the optimisation variables includes the flowrate of the organic, stripping and purge streams. Upper bounds on the optimisation variables v u correspond to maximum pump capacities. The non-linear programming problem is formulated as follows: Max
F e (v)
v
f ( ~ , x , v) I(x, v)
=
0
= 0
(Pl)
g(x, v) < 0 v c
_<
v
<
V
U
The solution of problem P 1 requires the use of nonlinear programming software able to handle differential equations, like the code gOPT of gPROMS (2000). In the following section, a particular environmental application of NDSX technology to effluent treatment and metal recovery is analysed. 5. E F F L U E N T T R E A T M E N T AND Cr(VI) R E C O V E R Y The removal of Cr(VI) from the effluent of surface treatment plants and the Cr(VI) recovery and concentration for recycling and re-use in the plant that generated the effluent, is analysed in this section. The aqueous and organic phases are contacted at the interface of the hollow fibre, where the extraction and stripping reactions between the solute Cr(VI) and the organic carrier (quaternary ammonium salt Aliquat 336) take place. The chemical equilibrium at the interface is represented by: CrO42 + 2A1C1
~
A12CRO4
+
2C1
(6)
The chemical equilibrium parameter for the extraction and stripping modules used in this work, are reported by Ortiz et al. (1996) and Alonso et al. (1997). The mathematical model used in this work was proposed and validated by Alonso et al. (1999). Details of the membrane modules can also be found in Alonso et al. (1999). The effluent treated has a nominal Cr(VI) composition of 1.23 mol m -3 that must be reduced to 0.00961 mol/m 3 before disposal, and simultaneously a minimum stripping Cr(VI) composition of 76 mol/m 3 must be reached for reuse. The optimisation problem P 1 was formulated and solved with the optimisation code gOPT of gPROMS (2000), under an NT operating system. The solution reported in Table 1, requires 5 iterations and 3.9 s of CPU time on a Pentium III 700 MHz workstation. Different initial points have been tried and the same optimum operating conditions have always been found. The two separation specifications, related to Cr (VI) compositions in the effluent and stripping phase, are always active constraints at the solution point. The organic and stripping
133 optimum flowrate values lie at their upper bounds. The steady state value of Cr (VI) composition in the organic phase is 82.44 mol/m 3 at the end of the extraction module, reaching the equilibrium value with the effluent Cr (VI) composition at this point. This value is similar to the optimum value of the organic Cr (VI) composition at the initial time, for the semicontinuous mode of operation. In the semicontinuous mode of operation, the Cr (VI) composition in the organic phase at the initial time is an optimisation variable. This is not the case for the continuous mode of operation, in which the Cr (VI) composition in the organic phase reaches a steady state value. Table 1 Optimal operating conditions. Flowrates Initial Solution (m3/h).10 -3 Point point
Fe Fs Fo Fp
67.8 50.0 50.0 1.038
75.4 200.0 180.0 1.215
Lower bound
Upper bound
20.0 20.0 20.0 0.5
200.0 200.0 180.0 3.0
The maximum effluent flowrate treated in the continuous mode is 75.4 l/h, whereas in the semicontinuous mode of operation it was found to be equal to 87.67 1/h for each batch, as reported by Eliceche et al. (2000). Although the effluent flowrate treated in the semicontinuous mode is slightly bigger, dead times between batches and start up procedures to reach the steady state conditions should also be considered to take a final decision. The stripping area and pump capacities are the bottlenecks for an increment in the plant capacity. Thus, an increment in these capacities would allow an increment in the effluent flowrate treated. The purge stream Fp finally carries all the Cr(VI) extracted from the effluent Fo. Both flow rates are correlated by a global Cr(VI) balance between the streams entering and leaving the plant. The purge flowrate is an order of magnitude smaller than the effluent flowrate. It always has the minimum required composition of 76 mol/m 3, when leaving the stripping tank. Thus, the flowrate of effluent treated is very sensitive to the purge flowrate. It has a monotonic increasing behaviour up to a point in which an unfeasible operation is reached because the stripping tank is diluted with the fresh stream Fp. This is the optimum value for the purge stream. An increment in the flow rates of the organic and stripping phases increases the amount of Cr (VI) extracted from the effluent and simultaneously increases the Cr (VI) recovered in the purge. For this reason, the optimum values of organic and stripping flowrate lie at their upper bounds. The same results are observed in the semicontinuous mode of operation. When the organic flowrate increases, the driving force for Cr(VI) extraction from the effluent to the organic phase increases due to the increment in the difference of Cr(VI) organic composition between the interface and the bulk, equation (2).
134 6. CONCLUSIONS The main contribution of this paper is the analysis of the steady state operation of membrane processes for effluent treatment and Cr(VI) recovery, that has not been addressed previously. The main tendencies are compared with the semicontinuous operation, reported previously by Eliceche et al. (2000). Preliminary results generated in this work show that the continuous mode of operating compares well with the semicontinuous mode of operation. At the conceptual design stage, the evaluation of different operating modes should be done with the purpose of choosing the best alternative. The steady state operation has the advantage of avoiding dead times and start up procedures associated with the semicontinuous operation. Further work is needed to address the synthesis, design and operability of the steady state operation of this membrane technology for industrial effluent treatment that leads to a new and cleaner technology. 7. NOTATION A C F
effective surface area, m 2 Solute Concentration, mol/m 3 Flow rate, m3/h Km Membrane mass transfer coef., m/h L Fibre Length, m z Axial distance, m
Superscripts E L S
Extraction module Lower Bound Stripping module
T U
Tank Upper Bound
Subscripts e in p s 0 oi out
Extraction phase Inlet composition Purge stream Stripping phase Organic phase Organic interface Outlet composition
REFERENCES
Alonso, A.I., Galan, B., Irabien A. & Ortiz I. (1997). Separation of Cr(VI) with Aliquat 336: chemical equilibrium modelling. Sep. Sci. Tech., 32, 1543. Alonso, A., Pantelides, C.C. (1996). Modelling and simulation of integrated membrane processes for recovery of Cr (VI) with Aliquat 336. Journal of Membrane Science, 110, 151. Alonso A., Galan, B., Gonzalez, M. & Ortiz, I. (1999). Experimental and theoretical analysis of a NDSX pilot plant for the removal of Cr(VI) from galvanic process wastewater. 1rid. Eng. Chem. Res., 38(4), 1666. D'Elia, N.A., Dahuron. L., & Cussler, E.L. (1986). Liquid-liquid extractions with microporous hollow fibres. Journal of Membrane Science, 29, 309. Eliceche, A., Alonso, A. & Ortiz, I. (2000). Optimal operation of selective membrane separation processes for wastewater treatment. Comp. & Chem. Engng, 24, 2115. gPROMS Technical Document-The gOPT Dynamic Optimisation Tool (2000). Process Systms Enterpris, Ltd. Ho, W.S.W., & Sirkar, K.K. (1992). Membrane Handbook, New York: Chapman & Hall Ortiz I., Galfin B. & Irabien A. (1996). Membrane mass transport coefficient for the recovery of Cr(VI) in hollow fibre extraction and stripping modules. Journal of Membrane Sci, 31, 46. Prasad, R., & Sirkar, K.K. (1988). Dispersion-free solvent extraction with microporous hollow-fibres modules. American Institute of Chemical Engineering Journal 34(2), 177.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
135
Bifurcation analysis of periodically forced systems via continuation of a discrete map V. Faraoni a, E. Mancusi a, L. Russo a and G. Continillo b* a Dipartimento di Ingegneria Chimica Universit/t "Federico II", Piazzale Tecchio 80, 1-80125 Napoli, Italia.
b Facolt/l di Ingegneria, Universit/t del Sannio, Piazza Roma, 82100, Benevento, Italia
In this work, we propose an innovative approach for the dynamical analysis of a periodically-forced system. It consists of constructing a discrete map that can be studied using a popular and robust continuation code (AUTO). The proposed method is applied to a system of two CSTR with a periodically inverted feed, previously studied via direct simulation by Zukowski and Berezowski. The new approach reproduces all previously identified behavior, and discovers new details of the system bifurcations. 1. INTRODUCTION Many processes in chemical engineering can be described as periodically-forced dynamical systems. In fact, it has been recently found that many processes, normally conducted in stationary conditions, can be made more efficient if forced to work periodically. A typical example of periodically driven reactive process of great interest is the so-called reverse-flow reactor [1]; in such system, the feed flow direction is periodically inverted. Recent studies have shown that, in reverse-flow reactors, complex regimes, like n-periodic, quasi-periodic and chaotic, can be achieved [2]. In order to properly design and control periodically forced processes, it is necessary to accurately describe the regime conditions when relevant operative or chemical parameters are changed. In the literature, the analysis of the regime conditions has mainly been conducted via direct simulation [3-4]; this approach is very time-consuming in periodically-forced systems, since the regime conditions are often reached after a time in the order of hundreds of flowinversion periods. Moreover, direct simulation cannot detect unstable regimes that can be of interest for many reasons. Shooting methods have been employed by Croft and Levan [5] in order to detect 1-periodic solutions for selected periodically-forced systems; this approach, however, is unsuitable when the regime solution is not periodic (i.e. quasi-periodic or chaotic). Clearly, the most comprehensive approach to accurately describe changes in stability and nature of regime solutions is the systematic application of bifurcation theory and of
Corresponding author Email:
[email protected](G.Continillo)
136 continuation. The main difficulties of this approach are the non-autonomous nature of the models and the presence of a forcing field that can be discontinuous. Salinger and Eigenberger [6-7], Khinast and Luss [8], and Khinast et al. [9] conducted a stability study of reverse-flow combustors. Both analyses are based on Floquet theory coupled with continuation techniques and employ "ad hoc" numerical codes. It would be desirable to develop a general approach to tackle the entire class of periodically forced systems, based on robust, widely- available tools. Our approach is based on the generation of an appropriate Poincar6 map, obtained via numerical integration of the continuous system with the well-known integration package VODE [10]. The resulting map is then studied via continuation with the software AUTO [11], that permits the stability characterization of the asymptotic regimes and the identification of bifurcation points.
2. CONSTRUCTION OF THE PERIOD MAPPING In principle it is possible to reduce the study of continuos time systems to the study of an associated discrete time system, such as a Poincar6 map. A Poincar6 map is related to a Poincar6 section Z, that is a hypersurface in the state space transverse to the flow of a given system of equations, that means:
(n(u),f(u)) ~ 0
(1)
( < , > defines inner product) where n(u) is a vector normal to the section located at u and f(u)is the vector field describing the flow. If the trajectory evolves in an n-dimensional space, it follows that the Poincar6 section is an (n - 1) dimensional surface, and that each point on this section is specified by ( n - l ) coordinates. The transformation that maps the current intersection to the subsequent intersection on a Poincar6 section is called a Poincar6 map. For a generic dynamical system, a Poincar6 map is defined only locally, around a point x* of a limit set, and it is not guaranteed that the trajectory emanating from any point on Z will intersect E. In the special case of periodically forced systems, it exists a Poincar6 map associated to a global cross-section that is a hypersurface transverse to the vector field for all orbits. For this reason it is possible to study the dynamics of continuos time system via a Poincar6 map for every initial conditions. This map merely tracks initial conditions after successive periods of the vector field. In this way, the dynamic behavior of the discrete system is equivalent to that of the continuous one. In fact, it is possible to show that fixed points of such a map are univocally correspondent to periodic orbits of the continuous system, and that the eigenvalues of the jacobian matrix of the map, and the related stability properties, are equal to the Floquet multipliers of the periodic orbits. Basically, the continuation algorithm implemented in AUTO can trace the fixed point locus of a discrete map f ( u k , ~ ) = U
TM,given an initial point
(~,~
of this locus, and detect
the bifurcations of the system. AUTO requires a functional representation of the discrete-time system. Since no explicit expression is available for the map, it must be provided via numerical computation. More explicitly, if the continuos-time forced system is: 2Xp
du/dt = f(u, t, I:o ), the map is: u TM = u k + ~ f(u, t,'cv )dt. The continuation of such a map is 0
137 conducted with calls from the AUTO main routine to an external integrator (VODE) which performs an accurate computation of the time integral from 0 to 2Zp; the numerical integrator substitutes the function on which AUTO performs the correction steps after a first prediction.
S U B R O U T I NE
FUNC
NUMERICAL
INTEGRATOR
f(u) AUTO
OTHER S!BROUTi NE Figure 1 - Schematic of the approach. The map is provided to AUTO within the subroutine FUNC, where AUTO holds all function definitions. Information between Func and the integrator travels via a vector. The vector state of the system is sent to the integrator, which sends it back after a time equal to 2~, for the one-iterate, and equivalently, after a time equal to 2nz~, for the n-iterate. This feature is particularly useful when studying period-doubling cascades.
3. AN EXAMPLE OF APPLICATION We have studied a system of two CSTR with heat exchange between reactors and surroundings and with a periodically inverted feed [12]. The model without feed-inversion is written in terms of mass and energy balances, and is given in dimensionless form by the following equations:
dXaXa+IO('C)Xbdr+Da(1-Xa ,nexp ~--' d J a - Ja+IO(,l:)Jb+Da(l_Xa dz
l+~3Ja
/
)nexP(l+~Ja 7~Ja ] +~(jH_Ja)
dXb / / dr --Xu+(1-IO(z))Xa+Da(1-xu)nexp 13'+1313J;
(2)
dJb -'Jb +(1-IO('Q)Ja+Da(l-Xb)nexp(713Jb ) I + bD]+~( j JH-J b dr where IO(1:)=int (Zh:p)-2int (a:/21:p), l:p being the inversion period. The volumes of the reactors and the heat exchange coefficients between the reactors and the surroundings are assumed equal for both reactors. This periodically-forced system (Eq. 2) has been chosen as a test case since it is lowdimensional, but shows, as reported by Zukowski and Berezowski [12], many dynamical features typical of the whole class of reverse-flow reactors.
138 The bifurcation study was conducted by considering as bifurcation parameter both the time of flow reversal (Zp) and the Damk6hler number (Da). The continuation results are presented in form of solution diagrams; such diagrams report a suitable norm or semi-norm of the state variables versus the parameter. Particularly, we represent the value of the conversion in the stream outcoming from the two CSTR system.
Figure 2 - The solution diagram for Da=0.04776. Solid lines: stable 1-periodic solutions; dashed lines: unstable 1-periodic solutions; filled squares: Neimark-Sacker bifurcations. Insets show details of delimited rectangular regions. Figure 2 shows a very complicated solution structure, with many bifurcations of various kind. Starting from low values of the inversion period, it is possible to observe a first bifurcation at point F1 on the period-1 low conversion solution branch: this is a subcritical pitchfork bifurcation. Following the branches stemming from F1, we encounter two saddlenode bifurcations (S1 and $2) and two secondary Hopfbifurcations (N-S1 and N-S2), leading to tori solutions for the continuous-time system. For Zp~[4.9, 8.4], we observe chaotic and pperiodic behavior, as described in Zukowski and Berezowski [12]; moreover, we also found another 1-periodic branch, on which more bifurcations where detected: a saddle-node ($5), a pitchfork (F2) and two more secondary Hopf (N-S3 and N-S2). In addition to what reported by Zukowski and Berezowski [12], we found a high conversion solution branch; this branch shows two more saddle-node bifurcations $3 and $4. The effect of the Damk6hler number on the reactor performance at fixed inversion period ( ~p = 1) is described in figure 3.
139
Figure 3 - The solution diagram for Zp=l. Solid lines: stable 1-periodic solutions; dashed lines: unstable 1-periodic solutions; filled squares: Neimark-Sacker bifurcations. Insets show details of delimited rectangular regions. The dynamic observed is extremely rich: lots of secondary Hopf, saddle-node and pitchfork bifurcations have been encountered; it is worth noting that a wide range of Damk6hler exists (N-S1 - N - S 7 ) in which possibly only chaotic and quasi-periodic solutions exist. 4. CONCLUSIONS This work shows how it is possible to reconstruct systematically the regime behavior of periodically-forced systems, by applying robust continuation algorithms to an associated discrete-time system, properly constructed and implemented numerically starting from a Poincar6 section of the underlying continuous time system. Moreover, the approach proposed permits to identify bifurcations and to automatically trace solution branches stemming from pitchfork bifurcations. Future applications include larger systems, such as those obtained by reducing distributed-parameter systems. Notations
A heat exchange area, m 2 Cp heat capacity, kJ/(kgK) C concentration, kmol/m 3 Da Damk6hler number (VRrin)/(FCin) E activation energy, (kJ)/kmol f vector function F volumetric flow rate, m3/s
h heat exchange coefficient, kJ/(m 2sK) AH heat of reaction, kJ/kmol IO switching function k Arrhenius constant koexp[-E/(RT)] n order of reaction R gas constant, kJ/(kmolK) r rate of reaction (kCn), kmol/(m 3s) t time, s
140 T temperature, K u vector state of the system V volume of the reactor, m 3
Greek letters c~ dimensionless degree of conversion (C0-C)/C0 13 dimensionless adiabatic increase of temperature (AH C0)/(T0pcp) 6 dimensionless heat exchange coefficient (Ah)/(pcpF[3) ~, dimensionless number related to activation energy E/(RT0)
;~ parameter | dimensionless temperature (T- T0)/(13To) dimensionless time F/(Vt) ~p dimensionless time of flow reversal
Subscripts and superscripts a,b refers to reactor a,b H refers to heat exchanger in feed k k-iterate out outlet from the system
REFERENCES 1. Matros Y.S., Unsteady process in catalytic reactor, Elsevier, Amsterdam, (1985). 2. Reh~t6ek J., Kubi~ek M., and Marek M., "Periodic, quasiperiodic and chaotic spatiotemporal pattems in a tubular catalytic reactor with periodic flow reversal", Comp. Chem. Eng., 22, 283-297 (1998). 3. Snyder J.D. and S. Subramanian, "Numerical simulation of a periodic flow reversal reactor for sulfur dioxide oxidation", Chem. Eng. Sci., 48, 4051-4064 (1993). 4. Van den Bussche K.M., Neophytides S.G., Zolotarskii I.A., and Froment G.F., "Modelling and simulation of the reversed flow operation of fixed bed reactor for methanol synthesis", Chem. Eng. Sci., 48, 3335-3345 (1993). 5. Croft T. D., and M.D. Levan, "Periodic states of adsorption cycles-I. Direct determination and stability", Chem. Eng. Sci., 49, 1821-1829 (1994). 6. Salinger A.G., and G. Eigenberger, "The direct calculation of periodic states of reverse flow reactor: I Methodology and propane combustion results", Chem. Eng. Sci., 51 4903-4913 (1996a). 7. Salinger A.G., and G. Eigenberger, "The direct calculation of periodic states of reverse flow reactor: II multiplicity and instability", Chem. Eng. Sci., 51 (1996b). 4915-4922. 8. Khinast J., and D. Luss, "Mapping Regions whit different bifurcation diagrams of a reverse-flow reactor", A IChE J., 43, 2034-2047 (1997). 9. Khinast J., Jeong Y.O., and D. Luss, "Dependence of cooled reverse-flow reactor dynamic on reactor model", A IChE J., 45, 299-309 (1999). 10. Brown P. N., G. D. Byrne, and A. C. Hindmarsh, "VODE: a variable coefficient ODE solver," SIAMJ. Sci. Stat. Comput., 10, 1038-1051 (1989). 11. Doedel E. J., Champneys A. R., Fairgrieve T. F., Kuznetsov Y. A., Sanstede B., and X. Wang, "A UT097: continuation and bifurcation software for ordinary differential equations", July (1997). 12. Zukowski W., and M. Berezowski, "Generation of chaotic oscillations in a system with flow reversal", Chem. Eng. Sci., 55, 339-343 (2000).
European Symposiumon ComputerAided ProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rights reserved.
141
Modelling and Optimisation of a High Density Fermentation Process Using Multi-Linear Models: An Application to a Bench Scale Bioreactor Omar Galfina, Ahmet Palazoglu b and Jos6 A. Romagnoli a a Laboratory of Process Systems Engineering, Department of Chemical Engineering University of Sydney, NSW 2006, Australia.
b Department of Chemical Engineering and Materials Science, University of California, Davis, CA 95616, USA. A multi-linear modeling approach is proposed to study a bench-scale bioreactor where the high-density sucrose fermentation to ethanol by Saccharomyces Cerevisiae takes place. Local linear models are used to determine the optimal sucrose profile that maximizes ethanol production. To assess the potential of multi-linear modeling, batch and fed-batch operational policies are compared. 1. I N T R O D U C T I O N The manufacture of many fine chemicals, pharmaceutical products, beverages, biofertilizers and many other industrial products involves fermentation. It is well known that biological processes present several operational challenges for process control engineers such as time-varying characteristics, nonlinear behavior, model inaccuracies, few specific measurements, constrained operation, presence of disturbances, irreversible behavior, limited corrective action, repetitive nature and slow process dynamics [1]. An application that exhibits all complexities mentioned above is the fermentation of sugars to ethanol via Saccharomyces Cerevisiae (yeast). Traditional brewing of sugars to ethanol is carried out in batch reactors with worts of 1112% (w/w) dissolved solids to produce beers of 4-5 % (v/v) ethanol. However, it is known [2] that high-gravity brewing at a limit of 16-18% (w/w) dissolved solids presents economic advantages. Yet, attempts to ferment worts above 18% (w/w) dissolved solids proved to be difficult, largely due to the high osmotic pressure acting on the yeast cells and ethanol toxicity [3]. Although yeast strains are unable to operate efficiently at high dissolved solids due to physical and physiological limitations, it is still possible to determine a sucrose profile that maximises conversion to ethanol. This feed policy helps cell adaptation mechanisms, overcoming the brewing limits over 18% (w/w) dissolved solids. In this study, we consider the batch and fed-batch fermentation processes, and given a model of the process, determine the optimal operating conditions that drive the system from a known initial state to the desired final state. A multi-linear model representation of the fermentation is used to determine the optimal sucrose profile.
142 2. MODELING OF FERMENTATION Fermentation processes involve a large number of biochemical reactions and interacting variables that complicate detailed process modeling. However, there are several models based on mass balances and Monod-like kinetics, which are a macroscopic scale representation of the process where a few parameters must be estimated from experimental data. Such mechanistic models have a nonlinear structure, making the parameter estimation process nontrivial [4]. An alternative to modeling systems with complex nonlinear dynamics uses the multilinear models [5,6]. Here, the operating range is divided into regions where a linear model with a suitable structure can approximate the system dynamics. The first step is to identify the number of regions where a linear approximation is feasible. Physical knowledge about the process often provides good insight on how many regions may be identified. The fermentation process involves six growth phases but it does not mean that six regions are to be identified. Each phase may exhibit linear or nonlinear behavior, thus, a single phase may require more than one linear model. In addition, the transition from one phase to another is not well defined. The second step is to estimate the parameters of the local models for the different regions. Finally, the third step is to incorporate a mechanism to blend the local models. The local linear models are described as below: :Ap~:
; ~(t0) :~:0 t ~ [ t o , t o + T )
;
(1)
p = l ..... m
z=~:
(2)
where ~: is the state vector, z is the output vector, which assume that all states are measurable, m the number of identified models, and A p is the process matrix, whose entries A 0 are the parameters to be estimated. Each linear model is identified along the process trajectory in the interval t e [ t 0, t o + T ) using the known initial conditions ~: (to) = ~0 9The states in these local models are the main variables for the fermentation process: ~1 = B i o m a s s , ~2 = S u c r o s e , ~3 = Ethanol and ~4 -" p H . The estimation of the process matrix A p is carried out by minimisation of the objective function, min
At,,'/
7
(3)
subject to Eqs. (1-2) and
~ [zj(k) - Zexp,j(k)]
j:l~:l where
t h e Zexp, j ( k )
2 < y
(4)
are the experimental profiles of the variables involved in the process, n
is the number of states and s is number of samples in t ~ [ t 0, t o + T ) t o perform the optimization. The model that represents the process in the whole operating range is given by:
143 m
~
m
= E ~p (~:; ~: )A p ~: p=l
;
E ~ p(~:;~: )= 1 p=l
(5)
where ~ p(~ ;~-) are the membership functions (Fig. 1) parameterized in the mean values of the trajectory ~- in t ~ [ t 0, t o + T ) .
_
o
.
.
.
46.64
.
.
m,..'
_
58.96 75.33 94.56
2
_
_
174.07
Figure 1. Membership functions parameterized in the mean values of the sucrose trajectory ~2Establishing the membership functions is a key step in multi-linear modeling. Most common approach is to identify the local regions and then establish a grade of membership for each value in these regions. We used a 'hat' function peaked at the most representative value (e.g. mean concentration of sucrose) in the region. 3. F E R M E N T A T I O N CASE STUDY Ethanol and carbon dioxide are the main fermentation products of sucrose by Saccharomyces Cerevisiae. The fermentation also results in an increase in the number of yeast cells (biomass). The experiments conducted in this study are performed at 30~ under anaerobic conditions. The experimental set-up consists of a 5-liter fermenter with pH and dissolved oxygen probes and a RTD sensor immersed in the fermentation broth. The pH and temperature are controlled with PI controllers, using a peristaltic pump and 250 W lamps respectively. The dissolved oxygen is controlled by an on-off solenoid valve and agitation speed. An I/O field bus collects sensory inputs directly from the fermenter and sends them to a PC. Outputs then respond to commands from the PC to correct deviations from the set points. Adequate conditions of temperature and nutrients were provided to isolate the effect of sucrose concentration on the fermentation. To avoid infections in this long-term fermentation (40 hours); the pH was not controlled, allowing it to reach values around 2 where competitive bacterial activity was reduced. 3.1 Medium and Inoculum Preparation The cultivation medium for Saccharomyces Cerevisiae (Dry Yeast) contained 120 - 200 g/L sucrose, 3.0 g/L (NH4)2504 , 0.7 g/L MgSO4, 0.5 g/L NaC1, 1.0 g/L KHEPO4, 0.1 g/L KEHPO4 and 0.2% yeast extract. Inocula (8-g dry yeast) were prepared in shake flasks containing sterilized medium held at 30~ for 20 minutes.
144 6 .
.
.
.
250-~
200
o
lO0
-
-
-
o Iw 0 60
10
20
30
-
-
-
time, hours
0
40
0
5
10
20
30
-
-
-
time, hours
40
pH
~ '~ c:
0
-
10
-
20
-
30
time, hours
40
I I 0
/ 10
. . 20
.
. 30
time, hours
40
Figure 2. Experimental profiles for a typical high density batch fermentation. These trends provide essential information for the identification of local models. 3.2 Experimental Results Experimental profiles of biomass, sucrose and ethanol concentrations and pH were obtained for 40 hours of batch fermentation. In each of the six phases, the cell nutritional and environmental requirements are different as illustrated by the experimental profiles in Fig. 2. Focusing on the biomass profile, the fermentation presents a fast cell growth and ethanol production in the first 12 hours. In this stage, the biomass concentration reaches its maximum value and then it drops in the subsequent 4 hours. After 16 hours of operation, over 60% of the ethanol were produced. The fermentation proceeds at a fixed rate of ethanol production, sucrose consumption and biomass extension. Therefore, a linear model every 5 hours is reasonable and enough to cover the rest of the fermentation until 40 hours. The next step is to identify the local linear models along the process trajectory. We identified 7 local models using dynamic optimization techniques (Eqs. 3-4), using the experimental data in Fig. 2. The multi-linear model is constructed using Eq. (5) and the membership functions displayed in Fig. 1. 3.3 Optimal Sucrose Profile Based on Multi-Linear Models In this section, we use the sucrose concentration as an input variable to maximize the amount of ethanol produced. To determine the optimal sucrose profile, the first step is to omit the sucrose equation from Eq. (5). The new variables are tagged as: Xl = ~1 , x2 = ~3 , x3 = ~:4 , and w = ~2 9The resulting reduced system is: m=7
m=7
Jc = ~ Ap x + ~ Bp w ;
p=l y = x2
x(O) = x o
(7)
p=l (8)
where the sucrose concentration w is the new input and y is the ethanol concentration. The initial conditions were the same as for the local models. The objective is to optimize a
145 function of the sucrose concentration w to track the ethanol production along a desired profile. The objective function is then given as: min J (w)
(9)
w~W
where N N )2 _ )2 ~ 2 k (yr,h: - yk + Z Y k (W~ wk_ 1 k=l k=l
J=
2k,Yk >0
;
m={w
1,W 2 , . . . , W N }
where Yk, r is the desired ethanol concentration, Yk is the actual ethanol concentration, ,;Ck , and y k are constant positive weights (chosen as 2, k = 10 and y ~ = 1 ), and w k are bounded inputs in the range 0 _< wk _< 130 g / L . The solution of the constrained optimization problem for the determination of the optimal sucrose concentration profile is shown in Fig. 3.
4. DISCUSSION OF RESULTS The fermentation process exhibits nonlinear behavior with strong interactions among the variables (Fig. 2). High-density batch fermentations present sucrose inhibition at early stages of the process and low conversion to ethanol. The way to overcome this drawback was by supplying sucrose to the fermentation following an optimal feed rate policy. The fermentation process was modeled as a combination of local models identified along the process trajectory. Using physical and experimental information (i.e. Fig. 2), 7 different operating regimes were identified. The parameters associated with the local models were estimated using dynamic optimization, which is a least-squares Levenberg-Marquart algorithm, coupled with a Runge-Kutta method.
i''000("
5
_1
@
.0 II1
@
0
10 20 30 time, hours
5OI 0
40
8o
I~
2'0
time, hours
3'0
4O
5
-J 6o ~40
pH
9
.r 20 UJ C 0
9
10 20 30 time, hours
40
1o
10 20 30 time, hours
Figure 3. Experimental profiles for the fed-batch operation with optimal sucrose profile. The process objective was to optimize a function of the sucrose concentration to track the ethanol concentration along of a desired profile (Fig. 2). The decision variable (sucrose concentration) was bounded in the range 0 to 130 g/L in order to avoid substrate inhibition
146 observed in batch fermentations. A subsequent optimization was performed to obtain the optimal feed flow rate without exceeding the pump and the reactor capacities, 0.1 L/min and 3.0 L respectively. It is important to mention that the sucrose concentration in the feed and the initial volume for a fed-batch process are two extra degrees of freedom that the control designer can manipulate to satisfy process requirements. Figure 3 shows the experimental profiles. The implementation of the optimal profile was open-loop; therefore it was difficult to track the system along the desired trajectory. The fedbatch operation exhibited high sucrose to ethanol yield in comparison with the batch operation (Fig. 2). REFERENCES
1. D. Bonvin, J. Proc. Cont., 8 (1998) 355. 2. G.P. Casey, C.A. Magnus, W.M. Ingledew, Applied and Environmental Microbiology, 48 (1984) 639. 3. F.H. White, Proceedings of the 15th Convention of the Society of Institute Brewing (Australia and New Zealand), (1978) 133. 4. F.S. Wang, and J.W. Sheu, Chem. Eng. Sci., 55 (2000) 3685. 5. O. Galfin, J.A. Romagnoli, A. Palazoglu, Chem. Eng. Sci., 55, (2000) 4435. 6. R. Murray-Smith, R., and T.A. Johansen (eds.), Multiple Model Approaches to Modeling and Control, Taylor & Francis, London, England, 1997.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) ~q-)2001 Elsevier Science B.V. All rights reserved.
147
Simulation of the FGD In-Duct Injection Technology using complex kinetic models A. Gareaa, J. A. Marqu6s a, T.L. Hechavarriaband A. Irabiena. a Dpt.
Quimica, E.T.S.I.I.y T., Universidad de Cantabria. Avda. los Castros, s/n, 39005 Santander, Spain.
b Facultad de Ingenieria Quimica. Universidad de Oriente, Santiago de Cuba, Cuba.
The aim of this work is the computer aided simulation of an entrained flow reactor operating under typical conditions of FGD in-duct injection at low temperatures. The modelling was performed considering the gas-solid reaction in a single particle of sorbent that is moving as well as the gas phase at any axial distance of the duct. The overall reaction rate is calculated from the transport resistance through the ash layer and the chemical reaction taking place at a sharp moving interface in the radial axis of the particle. The experimental profiles of the SO2 concentration in the extemal gas phase obtained in a pilot plant of In-Duct-Injection of Ca(OH)2 were introduced to the simulation in order to estimate the parameters of the reaction under study, working at different Calcium/Sulfur molar ratios. The proposed model allows to describe the SO2 removal and the solid conversion levels when varying the residence time in the duct, in good agreement with the experimental values.
1. INTRODUCTION Three main categories of Flue Gas Desulfurization (FGD) technologies are considered for controlling the emissions of sulfur dioxide in large coal power plants: dry sorbent injection, semi-dry and wet processes. The first option, Dry Sorbent Injection, provides a low-cost retrofit alternative for existing plants to meet emissions regulations, compared to the semi-dry and wet systems that require additional equipment and the subsequent sludge treatment in the case of wet processes. The FGD In-Duct-Injection technology at low temperatures involves the injection of a dry sorbent, typically Ca(OH)2, in conjunction with sprayed water in the ductwork ahead of the particulate collector [ 1-3]. The residence time of gas and solid phases in the duct is up to 3 seconds typically. The modelling of the desulfurization process at in-duct conditions is of practical relevance for the design and operation in large scales. It is required to describe the SO2 removal and the sorbent utilization levels at different operational conditions, being the Calcium to Sulfur molar ratio (Ca/S) the most important parameter.
148 The experimental results from the In-Duct Injection process at a pilot plant scale show that the coupling between gas and solid profiles can not be explained using a simplified model that only takes into account macroscopic balances for gas and solid [4,5]. Attemps to the modelling of the gas-solid reaction between the SO2 and calcitic sorbents at a microscopic level included the shrinking core model as well as the grain model in order to model experimental results from reaction systems based on fixed bed or thermobalance in laboratory scale [6-8]. The objective of this work is a better understanding of the gas-solid reaction that takes place in the duct section and the establishment of the desulfurization model, as a necessary tool for the design and optimization of the FGD In-duct injection process at low temperatures. The basis of the shrinking core was applied to a moving particle exposed to different SO2 concentration depending on its axial location of the duct section for describing the SO2 removal level and the sorbent utilization at the corresponding residence time. 2. M O D E L L I N G Since the pioneering work of Yagi and Kunnii in 1955, several particle structures were considered for the non-catalytic gas-solid reactions, such as the sharp interface or shrinking core model for solids assumed to be nonporous, and the grain/pore picture for porous structures [9]. The development of models involving solid reagents in some porous form was focused in the description of the structure evolution, in these cases, the effective diffusivity of gas in the ash layer and the surface kinetic constant may depend on the microstructure changes with conversion. Krishnan and Sotirchos pointed out for the direct sulphation of limestones that the reaction rate decreased faster during the reaction period than the predicted by the shrinking core model. It was required the modification of the efective diffusivity as an exponential function of the distance from the external surface of the particle [6]. Analogous conclusions were reported from other studies with calcium based sorbents, treating the effective diffusivity of gas in the product layer as dependent on solid conversion [10-12]. Taking into account the previous studies, the formulation of a model for the gas and solid behaviour in the in-duct injection process was based on the application of the shrinking core model to a single particle at any axial position in the duct. It is important to remark that the particle is in contact with a gas phase of variable SO2 concentration related to its position. The following assumptions were considered for modelling: An average particle diameter of the Ca(OH)2 sorbent. Isothermal conditions in the duct. Negligible external mass transfer resistance, being exposed the initial surface of the particle to the SO2 concentration in the external gas phase. - Steady-state profiles of SO2 concentration over the distance in the reacted layer, due to the shrinkage of the unreacted core is slower than the flow rate of SO2 toward the unreacted core. - The chemical reaction described as a simple first order reaction at the sharp interface which is moving inward toward the center of the particle during the reaction progress. -
-
-
149 Under these assumptions, the mass balance on the gaseous reactant
10[
OCt]: Or J
r 2 Or r 2 D e
(802)yields (1)
0
being concentration at any radial distance (r) in the reacted core; mol m 3. C~ the The superscript z accounts for any axial position in the duct. De the effective diffusivity in the reacted layer; m E S "1 .
$02
For providing the increase of the mass tranfer resistance with the progress of reaction, the diffusional parameter, D e , was also considered variable with the conversion of the particle as well as variable with the distance from the external surface of the particle. The boundary conditions required for solving equation (1) are - At the extemal surface of the particle: r=Ro
c t =C2
(2)
- At the reaction from: r=Rc
-Oe~
ac t r = rs
(3)
working with an average outer radius of particle Ro =2.75 10 .6 m. The chemical reaction rate per unit of surface area (rs) was defined as first order related to the SO2 concentration (Cd) corresponding to the radius of the unreacted core, Re r, : k, c~
(4)
The core of particle, R c , is shrinking with time as a result of the overall reaction rate that include both chemical reaction and mass transfer, being calculated as follows
- -dt
:
U so 2
ro . . . .
(5)
ll
where ps is the solid density (2.24 10"3 kg m -3) , Ms the molecular weight (80 g mol "1 of commercial Ca(OH)2 sorbent), and v the stoichiometric coefficients. The overall reaction rate equation is given by the following equation, in terms of mol
S "1
c~
ro. . . .
ll :
1
- - . + 4~rR2c k s
Ro-R~ 4z~R o R c De
(6)
150 The conversion of the particle is related to the variable Rc by the equation:
Xs= 1_ l R c l 3 t, Ro)
(7)
The simulated 802 concentration in the extemal gas phase at any axial position (Coz) becomes from the material balance x,
SR = 1 -
C---L~
(8)
C in.dua being introduced the operational parameter SR defined as the Calcium to Sulfur molar ratio (Ca/S stoichiometric ratio). The system of equations were transformed in order to work with the corresponding dimensionless variables of SO2 concentration, radius of the particle and total lenght of the duct. The parameters of the proposed model are the effective diffusivity in the reacted layer, De, and the surface reaction rate constant, ks. A commercial software, gPROMS (PSE Ltd.), was used for solution and the estimation of the parameters was based on the criteria of least squares minimization related to the experimental data of SO2 concentration in the gas phase at different locations in the axial axis of the duct (experimental values of Co). The experimental data were obtained in a pilot plant scale of own design detailed elsewhere [5]. The flow reactor was provided by different gas phase samplying points in order to analyze the SO2 concentration trend up to 3 seconds of residence time in the reactor, as the maximum value within the typical conditions of the in-duct injection process. The operating conditions were fixed to 1000 ppmv of SO2 concentration in the inlet gas phase, temperature of 60~ and relative humidity of 60%, varying the Ca/S parameter between the ratio of 4 to 18 in order to achieve a wide range of SO2 removal levels. 3. SIMULATION RESULTS The effects of mass transfer and chemical reaction on the overall reaction rate can be discussed from the obtained simulation results for three series of experimental data corresponding to the Ca/S molar ratios: 4, 9 and 18. The values of the estimated parameters, De and ks, in terms of time constants (DIRo 2) and (k/Ro), are shown in Table 1, with the values of the standard deviation of the estimation from each Ca/S ratio (crn_l). The estimation of the parameters for the whole range of CaJS was also performed and included in Table 1 as Global. The standard deviation values corresponding to the simulation of each serie of data Ca/S with the obtained parameters from the global estimation procedure were identified by O'n-l(global). The fitting to the experimental data is shown in Figures 1 and 2, being represented the trends of SO2 concentration in the gas phase and solid conversion up to the total residence time in the reactor or duct (3 s).
151 Table 1. Simulation results for diffusion and chemical reaction: estimated values of the effective diffusivity De and the kinetic constant k~, for Ca/S ratios 4, 9 and 18.
DIFFUSION AND CHEMICAL REACTION De~o 2 ($'9 k ~ o ( S ' 9 0"n-1[(Tn-l(global) Dan
DIFFUSIONAL CONTROL De/Ro 2 (s-t) Crn.l / O'n.l(globaO Ca/S 4 9 18 Global
83.39 46.35 36.76 48.98
0.060 / 0.094 0.084 / 0.092 0.063 / 0.087 0.089
93.48 47.05 36.54 48.55
4.31 10 4 1.54 107 4.23 10 6 1.3510 7
0.060 / 0.103 0.084 / 0.092 0.063 / 0.088 0.089
4.6 102 3.3 105 1.2 10 s 2.8 lO s
The dimensionless number DamkOeler 11, that accounts to the ratio of the chemical reaction velocity (k~(CoZ)n'l and the diffusion velocity in the particle (De/Ro)
Da• = ks (cz~)n-] Ro De
(9)
was calculated and included in Table 1 in order to quantify the relationship between these two mechanisms.
*n* e case of oo ,
rocess is con,
en, i ely
mass transfer
1~ 0 ,
that is, the concentration at the surface approaches zero. For the reaction order n=l and Dall > 102, implies that this dimensionless concentration is < 10-2 at the surface. This fact can be observed in Figure 3 showing the SO2 concentration profile in the reacted layer for different reaction times, which is consistent with the slight decrease of the unreacted core, represented in Figure 4.
{
1.0 0.8 o
. ~ ~ ' ~ ' ~ ~
0.6
j
SR4exp SR4est 9 StLOexp SR9est X SR18exp SR18est
O.4 0.2 0.0
r
0
0.5
i
1
i
1.5
i
t
2
2.5
3
x (s)
Fig. 1. S O 2 concentration in the gas phase in the duct for Ca/S: SR=4, 9, 18.
0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0
9 9 II
0
0.5
;~
1
1.5
x (s)
~
2
Fig. 2. Solid conversion in the duct for SR = 4, 9, 18.
..
2.5
3
152 1
0.8 .
.
.
.
3s /
/
.
0.6
.-2"...'2".---
3 0.4
0.99 o 0.98 ~0.97 0.96
0.2 0
,
0
0.2
i
0.4
i
r
0.6
i
0.8
Fig. 3. $ 0 2 concentration profile in the reacted layer at 0.1, 1.5, 3 s for Ca/S =4.
0.95 0
0.5
1
1.5
2
2.5
3
l:(s) Pig. 4. Unreacted core during reaction time (up to 3 s) for Ca/S =4.
4. CONCLUSIONS A computer simulation procedure has been developed to solve the differential equations of the proposed model allowing to fit experimental results from the pilot plant that reproduces the conditions of the In-duct injection process. The model was based on the overall reaction in a shrinking core of a particle located at any axial position in the duct section. From the estimation of parameters, De the effective diffusivity in the reacted layer of the particle, and ks, the surface reaction constant, it can be concluded with the diffusional resistance as the controlling step, being the surface reaction rate much faster. The simulated SO2 concentration and the solid conversion at any axial distance in the reactor under diffusional control were in a good agreement with the experimental data for the range of Ca/S molar ratio under study (4-18), given by a standard deviation lower than 10%. The comparison of the fitted parameter, De = 3.7 101~ (m 2 S'l), with the effective diffusivity of SO2 at the same temperature, 4.85 10-6 (m 2 s-1) shows, that the desulfurization process under study may not be described by gas diffusion in the particle. It can be related to a solid state diffusion process or to the SO2 diffusion in the microporosity of the small grains of sorbent in the particle. REFERENCES 1. A.L. Fenouil and S. Linn, Ind. Chem. Eng. Res., 35 (1996) 1024. 2. M.R. Stouffer, H. Yoon and F.P. Burke, Ind. Eng. Chem. Res., 28 (1989) 20. 3. R.W. Rice and G.A. Bond, AIChE J., 36 (1990) 473. 4. J.A. Marqu6s, J.L. Herrera, A. Garea and A. Irabien, Clean Air'99, (1999) Lisboa, Portugal. 5. J.A. Marqu6s, A. Garea and A. Irabien, CHISA 2000 (2000) Praga, Czech Republic. 6. S.V. Krishnan and S.V. Sotirchos, Can. J. Chem. Eng., 71 (1993) 734. 7. M. Maalmi, A. Varma and W.C. Strieder, Ind. Chem. Eng. Res., 34 (1995) 1114. 8. A.B.M. Heesink, W. Prins and W.P.M. Van Swaaij, Chem. Eng. J., 53 (1993) 25. 9. J.J. Carberry and A. Varma (eds.), Chemical Reaction and Reactor Engineering, Marcel Dekker, Inc., New York, 1987. 10. A.F. Shaaban, Thermochim. Acta, 180 (1991) 9. 11. A. Garea, J.R. Viguri and A. Irabien, Chem. Eng. Sci., 52 (1997) 715. 12. I. Fernfindez, A. Garea and A. Irabien, Chem. Eng. Sci., 53 (1998) 1869.
European Symposiumon ComputerAidedProcess Engineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rights reserved.
EQUISTAR:
153
R e l i a b l e S o f t w a r e for D e s i g n o f N o n i d e a l a n d R e a c t i v e S y s t e m s
S. T. Harding and C. A. Floudas I Department of Chemical Engineering, Princeton University Princeton, NJ 08544, USA
1
Introduction
Many commercial packages for process design and thermodynamic calculations are available today. Typically, these packages allow a broad variety of design problems to be solved and a large number of thermodynamic models from which to choose. Many design alternatives can be screened relatively quickly using such applications, and they greatly reduce the effort needed to design complicated processes. However, the solution methods employed by these packages frequently fail for systems that exhibit complex behavior. The reason for this failure is that local solution techniques are used to solve the nonlinear equilibrium equations that arise in the problem formulations. In this paper, a new tool for robustly and efficiently solving process design and thermodynamics problems is presented. This new tool is called EQUISTAR, which stands for E Q U i l i b r i u m Solution Toolkit for Azeotropic and Reactive Distillation Design. EQUISTAR addresses the need for a computational tool that can be reliably applied to highly nonideal and reactive systems and can solve the most commonly occurring thermodynamics problems. In section 2, the capabilities and design of EQUISTAR are presented. The global optimization algorithms that EQUISTAR employs are outlined in section 3.
2
O v e r v i e w of E Q U I S T A R
Local solution approaches have two features that make them attractive, especially for commercial implementations: they are fast, and they are relatively easy to implement. In order to achieve the kind of reliability that global optimization methods provide, one usually pays a premium in speed. In addition, global approaches are substantially more difficult to implement both in terms of the analysis of the mathematical equations, and in the actual programming effort. However, recent developments in global optimization approaches take a large step toward the practical implementation of global methods for process design. By analyzing the structure of the mathematical equations, it is possible to identify properties that can be exploited to greatly reduce the computational effort for guaranteed reliability. 2.1
EQUISTAR
Capabilities
EQUISTAR is a versatile package that incorporates global optimization methods in order to provide reliable solutions of many thermodynamic equilibrium problems that arise in the design and simulation of chemical processes. In addition, EQUISTAR can use this "toolkit" of robust equilibrium solution techniques to determine the design specifications for reactive and nonreactive distillation columns. The user is not forced to use the rigorous global optimization approach for any of the problems that EQUISTAR solves. In some cases, one may wish to do 1Author to whom all correspondence should be addressed.
154 a quick local search for a solution. EQUISTAR gives the user the option of specifying whether the problem is to be solved to global optimiality, or local optimality, and the user can specify the number of local searches that are performed. EQUISTAR can solve the following thermodynamic equilibrium problems: 1) phase and chemical equilibrium, 2) Gibbs free energy minimization, 3) phase stability (through Gibbs tangent plane distance minimization), 4) finding all homogeneous reactive and non-reactive azeotropes, 5) finding all heterogeneous reactive and non-reactive azeotropes, 6) isothermal or reactive flash calculation, 7) reactive or nonreactive bubble point calculation, and 8) reactive or nonreactive dew point calculation. The equilibrium conditions in each of these problems are only necessary conditions for the global equilibrium solution. Therefore, the solution that is obtained may correspond to a thermodynamically unstable system. EQUISTAR allows the user to specify whether or not to check the stability of each equilibrium solution that is obtained. The stability check is performed by solving the tangent plane distance minimization problem to global optimality, or until a negative tangent plane distance is located. In addition to solving stand-alone thermodynamics problems, EQUISTAR incorporates its solution algorithms into the capability of solving reactive and non-reactive distillation design problems. EQUISTAR provides the user a choice of algorithms for reactive or non-reactive distillation design algorithms, 1) a modification of the Inside-Out Algorithm, and 2) a modification of the Bubble-Point algorithm. EQUISTAR allows the user to choose from a wide range of thermodynamic models for representing the system's physical behavior. A number of equations of state axe available: 1) the Redlich-Kwong equation, 2) the Soave-modified Redlich-Kwong equation, 3) the PengRobinson equation, and 4) the van der Waals equation. In addition, several activity coefficient equations are available: 1) the Wilson equation, 2) the NRTL equation, 3) the UNIQUAC equation, and 4) the UNIFAC group-contribution method. 2.2
EQUISTAR
Software Design
The EQUISTAR program primarily consists of optimization problem formulations and highlevel optimization algorithms. The program is written in C. These formulations and algorithms are based on novel analysis by several authors and are described in section 3. EQUISTAR automatically generates problem formulations in the c~BB problem format. c~BB is a global optimization approach for solving general twice-continuously differentiable nonlinear programming problems developed by [2, 1]. c~BB is based on a branch-and-bound framework coupled with novel convex underestimators and the program manages the variable branching and the formulation of the upper and lower bounding problems. In order to solve the actual optimization problems, MINOPT is called. MINOPT is a Mixed-Integer Nonlinear OPTimization modeling language and solver developed by [10]. MINOPT converts the formulation of the optimization problem into a format that can be sent to an equation solver. MINOPT has interfaces with a number of linear, nonlinear, mixed-integer linear, mixed-integer nonlinear, and differential and algebraic equation solvers. Depending upon the type of problem that is passed, MINOPT converts the formulation into the correct format, sends it to the appropriate solver, and passes the solution back to the program that called it. Through this structure, the implementation of EQUISTAR and the formulation of its problems are independent of the local optimization method. The algorithms within EQUISTAR possess their own interdependencies, as shown in figure 1. The middle layer of the figure are the basic global optimization algorithms for solving
155
equilibrium problems. Note that each of these algorithms may call the phase stability problem to verify the stability of the solution. These algorithms call c~BB as the general global optimization solver. At the top of the figure are the highest level algorithms: distillation design and phase and chemical equilibrium. Each of these top level algorithms require the repeated solution of the equilibrium algorithms.
Figure 1: Relationship between EQUISTAR components
3
R e v i e w of S o l u t i o n M e t h o d s
Each of the problem types addressed by EQUISTAR has its own solution algorithm. section provides a summary of each of the algorithms. 3.1
This
Gibbs Free Energy Minimization
The minimization of the Gibbs free energy of the system is a fundamental approach for determining the equilibrium state of a system. A necessary and sufficient condition for equilibrium is that the Gibbs free energy of a system at constant temperature and pressure be at its global minimum. The general problem referred to as the Gibbs Free Energy Minimization Problem (GMIN) is defined as follows" Given N components with initial moles {sT1, sT,... ,n T } participating in up to P potential phases and R chemical reactions at constant temperature and pressure, find the mol vector n that minimizes the value of the Gibbs free energy function and satisfies the material balance constraints. The algorithm that EQUISTAR uses to determine the global minimum Gibbs free energy is based on the approach developed by [8] and is outlined below. 1. The user provides the system temperature and pressure and overall composition and the number and type of phases. 2. The GMIN problem is solved locally to generate an upper bound. 3. A branching variable is chosen and the current domain is partitioned by bisecting the bounds of the branching variable.
156
4. In each new domain a convex lower bounding problem is solved and the domain is discarded if the solution is greater than the current best upper bound. 5. Return to Step 2 and repeat until the best upper and best lower bounds converge. 6. The solution of the problem provides the composition of each phase.
3.2
Tangent Plane Distance Minimization
[3] and [11] have proved that a necessary and sufficient condition for a candidate equilibrium solution to be the true equilibrium solution is that the tangent plane distance function be nonnegative for all phases in the candidate solution. The tangent plane distance function is defined as the distance between the Gibbs free energy surface for the new phase, and the tangent plane to the Gibbs energy surface constructed at the candidate equilibrium solution. Based on the work of [9] and [4], the tangent plane distance minimization problem (TED) is solved in EQUISTAR using the following algorithm: 1. The user candidate 2. The T P D 3. Check the
provides the system temperature and pressure, and the composition of the phase. problem is solved locally to generate an upper bound. current best upper bound. I f it is less than zero, then stop the algorithm
because the candidate phase is unstable. 4. A branching variable is chosen and the current domain is partitioned by bisecting the bounds of the branching variable. 5. In each new domain a convex lower bounding problem is solved and the domain is discarded if the solution is greater than the current best upper bound. 6. Return to Step 2 and repeat until the best upper and best lower bounds converge or the best upper bound becomes negative. 7. The solution of the problem determines the stability or instability of the candidate phase. Enclosing All Azeotropes
3.3
The phenomenon of azeotropy occurs in many industrial applications. Azeotropes restrict the amount of separation of a multicomponent mixture that can be achieved by distillation. The ability to predict whether a given mixture will form one or more azeotropes and to calculate the conditions and compositions of each azeotrope is essential if one wants to model separation processes. An azeotrope is defined as a liquid mixture that boils at a constant temperature where the composition of the vapor phase is identical to the composition of the boiling liquid. When the boiling liquid contains a single phase, this phenomenon is called a homogeneous azeotrope. If the liquid consists of two or more phases it is classified as a heterogeneous azeotrope. Azeotropes may also occur in systems where one or more chemical reactions are occurring. These are called reactive azeotropes, and may be classified as homogeneous reactive azeotropes or heterogeneous reactive azeotropes, depending upon the number of liquid phases.
3.3.1
Enclosing All Homogeneous Azeotropes
The algorithm presented below for the location of all homogeneous non-reactive and reactive azeotropes is based on the work of [6]. 1. The user provides the system pressure.
157
2. A branching variable is chosen and the current domain is partitioned by bisecting the bounds of the branching variable. 3. In each new domain a convex lower bounding problem is solved and t h e d o m a i n is d i s c a r d e d if the solution is greater t h a n zero. 4. Return to Step 2 and repeat until all domains have been eliminated, or the size of the remaining domains are within a given tolerance. 5. The solution of the problem determines the temperature and composition of all homogeneous azeotropes in the system.
3.3.2
Enclosing All Heterogeneous Azeotropes
The algorithm presented below for the location of all heterogeneous non-reactive and reactive azeotropes is based on the work of [5]. 1. The user provides the system pressure. 2. A branching variable is chosen and the current domain is partitioned by bisecting the bounds of the branching variable. 3. I n e a c h n e w d o m a i n , t h e p o s s i b i l i t y of a t r i v i a l s o l u t i o n is c h e c k e d . If a t r i v i a l solution is possible, the d o m a i n is k e p t , b u t S t e p 4 is skipped for the d o m a i n . 4. In each new domain a convex lower bounding problem is solved and the domain is discarded if the solution is greater than zero. 5. Return to Step 2 and repeat until all domains have been eliminated, or the size of the remaining domains are within a given tolerance. 6. The solution of the problem determines the temperature and composition of all heterogeneous azeotropes in the system. 3.4
Flash Calculation
Unlike Gibbs free energy minimization, the flash calculation takes an equation-solving approach to the determination of phase and chemical equilibria. The solution of the isothermal flash and reactive flash problems in EQUISTAR provides all compositions that satisfy the flash equations. The algorithm is based on the approach for finding all solutions to nonlinear systems of equations developed by [7]. 1. The user provides the system temperature and pressure and feed rate and composition. 2. A branching variable is chosen and the current domain is partitioned by bisecting the bounds of the branching variable. 3. In each new domain a convex lower bounding problem is solved and the domain is discarded if the solution is greater than zero. 4. Return to Step 2 and repeat until all domains have been eliminated, or the size of the remaining domains are within a given tolerance. 5. The problem solution specifies the composition and flowrate of the vapor and liquid phases. 3.5
Bubble
Point and Dew Point Calculation
The calculation of bubble point temperatures and dew point temperatures is a natural extension of the flash calculation. These calculations are commonly encountered in the design and simulation of distillation columns. The bubble and dew point calculations are phase a n d / o r chemical equilibrium calculations.
158 In these formulations, the phase equilibrium condition is represented as the equality of chemical potentials for all components. The difference between the bubble and dew point problems and the flash problem is that the composition of only one phase is being determined, and the temperature of the equilibrium state is being determined. 1. The user provides the system pressure and liquid (vapor) composition. 2. A branching variable is chosen and the current domain is partitioned by bisecting the bounds of the branching variable. 3. In each new domain a convex lower bounding problem is solved and the domain is discarded if the solution is greater than zero. 4. Return to Step 2 and repeat until all domains have been eliminated, or the size of the remaining domains are within a given tolerance. 5. The solution of the problem determines the bubble (dew) temperature and the composition of the vapor (liquid) phase.
4
Conclusion
Based on significant developments in global optimization approaches developed over the past several years, EQUISTAR provides a suite of algorithms for the reliable and efficient solution of process design and thermodynamic equilibrium problems.
References [1] Adjiman C.S., Androulakis I.P., and Floudas C.A., 1998b, A global optimization method, aBB, for general twice-differentiable N L P s - II. Implementation and computational results. Comput. Chem. Eng. 22, 1159-1179. [2] Adjiman C.S., Dallwig S., Floudas C.A., and Neumaier A., 1998a, A global optimization method, aBB, for general twice--differentiable N L P s - I. Theoretical advances. Comput. Chem. Eng. 22, 1137-1158. [3] Baker L., Pierce A., and Luks K., 1982, Gibbs energy analysis of phase equlibria. Soc. Petro. Eng. J. (p. 731). [4] Harding S. and Floudas C., 2000a, Phase stability with cubic equations of state: A global optimization approach. AIChE J. 46, 1422-1440. [5] Harding S. and Floudas C., 2000b, Locating all heterogeneous and reactive azeotropes in multicomponent systems. I~EC Res. 39, 1576-1595. [6] Harding S.T., Maranas C.D., McDonald C.M., and Floudas C.A., 1997, Locating all azeotropes in homogeneous azeotropic systems. ISJEC Res. 36, 160-178. [7] Maranas C.D. and Floudas C.A., 1995, Finding all solutions of nonlinearly constrained systems of equations. Journal of Global Optimization 7, 153-182. [8] McDonald C. and Floudas C., 1994, Decomposition based and branch and bound global optimization approaches for the phase equilibrium problem. Journal of Global Optimization 5, 205-251. [9] McDonald C. and Floudas C., 1995a, Global optimization for the phase stability problem. AIChE J. 41, 1798. [10] Schweiger C.A. and Floudas C.A., 1998c, MINOPT A Modeling Language and Algorithmic Framework for Linear, Mixed-Integer, Nonlinear, Dynamic, and Mixed-Integer Nonlinear Optimization. Kluwer Academic Publishers, in preparation. [11] Smith J., Missen R., and Smith W., 1993, General optimality criteria for multiphase multireaction chemical equilibrium. AIChE J. 39, 707.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jargensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
159
CFD Modeling of Fast Chemical Reactions in Turbulent Liquid Flows L.K. Hjertager, B.H. Hjertager and T. Solberg Chemical Engineering Laboratory, Aalborg University Esbjerg, Niels Bohrs vej 8, DK-6700 Esbjerg, Denmark. Many industrial processes involving chemical reactions happen in turbulent flow. For fast reactions the reaction rate is limited by the micromixing rate which is controlled by turbulence. Micromixing directly influences the reaction and may lead to changes in both conversion and selectivity. This paper will discuss and present results from various models including the so-called eddy dissipation concept (EDC) and the presumed probability distribution (PDF) models. The simulation results will be compared to experimental data from chemical reactions in liquid systems. 1. INTRODUCTION Many industrial processes involving chemical reactions happen in a turbulent flow. For infinitely fast reactions the reaction rate is limited by the micromixing rate which is controlled by the turbulence. Micromixing directly influence the reaction and can lead to changes in both conversion and selectivity [ 1]. Pohorecki and Baldyga [2] have performed experiments in a tubular reactor where they found the conversion length for an acid-base neutralisation at different Reynolds numbers. Hannon et. al [3] have tested two different models for chemical reactions; the finite rate combustion model and a presumed PDF multi-scale mixing model. The results from these numerical simulations were compared to the experimental results by Pohorecki and Baldyga [2]. Hannon et. al [3] showed that the finite rate combustion model was not able to predict the conversion length, and argued that this was because the finite rate combustion model contains no information on mixing at scales where viscous/diffusive effects are important. They also pointed out that the model defines the covariance of concentration fluctuations on reaction rate as a direct function of local k and ~ values [3]. They found that the multi-scale mixing model with a beta-PDF (instantaneous concentrations) could predict the length within a reasonable accuracy [3]. The Eddy dissipation concept (EDC) was developed for prediction of gaseous combustion reactions in turbulent flows. It is based on the assumption that the reaction time scales can be related to the dissipation of turbulent eddies which contains the reactants and products [4]. There are two main differences between mixing of reactants in a gas phase and a liquid phase, which are of importance when reacting flows are to be modelled. The first difference is that the coefficient of molecular diffusion is much higher in gases than in liquids, meaning that the Schmidt number in the gas phase is much smaller (So-l) than in the liquid phase (Sc>>I). The second difference results from the density variation of the gas phase and the resulting sensitivity of the gas phase density to pressure and temperature variations [5].
160 It is the objective of this paper to present results from various models including the EDCmodel and PDF-models. The simulation results will be compared to experimental data from chemical reactions in liquid systems. 2. REACTION MODELS
2.1 The Eddy Dissipation Concept The EDC model was developed for combustion reactions. These reactions can often be treated as a single-step irreversible reaction with finite reaction rate [4]: 1 kg A + s kg B -~ (1 +s) kg C
(1)
This simple reaction scheme results in mixture composition being determined by solving for only two variables, the mass fraction of species A, YA, and the mixture fraction, f. These equations read:
at (p. Y~). +
.
a
.
a
at (p "f) + ~(P
(p .u, Y~. )
.
~
" U' " f ) = cgx, L
.
Sc~ ax,
as] ax,
+R a
(2)
(3)
Here RA is the time mean reaction rate, PT is the turbulent viscosity and Sc~ is the turbulent Schmidt number. The basis for this to be valid is that the turbulent Schmidt numbers are equal for all species, an approximation, which is often found to be valid in turbulent flows. A transport equation for the mass fraction of species A is solved (2), where the reaction rate of species A is taken as the smallest of the turbulent dissipation rates of species A, B and C. R A = - A . P . - k . min Y A , - - , B . s
(4)
The constant A in equation (4) is found to be 4 and B is 0.5 for gaseous combustion reactions [4]. 2.2 Scalar mixing/dissipation theory Since the EDC-model is stricly valid only for Sc-~l, an algebraic expression for the scalar dissipation timescale could be used to take the effect of higher Sc-numbers (Sc>>l) into account. From Fox [6] the scalar dissipation rate is given as:
. . . ?' ~'2 /~ \r
.
2
.
2c
+-ln(Sc) 2
(5)
Here % is the dissipation rate of a scalar variable ~ and <(~'2> is the variance of the scalar variable.
161 The EDC-model for infinitely fast reaction assumes that the reaction is limited by micromixing. For gases (Sc-~l) this mixing rate is said to be proportional to the dissipation rate of turbulent kinetic energy. However, for liquid reactions (Sc>>l) the mixing rate must be related to dissipation of scalar variables. One alternative is to replace the timescale of fluid dynamic dissipation (k/e) with the time scale of scalar dissipation (<~)'2>/~;~). If we use the expression in (5), we get the modified time mean reaction rate for chemical species A (EDCSDT) as:
<~b.~ > .min YA,
R A =-A.p.
(6)
In the standard EDC-model the dissipation of the product C is also included, this is because combustion reactions are strongly exothermic reactions. For liquid reactions, which are isothermal the dissipation of the product C does not have to be considered. The above mentioned scalar time scale is valid for a fully developed turbulent structure. It has been found that the development in liquid mixtures takes longer time. The multiple time scale turbulent mixer model of Baldyga [7] takes account of these processes.
2.3 Multiple-time-scale turbulent mixer model (MTS) In liquid mixtures (Sc>> 1), the local value of the concentration variance O'S2, Can be divided into three parts according to the scale of segregation and the related mechanism of mixing [7]: 2
2
2
o-s = o-~ + o-2 + o'3
(7)
where cr 12, (y22 and (Y32 is the variance in the inertial-convective, the viscous convective and the viscous diffusive subrange, respectively. The inertial convective variance is produced from the macroscopic inhomogeneity of the mixture fraction, f, as a result of velocity fluctuations. Turbulent diffusivity is expressed as: DT = ~VT ; Scr
vr
= ~jUT p
(8)
Engulfment parameter, E, is given by: E = 0.058
(9)
Decreasing the scale of segregation by viscous deformation increases the wave numbers and conveys the variance into the viscous-diffusive subrange of the concentration spectrum where mixing on the molecular scale occurs by molecular diffusion in deforming slabs [7]. G ~ (0.303 + 17050/Sc). E
(10)
162 The evolution of the complete crl2, or22and era2 [7]:
variance (~s2) becomes, when summing up the equations for
COx,J
(11)
Another alternative for the time mean reaction rate presented in part 2.2 is to take account of the Multiple-time-scale turbulent mixer model given by Baldyga [7] and described above. Since for infinitely fast reactions the reaction is limited by the micromixing rate (pGcr3e) the modified form of the EDC-model (EDC-MTS) will be expressed as: RA = - A - p - G . o - 3
min/,
(12)
2.4 Presumed PDF methods
Presumed PDF's are the probability density function of the passive scalar concentration distribution. The concentration may be expressed by the mixture fraction, f [5]. Transport equations are solved for the time-mean of the square of the concentration fluctuations, (rs2, and from the calculated values of that quantity and assumptions regarding the shape of the instantaneous concentration-time profile, a hypothetical distribution of the instantaneous concentrations with time is derived [8]. Two versions of the presumed PDF will be described, namely the battlement and the beta PDF. 2.4.1 Battlement probability density function The instantaneous concentrations at a point is assumed to follow "battlement shaped time variation" with allowance for )max>f>)mi. when )m~x and )minare the maximum and minimum possible values of the instantaneous concentrations, respectively [8]. Based on this the time average of the mass fraction may be calculated by the following expression:
: a. r;, (z+)+ o-
rrA 0"_)
(13)
2.4.2 The beta probability density function The beta function written as a PDF has the following form [9]:
=
B(v, w)
Based on this the time average of the mass fraction may be calculated by the following expression: 1
0
This model was proposed and validated by Hannon et. al [3].
(14)
163
3. NUMERICAL CONFIGURATION The various reaction models will be tested against the experimental data of Pohorecki and Baldyga [2], where they found the reaction-zone length for 95 % conversion of species A (base), for a simple acid-base neutralisation expressed as: A + B ~ C. The tube arrangement that was simulated had an outer diameter of 0.04 m, and an inner tube with a diameter of 0.0052 m. Acid was fed into the outer tube which had a length of 2 m, and the base was fed into the inner tube which was 1 m long. The simulations were performed at three different Reynolds numbers; 13000, 20000 and 25000, which is the same as the ones used in the experiments by Pohorecki and Baldyga [2] and the numerical simulations of Hannon et. al [3]. The simulations were performed with the standard EDC-model, the EDC-model with an algebraic expression for the scalar dissipation time scale (EDC-SDT), and the EDC-model with the multiple-time-scale turbulent mixer model (EDC-MTS). Simulations were also performed with the use of two PDF models, the battlement PDF and the beta PDF. The Schmidt number was set to 800. The reaction rate constant, A, was set to 4 in the first cases and calibrated to 1 in the EDC-MTS case. Both the standard k-~-model and the RNG-k<:-model was used in the simulations. The simulations was performed in a 2D-axis-symmetric CFD in-house code, which solves the RANS equations including the k-e-model equations. 4. RESULTS AND DISCUSSION It can be seen from figure 1 that the standard EDC-model is not able to predict the reactionzone length found in the experiments of Pohorecki and Baldyga [2]. The short reaction-zone length may be caused by the fact that the EDC-model contains no information about mixing at the scales where viscous/diffusive effects are important. The EDC-SDT-model does not give a much better prediction of the reaction-zone than the standard EDC-model. The EDC-MTSmodel is able to predict the reaction-zone length. The reaction-zone length is approximately the same, indicating that the micromixing is taken account for by having a multiple-time scale.
Fig. 1 Comparing the results from the different EDC-models
Fig. 2 Comparing the results from the different PDF-models with MTS
164 As can be seen from figure 1, none of the EDC-models are able to predict the effect of increasing Reynolds number. This is the case even for the EDC-MTS model where the effect of mixing at scales where viscous/diffusive effects are important are taken into account. From figure 2 it can be seen that both the battlement-PDF and the beta-PDF predicts too short reaction zone length. The beta-PDF is the only model which is able to predict the right effect of increasing Reynolds number. The same type of model used by Hannon et. al [3] gave very satisfying results using the commercial CFD code, Fluent. 5. CONCLUSIONS The EDC-model has been examined and the results showed that the standard-EDC model is not able to predict the reaction-zone length. The modification of taking account of multiple time scale, the EDC-MTS-model gave results that were comparable with the experimental results. None of the EDC-models were able to predict the effect of increasing Reynolds number, not even the EDC-MTS model where the effect of mixing at scales where viscous/diffusive effects are important are taken into account. This may be caused by the fact that the EDC-model uses time-averaged concentrations instead of instantaneous concentrations. The PDF-models both predicts too short reaction zone length. The beta-PDF is the only model which is able to predict the right effect of increasing Reynolds number, this is probably caused by the fact that this model use instantaneous concentrations. Further work is needed where the EDC-MTS model and the beta-PDF model is tested against other geometries, conditions and chemical reactions. REFERENCES
1. J. Baldyga and R. Pohorecki, The Chemical Eng. Journal, No. 58 (1995) 183. 2. R. Pohorecki and J. Baldyga, Ind. Eng. Chem. Fund. Vol. 22 (1983) 392. 3. J. Hannon, S. Heam, L. Marshall and W. Zhou, Assessment of CFD approaches to predicting fast chemical reactions, Prepared for Presentation at the 1998 Annual Meeting, Miami Beach, FL, Nov. 15-20. 4. B.F. Magnussen and B.H. Hjertager, 16th Int. Symposium on Combustion, The Comb. Inst., (1976) 719. 5. J. Baldyga and J.R Bourne, Turbulent mixing and chemical reactions, John Wiley Sons, 1999. 6. R.O. Fox, Revue de l'Institut francais du petrole, Vol. 51, No. 2 (1996) 215. 7. J. Baldyga, Chem. Eng. Sci., No. 44 (1989) 1175. 8. M.A.S. Serag-E1-Din, The numerical prediction of the flow and combustion processes in a three-dimensional can combustor, PhD-Thesis, University of London, 1977. 9. H.K. Li and H.L. Toor, Ind. Eng. Chem. Fund., Vol.25 (1986) 719.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
165
Symbolic Discretization of Distributed Parameter Process Models on SelfAdaptive Moving Grids R. K6hler, J. Rieber, M. Zeitz Institut ftir Systemdynamik und Regelungstechnik, Universit~it Stuttgart, Pfaffenwaldring 9, D-70550 Stuttgart Germany First principle modeling of chemical processes with distributed parameters leads to partial differential equations (PDEs), which must be preprocessed for application of numerical simulation or optimization algorithms based on differential algebraic equations (DAEs). This contribution presents the symbolic preprocessing tool SYPPRoT of the simulation environment DIVA in order to apply the Method-of-Lines (MOL) approach to transform PDEs into DAEs. A self-adaptive moving grid method is implemented for MOL discretization by means of finitedifference and finite-volume schemes. The PDE preprocessing is illustrated by an adsorber model. 1 INTRODUCTION The partial differential equations (PDEs) and the related boundary conditions of distributed parameter models can be transformed into differential algebraic equations (DAEs) using the Method-of-Lines (MOL) approach [1]. For automatic generation of DAEs out of PDE models by means of MOL approach, the SYmbolic P_re PROcessing Tool SYPPRoT [2] has been designed and implemented for the simulation environment DIVA [3]. DIVA is a flowsheet simulation tool for chemical processes and contains efficient numerical routines for the resulting overall DAEs. The preprocessing tool SYPPRoT provides configurable conventional discretization methods like finite-difference- and finite-volume-schemes, which can be applied on 1dimensional spatial domains represented by uniform, non-uniform, and self-adaptive moving grids. The latter refers to grid adaptation, which is based on the equidistribution of an arclength monitor function [4, 5]. In this contribution, the functionality of SYPPROT concerning the MOL discretization on self-adaptive moving grids will be explained. The symbolic model representation and specification of discretization parameters are illustrated for a countercurrent adsorber. Simulation results obtained by DIVA show the increased performance of the selfadaptive moving grid method compared to classical discretization schemes on fixed grids. 2
ARCHITECTURE OF THE SYMBOLIC PREPROCESSING FOR THE SIMULATION ENVIRONMENT DIVA
The DIVA architecture (Fig. 1) comprises three layers each of which is accessible by the user for editing and debugging. The bottom layer contains the DIVA kernel with a library of generic process unit models and numerical methods. A process unit model consists of several FORTRAN subroutines collected in the model library and a data file for model parameters and initial values. The process unit model as well as the data file are automatically generated by the code generator (CG), which builds the next layer of the architecture. The CG requires the complete model information written in a CG input file. The CG input file and the numerical subroutines of
166
Fig. 1: DIVA architecture with the Symbolic Preprocessing Tool SYPPRoT, the Code Generator, the DIVA Model Library, and the Numerical DAE-Methods [3, 6].
DIVA use the linear implicit DAE model representation
B ( x , u, p, t) . -dx ~ = f(x,u,p,t)
t > to,
x(to) - Xo,
y = H.x
(1)
with a differential index of one. B denotes the usually singular left-hand-side matrix and f the right-hand-side vector. The symbols x(t), u(t), y(t), p, and x0 are vectors of the states, inputs, outputs, parameters, and initial values, respectively. The output vector y is obtained via matrix H as a subset of the state vector x. The CG input file is either written by the user or is the result of the symbolic preprocessing of a PDE model, which is the top layer in the DIVA architecture. The preprocessing steps concern MOL discretization of PDEs into DAEs, index analysis and reduction of DAEs, as well as DAE transformation into the linear implicit form (1). The preprocessing functions are implemented by means of the computer-algebra-system MATHEMATICA. For definition of a mixed PDE and DAE model, the user writes an input file using the MnTHEMnTICn-Data-Structure (MDS). A MDS input file consists of different sections to define parameters, variables, and equations as well as discretization methods and grids for 1dimensional spatial domains. First step of preprocessing concerns MOL discretization of PDEs into DAEs by means of available finite-difference and finite-volume schemes applied on fixed or self-adaptive moving grids. In a second step, the resulting DAEs are transformed into form (1) required by DIVA. Finally, the DAE-Writer of SYPPROT generates the CG input file. In the following chapters, the discretization capabilities of SYPPROT using self-adaptive moving grids will be presented and illustrated by an adsorber model. 3 M O L DISCRETIZATION OF PDE Typical distributed parameter models of chemical processes are coupled PDEs up to second order in space z and of first order in time t. The related boundary conditions (BC) depend on input vectors Vo,~(t), and the initial conditions (IC) are specified by profiles x0(z):
AOX Ot
_
o 9(z, to)
=
c02X O F ( x , u , p , z , t ) + S(x, u, p, z, t) ~ + Oz C Ox o,~-5-;z+ Fo,~(x, Vo,~,p, t)
~o(z)
t > to, z e (o, l)
(2)
t > to, z e {0, l}
(3)
z E [0, l]
(4)
167
The matrices A(x, u, p, z, t) and C(x, u, p, z, t) as well as the source vector S(x, u, p, z, t) may depend on state x(z, t) E N", inputs u(z, t), parameters p, and on z and t. The flux function F(x, u, p, z, t) = w(x, u, p, z, t) 9x(z, t) represents the convective transport with flow velocity w(x, u, p, z, t). This general form of PDEs and BCs of parabolic as well as of hyperbolic type is handled by the symbolic preprocessing tool SYPPROT. It is assumed that the model equations are well posed and have a unique solution. For discretization of PDE models (2)-(4) by the MOL approach, the continuous coordinate z is replaced by discrete grid points z~ for static grids or zk(t), k = l(1)kma~ for moving grids. In Fig. 2, this is illustrated for a static grid. The grid partition also includes segmentation by means of continuous control volumes (CVs). In this case, grid points represent the cell center points of CVs. On the discretized domain of z, the partial derivatives Oix(z, t)/Oz i, i = 1, 2 are approximated by functions of the state variables x~(t) at grid points z~. For these approximations, SYPPRoT provides configurable finite-difference and finite-volume schemes. The result of the MOL discretization are ODEs at the inner grid points z~, k = 2(1)kma~ - 1, and algebraic equations (AEs) for finite-difference schemes or ODEs at the boundary grid points Zk, k = 1, kin,= for finite-volume schemes. Furthermore, the initial profiles x(z, to) are transformed into xk(to), k = 1(1)kraal, which are consistent with the AEs ifBCs (3) and ICs (4) are formulated consistently.
Self-adaptive Moving Grid Method The considered moving grid method uses the Lagrangian approach of moving the grid nodes continuously along with the solution [4, 5]. The grid consists of moving grid nodes zk(t), k = 2(1)kma~ - 1 for the inner spatial domain and fixed grid nodes zk(t), k = 1, kr, a~ at the boundaries. This approach reduces rapid changes of steep front solutions at fixed grid points leading to small local discretization error and small time steps during numerical simulation. Thus, the effort of solving additional moving grid equations is partly compensated. Furthermore, the number of grid nodes can be reduced considerably compared to equidistant grids which results in less computational effort. Using the Lagrangian form of the partial time derivative at a grid node Ox/Otlzk(t) = dxk/dt - (Ox/Oz .dz/dt)[zk(t ) , the semi-discrete form of PDE (2) is written as
A
--~-
(ox z) -~z-d~
02 x
+ ~k(t)
OF(z, u, p, z, t) + S(x~, uk,p, z~, t) Oz Zk(t)
t > O. (5)
168 No transformation is necessary for the BC (3), because the boundary grid points are fixed. Additional equations are required for moving the grid nodes zk(t), k = 2 ( 1 ) k , n a , - 1. The moving grid equations are based on the spatial equidistribution property [7]
f zkt-1 M ( z , ~'Zk
fzl :k M ( z , --1
z)dz -
k = 2(1)kma~ - 1
x)dz
(6)
with the arclength monitor function
M (z, x) =
1+ n
" Oz
(7)
"
i=1
considering the spatial gradients of the state variables x E ]~n [4]. The parameters L and Xi, i - 1(1)n are used for scaling of the spatial domain and the states. Furthermore, it is possible to restrict the monitor function (7) to a subset of state variables x ( z , t). Due to numerical stability reasons, the smoothing procedures following Dorfi and Dmry [4, 5] are considered for implementation within the symbolic preprocessing tool. The general form of the moving grid equations resulting from (6) reads "rE(x, z, ,~) dz = g(x, z, ~)
(8)
The matrix E and the function vector g depend on smoothing techniques determined by the temporal and spatial regularization parameters T and ,~, respectively. The parameter T mainly improves the stability of the numerical solution and results in a delay of grid adjustment, whereas ,~ is used to control the grid expansion such that ,'; ,~ + 1
< Zk -- Zk-1 < '~ + 1 - Zk+l -- Zk --
k -- 2(1)km~ - 1.
(9)
For many models, the smoothing parameters can be chosen according to a default setting ,~ -- 1 and T = 0. But certain model equations may cause difficulties like non-physical oscillations of the solution or displacement of the front location. The discretization of integral (6) and function (7) depends on the chosen discretization of the spatial domain according to finite-differences or finite-volumes as well as on stability criteria [7]. 4
SYMBOLIC DISCRETIZATION OF A COUNTERCURRENT ADSORBER
For the illustration of the application of SYPPRoT, a countercurrent adsorber model is considered. This model gives a simplified and scaled description of the component mole fractions x ( z , t) and y ( z , t) of a binary mixture in the adsorber: Ox O---t =
(gx --O--z - - J ( x ' Y )
Oy .-Ot x(O,t)
=
p Oz xi,(t),
x(z,O)
-
x0(z),
=
10y
+j(x,y)
y(1,t) = y i , ( t ) y ( z , O ) = yo(z)
t > 0,
z E (0,1)
(10)
t>0,
zE(0,1)
(l l)
t > 0
(12) z E [0,1]
(13)
169 The component with mole fraction y(z, t) is considered inert. The parameters v and # denote ratios of molar mass and molar convective flows, and j(x, y) represents the nonlinear mass exchange function between the phases. In the following, the encoding of parts of the adsorber model by means of the MDS is shown. The definition of state variable x(z, t) contains minimum and maximum values, which are required by the numerical routines of DIVA. Moreover, these values are used to determine the parameters Xi in (7). The D o m a i n statement assigns a spatial domain to the state variable. Here, x(z, t) is declared on the domain named MovGrid [ z ] of the spatial coordinate z: Scalar[
x[z,t],
Comment MinValue NumericalErrorValue
-> "mole fraction", Domain -> 0.0, MaxValue -> 0.0001, InitialValue
-> -> ->
"MovGrid[z]", 1.0, 0.0 ]
The PDE (10) and the corresponding BC in (12) are defined in a common MDS object S c a l a r [ . . . ] for single equations. The D i s c r e t i z a t i o n statement assigns a user parametrized finite volume discretization scheme FVupwind to the PDE and the BC: Scalar[
D[x[z,t],t] ==-D[x[z,t],z]-j[z,t], LowerBound -> x [ z , t ] - x i n = = O , D i s c r e t i z a t i o n -> "FVupwind", Name -> "PDE(1)", Comment -> " m a t e r i a l b a l a n c e
for x(z,t)"
]
The grid definition for the spatial domain M o v G r i d [ z ] shows statements for initial grid point distribution ( C o m p u t a t i o n ) and specification of moving grid parameters n ( S p a c e S m o o t h i n g ) and "i- (TimeSmoothing). Furthermore, only x(z, t) is selected to be considered within the monitor function ( M o n i t o r S t a t e s ) : Grid[
MovGrid[z], Computation Granularity MovingGrid
-> F u n c t i o n [ { k } , (k-l) / (kmax-l) ], -> "kmax", -> { S p a c e S m o o t h i n g -> 1.0, TimeSmoothing -> 0.i, MonitorStates -> {x[z,t] } } ]
Calling the automatic discretization of SYPPROT transforms the MDS input file into a CG input file by performing MOL discretization, DAE transformation and translation of MDS into CGlanguage, shown in Fig. 1. These three preprocessing steps are performed by executing a few SYPPRoT commands within a usual MATHEMATICA session, see [3] for a more details. 5
B E N C H M A R K RESULTS In Fig. 3, simulation results for the countercurrent adsorber model (10-13) are depicted. Characteristic for this process is a steep front moving from z = 0 to z = 1. This behavior is reflected in the trajectories of moving grid nodes on the fight in Fig. 3. Nodes crowd together at the position of the front to achieve a small spatial discretization error. For evaluation of the numerical performance, benchmark simulations on equidistant grids and moving grids are compared. The solutions obtained on an equidistant grid of 1000 nodes and on a moving grid of 80 nodes with parameters ~ = 1 and ~- = 0.1 show no significant differences. The improved performance is obvious, regarding the CPU-times on a SUN Ultra 60 machine of 2.0s and 10.0s for the moving grid and equidistant grid simulations, respectively.
170
Fig. 3: Simulation results of a countercurrent adsorber on a self-adaptive moving grid with kmax = 80, t~ = 1, r = 0.1, and considering only x(z, t) within the monitor function (7). Other benchmark models like a convective transport problem, a flame propagation model, and a circulation-loop reactor also confirm the increased perfomance for moving grid nodes instead of equidistant grids. Comparisons with equidistant grids show that only 2-20% of the grid nodes are required to compute results of the same accuracy in 15-50% of CPU-time. 6 CONCLUSIONS In the symbolic prepocessing tool SYPPRoT of the simulation environment DIVA, the automatic generation of DAEs for distributed parameter models has been extended by a moving grid technique. The grid movement is controlled by equidistribution of an arc-length monitor and is regularized by two smoothing parameters. Parametrization and change between provided grids and discretization schemes are very simple and allow a rapid test of various MOL approaches without model reimplementation. Benchmark simulations show the increased performance with steep moving spatial fronts. REFERENCES [1 ] W.E. Schiesser. The numerical method of lines: Integration of PDEs. San-Diego, 1991. [2] R. Ktihler, et al. Symbolic Preprocessing for Simulation of PDE Models of Chemical Processes. Special Issue Method of Lines in Journal of Math. and Comp. in Sim. (accepted). [3] R. Krhler, et al. Method of lines within the simulation environment DIVA for chemical processes. In A. Vande Wouwer, P. Saucez, W. Schiesser, editors, Adaptive Method of Lines. CRC Press, 2001. [4] E.A. Dorfi and L. O'C. Drury. Simple adaptive grids for 1-d initial value problems. J. Comp. Phys., 69:175-195, 1987. [5] L.G. Verwer, et al. A moving grid method for one-dimensional PDEs based on the method of lines. In J.E. Flaherty, P.J. Paslow, M.S. Shepard, and J.D. Vasilakis, editors, Adaptive Methods for Partial Differential Equations, 160-175. SIAM, Philadelphia, 1989. [6] A. Krrner, et al. DIVA - An open architecture for dynamic simulation. In R. Eckermann, editor, Computer Application in the Chemical Industry, 485-492. VCH, Weinheim, 1989. [7] S. Li, L. Petzold, and R. Yuhe. Stability of moving mesh systems of partial differential equations. SlAM J. Sci. Comput., 20(2):719-738, 1998.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
171
Computational tools for nonlinear dynamical and bifurcation analysis of chemical engineering problems M. Kohout, I. Schreiber and M. Kubf6ek Department of Chemical Engineering, Department of Mathematics, Center for Nonlinear Dynamics of Chemical and Biological Systems Prague Institute of Chemical Technology, Technick~i 5, 166 28 Prague 6, Czech Republic We present a program package CONT for modelling and analysis of nonlinear problems in general, and chemical engineering problems, such as chemical reactors or adsorption columns, in particular. Model should be in the form of ordinary differential equations, spatially distributed systems described by partial differential equations may also be treated upon transformation to ODEs by a built-in discretization. A basic method used in the program is continuation of a steady state, periodic solution or another boundary value problem with respect to a parameter which provides a solution diagram. Simultaneous stability analysis helps to identify bifurcation points in such diagrams and these may be continued with respect to another parameter to obtain a bifurcation diagram. Additionally, the program performs a direct integration of ODEs, calculates Poincare orbits and Lyapunov exponents, which are useful for analysis of complex dynamics. 1. CONTINUATION, BIFURCATIONS AND NONLINEAR DYNAMICS Dynamical systems are commonly represented by differential equations defined on a finite or infinite dimensional state space. Let us consider the finite dimensional case, leading to a set of n ordinary differential equations (ODEs)
dx d t = f(x;[t, ~),
x E 9~n,
(I)
where x is the vector of dependent variables, t is time and tx, 13are scalar parameters. Nonlinear dynamical systems described by Eq. (1) can be explored by a large number of methods [1--4] ranging from finding steady states and determining their local stability to an analysis of complex nonperiodic solutions involving calculation of a fractal dimension of a corresponding attractor (geometric complexity), Lyapunov exponents (temporal instability) and various other measures of complexity. Of particular interest for engineering purposes are special solutions to Eq. (1) subject to various constraints, such as boundary conditions, and variations of these solutions with parameter(s). The relevant method here is that of numerical continuation with respect to a parameter [1,4-7]. One-parameter families of solutions may contain bifurcation points, where phase portrait undergoes a qualitative change; such points can be localized and traced by continuation when another parameter is added. Although this procedure can be repeated again, providing
172 thus bifurcations of codimension higher than two, in practice it is often sufficient to define two distinct parameters, say o~ and ~, and construct either one-parameter solution d i a g r a m s or two-parameter bifurcation diagrams. To accomplish some of the tasks outlined we created a software computational tool CONT whose functionality is described below and illustrated by simple problems from reaction kinetics and adsorption. The program CONT can perform the following tasks: 1. one-parameter continuation of steady states or periodic orbits and determination of local stability 2. two-parameter continuation of local bifurcation points for steady states or periodic orbits (such as a Hopf bifurcation, limit and branch points, period doubling and torus bifurcation) 3. one- or two-parameter continuation of solutions subject to general nonlinear boundary conditions (other than periodic ones) 4. two-parameter continuation of homoclinic and heteroclinic orbits (i.e. a special boundary value problem) 5. direct numerical integration of ODEs providing general orbits or Poincare-discretized orbits including automatic search for steady states/periodic orbits and repeated incrementation of parameter(s) 6. Lyapunov exponents for general orbits including repeated incrementation of parameter(s) All the functions (when they are relevant) can also be applied to periodically forced ODEs (via a smooth function or pulses) and discrete iterated maps. Also, a built-in discretization scheme for partial differential equations on an interval makes it possible to apply the analysis to PDEs, provided that sufficient computational power is available. In our previous work [8-10] and below we provide some examples clarifying the functions of CONT and showing how the results can be presented. Comparison with similar software is made. 2. E X A M P L E 1 - AN AUTOCATALYTIC B I O C H E M I C A L R E A C T I O N Redox bioconversion of thiol groups (2 SH ~ S-S) in proteins within cytosol may be attributed to a variety of biological functions such as biological clocks, cell division and carcinogenesis [ 11 ]. A simple biochemical mechanism describing the kinetics in terms of dimesionless concentrations x (of the reduced "SH" proteins) and y (of the oxidized "S-S" proteins) is as follows: dx v0 + x ~t d t = f ( x , y ) = ct 1 +-----~ - x - x y ,
dy d t -- g(x,y) - ~ x + x y - S y
(2)
where tx, 8, v0 > 0 and 13,T > 1 are parameters. We fix v0 = 0.01, [3 = 1.5 and ), = 3 and use tx and 8 as bifurcation parameters. The variable x represents the autocatalyst and tx is the rate coefficient of the autocatalytic step while y represents the inhibitor and 8 is the rate coefficient of the degradation of the inhibitor.
2.1. Periodic pulsed forcing of the homogeneous reaction system Assuming for simplicity that any transport within the cytosol is effectively fast compared to chemical processes Eqs. (2) describe a spatially homogeneous system equivalent to a CSTR. From outside the cellular environment the SH ~ S-S dynamics can be controlled by periodic signalling which may cause sudden pulse-like increase/decrease of the concentration of the
173
autocatalyst or inhibitor. For instance, a signalling agent can rapidly consume the autocatalyst. This situation would correspond to adding a periodic delta function x(e -a - 1) ~, 8(t - kT) to f ( x , y ) in Eqs. (2), where A > 0 defines a forcing amplitude and T is the forcing period. Choosing tx = 25 and 8 - - 1.6, Eqs. (2) possess a unique steady state which is, however, excitable with respect to a pulsed removal of the autocatalyst. Applying repeatedly the pulses we obtain a temporal dynamical pattern of alternating small- and large-amplitude responses. These firing patterns may be periodic or not, depending on the choice of A and T. For periodic responses with period qT, q -- 1,2, 3, 4, CONT-generated branches denoted Pq for fixed A = 0.5 and varying T including stability changes and associated bifurcation points are plotted in Fig. 1a. Simultaneously, in Fig. lb the maximum Lyapunov exponent within the same range of T is plotted, indicating chaotic dynamics of responses (positive values) fitting the interval of unstable periodic orbits in Fig. 1a. Finally, taking the bifurcation points on the period one branch (i.e. q = 1) from Fig. l a and performing continuations in A - T plane generates the amplitude-period bifurcation diagram in Fig. 2, delineating the regions of multiple period one orbits (bounded by limit point curves), a central region of no stable period one orbits (bounded partly by period doubling curves and partly by limit point curves) and regions of stable orbits (bounded in part by torus bifurcation curves, and the above two other kinds of curves). As suggested by Figs. 1a,b, the central region is filled with periodic orbits with q larger than one (these can be further studied by continuation) mingled with chaotic orbits.
~
6
~
.J,# P,'~/;:" ,/
m
~,
4
2
p~/.,;,.~..,:
..--'" -"
00
,
1
2
.--,-----
o<
.... ..... . Z
,
,
3
,
4
5
forcing period T
t~
2
"t~
0 ~,- -1 -2 0
Fig.
1
2 3 forcing period T
4
5
1. Periodicaly forced homogeneous system; a
- solution diagam for periodic orbits; full/dashed line
0
0
Fig. 2.
2 forcing period T
4
Bifurcation diagram in T - A plane: full line-
limit point, dashed line - period doubling, dotted line -
- stable/unstable orbit, full square - period doubling,
torus bifurcation, full square - degenerate period dou-
empty square - torus bifurcation; b - plot of maximum
bling, empty s q u a r e - Bogdanov-Takens point.
Lyapunov exponent X1.
174
2.2. W a v e s in the r e a c t i o n - d i f f u s i o n s y s t e m
In fact, diffusion of both the autocatalyst and the inhibitor in cytosol is likely to occur on time scales comparable to those for reaction, thus resulting in reaction-diffusion patterns such as waves. Assuming 1D spatial extension the dynamical equations are: clX
c)2X
=
Dx-~z 2 + f ( x , y ) ,
(3)
~gy igt =
~2y D y ~ + g(x,y).
(4)
at
Chemical waves with a constant velocity u on an unbounded interval can be studied upon coordinate transformation ~ - z - ut bringing the partial differential system (3),(4) into ODEs:
dx
dy
dv
d---~ = V,
d---~ -'- W,
d---~ ~-"
uv 4- f (x, y) Ox
dw '
uw 4- g(x, y)
d---~ - - --
Or
whose Z-periodic solutions correspond to periodic wave trains of the same wavelength, and homoclinic/heteroclinic solutions correspond to solitary pulse/front waves. We choose o~ = 10 and use u and ~5 as bifurcation parameters. Since a homoclinic orbit is a limiting case of a family of periodic orbits the loci of homoclinic orbits (i.e., solitary pulse waves) delimit a region of periodic waves, the other delimiting lines being the loci of Hopf bifurcation and limit points on periodic orbits, see Fig. 3. Typical profiles of the waves are displayed in Fig. 4. The calculations show that stable waves with a definite velocity exist within an interval of ~5E [0.75,1.57]. Below the system provides only spatially homogeneous solutions, while above a complex spatiotemporal dynamics occurs.
15 .~, "5 10 o >
2.5
j=. i
1.5
..t
~s,
0 0.75
0.5
~f
stable per.waves...~..~....--~_...:%................. ,
|
1
,
_..I . . . . . . . . .
1.25 8
,
|
1.5
C
1
/
5
b
2
9
i o.
,
1.75
Fig. 3. Bifurcation diagram in 5--u plane; full line - pulse waves, dashed line - Hopf bifurcation points, dotted line - limit points on periodic orbits; Dx = Dr -1.
0
\
|
0
10
30
20
40
Z
Fig. 4. Selected profiles of pulse waves from Fig. 3" a)5=1.58096, u=2.51892, b)~5=1.2, u=2.35954, c)~5=0.85118, u=0.8, full/dashed line- stable/unstable wave.
This phenomenon can also be examined with CONT by using a built-in spatial discretization on a finite interval with, say, no-flux boundary conditions. A direct solution of Eqs. (3),(4) using 400 mesh points on an interval of length L = 200 for ~5= 1.58 provides a chaotic wave pattern as shown in Fig. 5. To get a more condensed information one can generate a Poincare map, for example by taking spatial profiles each time x(t,z = L) passes through a maximum. Fig. 6
175 represents the Poincare map revealing a fractal structure which is likely of a high dimension. One could proceed further in examining complexity of the pattern by calculating Lyapunov exponents.
Fig. 5. Space-time plot of a chaotic wave pattern.
Fig. 6. Poincare plot corresponding to the chaotic pattern in Fig. 5.
3. E X A M P L E 2 - F R O N T WAVES IN A D S O R P T I O N S Y S T E M
We examine adsorption of anthracene from cyklohexane onto activated alumina in a fixed bed. This system can be described by local mass balances in both phases [ 12]
~C . bt
.
~)q _ i)t -
kmap(C - C*) ~)c ~)2c . v-- + Ee bZ Ea bZ2 , kmap(c-c*) q pp(1 - ee) ' c* - a - bq' .
(5)
(6)
where c is the solute concentration in liquid, q is the adsorbate concentration in the solid, ee is the bed porosity, km is the mass transfer coefficient, v is the interstitial velocity and Ed is the axial dispersion coefficient. For equilibrium we use the Langmuir isotherm. We choose fixed values [12] ee = 0.4, kmap = 0.036 s -1, Pp = 1.47 kgdm -3, a = 22.0 m3kg -1, b = 375.0 m3mo1-1, Ecl = 0.1 cmZs -1. Upon coordinate transformation ~ = z - ut applied to (5), (6) we obtain a system of three ODEs and the problem of finding the travelling shock wave (adsorption front) is converted to finding a heteroclinic orbit. Results of the continuation of heteroclinic orbits are shown in Figs. 7 and 8. We note that the curve in Fig. 7 has a minimum due to the combination of axial dispersion and mass transfer effects implying that an optimal input flow corresponding to a minimal width of adsorption front exists. 4. C O N C L U S I O N S We introduced the software tool CONT for continuation and bifurcation analysis of nonlinear dynamical systems. The use of such a tool allows for a detailed insight into complex dynamics of the systems studied. The continuation part of our software solves boundary value problems
176
E 500 .~. o
400
8
300
-.~
,..-,
E
8
f
i..
8 200
"5 100 ~9
0
0
f
J
1
f J
f
0.01
J t...
0.005
~.
8
2
3
4
5
interstitial velocity [cm/s]
Fig. 7. Width of adsorption zone vs interstitial velocity v obtained by the continuationof heteroclinic orbits.
-~
. of motion
o
0
1O0
200
300
400
500
length [cm]
Fig. 8. Profiles of selected front waves in adsorption column corresponding to the curve in Fig. 7.
by multiple shooting method (with an improved numerical stability by using an adaptive mesh) which enables us to handle discontinuous jumps occurring in pulsed systems. These problems are hard to solve by methods relying on some form of discretization such as orthogonal collocation [6]. Another advantage may be that also dynamical analysis is conveniently built in. Acknowledgments: This work has been supported by the project No. VS96073 and fund MSM 223400007 of the Czech Ministry of Education and grants No. 201/98/0220 and 203/98/1304 of the Czech Grant Agency. REFERENCES
1. M. Kubi6ek, M. Marek, M. Computational Methods in Bifurcation Theory and Dissipative Structures, Springer Verlag, New York, 1983. 2. M. Marek, I. Schreiber, Chaotic Behaviour of Deterministic Dissipative Systems, Cambridge University Press, 1995. 3. P. Gray, S.K. Scott, Chemical Oscillations and Instabilities, Clarendon Press, Oxford, 1990. 4. Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer Verlag, New York, 1995. 5. R. Seydel, Practical Bifurcation and Stability Analysis, Springer Verlag, New York, 1994. 6. E.J. Doedel, AUTO: Software for continuation and bifurcation problems in ordinary differential equations, Applied Mathematics, Pasadena, 1986. 7. E.L. Allgower, K. Georg, Numerical Continuation Methods, Springer Verlag, New York, 1990. 8. M. Kubi6ek, I. Schreiber, ZAMM 77, Suppl. 2 (1997) 603. 9. M. Kubf6ek, I. Schreiber, ZAMM 78, Suppl. 3 (1998) 981. 10. I. Schreiber, M. Kohout, M. Kubf6ek, In Scientific Computing in Chemical Engineering II, F. Keil et al. eds., Springer, Berlin 1999, Vol. 2, 200. 11. E.E. Selkov, Biofizika 15 (1970) 1065. 12. P.C. Wankat, Rate-controlled separations. Elsevier, New York, 1990.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
177
Multi-scale modelling of growing polymer particles in heterogeneous catalytic reactors J. Kosek*, F. Stepanek, A. Novak, Z. Grof and M. Marek Department of Chemical Engineering, Prague Institute of Chemical Technology, Technicka 5, 166 28 Praha 6, Czech Republic
The problem of polyolefine particle morphogenesis in a heterogeneous gas or slurry catalytic reactor is considered. A conceptual modelling approach is proposed, allowing for the multiple time- and length-scales on which polymerisation processes typically occur. Models of polymer growth and flow in the pores of a catalyst support, catalyst particle fragmentation, and the evolution of a polymer macro-particle are described as well as physical characteristics of key objects forming the particles. 1. INTRODUCTION In 1999, the production of polyethylene (HDPE, LLDPE) and polypropylene (PP) in the world amounted to 58 Mt, representing a business of approximately US$ 55 bn. The introduction of new, highly active supported metallocene catalysts into polyolefine production has opened up new opportunities for the manufacture of polymers with tailored properties, but it has also brought new challenges in terms of process control, processing of product and an increased need to understand the relationship between catalyst support structure, process conditions and final product properties - the polymer particle size, shape and internal morphology being particularly important parameters. The structure-property relationship is rather complex, as phenomena occurring during polymerisation at various time- and length-scales jointly determine the final polymer particle size and morphology [1]. The length-scales involved in a typical heterogeneously catalysed fluid-bed polymerisation reactor are shown schematically in Figure 1. For example, the polymerisation kinetics at the molecular level determines the polymer chain-length distribution, tacticity, branching and composition, which determines the visco-elastic properties of a polymer melt, its melting temperature, etc. The properties of molten and semicrystalline polymer together with the architecture of a catalyst support then determine the catalyst fragmentation mechanism which in turn affects the structure of a growing polymer macro-particle, thus its heat- and mass-transfer characteristics. These - via the material and energy balance of the particle - reflect back on the polymerisation kinetics [2,3,4]. As a consequence of such coupling, a multi-scale model has to be adopted if the kinetics of polymer particle growth and morphogenesis is to be quantitatively modelled. Currently, no predictive model for particle structure development is available. Our ambition is to develop one, and in the present contribution we describe its main features and implementation. *Corresponding author. Phone: +420 2 2435 3296; Fax: +420 2 311 7335; E-mail:
[email protected]
178
Figure 1: The length-scales involved in a fluid-bed heterogeneous polymerisation reactor. 2. METHODOLOGY, IMPLEMENTATION AND RESULTS
The basic idea behind our approach is to identify real-world objects of interest, such as a catalyst carrier, polymer micro-grain containing a fragment of a catalyst support, or a polymer macro-particle, as well as physico-chemical processes acting upon these objects and transforming them. Data structures in the software then correspond to these real-world objects, and individual software modules implement more-or-less rigorous models of the realworld processes. The objects and processes that we currently consider are shown in Figure 2, their description follows below. A simulation of the particle growth and morphology evolution starts by the selection of an initial object (e.g., a catalyst support of certain porosity, pore-space correlation length, catalyst site distribution, etc.), the specification of external conditions to which the initial object will be sequentially exposed (e.g., temperature, pressure, and species concentrationsall can be a function of time), and then letting the object evolve in time until a stop condition is encountered, e.g., the residence time of the particle in the reactor or a model constraint such as particle overheating. /
2D or 3D images of catal, supports
Polymerisation conditions (T, P, Cspecies)
/ 1
| 1
SEM/TEM images of polym, particles
" ~ ~ f r agments Catalystwith ~ Pore filling, Multi-grai n ~ ( Polymer Fragmentation7 ~ a c h e d polymer) growth / ~,,macro-particle/)
Figure 2: Model representations of real-world objects, and the processes transforming them.
179 2.1.
Objects
2.1.1. Catalyst support particle The structure of a catalyst particle can either be obtained from SEM or TEM images [5] by the method of reconstructed porous media [6] or directly from 3D images (e.g., X-ray tomography). It is encoded in the form of the so-called phase function f: 9~3 --~ {0,1 }, defined as fix) = 1 if point x belongs to the pore space and fix) = 0 otherwise. An example of such a binary-encoded porous particle is shown in Fig. 3a. An alternative way of encoding the porous particle is the so-called skeleton, obtained from the phase function by a technique called conditional thinning [6] which reduces the solid phase into a network of branches in which every point is assigned a value corresponding to the number of material layers that had to be removed in order to reduce the solid volume into a thin filament. These values thus measure the local strength of the skeleton.
Figure 3: (a) A reconstructed porous catalyst particle. (b) Cross-sections of a porous medium being filled by a growing polymer layer in the limiting case of a fast reaction and slow polymer flow (simulation results).
2.1.2. Micro-grains A micro-grain is a fragment of the original catalyst particle surrounded by a polymer. Micro-grains arise from the fragmentation of catalyst support particles [1,5] as a result of tensile and swelling stresses induced by a growing layer of polymer that fills the particle pores. In the current implementation, each micro-grain is assigned an activity, proportional to the surface of the fragment containing an active catalyst (i.e., forming a pore wall before fragmentation has occurred), and an equivalent radius, defined as a radius of a sphere containing the same amount of polymer as the micro-grain. The physico-chemical properties of the polymer forming the micro-grain are a function of temperature and thus depend on the position of the micro-grain in its parent macro-particle, given by a positional vector in spherical coordinates. The rate of micro-grain growth is proportional to its activity, temperature, and monomer concentration at its center (in the current implementation, stationary monomer concentration profile in the micro-grain is assumed, but a fully dynamic case with micro-grain diffusion can also be considered). The monomer concentration at the surface of the micro-grain again depends on the position within the macro-particle.
180
2.1.3. Polymer macro-particle A macro-particle is an agglomerate of the above described micro-grains, specified by their count and position vectors. The representation of the particle as a heterogeneous medium carries full information about its morphology; however, it is not always necessary or computationally feasible to solve the material and enthalpy balances at the macro-particle scale in a full 3D representation of the particle. An "effective medium" twin of the macroparticle can thus be simultaneously maintained, which simply carries information about the radial profiles of effective diffusivity, permeability, and thermal conductivity of the particle. These quantities can readily be calculated when a "unit cell" (i.e., a cubic volume element) is extracted from the 3D macro-particle at a given radial position and passed into existing subroutines enabling the calculation of these effective quantities for an arbitrary porous medium [6]. The effective-medium model then supplies back the temperature, pressure, and species concentration profiles within the macro-particle, which are needed for the modelling of micro-grain growth.
Figure 4: Catalyst particle fragmentation mechanisms (left) and polymer growth in a single pore (right - monomer concentration is indicated by the colour levels); (a) the limiting case of a fast monomer diffusion and a slow reaction, leading to gradual pore filling and the "bisection" fragmentation mechanism; (b) fast reaction leading to a "bottle-necking" effect during a single-pore filling and a subsequent "shrinking core" fragmentation. 2.2. Processes
2.2. I. Polymer growth and pore filling The initial stage of polymerisation is the growth of polymer layer in the pore space of a catalyst support particle and the re-distribution of the polymer inside the pore space. A thorough description of the polymerisation kinetics is given in [7]. As far as polymer flow and monomer diffusion in the fluid and polymer phase are concerned, the present implementation assumes a simple Fickian diffusion in both phases (absorption equilibrium is considered locally at the fluid-polymer interface), and treats the polymer phase as a swelled, viscous fluid. An example of simulation results from the filling of a single cylindrical pore are shown in Figs. 4a,b; the growth of polymer in the pore space of a porous medium with somewhat more complex geometry is shown in Fig. 3b. The redistribution of the polymer melt in the pore space of the catalyst is a problem of free interface propagation, addressed recently in [8].
181
2.2.2. Fragmentation of catalyst support As the polymer growth continues, the catalyst particle eventually breaks up into smaller fragments as the consequence of stresses induced by the polymer expanding in the confined pore space. Two limiting fragmentation mechanisms are shown schematically in Figs. 4a,b: the so-called bi-section mechanism which occurs when the pore space is first homogeneously filled by the polymer, and the so-called shrinking core mechanism, arising in situations of fast polymerisation and slow polymer flow. Three possible methods can be used for the simulation of the fragmentation process, briefly described here in the order of decreasing complexity. (i) Knowing the distribution of stresses on the pore walls and the geometry of the solid phase, it is in principle possible to use a commercial FEM code for finding the fracture zones. However, this approach is not feasible given the size of a typical particle. (//) The second method is based on dividing the porous particle into convex fragments so as to minimise the total newly created surface. This represents a certain thermodynamic limit and a method based on Delaunay triangulation generalised into a 3D space with non-Euclidean metrics can be used for this purpose. (iii) Finally, one can use the skeleton representation of the porous solid by simply disjoining the skeleton at its weakest points and then re-assigning the solid phase removed during conditional thinning to the discrete branches. This method so far appears to be computationally the most feasible.
Figure 5: (a) Force-distance relationships F(r) for a binary interaction of "hard" (highly crystalline) and "soft" (less crystalline, partially molten) micro-grains. (b) Visualisation of a simulated multi-grain growth of a polymer macro-particle composed of"soft" micro-grains.
182
2.2.3. Multi-grain growth of a macro-particle Once the fragmentation phase is complete, a coarse-graining step is performed in order to render further simulation computationally feasible: the catalyst fragments are considered as zero-measure points and the amount of polymer attached to each of them determines the effective radius of the newly created micro-grains. The micro-grain positions are then updated according to the Newton's law; their mutual interactions depend on the polymer properties and are shown for a pair of crystalline and partially molten particles in Figure 5a. In every time step, the radii of micro-grains are updated as polymer is produced while material and enthalpy balances on the macro-particle scale are solved simultaneously. An example of a growing macro-particle is shown in Figure 5b. 3. CONCLUSIONS AND PROSPECTS The methodology and software for the simulation of polymer particle growth in heterogeneous catalytic reactors, as described in the present work, represent one of the first attempts to address the problem of multi-scale modelling of polymer particle morphogenesis as a whole. A computationally feasible way of spanning the gaps existing so far between models focusing on particular aspects of particle growth is proposed, allowing for phenomena occurring at several length-scales to be linked. The modular concept of our approach allows for further independent refinement of individual sub-models, while the overall structure of the information propagation in the direction: pore filling - catalyst fragmentation - macro-particle growth remains unchanged. The following two areas are currently the bottlenecks: (1) The visco-elastic interactions between micro-particles in the multi-grain model. Only binary interactions are considered so far, and only the normal component of the force vector is taken into account (also the tangential and angular components of the inter-particle interaction should be accounted for). The interaction h i s t o r y - in the form of dilatational hysteresis - also plays a role in real systems, while in the current model we consider the force-distance relationship to be a state quantity. (2) More insights into the mechanisms involved in catalyst fragmentation are still needed. The role of thermal effects in catalyst carrier fragmentation as well as the importance of solid-phase microstructure on crack formation and propagation have not yet been investigated. Strong experimental input is required in this area. REFERENCES
1. G. Weickert, G. B. Meier, J. T. M. Pater and K. R. Westerterp, Chem. Eng. Sci., 54 (1999) 3291. 2. J.A. Debling and W. H. Ray, Ind. Eng. Chem. Res., 34 (1995) 3466. 3. E.L. Hoel, C. Cozewith and G. D. Byrne, AIChE J., 40 (1994) 1669. 4. R.A. Hutchinson, C. M. Chen and W. H. Ray, J. Appl. Poly. Sci., 44 (1992) 1389. 5. M. Kakugo, H. Sadatoshi, J. Sakai and M. Yokoyama, Macromolecules, 22 (1989) 3172. 6. P.M. Adler and J.-F. Thovert, Appl. Mech. Rev, 51 (1998) 537. 7. J. Kosek, Z. Grof, F. Stepanek, A. Novak and M. Marek, Chem. Eng. Sci., submitted. 8. F. Stepanek, M. Marek and P. M. Adler, AIChE J., 45 (1999) 1901. D
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V.All rightsreserved.
183
Semi-Batch Emulsion Copolymerization: A General Model for a Copolymer Formed from n Monomer Units Stefan Kr~imera*, Ralf Gesthuisen a aLehrstuhl ftir Anlagensteuerungstechnik, FB CT, Universit~it Dortmund, Emil-Figge-Str. 70, D-44221 Dortmund A detailed, general monodisperse mathematical model for seeded semi-continuous emulsion copolymerisation (SCEP) with n comonomers is presented. The presented model describes the SCEP process in detail using radical, monomer and particle balances in water and particle phase and employing a phase equilibrium as the connection. This approach, though in parts known from other modelling approaches, is new in incorporating the latest research on dynamic radical balances, particle nucleation, the gel effect and the avarage number of radicals per particle. It is, however, limited to conditions without a droplet phase. Every conceivable SCEP process can thus be described once the physical data has been provided and the model can easily be extended by a batch-reactor heat balance. Results have been calculated for the pairs and triples (Vinyl Acetate-Butyl Acrylate), (Vinyl Acetate-Butyl Acrylate-Methyl Methacrylate), (Butyl Acrylate-Methyl Methacrylate), (Methyl Methacrylate-Styrene). The model works also for the homopolymerisation for Methyl Methacrylate, Styrene and Vinyl Acetate. The results for Vinyl Acetate and Butyl Acrylate show reasonable agreement with published experimental data. 1. INTRODUCTION SCEP is a major industrial process for latex production yet a very complex process from a reaction engineering and modelling viewpoint. Understanding, modelling and controlling the process has been researched since the 1940s. Due to its multiphase and compartmentalised nature it offers the possibility of preparing polymers with unique properties and small breadth of chain length distribution and allows for fast and efficient heat removal. In this three-phase process (droplet, particle and water phase) the reaction takes place in two of the three phases (particle and water phase). The main reaction phase depends strongly on the batch time, the comononers used and the water solubility of the monomers. Mass transfer and phase distribution thus play a major role in the system. Further complicating factors are the cross-propagation and cross-termination reactions of the monomers. Many SCEP models exist, most of them developed for a specific monomer pair or triple. They often use specific assumptions in equation development. *corresponding author. Partially funded by Max-Buchner-Stiftung,KennzifferMBFSt2087
184 A model describing the effects globally will become quite large as all contributions have to be taken into account.
1.1. Modelling history (Smith and Ewart, 1948) published a model for a batch homopolymerisation. Since then, modelling has been extended to semi-batch polymerisation, copolymerisation and semi-batchcopolymerisation. Polydisperse models using moments or full population balances and simpler monodisperse models have been shown to give reasonable results. The first complete model for homopolymerisation using population balances was developed by (Min and Ray, 1974). Monodisperse models are generally based on the Long Chain Hypothesis. Distributions of chain length and particle size are not a result of these modelling approaches. A great number of publications deals with monodisperse models for batch-copolymerisation and semi-batchcopolymerisafion, two of the extensive ones are by (Richards et al., 1989) and (Forcada and
Asua, 1990). 2. MODEL DEVELOPMENT
2.1. Propagation Using the Long Chain Hypothosis and symbolising a polymer chain with a specific endgroup M i.] propagation can generally be expressed as
as [~
,,~ M . + M
or
kP >,,~ M .
(1)
rp - ke [,,~ M.] [M]
If simplifications such as the mean number of radicals per particle (~) and the number of particles (NT) are used and the rate of formation equation is extended to n monomers, the rate can be expressed as in eq. 2. For the formation of a polymer a probability (P) that the monomer hits a certain polymerradical needs defining. This is done using the concentrations of the endgroups: Ri-
kejiPj [M i] p -~Nr
NA
j=l
with
Pj =
(2)
i e [1..n]
~
E ['" Mi'] p
i=1
Using the Quasi Steady State Approximation the concentrations can be identified. The rate of formation of a polymerradical can be expressed as such: dt
= -
kpji [" MJ'] p [Mi] p Qss=A0
kpij ["-' Mi'] p [MJ] p +
j=l
jTAi
Vi E [1..n]P,j r i (3)
j=l
jT~i
The resulting equation system can be solved generally for [~ Mi'] p and the solution can be shown to be correct. Such, the probabilities can be defined. An example solution for three monomers is given as:
[,',-'M 2-] [~ M 3.]
--
[M23]~(kp12kp31[M1] p + kp12kp32 [M2] p -~-kp13kp32 [M 3] ~) [M ] (kp13kp21 [Me] p at- kp12kp23 [M2] p at- kp13kp23 [M3] )
(4)
185
2.2. Phase equilibrium The monomers are distributed in all three phases of the system. To determine the concentrations of the monomers in the phases a phase distribution algorithm used by (Urretabizkaia and Asua, 1994) is reduced to water und particle phase. It employs constant phase partition coefficients (#//) for the equilibrium between the phases, where i defines the considered monomer, j the phase. The phase distribution algorithm calculates the volumes of the monomers in the phases and the total volume of each monomer in the reactor. The interested reader is referred to Urretabiskaia et al. for a detailed description of the algorithm. This approach gives good results and converges quickly. 2.3. Material Balances Using the above derivation and balances for monomer and initiator are given as:
dM i -- -Ri - VWkpii[Mi]W[R]w-+-hMi i, j E [1..n] dt dI w -- hi -- klI - h i - fkd[l]WV w where M i = [Mi]VR dt
(5)
(6)
I w -- [l]Wy w
2.4. Particle Balances The following particle balance can be developed. It covers particle formation by exceeding the critical chain length, agglomeration, termination and micellar absorption, if the number of micelles is known:
dNr dl
~
--
i= 1
iw w kam[R]WNm kpii[M ] Rjcrit +
N2 jcri, Jcri, T .+_NA VW-kT E [RJ] w E [ei]w - kagl NA VR j-- 1 i--jcrit-j
(7)
A micelle balance is necessary. Emulsifier is not fed to the system and such micelles are only depleated by particle formation and particle stabilisation.
Nm = as(ST -CMC) - a p
ap -- (367~)2 (VP) 2/3
(8)
Am
2.5. Radical Balances A differential equation model developed by (Li and Brooks, 1993b) based on (Smith and Ewart, 1948) will be used for the radical balance in the particle phase. d~ = kap[R] w _ -kdn-n -- tp
dt
kr
~2
where
tp-
( -
2(2kap[R] w -k-_ kdn) ( -s 2kap[R] w q- kdn -k- -
_
VPNA/NT )
(9)
The initiator-radicals in the water phase are developed as:
d[Rl]W = 2 fkd[I] _ ~ kpii[Mi]W[Ri]W_-~T[Ri]W[R]W_ (kam Nm NT ) dt NA V w + kap NA VR [RI]W (1 O) i=1 Radicals made of one monomeric unit are given as type separated equations:
d[R~ ]w dt
--
-i -nNT kdnNAVW
(
Nm
NT )
kamNAVW -~- kaPNAV R
[R~]w -~- kpii[Rl]W[Mi]
- ~ kpij[MJ]W[g~] w - kTii[R~]W[R] w
j=l
(11)
186 For radicals of more than one monomeric units, a type separation is not made:
IvANmw + kap ,Vnr. AvRNT kpii[Mi]w([ej-1] w - [Rj] w) - -w kT[gj]w[e] w - ( kamnr.V ~L ) [Rj] w
d[ej]w i=1
VjE[2;3;..;jcrit]
and
[R]W-E[Ri]W+E[R~]W+[RI] i
and
Rw [R]W=NAVW
(12)
i
2.6. The Gel Effect A detailed physical model for the gel effect has been developed by (Soh and Sundberg, 1982) and simplified by (Li and Brooks, 1993a). This model- extended to copolymerisation by a probability and fractional conversion weighted average approach- is used here. The interested reader is referred there. 2.7. Volume Balance A total volume balance is necessary to account for all reactions and find the relevant concentrations in the material balances and the phase algorithm.
dVn dt
nI "MI + PI
i:1
DMi
--VW E i=1
--
RpiMmi i=1
kpii[Mi]W[R]WMMi
PMi 1 PMi
01)) (13)
2.8. Parameters Most parameters can be found in the cited publications. Propagation constants (kpii) , initiator decomposition (f, kd), micellar surface area and absorption and particle surface coverage (Am, kam, as, CMC, Ap), termination coefficients (kTii, kT), agglomeration and critical chainlength (kagl, jcrit) are literature values for the relevant monomers. Desorption coefficients have been found by an extension of the approach presented by (Asua et al., 1989). 3. SIMULATION Figure 1 shows simulation results. The results for Vinyl Acetate and Butyl Acrylate show good agreement with published experimental data shown by (Dimitratos, 1989). The gel effect can be seen in a very pronounced manner in all simulations, namely when conversion increases strongly. This is caused by the decrease in the termination rate leading to an increase in the average number of radicals per particle. As can be seen in the first three subfigures of figure 1, the gel effect parameters need further adjustment, as (Soh and Sundberg, 1982) published it for bulk polymerisation. Although literature data for the simulated systems could not always be identified, it can be stated that the shown curves qualitatively depict expected results. 4. SUMMARY, CONCLUSION UND O U T L O O K A complete monodisperse model for a semi-continuous emulsion copolymerisation for n monomers has been developed. Different runs show that the model copes with ter-, co- and homopolymerisation of different species. The runs have all been simulated in the same manner, the monomers are added with a constant flowrate. At a certain point in time (where the discontinuity in the curve can be seen), dosage is stopped. The gel effect- using parameters
187 from (Soh and Sundberg, 1982) - has a strong effect at a high conversion level. For emulsion polymerisation it is thought that the parameters will need adjustment with experimental data. 5. NOTATION m
[.]
Concentration
p R
Density Reactor CMC Critical micelle conc. jcrit Critical chainlength kag l Agglomeration
M
NA Pj rp
Monomer Avogadro's Number Probability Propagation rate
kt
Averagetermination rate contant P Particlephase A Surfacearea f Efficiencyfactor kdn Radicaldesorption kai Absorption rate
M i Monomer i Nm Number of micelles R Sr
Radical Surfactantconcentration
w
as I kpi j
kd Mi Nr Ri V
Average number of radicals per particle Water phase Surfactant area parameter Initiator Propagation rate constant Initiator decomposition rate constant Mol. mass monomer i Number of particles Rate of reaction M i Volume
REFERENCES Asua, J. M., E. D. Sudol and M.S. E1-Aasser (1989). Radical desorption in emulsion polymerization. Journal of Polymer Science: Part A: Polymer Chemistry 27, 3903-3913. Dimitratos, Y. N. (1989). Modeling and Control of Semicontinuous Emulsion Copolymerisation. PhD thesis. Lehigh University. Forcada, J. and J. M. Asua (1990). Modeling of unseeded emulsion copolymerisation of styrene and methyl methacrylate. Journal of Polymer Science: Part A: Polymer Chemistry 28, 9871009. Li, B. and B. W. Brooks (1993a). Modeling and simulation of semibatch emulsion polymerization. Journal of Applied Polymer Science 48(10), 1811-1823. Li, B. and B. W. Brooks (1993b). Prediction of the average number of radicals per particle for emulsion polymerization. Journal of Polymer Science: Part A: Polymer Chemistry 31, 2397-2402. Min, K. W. and W. H. Ray (1974). On the mathematical modeling of emulsion polymerization reaction. Journal of Macromolecular Science - Reviews of Macromolecular Chemistry Cl1(2)(177), 177-255. Richards, John R., John P. Congalidis and Robert G. Gilbert (1989). Mathematical modeling of emulsion copolymerization reactors. Journal of Applied Polymer Science 37, 2727-2756. Smith, W. V. and R. Ewart (1948). Kinetics of emulsion polymerization. The Journal of Chemical Physics 16(6), 592-599. Soh, S. K. and D. C. Sundberg (1982). Diffusion-controlled vinyl polymerization. I. to IV. Journal of Polymer Science 20, 1299-1371. Series of 4 articles. Urretabizkaia, A. and J.M. Asua (1994). High solids content emulsion terpolymerization of vinyl acetate, methyl methacrylate, and butyl acrylate. I. Kinetics. Journal of Polymer Science, Part A: Polymer Chemistry 32(9), 1761-1778.
188 Vinyl Acetate
Butyl Acrylate
Conversion
0.02 O Dimitratos [ Model
= 0.8 o =~0.6
8
0.
.~0.015
0.4
0
5000
0.01 ~0. r..)
~
%
0.2
o=0.
~
O
0.005
10000
0
15000
0.
0
5000
Triple (VAC, BA, MMA)
10000
5000
Pair (VAC, MMA)
1
0.7
I
---- VAC I BA MMA
=00.8
15000
Pair (MMA, STY)
0.5
0.6
0.4
~0.8
0.4
0.3
0.6
o
0.2 a~ 0.2 0
0
1
2
0
3 X 104
0.2 0
0
1
2
3
4
"" ~k___ 0
5000
10000
15000
0
50;0
~o/~o
15000
x 104
0.8
0.8
o= 0.6
=o 0.6
=o 0.6
o= 0.4
"i 0.4
0.2
0.2
2
r..)
0.4 0.2
i
3 x 104
Homopol., STY
2
;
0
4 x 104
Homopol., VAC 2.5
1
=o0.8
.o
"~
Homopol., MMA t
.S~
2
~.o 1.5
0.6 o "~ 0.4
i
[m 1,_
2.
,,'~.
Conversion I CONCVAC.~
i
~
.~
i
1
---- ConversiOnSTY[ o 0.5
rjo 0.2 o
MMA STY ]
1
0.8
1
[~
0.4
/
0.1
A
1.2
o=9 =~
~
15000
1.4
,- vA ]
0.6
10000
o
1
2
time(s)
3 x 104
0
I i I m ~ i ii
0
5000
10000
time(s)
15000
0
5000
10000
15000
time(s)
Fig. 1. Simulation run summary: VAC-Vinyl Acetate, BA-Butyl Acrylate, MMA-Methyl Methacrylate, STY-Styrene
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
189
Computer Aided Continuous Time Stochastic Process Modelling Niels Rode Kristensen a, Henrik Madsen b and Sten Bay JCrgensen a aComputer Aided Process Engineering Center (CAPEC), Department of Chemical Engineering bSection for Mathematical Statistics, Department of Mathematical Modelling Technical University of Denmark, DTU, DK-2800 Lyngby, Denmark A grey-box approach to process modelling that combines deterministic and stochastic modelling is advocated for identification of models for model-based control of batch and semi-batch processes. A computer-aided tool designed for supporting decision-making within the corresponding modelling cycle is presented. 1. INTRODUCTION With the development and increasing number of possible applications of advanced model based process control schemes, e.g. model predictive control (MPC), more and more rigorous demands are placed on the quality of available dynamic process models. Model quality measures the ability of the model to predict the future evolution of the process, so in order to obtain good prediction performance, these models must be able to capture the inherently nonlinear behaviour of many process systems, such as batch and semi-batch processes. Furthermore these models must be able to provide predictions in the presence of noise, i.e. process noise due to approximation errors, unmodelled inputs and plant-model mismatch and measurement noise due to imperfect measurements. Meeting both demands with the same model is difficult, so there is a tendency in litterature to use either a deterministic approach or a stochastic black-box approach to process modelling. The deterministic approach is based on using first engineering principles to derive ordinary differential equation (ODE) models. These models are well-suited for describing nonlinear behaviour, but they lack the desired predictive capabilities in the presence of noise, because they do not encompass a noise model and because unknown parameters are estimated in an output error (OE) setting, which tends to emphasize the pure simulation capabilities of the model instead of the predictive capabilities, cf. Young (1981). The stochastic black-box approach, on the other hand, is based solely on using time series data for identifying a model, usually in the form of a discrete time transfer function model. These models usually have very nice predictive capabilities because of their inherent noise model and because unknown parameters are estimated in a prediction error (PE) setting, cf. Young (1981). Unfortunately these models are not equally well-suited for describing nonlinear behaviour, especially not outside the (possibly narrow) operating region, within which the time series data for identification is obtained.
190 In this paper an alternative grey-box approach to process modelling is advocated. This approach combines the deterministic approach and the stochastic black-box approach in a way that seeks to combine their respective strengths, i.e. from ODE models the intuitive appeal of their derivation from first engineering principles and their ability to describe nonlinear behaviour, and from stochastic black-box models the nice predictive capabilities and their ability to handle both process and measurement noise. The aim of this paper is to describe the grey-box approach and outline its advantages. This is done in Section 2, where a computer aided tool that aims to support decision-making within this approach is also presented. In Section 3 a small example is given to illustrate one of the advantages of this approach and the conclusions are presented in Section 4. 2. A GREY-BOX APPROACH TO PROCESS MODELLING
A very appealing way of combining the deterministic and the stochastic approaches to process modelling is to use stochastic differential equation (SDE) models as shown by Astrrm (1970). The grey-box approach advocated in this paper is therefore based on SDE models in the It6 sense or, to be more specific, on the continuous-discrete stochastic state space model dxt = f (xt , ut,t, O)dt + ~(t, O)dcot Yk = h(xk, uk, tk, 0) + ek
(1) (2)
where t E ~ is time, xt E X c Nn is a vector of state variables, ut c U C Nm is a vector of input variables and Yk E y C R l is a vector of measurements, xk = Xt=tk and uk = Ut=tk. 0 C 19 C RP is a vector of parameters, and f(.) C R n, or(.) C ]t~n x q and h(-) E R l are nonlinear functions, cot is a q-dimensional standard Wiener process and ek C N (O,S(tk, O)) is an/-dimensional white noise process.
Fig. 1. The modelling cycle for control which constitutes the core of the grey-box approach to process modelling.
Figure 1 shows a modelling cycle based on this model, which describes the grey-box approach, and by means of which some of its advantages can be outlined. 9 The continuous time system equation (1) allows the initial structure of the model to be determined from first engineering principles in the form of an ODE model, which is intuitively appealing, since any prior physical knowledge can be included and because
191 the parameters of the model can easily be given a physical interpretation. Furthermore, most chemical and process systems engineers are familiar with this way of constructing a model. 9 When subsequently determining unknown parameters of the model from a set of data, the continuous time system equation (1) and the discrete time measurement equation (2) make the model flexible by allowing varying sample times and missing observations. 9 The model provides a separation between process and measurement noise, which along with the stochastic nature of the model allow the parameters to be estimated in a PE setting using a statistically sound method, e.g. m a x i m u m likelihood (ML). 9 For the same reasons statistical tests and residual analysis can subsequently be applied in a systematic manner to validate the model, and if it is found that the model is not valid these tools also provide information on how to alter the model to improve its quality. In the following the individual elements of the modelling cycle are explained in more detail. Once a model structure has been determined from first engineering principles, unknown parameters of the model can be estimated from a set of data. Nielsen et al. (2000) have recently reviewed the state of the art with respect to parameter estimation in discretely observed It6 SDE's and found that only methods based on nonlinear filtering provide an approximate solution to the full problem of determining ML estimates of the parameters of the continuous-discrete stochastic state space model. Unfortunately, applying nonlinear filtering is difficult, so in order for the grey-box approach to be feasible, extended Kalman filtering (EKF) is used instead as shown in the following. Determining ML estimates of the parameters means finding the parameters 0, including the initial conditions x0, that maximize the likelihood function with respect to 0 given a set of measurements Yo, yl . . . . . y~ . . . . . YN. By introducing 9~ = [Yk,Yk-1, . . . ,Yl ,YO] and ~klk_ 1 = E { y k l ~ k - 1 , 0}, Rklk-1 = V { y k l ~ k - 1 , 0 } and ek -- Yk -- 33klk-1 and by assuming that the conditional probability densities are Gaussian, the likelihood function becomes
L(YNI 0 )
--
fip(ykl~k_l,O) k=l
p(yolO)
--
f i exp (--
e'k P~kk-
l13k) l
p(yolO
)
(3)
k=l v/det (Rk k-l) (V/~)
where, for given parameters 0, ek and Rklk-1 can be computed by using a continuous-discrete EKF. If prior information is available in the form of an a priori probability density function p(0) for the parameters, Bayes rule can provide an improved estimate of the parameters by forming the posterior probability density function, i.e. (4) p(0[YN) -- L(YNIO)p(O) o, L(YNlO)p(O) P(YN) and subsequently finding the parameters that maximize this function, i.e. by performing maximum a posteriori (MAP) estimation. By assuming that the prior probability density of the parameters is Gaussian, and by introducing/1o = E{O}, E0 = V{O} and e0 = 0-/1o the posterior probability density function becomes p(OiYN)~
1-NI exp( - l e r O - 1 l
k=l v/det (Rk k- 1) (V/~)
p(yolO)
p
v/det (E0) ( v ~ - )
(5)
192 If, instead of a single set of measurements, several consecutive, but yet separate, sets of measurements, i.e. y11, y22 ..... YNi ..... ys s, possibly of varying length, are available, a similar estimation method can be applied by expanding the expression for the posterior probability density function to the general form p(OIY)~
l~I i=1
exp(--l(e~)T(R~lk-1)-l(eik)) k=l ~//det(Rikk_l)(V/~) l fi
P(y~lO)
e x p ( - 12 e T~ 1 -1 76176 v/det ( l ~ 0 ) ( v ~ ) p
(6)
I
where Y - [yll, y22,... , Y/vi,'", ySs]" Finding the estimates of the parameters 0 is now a matter of further conditioning on Y o - [yl,y2,... ,rio,..., ySo] and applying nonlinear optimisation to find the minimum of the negative logarithm of the resulting posterior probability density function, i.e. t~ - a r g m ~ - In (p(01Y , Yo)) (7) ttEt~
With this formulation it is possible to perform MAP estimation on several data sets of varying length, but as special cases it is also possible to perform ML estimation on several data sets (with p(0) uniform), MAP estimation on a single data set (with S - 1) and ML estimation on a single data set (with p(0) uniform and S = 1). When the unknown parameters of the model have been found using one of the above estimators, statistical tests and residual analysis can be performed. First of all, since the estimators are all asymptotically Gaussian the parameter estimates and their standard deviations can be used to perform marginal t-tests for parameter significance, i.e. to test if the parameters are significantly different from zero. This is particularly important for the process noise parameters, because parameters that are significantly different from zero indicate that the model structure is not perfect, i.e. that there may be approximation errors, unmodelled inputs or plant-model mismatch. It is an inherent assumption of the above methods for estimation of parameters that the conditional probability densities are Gaussian, and for nonlinear systems this assumption is only likely to hold when small sample times are used, so the validity of this assumption should also be tested by performing a test for Gaussianity. Finally it is possible to test if the model is correct by performing a goodness of fit test as shown by B a k e t al. (1999) and by performing residual analysis. For the latter purpose both standard linear methods and nonlinear methods based on nonparametric modelling are available, cf. Nielsen and Madsen (2001). For supporting decision-making within the modelling cycle a computer aided tool, CTSM, has been developed, cf. Kristensen and Madsen (2000). Within this tool a number of program units corresponding to the individual elements of the modelling cycle have been or will be implemented, including a graphical user interface for setting up the model structure and algorithms for estimating parameters and performing statistical tests and residual analysis. Altogether these program units aid the chemical or process systems engineer when setting up models. 3.
-
EXAMPLE
The following is an example, which illustrates an important feature of the grey-box approach the possibility of determining whether a given model structure is correct from estimates of the
193
process noise parameters. The process considered is a simple fed-batch fermentation process described by an unstructured model, i.e. ds
yS
-
=
\y~,]
8 )
/-~(s)x/
S/<
Vk
-I--
-I-
(1
)
F
[o! 0 0]
dt +
(y2
1
0
dot , t C [0, 3.8]
(8)
(y2
e~
(9)
\e v
where X and S are the concentrations of biomass and substrate, V is the volume of the fermenter and F is the feed flow rate, and finally e~k C N (0, 0.01), e s C N (0, 0.001) and e kv C N (0, 0.01). For the growth rate/~(S) two different cases are considered, namely 9 A correct model structure with lu(S)
s
-- ~max K1S2+S+0.5
9 An incorrect model structure with p(S) - / U m a x
9
S s-4E"
corresponding to biomass growth with Monod kinetics and with and without substrate inhibition respectively. Using the true parameter values in Table 1, 10 sets of simulation data (100 samples each with a sample time of 0.038) have been generated by perturbing the feed flow rate along an analytically determined optimal trajectory, and all the results mentioned in the following correspond to 2S, (Ye 2v and the initial conditions ML estimation of ]-/max (or/Umax), g l (or/~1 ), (y2, (y2, (y2, (YS' (Ye using all 10 data sets.
Table 1
True and estimated values of the parameters of the fermentation process model. Upper part: Case 1- correct structure of ll(S). Lower part: Case 2-incorrect structure of l2(S). Parameter ]-/max K1 o2 o2 c~2 ~max
True value 1 0.03 0 0 0 -
/~1
-
c~2 o2 c~
0 0 0
Estimated value 1.021 0.03005 4.026e-4 1.365e-5 3.100e-4 0.7661 0.01066 0.05687 0.08714 0.002089
Standard deviation 0.0044 0.00139 1.270e-4 1.391e-5 1.298e-4 0.0066 0.00007 0.00369 0.00935 0.000167
Significant YES YES NO NO NO YES YES YES YES YES
With the correct model structure, the parameter estimates in the upper part of Table 1 are obtained. The estimates of ,t/max and K1 are very accurate, and the estimates and standard deviations of (y2, (y2 and (y2 indicate that these parameters are not significantly different from zero.
194 This is subsequently confirmed by performing t-tests, and this indicates that the model structure is indeed correct. With the incorrect model structure, on the other hand, the parameter estimates in the lower part of Table 1 are obtained. Now 6~, 6 2 and 6 2 are all significantly different from zero, indicating approximation errors, unmodelled inputs or, as in this case, plant-model mismatch. 4. CONCLUSION A grey-box approach to process modelling that combines deterministic and stochastic modelling is advocated for identification of models for model-based control of batch and semi-batch processes, and a computer-aided tool designed for supporting decision-making in the corresponding modelling cycle has been presented. The grey-box approach is based on flexible and statistically sound continuous-discrete stochastic state space models, which have the same appeal as ODE models with respect to their derivation from first engineering principles. One of the most important advantages of the approach is its built-in features for performing model validation by means of statistical tests and residual analysis, e.g. that the significance of the parameters of the process noise term may provide information about the validity of a proposed nominal model. REFERENCES Bak, Jakob, Henrik Madsen and Henrik Aalborg Nielsen (1999). Goodness of fit of stochastic differential equations. In: Symposium i Anvendt Statistik (Peter Linde and Anders Holm, Eds.). Kristensen, Niels Rode and Henrik Madsen (2000). CTSM - Continuous Time Stochastic Modeling - Version 1.0. IMM, DTU, Lyngby, Denmark. Nielsen, Henrik Aalborg and Henrik Madsen (2001). A generalization of some classical time series tools. Computational Statistics and Data Analysis. To appear. Nielsen, Jan Nygaard, Henrik Madsen and Peter C. Young (2000). Parameter estimation in stochastic differential equations: An overview. Annual Reviews in Control 24, 83-94. Young, Peter C. (1981). Parameter estimation for continuous-time models - a survey. Automatica 17(1), 23-39. Astr6m, Karl Johan (1970). Introduction to Stochastic Control Theory. Academic Press, New York, USA.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
195
Assumption retrieval from process models R. Lakner a, K.M. Hangos b'a, I.T. Cameron c aDept, of Computer Science, University of Veszpr6m, 8201 Veszpr6m, POBox 128, Hungary bSystems and Control Laboratory, Computer and Automation Research Institute HAS, 1518 Budapest, POBox 63, Hungary CCAPE Centre, Dept. of Chemical Engineering, The University of Queensland, Brisbane, QLD 4072, Australia Process models of lumped systems are considered in this paper in their "canonical" form where the equations and variables are classified, the "natural" set of design variables and "natural" assignment are selected. An efficient intelligent algorithm is proposed to generate the assumption sequences leading from one model to another in an automated way. The algorithm has been implemented in PROLOG within our intelligent model editor. Two simple assumption retrieval examples are also presented and discussed for analyzing and comparison purposes. 1. INTRODUCTION The automated generation of process models from the engineering system description and modelling assumptions is one of the most important and challenging tasks in computer aided modelling (CAM). Efficient algorithms solving this task are essential for constructing automated modelling tools, which can be regarded as intelligent front-ends for dynamic simulators. There are a number of important contributions reported in the literature (see e.g. [3], [4]) toward formal description of process models for their automated generation. The syntax and semantics of process models are defined in both of these papers using an object-oriented software language. A formal representation of assumptions in process modelling together with a systematic methodology of process model development is reported in our earlier paper [ 1]. Here the modelling assumptions acted as formal transformations on process models. The retrieval of modelling assumptions from a process model for analyzing and comparing purposes is an equally important but unusual problem where not only efficient algorithms but the engineering understanding is lacking. The retrieved assumptions can be used to advise the modeller to improve the model in case of any solution or control problems. The present paper deals with the inverse task of model transformations: with assumption retrieval from two related process models of the same process system but with different modelling assumptions. The aim is to construct efficient intelligent algorithms to generate the assumption sequences leading from one model to another in an automated way. This research has been supportedby the Hungarian Science Fund through grant No. T026575 and also through the Australian Research Council Grants A89530349 and A89937094.
196 2. PROCESS MODELS IN CANONICAL FORM Process models of lumped systems are DAEs with a well-defined structure dictated by the underlying physics and chemistry of the system. It is advisable to follow a systematic modelling procedure to develop the model equations, which results in a set of structured model equations obeying the hidden syntax and semantics of process models [ 1]. In order to solve the assumption retrieval problem efficiently, the process models in "canonical" form are considered where the equations and variables are classified, the "natural" set of design variables and "natural" assignment of the algebraic equations to algebraic variables are selected. It is important to note that in our case the detailed lumped dynamic model equations and the variable set are the result of the model builder of our intelligent model editor [2], which always produces them in canonical form.
2.1. Equation categories There are three types of model equations in a process model according to their origin: 9 balance equations (in differential or algebraic forms) 9 transport equation terms (parts or terms of the balance equations) 9 constitutive algebraic equations (intensive-extensive relationships, equations of state, transfer rate expressions, thermodynamic property relations, reaction rate expressions, balance volume constraints and equipment and control relations) The main characteristics of the equations are the equation identifier, the variable the equation is assigned to, the equation type and the equation itself stored in the form of a binary tree. 2.2.Variable categories Driven by the role of the variable in the process model, we can distinguish the following variable categories: 9 balance volumes 9 conserved extensive quantities (for each balance volume) 9 physico-chemical property variables 9 transfer and reaction rate variables 9 thermodynamical state variables 9 design variables The main characteristics of the variables are the variable name, the name of the balance volume when necessary, the type of the variable, the identifier of the equation which is assigned to the variable if it exists, and a list with the other identifiers of the equations containing that same variable. The structure of the DAE set forming the process model is used to classify the variables into differential and algebraic ones. The variables can be further classified according to three ways they are specified: 9 defined by an equation (either differential or algebraic) 9 defined as constant 9 defined as unspecified (design) variable The above classification assumes to have an assignment of the variables to the equations. Naturally in the end of the model building and simplification the unspecified variables will form the set of design variables.
197 3. MODELLING ASSUMPTIONS Modelling assumptions can be regarded as representations of the engineering activity and decisions during the whole modelling process in constructing, simplifying and analyzing process models. Assumption-driven modelling works directly with modelling assumptions thus enabling the definition and handling of process models as structured text with defined syntax and semantics. Algebraic manipulations are then described as equivalence transformations, and model simplification and enrichment assumptions as general modelling transformations acting on process models.
3.1. The syntax of model simplification assumptions Model simplification assumptions can be formally defined as triplets [ 1]" model_variable_name relation keyword where model_variable_name is a model variable identifier or identifier of a group of model variables, relation is a relation sign, usually an equality (=) and keyword is either a constant (numerical or symbolic, like nil), or another model_variable_name. Because of the syntax of the model simplification assumptions, the result of any assumption retrieval is a (not necessarily unique) sequence of modelling assumptions in the form of triplets as given above. Model simplification assumptions can be either elementary (or atomic) or composite composed of a conjunction of elementary assumptions. We can formally associate a model transformation to each simplification assumption. Quite often, but not always, the simplification transformation is performed in two substeps: 1. Add the equality describing the assumption to the already existing set of model equations and perform algebraic transformations (for example substitutions) to get a more simple form. 2. Adjust the set of differential, algebraic and design variables to satisfy the degree of freedom requirement. 3.2. The semantics of modelling assumptions If one performs a sequence of model simplification transformations, the resultant model may be different if the order of the assumptions is changed. The reason is that model simplification assumptions may be related and non-commutative [1]. Moreover, model simplification transformations are projections in mathematical sense, therefore it is not possible in general to retrieve fully the original model from the simplified one and from the simplification transformations. Because of the above properties, the result of an assumption retrieval from the original and the simplified models may not be and in general will not be unique. 4. ASSUMPTION RETRIEVAL The task of assumption retrieval is formulated in the form of standard problem formulation of computer science in order to prepare its rigorous formal algorithmic solution as follows. Given: two process models from the same model hierarchy both in canonical form. Generate: the assumptions, which lead from the more complex model to the simpler one. Because of the non-uniqueness of the assumption retrieval task, an intelligent exhaustive search algorithm is needed and proposed for its solution.
198
4.1. The assumption retrieval algorithm The algorithm consists of three main phases: 1. Clumsy comparison of the two process models (comparison of the number and type of variables and equations) in order to examine whether the models are in the same model hierarchy and both are in canonical form. 2. Retrieval of the unambiguous and reversible transformations by comparing the two models to find a better starting point for the next heuristic retrieval step. The result of this deterministic step is a (or in case of non-commutative assumptions several) partially retrieved process model(s). 3. Further retrieval by heuristic search starting from the detailed model toward the (or a) partially retrieved model with the following steps: a. Collect the list of possible assumptions on the basis of the differences of the two models. b. Execute a breadth first search by transforming the detailed model using the list of possible assumptions, and by comparing the resulted simplified models with the partially retrieved model. c. Continue with step 3. until all partially retrieved models are examined. 5. RESULTS AND DISCUSSION We have implemented the algorithm in PROLOG within our intelligent model editor [2]. The model builder module of the editor constructs the free models in canonical form. Thereafter the model simplifier module is used to obtain simplified models. The assumption retrieval module has been implemented as an independent additional module. In order to verify and validate the assumption retrieval module properly, several case studies have been performed. The "free model", that is the more complex one is build by the model builder which is simplified by the model simplifier in a specified way. Then the assumption retrieval module was used to generate all possible assumption sequences which may lead from the free model to the simplified one. It is important to note that the retrieval algorithm uses the same set of possible assumptions as the model simplifier and the number of assumptions in the resulted sequence is set by the user from 1 to 3. The two assumption retrieval examples presented here use the same simple process below.
5.1. A simple phase equilibrium process model A simple single component phase equilibrium system is considered as an example [1]. Vapour (denoted by variable ending "v") and liquid ("l") are taken from a vessel, whilst energy is supplied via a heater. Inside the vessel we have two holdups Mv and M/and temperatures Tv and Tl. A feed with mass flow rate F comes into the system. Two assumption retrieval results of this system are seen in the figures (Figs. 1. and 2.) below. Both the free and the simplified (transformed) model equations are shown in separate windows together with all the assumption sequences found by the assumption retrieval algorithm.
5.2. Assumption retrieval from the simple phase equilibrium process model The two assumption retrieval examples have been selected in such a way that both the properties of the proposed algorithm and the assumption retrieval problem are highlighted.
199 Example 1. Non-related assumptions The first simplified model has been generated by a sequence of two non-related assumptions: "My differential-var is constant" and "all physico-chemical-var is constant" (1) Figure 1. below shows the result of the assumption retrieval with maximum 2 assumptions in the sequences. It is seen that 4 possible assumption sequences are found, all of them consist of precisely 2 assumptions. There are two pairs, which only differ in the order of the two assumptions. This shows that these two assumptions are indeed non-related, therefore commutative. Besides the original generating sequence (1), another variant with a different assumption "Uv differential-var is constant" and "all physico-chemical-var is constant" (2) is also generated. The two assumptions "My differential-var is constant" and "Uv differential-var is constant" (3) are clearly related, which is the reason of this multiplicity. Example 2. Related and equivalent assumptions The second example is generated by two assumptions: "Vv balance-volume is nil" and "all physico-chemical-var is constant" (4) but the retrieval algorithm is run with the maximum 3 assumptions in the sequence. Part of the resulted assumption is shown in Fig. 2. below. It is seen that a large number of possible assumption sequences are obtained which are even qualitatively different, not only the order of the assumptions changes. The algorithm produces all of the possible sequences consisting of two and three assumptions, starting with the true generating sequence as the only twoassumption sequence. 5.3. Lessons learned
The above retrieval case studies indicate and the detailed results show, that there exist an ordered minimal assumption sequence leading from a process model to a simplified one which is unique in a certain sense. This can be obtained by the proposed algorithm such that: 9 candidate assumptions are generated in their order in the assumption hierarchy, 9 only the preferred form of equivalent assumptions is taken into account, 9 all the consequences of an assumption in the sequence are suppressed, 9 a predefined sequence of the commutative assumptions of the same order is considered only. REFERENCES
1. K.M. Hangos and I.T. Cameron, A Formal Representation of Assumptions in Process Modelling, Comput. Chem. Engng., (2001) in print 2. R. Lakner, K.M. Hangos and I.T. Cameron, An Assumption-Driven Case Sensitive Model Editor. Comput. Chem. Engng. (SupplemenO, 23, $695-$698 (1999) 3. P.C. Piela, T. G. Epperly, K. M. Westerberg and A.W. Westerberg, ASCEND: An Object Oriented Computer environment for modeling and analysis: The modeling language. Comput. Chem. Engng., 15, 53 (1991). 4. G. Stephanopoulos, G. Henning and H. Leone, MODEL.LA, A Modeling Language for Process Engineering: The Formal Framework, Comput. Chem. Engng., 14, 813 (1990).
200
Fig. 1.
Assumption retrieval example 1.
Fig. 2.
Assumption retrieval example 2.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
201
Dynamic Simulation of Batch Crystallization Process by Using Moving Finite Difference Method Y. I. Lim*, J. M. Le Lann ~, X. M. Meyer ~, and X. Joulia w Laboratoire de G6nie Chimique (LGC, UMR-CNRS 5503), INPT-ENSIGC 18 Chemin de la loge, F-31078 Toulouse, France The moving finite difference method combined with Weighted Essentially Non-Oscillatory (WENO) scheme is addressed for the dynamic simulation of batch crystallization processes described by hyperbolic-like PBE (Population Balance Equation) with a discontinuous initial condition. The accurate and stable WENO scheme shows an improvement of numerical results over conventional discretization methods (backward or central) on fixed grids as well as on moving grids. Owing to the moving grid method with the WENO scheme which tracks well a steep front or shock, the physical model of PBEs could be numerically represented more exactly. For illustration, numerical results are compared with experimental results for the crystallization process of the potassium sulfate (K2SO4/H20). The new approach is considered as an efficient numerical solution procedure for the verification of models described by hyperbolic-like PDEs (Partial Differential Equations). 1. INTRODUCTION The Population Balance Equation (PBE) has been enlightened to describe the Particle Size Distribution (PSD) in chemical engineering practice such as crystallization, polymerization, emulsion, L-L/L-G dispersion and microbial culture, since modeling with the PBE provides a good description for parameter identification, operating conditions, process control and design. In crystallization processes, the PBE, which governs the size variation of crystals, is solved together with mass/energy balances and crystallization kinetics such as nucleation, crystal growth, breakage and agglomeration. The system, which often leads to hyperbolic-like IPDEs (Integro-Partial Differential Equations), is complex due to a lot of feedback relationships between the different equations. Moreover, the hyperbolic-like PBE could cause much numerical error and instability in its numerical solution for the density function. In recent years, several sophisticated packages based on the method of lines (MOL) have been developed for the automatic numerical integration of time-dependent problems in PDEs on fixed uniform/nonuniform grids. These packages greatly benefit from the successful developments of automatic stiff ordinary differential equation (ODE) solvers. However, from the PDE point of view, they integrate only in a semi-automatic way in the sense that they automatically adjust the time-step sizes, but use just a fixed space grid, chosen a priori, for the entire calculation. For PDE solutions possessing moving sharp spatial transitions, a fixed grid e-mail:
[email protected],phone: +33 5 6225 2424, fax: +33 5 6225 2318 , e-mail:
[email protected], phone: +33 5 6225 2357, e-mail:
[email protected],phone: +33 5 6225 2358 ~To whom correspondence shouldbe addressed, e-mail:
[email protected],phone: +33 5 6225 2355
202 is computationally inefficient, since for an accurate solution this grid often must contain a very large number of nodes. In such cases methods which automatically adjust the size of both the space and the time steps are likely to be more successful in efficiently resolving critical regions of high spatial and temporal activity. Methods and codes that operate this way belong to the realm of adaptive or moving mesh methods. Static adaptive mesh methods are incorporated to add new mesh points to maintain the accuracy or delete unneeded mesh points to improve the efficiency. The addition or deletion of mesh points can interrupt the temporal integration, and must be carefully done. Therefore, these are not easy to use. Whereas, in dynamic adaptive mesh methods (or moving mesh methods), a fixed number of grid points is used and the nodes move dynamically in such a sense that the nodes are concentrated on regions of high spatial variation. Often the moving mesh method [11 works quite well with central spatial discretization. However, for some problems involving discontinuities (such as discontinuous initial condition), very steep fronts or non-convex flux functions, the use of higher order upwinding schemes such as ENO (Essentially Non-Oscillatory) schemes can yield better results [21. Lim et al. TM postulated a moving mesh method based on EP (Equidistribution Principle that means placing grid points along uniform arc-length intervals) combined with a WENO (Weighted ENO) discretization method [4] for tracking a steep moving front or shock. To more correctly represent physical models of the PBE into numerical solutions, different numerical procedures have been proposed in areas of the Finite Difference Method (FDM): first-order backward FDM [51, self adaptive FDM [6], adaptive FDM [7], Essentially NonOscillatory (ENO) discretization method [8]. But it is difficult to judge numerical solution quality due to confusion between a unreliable model (PBE) and truncation error (in the numerical procedure). This study attempts to minimize numerical error and aims to obtain more accurate numerical solution of the physical model such as PBEs. In the next section, the moving mesh strategy combined with the WENO scheme is addressed in order to obtain an exact numerical solution of the PBE crystallization model containing nucleation, growth and agglomeration kinetics. In the third section, a simple PBE model is tested and compared with its analytic solution, and then a practical model in a well-mixed batch crystallizer with seed for a K2SO4-water system is solved and compared with experimental results. 2. D I S C R E T I Z E D P B E
To know the crystal size distribution defined by the density function (n) with respect to time (t) and crystal size (L), the PBE [91 is usually expressed in terms of birth of nuclei (BN), their growth (G), and birth (BA) or death (DA) caused by agglomeration, regardless of breakage term. nt+(nG)L=BN.8(L-L0)+ BA + DA (1) where subscripts t and L denote the time and space partial derivatives and kronecker delta, 8(L-L0), means that nucleation takes place only at a minimum size L0. These three kinetics terms are often explicitly expressed
O=O(t,L)
BN=BN(t, Lo)
(2) (3)
BA=-~ ~I3(L- Li,Li)" n ( L - Li)-n(Li)dL i
(4)
DA= n(L)Ig 13(L,L i ). n(L i )dLi
(5)
1
203 The hyperbolic-like integro-PDE is reformulated within the moving mesh method Ill as following: f i - L - n L +(nG)L=BN-8(L-L0)+ BA + DA (6) where, fi and I~ denote time derivatives of n and L, respectively. A moving mesh equation (MMPDE) such that meshes are concentrated on steep regions is also added : ni+,/2
ni_,/2 _ 0,
i=2 .... , N-2.
(7)
Mi+l/2 Mi-l/2 where, ~+1/2= fii+,/2-'/(1 +3' )( fi~+3/2-2 fii+,,~+ fii-,/: ),
i = 1, ..., N-2.
(8)
^ n~+,,~=/~ +n/x -
h h(Li+l - l~i) + -1 , i=0, 1, ..., N-1. (9) (Li+ 1 - Li) 2 x (Li+ 1 - Li) 3' denotes the global smoothing parameter for stability of the grid spacing (in general, ~/=1 or 2) and h (=l/N) is a uniform step size on the computational coordinate. The temporal smoothing parameter, x, could be chosen near the integration time step. The monitor function M(t,L) that measures solution variation is defined with first-order derivatives : Mi+1/2= _]or + ni+l - n~i (10) Li+ 1 - L i The parameter (x is usually determined as the median value of first-order derivatives and it is not sensitive to the solution. Four boundary conditions for the M M P D E of global smoothing are imposed: fi,/2 = fi3/2, fiN-3/2= fiN-,/2, L0=a, LN=b (11 ) Globally speaking, the equation (7) means that the grid concentration ( ~ ) increases as arclength (M) increases. The extended PBEs (physical PDE and MMPDE) must be discretized in the spatial direction (L) to solve it in the framework of MOL (Method of line), using DAE integrator. In this study, the hyperbolic term, (nG)L, is discretized by the third order WENO scheme la] socalled WS3 scheme that can track well steep or discontinuous fronts. The first order derivative term (nL) of the monitor function (M) is usually discretized by the second order central method (FS2 scheme). 3. C A S E S T U D I E S 3.1 Simultaneous nucleation and growth without agglomeration Supposing that the growth rate, G=I.0, the birthrate, Bn(v)=1000xe (-l~176176 and an initial condition is for the density function based on the crystal volume (v) : n(0,v)=500xe (-l~176176 10 s < v < 0.3 (12) the analytic solution 9 is known as : ..... e -'~176176 ] (13) n(t,v) = 500e -'~176176 + [e-'~176176 where Vmax=max(v-t, 105). In this study, the first-order backward method (FS1) and third order W E N O scheme (WS3) are tested for the discretization method of first order derivatives. In Fig. 1, the analytic solution at t=0.01 is compared with the numerical solutions of the FS1/WS3 of fixed 200-grid and the FS1/WS3 of moving 40-grid. The results on 40 moving grids show an improvement over those on fixed 200 grids. However, examining the results of the moving mesh in the logarithmic scale, there is no available information within the small crystal region because solution variation is very small. In fact, rare mesh numbers in smooth regions are inevitable for the normal moving mesh strategy.
204 500
.......
- '~,
\~
400
"~ =
~,, 400 ~ ",
FS1 200-gridAnalytic fixed solution
,. ,
-o
300 -
E
"-
500 q, ..
Initial Condition
L tt~ ~'~
. . . . . . . Initial Condition .-------..-Analytic solution -'-
FS1 moving 40-grid
300
, == 200
200
r~
1,:~1 100 r~
_
'
~f,,,,
100 0
0 0
0.01
0.02
0.03
Crystal volume (v)
0.04
0.05
0
0.01
0.02
0.03
0.04
0.05
Crystal volume (v)
(a) Fixed grid (b) Moving grid Fig. 1. Population density distribution on fixed 200-grid and on moving 40-grid. 3.2 Potassium sulfate (K2SO4) crystallization process in a well-mixed batch cooling crystallizer. With natural cooling in MSMPR (Mixed-suspension, Mixed-product-removal) vessel, the PBE is for the crystal density distribution (n, number of crystals/kg-solution/m) 0n(L, t) aG(L, t). n(L, t) ~ + = Bn6(L-L0) (14) 0L Assuming that an effective size for crystal birth (L0) is equal to 50~tm, the kinetic parameters according to Jones et al. (1986) l~ are defined as 9 Bn(L0, t)=2.0x 107.e'93OO/RT.(MT/Ps).~ (15) G(L<700~tm) = 1.44. e-a0a~ (1 +2L2/3).~2 (16) (L] ~ G(L>700 gm)=G(700)" L~ ) (17) Where the magma density (MT) and relative supersaturation (a) are defined as MT= pc.(W-W*)/(pc/Ps-W*)
~-
W -W* W*
(18) (19)
where the crystal density pc=2660 kg/m 3 and Ps denotes the bulk solution density 5. w (kg] K2SO4/kg H20) is the prevailing solution concentration and the equilibrium solubility (w*) is expressed as the function of temperature (T, ~ The temperature profile of natural cooling ~ is changed along with time for 180 minutes. T(t) = 15+45e -835599t (20) To obtain more large crystals, the seed is usually put as a discontinuous initial condition. n(L,0)=5.34• 107 (no./kg/m) 500~tm_
205 dw/dt=-pckv( ~dm3 )
(23)
Lmax
dm3/dt=3 ~LZ(t).n(L,t).G(L,t)dL
Lmin=50~tm, Lmax=1900gm
(24)
Lmin
where the volumetric shape factor 12 (kv) is defined according to the crystal size (L). For the initial condition of equation (23), w(0)=w*(0)+7.32• 10.3 (kg/kg). Finally, the batch crystallization model aims to solve one PBE as well as one mass balance equation associated with birth and growth kinetics subject to the cooling strategy (T(t)). Useful statistical quantities for representing a distribution of crystal sizes about the weightmean such as the weight-mean size (Lwm)and Coefficient of Variation (CVwm)11 of weightmean are compared with those of the experimental results in Table 1, which also shows numerical results of previous works and the present study for the batch natural cooling crystallization of KzSOa-water system. The Lwm and the CVwm of this study of the fixed 40grid well agree with those of the Jones & Mulin (1974). However, it is unable for the equation (14) to predict exactly the final crystal density profile, as shown in Fig. 2. Indeed, it is shown in Fig. 2 that less accurate solution procedures (less grid number or lower-order discretization) give better agreement with the experimental results to some extent. As shown in Fig. 1, the WS3 scheme gives always more reliable numerical solution. The numerical solution by using logarithmic monitor function, M= ~/a + (0 In(n) / c3L)2 on 40 moving grids is still insufficient to predict the final particle distribution in Fig 2. Therefore a new PBE containing agglomeration kinetics is needed in this case of the KzSO4/H20 system to fit the physical phenomena. Table 1. Comparison of experimental and simulation results for the batch natural cooling crystallization of K2SO4-water system. Simulation Experiment Fixed grid Fixed 40-grid Moving 40-grid 594 a Lwm 792 a 940 b (gm) 648 c 540 c 681 c 75 a CVwm 58 a 70 b (%) 86 c 77.4 c 85.4 c o
a:
Jones & Mulin (1974), b: Rohani & Bourne (1990), c: present study (with WS3)
4. CONCLUSION AND PERSPECTIVES
The moving finite difference method is applied to the dynamic simulation of the batch crystallization. Steep moving fronts appearing in the solution of the crystallization model are well captured. In particular, the moving grid method combined with the WENO scheme can efficiently solve the discontinuous-continuous particulate process in a stable manner. The method takes a reasonable calculation time, even through overhead for grid position location, since a small number of nodes is used. However, a crystallization model without agglomeration and breakage mechanisms may be limited to obtain a realistic numerical solution of the crystal density distribution at a final crystallization time. It is so planed to numerically solve a more complete PBE model with agglomeration and breakage of particles.
206
Owing to the accurate numerical procedure of the moving m e s h m e t h o d for PBEs, parameter identification of agglomeration kinetics will be encouraged in the future work using optimization loop. However the C P U time to calculate m e s h m o v e m e n t should a priori be diminished to accelerate optimization. A two-step solution procedure where the m e s h generation part and the physical model solution part are sequentially connected is also planed to diminish the calculation time. 1E+ll 1E+10
:
Experiment(Jones, 1974) Initial condition ~• WS3 fixed 40-grid FS1 moving 40-grid o WS3 moving 40-grid
1E+ll
o Experiment(Jones, 1974) Initial condition ,, FS1 fixed 200-grid
1E+10
1E+09
1E+09
~9 IE+08
.,~ 1E+08
1E+07
~ 1E+07
1E+06
r,,)~ 1E+06
100000
100000
o
[.
oooO D
10000 0
,
i
200
400
9
9
600
,
800
--
,.,
i~
i
,
g,
,
1000 1200 1400 1600 1800 2000
C r y s t a l size (L, ~tm)
0
0
10000 0
,
,
200
400
0
o 600
800
o
.
.
.
1000 1200 1400
.
-~
1600 1800 2000
C r y s t a l size (L, lttm)
Fig. 2. Comparison of simulation results with the experimental results for K2SO4/H20 system
REFERENCES [1] Huang, W. & R. D. Russell (1997), Analysis of moving mesh partial differential equations with spatial smoothing, SIAM J. Num. Anal., 34, 1106-1126. [2] Li, S. & L. Petzold (1997), Moving mesh methods with upwinding schemes for time-dependent PDEs, J. Comput. Phys., 131,368-377. [3] Lim, Y. I., S. H. Jeong, J. M. Le Lann & X. Joulia (2000), Moving finite difference method for tracking a shock or steep moving front, ESCAPE10, Italy, 55-60. [4] Jiang, G. & C. W. Shu (1996), Efficient implementation of weighted ENO schemes, J. Comp. Phy., 126, 202228. [5] Rohani, S. & J. R. Bourne (1990), A simplified approach to the operation of a batch crystallizer, Can. J. Chem. Eng., 68(10), 799-806. [6] Muhr, H. R. David & J. Villermaux (1996), Crystallization and precipitation engineering-VI. Solving population balance in the case of the precipitation of silver bromide crystals with high primary nucleation rates by using the first order upwind differentiation, Chem. Eng. Sci., 51(2), 309-319. [7] Lee, G. B., X. M. Meyer, B. Biscans, J. M. Le Lann & E. S. Yoon (1999), Adaptive finite difference method for the simulation of batch crystallization, Comp. Chem. Eng. sppl., s363-s366. [8] Brahmadatta, M., R. Kohler, A. Mitrovic, E. D. Gilles & M. Zeitz (2000), Symbolic discretization of population models for process simulation, ESCAPE 10, Italy, 547-552. [9] Kumar, S. & D. Ramkrishna (1997), On the solution by discretization balance equations by discretization-III. Nucleation, growth and aggregation of particles, Chem. Eng. Sci., 52(24), 4659. [10] Jones, A. G., J. Buds & J. W. Mullin (1986), Crystallization kinetics of potassium sulfate in an MSMPR agitated vessel, AIChE J., 32(12), 2002-2009. [11] Jones, A. G. & J. W. Mullin (1974), Programmed cooling crystallization of potassium sulphate solutions, Chem. Eng. Sci., 29, 105-118. [12] Budz, J., A. G. Jones & J. W. Mullin (1987), On the shape-size dependence of potassium sulfate crystals, Ind. Eng. Chem. Res., 26, 820-824.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
207
Effective Model Reduction for Analysis of Distributed Parameter Systems Yi Liu a* and Elling W. Jacobsen a* as3-Process Control, Royal Institute of Technology, S-100 44 Stockholm, Sweden The first step in solving PDE models is typically model reduction through discretization of the spatial variables. When the reduced model is intended for analysis, e.g., in control design, parameter fitting or nonlinear analysis, it is crucial to obtain a reduced model of low order. In this paper we evaluate different methods for obtaining low order models of infinite dimensional models. The focus is on so-called moving mesh methods, in which the grid used for discretization is made dynamic in order to adapt to the solution. By employing results from feedback control theory, we show how the mesh controller should be designed in order to minimize the error introduced by the model reduction. Both finite difference methods as well as a finite element method, orthogonal collocation on finite elements, are considered. The results are illustrated by application to a 1-D model including convection and reaction. 1. I N T R O D U C T I O N In order to solve nonlinear partial differential equations (PDEs) numerically, model reduction is usually required, that is, discretization of the partial differential equations to a set of ordinary differential equations. Some common factors such as accuracy, efficiency and stability are usually considered to determine whether a discretization method is suitable to solve a specific problem. For problems which do not have a critical requirement on efficiency, but rather on accuracy, such as in specific simulations, fine grids can be used to achieve acceptable accuracy. However, in many problems it is crucial that the reduced model is of a reasonably low order, e.g., in parameter fitting and model analysis. One class of methods which aims at low order models in model reduction of PDEs is the so-called moving mesh methods, e.g. [3], in which a mesh equation involving node speed is employed to calculate the meshpoint locations simultaneously with the solution of the differential equations. In principle, the idea is to concentrate a mesh, which has a fixed number of nodes, in regions of rapid variation of the solution. During the last two decades, moving mesh methods have been the subject of significant research, and a large number of different methods have been proposed in the literature, e.g., [ 1], [3], [5], [6]. The methods can roughly be divided into two categories, moving finite difference methods (MFD) and moving finite element methods (MFEM), depending on the spatial discretization method employed. Although the principle idea of moving mesh methods is relatively simple, there exist a number of problems with the proposed methods. These are mainly related to stability robustness *e-mail: liuyi @ s3.kth, se re_mail: jacobsen @s3.kth.se
208 with MFDs, and a high sensitivity to user defined parameters with MFEMs. In this paper, moving mesh methods are studied from a control point of view. The algorithm determining the mesh movement can be interpreted as a feedback controller. By analyzing the control problem, we determine a reasonable structure of the controller. Furthermore, we show that the control algorithm employed in most methods corresponds to pure I-control. Based on this, we provide a plausible explanation for the robustness problems encountered in many methods. We start by analyzing finite difference methods and then continue to develop a relatively simple moving mesh controller for a finite element method with orthogonal collocation on elements. 2. MFDS FROM A CONTROL POINT OF VIEW Consider of a general one-dimensional PDE :
Ut -- f(u,x,t),
x C ~,
0 < t < T,
(1)
with boundary and initial conditions u(x, to) = uo(x), b(u,x,t) = O. Introduce the mesh equation g(x(~,t),u) = O, in which x and ~ denote the physical and computational coordinates respectively. By employing the total derivative, (1) can be rewritten as
U = Uxk + f(x,u,t)
(2)
The equidistribution principle, employed in most moving mesh methods, involves determining a positive monitor function M(x,t), which provides some estimate of the computational error in the solution of the underlying PDE, and then equidistributing M(x,t) over the spatial domain for all t. Mathematically, it can be expressed as
fJo x ( ~ ' t ) M ( x , u ) d x -
~
/o'
(3)
M(x,u)dx
If differentiating (3) with respect to ~ twice, the right-hand side of the differential form vanishes. Hence it is natural to define the error measure as the left-hand side of the differential form of the EP
E(t)--
M(x(~,t),t)
~
(4)
The discretized form of (4) is
Ei(t) _ Mi+l(t)+Mi(t) 2(l/n) 2
(Xi+l(t)-xi(t))-
Mi(t)+Mi-l(t) 2(l/n) 2
(xi(t)-Xi-l(t))
(5)
In the following, we shall consider the equidistribution problem from a control point of view. Considering the error measure E as the output and the moving mesh x as the input signal, the problem is to determine a controller C which gives a small error signal E, employing feedback control with the node locations x as manipulated variables. Now write the system on a time-dependent matrix form
E(t)
= a(t)x(t)
where A(t) is a tri-diagonal matrix whose elements are given by (5).
(6)
209 Keeping the error measure E equal to zero for all t corresponds to perfect control. Therefore ideally A(t)x(t) = 0 holds. This is in fact equivalent to the DAE formulation discussed in [4]. As discussed in [4], there exist a number of numerical difficulties in solving the DAE problem, and some kind of relaxation is therefore required. To relax the bandwidth requirement, which corresponds to the relaxation employed in most existing MFDs, the error is forced to decay with a time constant Zl. Specify the closed-loop system as (7)
E(t) -- - L E ( t ) T,I
Combining (7) with the differential form of (6) yields the "ideal" moving mesh controller
A(t)2(t) - - 1 E
(8)
(t) - A(t)x(t)
"~I
It is computationally difficult to obtain the dynamic term A(t). Therefore, we neglect this term to obtain the mesh controller
1E(t) a(t)~(t) -- -'el
(9)
which is a pure 1-controller. Equation (8) in fact corresponds to the method MMPDE2 proposed by Huang et al. [3] which hence appears to be the "optimal" structure for mesh control. The mesh controller (9) is equivalent to the method MMPDE4, also proposed by Huang et al. [3]. Two types of monitor functions, arclength and curvature, are commonly used in MFD methods. The arclength monitor function is defined as M(x, t) -
1 § o~
()u(x,t)
~x
where tx is a
parameter affecting the grid density around the region of rapid solution variation. A spatial smoothing technique is often employed to reduce the possibility that adjacent intervals of the mesh nodes are highly different, which will affect the accuracy of the solution. i+p. (k/k + 1)i-JMj(t) where k > 0, p is often chosen to be 1 or 2. The final discrete l~4i(t) -- V "--'j--t-p moving mesh equations are obtained by replacing Mi(t) by A4i(t). A large number of different MFD methods have been proposed in the literature, see e.g., [1], [3]. The main difference between the methods stems from different approximations and discretizations of the EP. As pointed out by Li et al. [4], all available MFD methods can be seen to be regularizations of a DAE system involving the semi-discretized forms of the underlying PDE and the EP constraint. However, they often give poor robustness, are problem dependent and it is in general difficult to choose the regularization parameters for the resulting ODE. Except for MMPDE2, discussed above, which has computational difficulties in calculating the time-derivative of the monitor function, all other available MFD methods can be seen as model based controllers in which some dynamic terms have been left out. Furthemore, the controllers employed usually correspond to pure I-control. From control theory, it is well known that pure I-control may give rise to significant oscillations, or even instability, if the control bandwidth is pushed to high. This may explain why many MFDs experience oscillations and instability, depending on the underlying problem and choice of control parameters. In particular, at frequencies in which the term A(t), which is not considered in most mesh controllers, is significant one should expect problems using pure I-control. A possible remedy is to employ a controller which provides a phase-lift, e.g., ID-control. Due to space limitations we do not pursue this here.
210 3. O R T H O G O N A L COLLOCATION ON MOVING FINITE ELEMENTS A moving mesh may significantly increase the efficiency of finite difference approximation. However, finite differences are as such relatively inefficient for many problems. In general, discretization based on finite elements provide a higher accuracy over finite differences for a given model order. Thus, one might expect MFEMs to be more efficient than MFDs. A number of MFEM methods have been proposed in the literature. The pioneering work was done by Miller and Miller [5] who employed piecewise linear approximations in each of the finite elements. In this case, the mesh movement is based on minimizing the residual of the original equations written in finite element form, and can be closely associated with the equidistribution principle. As for the case of MFDs, the resulting set of DAEs needs to be regularized [5], thereby introducing a number of control parameters. The MFEM methods, while cited as highly efficient for many problems, have often been criticized for their complexity and sensitivity to the users choice of the control parameters, e.g., [2]. Sereno at. al. [6] proposed a method similar to that of Miller and Miller, but based on orthogonal collocation within the elements. The method requires the user to choose 6 control parameters, and no guidance is provided as to how these should be chosen. However, from the examples presented in [6], it appears that the choice of the parameters is critical. The main reason for the complexity and large number of parameters in available MFEM methods is probably that the authors derive a model based control algorithm, based on a model which in itself is highly complex. However, from control theory, it is well known that relatively simple controllers may provide good performance even on complicated processes when the principle of feedback is employed. We here propose to employ a relatively simple feedback controller for MFEMs, similar to the controller derived for MFDs above. As the underlying discretization method, we employ finite elements with orthogonal collocation within each element. Introduce the notation xij, i = 1,... n, j = 1,--. m + 2 for the computational mesh, in which n and m denote the number of elements and their interior nodes, respectively. Apply the collocation method within each element, i.e., Uij(t) de__fU(Xij t) = v'm+2 Z_.p= 1 lip(Xij)Uip(t) where li i s the (m + 2)-th order Lagrange polynomial in element i. The elements are connected by letting
Uil z Ui-l,m+2. To distribute the elements, we control the mesh according to the equidistribution principle. The location of the interior nodes of the elements are kept constant relative to the normalized length of the elements. The approximation error within each element is determined from residual computations, i.e., by substituting the approximate solution into the original PDE at some non-collocation points. We here evaluate the residuals at the points Sir, r = 1,..., m + 1, consisting of the midpoints between the collocation points. Define a (m+l) by (m+2) m a t r i x Orj- lij(Xir). We then have
def
m+2
m+2
air- U(Xir,t)= E lij(Xir)Uij- E OrjUij j-1
j-1
From the formulation (3), the residuals at Sir c a n then be computed directly from values of
211 l,tij on the computational grid
m+2 Rir -- ~ l i r - ( l t x ) i r ~ i r - fir -- E
OrjfliJ -- ( U x ) i r ~ i r - fir
j=l Hence the residual-based monitor function is defined as m+l
Mi -
~
IRirl (Xi,r+l -- Xi,r) + ~,
i -- 1 , " " , n
r= 1
Here 15 is a small constant to avoid numerical problems when the residuals approach zero. Finally, we employ a simple I-controller to asymptotically achieve equidistribution
1 (~f)i----~(y,~
Mi
1
t= l M i
n '
)
where (/~f)i
--
Xi,m+2--Xi,1
(10)
Note that only a single parameter, z, needs to be specified by the user. From a number of examples, we have found this relatively simple method to be both efficient and robust. However, more robust controllers including phase-lift will be considered in the future.
4. NUMERICAL EXPERIMENTS We consider here the reaction-convection problem
~~),rt ~i)x ~0 ~0 ~'c t ~)x
=
Da(1-ct)nexp
=
Da(1-~)nexp
(150) ~tl+~0
( ~0 ) ~'1+]30
(11) +6(0/4-0)
(12)
where o~ is conversion, 0 temperature, I: time and x position, all dimensionless. We discretize the model using moving finite differences and moving finite elements, respectively. The mesh controller is given by (9) and (10), respectively. To test the accuracy of the resulting reduced order models, we perform a simulation with the following parameters: 8 = 3.0, D a - - 0.15, 13= 2.0, n = 1.5 and 3(= 12.0. The initial profiles are steady-states obtained with coolant temperature 0n -- 0.3, and at t = 0 + 0n is changed to - 0 . 3 . A "reference solution" was obtained using moving finite elements with a very fine grid, corresponding to a model with approx. 400 ODEs. Figure 1 shows the solution obtained with the MFD using 17 internal nodes, i.e., a reduced order model with a total of 53 ODEs. The discretization was performed using a 1st order upwinding scheme. The control parameters were chosen as %1 = l e -- 3, ~ = 100, k = 0.1, p = 1. Also shown is the reference solution and a solution obtained with finite differences on a fixed uniform mesh with 100 nodes (200 ODEs). From the figure we see that the MFD model with 53 ODEs provides a significantly better approximation than the FD model with 200 ODEs. However, we note that a "tail" appears also in the MFD solution. Figure 2 shows the solution obtained with MFEM using 4 elements, each with 5 internal collocation points (51 ODEs). The control parameters were chosen as ~I = 1 . e - 3 and [3 = l e - 3. As seen from the figure we obtain a solution which is very close to the reference solution. Thus, we find that the MFEM is superior to the MFD for this problem.
212
0.2
0.4 x 0.6
0.8
1
Fig. 1. Conversion for x = 0 : .2 : 1 Solid - MFD, Dashed - reference, Dotted - FD
0.2
0.4 x 0.6
0.8
1
Fig. 2. Conversion for z -- 0 : .2 : 1 Solid - MFEM, Dashed - reference
5. C O N C L U S I O N S AND C O M M E N T S We have in this paper analyzed moving mesh methods from a feedback control perspective. Based on this, a reasonable structure of the multivariable mesh controller was developed, and it was shown that most controllers employed in MFD methods are essentially pure I-controllers. Based on the latter, we proposed a plausible explanation for performance and stability problems experienced when using existing MFD methods for some problems. Available MFEM methods are based on highly complex controllers, which furthermore contain a large number of control parameters that must be decided by the user. In this work we therefore proposed a relatively simple control algorithm for MFEMs which only contains two control parameters with a clear interpretation. A monitor function based on estimated residuals in the elements is developed to gain a more accurate error estimation.
REFERENCES [1] E.A. Dorfi and L.O'C. Drury. Simple adaptive grids for 1-d initial value problems. Journal of Computational Physics, 69:175-195, 1987. [2] R.M. Furzeland, J.G. Verwer, and P.A. Zegeling. A numerical study of three moving grid methods for one-dimensional pdes. J. Comp.Phys., 89, 1990. [3] W. Huang, Y. Ren, and R.D. Russell. Moving mesh methods based on moving mesh partial differential equations. Journal of Computational Physics, 113:279-290, 1994. [4] S. Li, L. Petzold, and Y. Ren. Stability of moving mesh systems of partial differential equations. SIAM J.SCI.COMPUT.., 20(2):719-738, 1998. [5] K. Miller and R.N. Miller. Moving finite elements, i*. SIAM J. Numer. Analysis, 18(6):1019-1032, 1981. [6] C. Sereno, A. Rodrigues, and J. Villadsen. The moving finite element method with polynomial approximation of any degree. Computers Chem. Engng., 15(1):25-33, 1991.
European Symposiumon ComputerAided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
213
Global Terrain Methods for Chemical Process Simulation A. Lucia and F. Yang Department of Chemical Engineering, University of Rhode Island, Kingston, RI 02881-0805, USA This paper briefly describes a completely different, novel and general approach to finding all physically meaningful solutions and singular points to chemical process models based on moving up and down the landscape of the least-squares function. The resulting algorithms are called Global Terrain Algorithms and consist of a series of downhill, equation-solving computations and uphill, predictor-corrector calculations from one stationary point to another. Key concepts and numerical results are illustrated using chemical process examples.
1. PREAMBLE Look at Fig. 1. Your eyes are immediately drawn to the lower portion of the figure and the curved valley that 'connects' the two singular points (i.e., a local minimum and saddle point in the least-squares function) and the solution (or global minimum of FTF). This valley and the associated implied connectedness is not a coincidence or something specific to the example shown in Fig. 1. It is a general mathematical property, independent of dimension, that holds for many, many models of physical systems under very weak conditions of twice continuous differentiability. In fact, distinct features of the least-squares 400 landscape (i.e., valleys, ridges, ledges, etc.) have solid theoretical foundation with clear algorithmic, 38o implementation and numerical implications! The main focus of this paper is the development of 360~ ~ ~ Z intelligent, reliable and efficient global terrain methods for finding all E 340 -Y. physically meaningful solutions and singular points of mathematical / models of physical systems. Think of valleys, ridges and ledges as 'lines with special properties' that connect solutions and singular points and the 0.0 0.2 0.4 0.6 0.8 1.0 concept of terrain-following as the conversion task of tracing out these 'lines' in a Figure 1" Connectednessof StationaryPointsfor a NonlinearCSTR multidimensional space.
i'Oi
,!
214
2. M A T H E M A T I C A L F O U N D A T I O N . Consider two neighboring level curves along the curved valley in Fig. 1. Note that the distance, say A, between any two neighboring level curves in the normalized gradient direction is largest exactly in the valley and that this distance decreases in magnitude as points move out of the valley along the same neighboring level curves (i.e., the level curves become more tightly packed together). Therefore the norm of the gradient of the least squares function must be smaller at any point in the valley than at any neighboring point on any given level curve since the same change in the least-squares function results from the largest change in distance. Thus the valley connecting the stationary points shown in Fig. 1 can be characterized as the collection of local minima in the gradient norm over a set of level curves. This same constrained minimum gradient norm property also characterizes ridges, ledges and other distinct features of the landscape. Mathematically, valleys, ridges, ledge, etc. can be defined as a collection of solutions, say v, to a set of general nonlinearly, constrained optimization problems v = {min gTg such that FTF = L, for all L e L
}
(1)
where F is a vector function, g = 2jTF, J is the Jacobian matrix of F, and where L is any given value (or level) of the least-squares objective function and L is some collection of level curves. That is, for any given level curve, we find the point on L that corresponds to a local minimum in the gradient norm. The collection of minima for all levels gives all (or part) of a valley, ridge, or ledge. Equation 1 forms the backbone for Global Terrain Methods and plays an important role in the development of predictor-corrector algorithms used to implement those ideas. Moreover, it is not necessary to know the set of level curves, L, a priori. In fact, L is actually a computational by-product of terrain-following. 3. T E R R A I N - F O L L O W I N G . Reliable and efficient downhill movement is easily resolved by a trust region or equivalent norm-reducing approach [1]. Uphill movement, on the other hand, is challenging because valleys, ridges and ledges do not define the only uphill directions and because numerical difficulties can occur near singular points. We suggest uphill exploration be done using a bounded predictor-corrector algorithm in which predictor steps are taken in uphill Newton-like directions and corrector steps are used to return iterates to a valley, ridge or ledge.
3.1. Predictor Steps and Uphill Newton-like Vector Fields. Uphill Newton-like steps are given by AZN = j-1F
(2)
where z is a vector of unknown variables. Properly bounded uphill Newton-like movement tends to naturally move along a valley, ridge or ledge - a fact that is clearly illustrated by the flow of the portion of the normalized uphill Newton-like vector field shown in Fig. 2. Suitable increase in the least-squares function can be obtained by controlling the size, %, of the
215
uphill Newton-like step. Using a Taylor series expansion of FTF, we can arrive at the rule (3)
t~ = min { II (j-IF)II, 2el (JTF)TAN I / [ I ANT(E FiHi)AN [ ] }
where AN is the normalized uphill Newton-like direction, Fi is the ith component of F, Hi is the Hessian matrix of the ith component function, and ~ is some tolerance used as a measure of 'truncation error'. From Eq. 3, predictor (or uphill Newton-like) steps are given by Azp = %AN
(4)
3.2. Corrector Steps. To ensure that predictor iterates do not wander too far, Eq. 1 is solved intermittently to return iterates to a valley, ridge, etc. When SQP methods are used, the solution is computed by iteratively solving min gTBAc + V2mcTlvI ~ such that gTAc = - (FrF - L)
(5)
where B = (jTj + • FiHi), Hi is the element Hessian matrix of the component function Fi, M is the Hessian matrix of the Lagrangian function defined by L = gTg _ )~(FrF - L), )~ is a Lagrange multiplier, and where FTF, g, J, B, etc. all depend on z. Physical bounds on variables or other inequalities are easily added to eithex Eqs. 1 or 5. Line searching or trust region methods can also be used to control the size of the corrector steps, ~ , by forcing reduction in the gradient norm to give Azc = ~ c
(6) Since gradient directions are orthogonal to the tangent to any level curve and the Newton-like and gradient directions are only exactly collinear in valleys, on ridges, etc., one simple but useful criterion for invoking corrector steps is
400
360
.r
340 i i o c a l ~ l n i ~ ~ _ . . , . ~ ~ . ~
y
320
301~.0
0.2
0.4 0.6 conversion
0.8
1.0
Figure 2: Flowof Uphill NewtonVector FieldAlongValleyfor a NonlinearCSTR
cos 0 = AgTAN < O
(7)
where 0 is the angle between the normalized gradient, Ag, and the normalized Newton-like step and | is some number very close to 1. We typically use 0.99 for the value of |
216
3.3. Detecting the Presence of a Singular Point. One very general criterion for terminating uphill exploration is given by the condition ANTjTJAN << ANT(E FiHi)AN
(8)
which is a necessary and sufficient condition for identifying any singular region, relies on the fact that normalized Newton-like steps align with the null space of the Jacobian matrix, and thus works equally well for minima, saddle points or maxima. When uphill exploration indicates the presence of a 'nearby' singular point, the overall algorithmic logic switches back to 'equationsolving' to look for that singular point using an acceleration technique - either quadratic acceleration, Krylov subspace methods like conjugate gradient or null space rotation techniques. Uphill exploration provides a 'good' initial guess for each subsequent singular point calculation.
3.4.Boundary Collisions. Uphill movement can also result in collision with a boundary of the feasible region. Once a boundary collision occurs, the global terrain algorithm returns to a previously computed singular point or solution and 'explores' the landscape in the 'opposite' direction. Thus collisions with a boundary of the feasible region are used to signal an end to the usefulness of exploration in a particular (eigen)direction. However, regardless of the complexity of the landscape boundary collisions can occur and can be used in a positive way to understand the underlying geometry of the connectedness of stationary points. 3.5. Initiating Movement Using Eigen Information. Initiating downhill movement from a saddle point or maximum to either a singular point of lower norm or a solution always takes place along an eigendirection, v, of negative curvature, which can be calculated from efficient eigendecomposition of the Hessian matrix of the leastsquares function using Lanzcos or some other eigenvalue-eigenvector technique [1]. The magnitude of the step in this negative eigendirection is determined from the expression o: = (-2FTF/s w
(9)
where here ~ is the largest negative eigenvalue of the Hessian matrix of the least squares function. An additional safeguard is used to ensure that norm reduction occurs on the initial downhill movement from any singular point. Initial uphill movement from a saddle point, local minimum or global minimum of the leastsquares function always takes place along a direction of smallest positive curvature (see Fig. 1). Here again, efficient eigendecomposition can be used to compute any necessary directions of positive curvature and the corresponding magnitude of the step is determined by choosing the largest cz such that the actual nonlinear value of the least squares function, say F~F(z + czv), is close to that given by FrF + ~2~L2vT(jTj + • FiHi)v, where v is an eigenvector of positive curvature.
3.6 Terminating the Global Terrain Algorithm. Think of the task of computing solutions and singular points as a tree structure in which the nodes of the tree are the stationary points and the branches of the tree are the equation- solving
217
n-pentane@T=360K,P=13.4189bar
and/or uphill climbing tasks needed to move from one stationary point to another. Under assumptions of twice titllltltl ~saddl soluetio, nP~ continuous differentiability stationary points are really only smoothly connected along valleys, ridges, etc. to 'neighboring' stationary points. Because downhill and uphill movement takes place in eigendirections of greatest negative curvature and least positive curvature respectively, the number of connections is limited to four or less depending on the nature of the Figure3:Terrain-Followinfor g theSimplifiedSAFTEquation stationary point. These connections can be easily catalogued so that cycling is avoided (i.e., previously computed stationary points are not computed again), symmetry is exploited (e.g., in finding complex conjugate solutions) and the maximum amount of available irfformation is used in initiating the next move. This limited connectedness makes it possible to determine when the global terrain algorithm has finished. ..
4. NUMERICAL ILLUSTRATION. Consider the task of finding the physically meaningful compressibility roots of the simplified SAFT (Statistical Associated Fluid Theory) equation of state of Fu and Sandier [2] for n-pentane at T= 360 K and p = 13.4189 bar. Because n-pentane is nonassociating, there are three parameters needed for the S AFI" equation - the square well depth (kt~ the molar segment volume at 0 K (v~176 and the number of segments (m). The parameters used in this illustration are taken from Luo [3] and have values 101.564 K, 18.574 ml/mol and 3.708 respectively. At the given conditions, there are seven compressibility roots and six singular points. These are shown in Table 1. Table 1 Roots and Singular Points of Simplified SAFT Equation for n-Pentane @ 360 K & 13.4189 bar Roots 0.482826 +/- 0.001293 (vapor-like) 0.057603 (liquid) 0.014399 0.008106 0.008089 +/- 0.002508
Singular Points 0.482817 0.350737 0.047699 0.012546 0.008217 +/- 0.001481
There is an easily computed lower bound on the liquid compressibility similar to the molecular co-volume, b, in cubic equations of state that is given by the expression ZOO= p(m~)~176 which in this example has a value of 0.2301. In practice, one would normally be interested in calculating only the physically meaningful roots; thus the computed lower bound is useful in this
218
regard. However, calculating a specific (group) of roots is complicated by the fractal nature of the basins of attraction which can cause iterates close to one root to 'jump' to another, perhaps physically meaningless root. Moreover, because the level curves for the simplified SAFT equation are extremely fiat, the valleys, ridges, etc. are very weakly defined except near the stationary points and this presents an additional challenge for the terrain-following approach. See Fig. 3. Using an initial compressibility estimate of Zo = 0.9, our Global Terrain Algorithm first finds the singular point, Z' = 0.482817 in 31 function calls (iterations). Performing eigendecomposition at the singular point [ 1] gives the canonical eigenvectors Vl = 1 + 0i and v2 = 0 + i, where v~ is an uphill direction and v2 represents a downhill direction. So in principle, the directions +/- Vl and +/- v2 need to be 'explored'. Downhill equation-solving calculations using a complex domain trust region strategy described in [ 1] initiated in either imaginary eigendirection +/- v2 finds the complex-valued solution, Z = 0.482826 +/- 0.001293i, in an additional 4 iterations. Because these complex-valued roots are known to be conjugate, since the temperature and pressure specifications are real-valued, only one needs to be computed. Eigenvector initiation in the uphill direction -Vl, since it is in a direction away from the initial guess Z0 = 0.9, and uphill predictor-corrector calculations find a singular region in 20 iterations. Quadratic acceleration to the singular point, Z' = 0.350737, which is a maximum on the real line, takes 4 more iterations. Eigen initiation in the direction Vl results in a boundary collision at Z = 1 in 5 iterations. Note at this stage of the calculations, the first singular point Z' = 0.482817 has four connections (a pair of complex conjugate roots, another singular point, and a boundary) while the singular point Z' = 0.350737 has one connection (the singular point at Z' = 0.482817). Thus we conclude that the complex conjugate roots are vapor-like roots. Eigendecomposition at the singular point Z' = 0.350737 gives the downhill direction, Vl = 1 + 0i. Subsequent equationsolving calculations, initiated in the direction - Vl since again it is in the direction away from the singular point at Z' = 0.350737, find the real-valued liquid root in 6 iterations and the algorithm actually finishes since all physically meaningful roots have been found. Thus a total of 70 iterations are required to find all physically meaningful, albeit some complex-valued, compressibility roots and singular points.
Acknowledgement. The authors would like to thank the National Science Foundation for support of this work under Grant No. CTS-9818130.
REFERENCES. 1. A. Lucia, I & EC Res., 39 (2000) 1713. 2. Y. H. Fu and S. I. Sandier, I & E C Res., 34 (1995) 1897. 3. Q. Luo, M.S. Thesis, University of Rhode Island (1999).
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
219
Estimation of a deactivation model for the methanol synthesis catalyst from historic process data. Ingvild LCvik
a, Morten RCnnekleiv b, Ola Olsvik band Terje Hertzberg a
aDepartment of Chemical Engineering, Norwegian University of Science and Technology N-7491 Trondheim, Norway bStatoil R&D Center, N-7005 Trondheim.. Norway The scope of this work was to develop a deactivation model for the methanol synthesis catalyst that includes the effect of temperature and water, based on historic process data from a methanol plant. A model on the generalized power law form was successfully fitted to process data from a limited period of time. The estimated model is of second order. No measurable effect of water was found, probably because the variations in the feed compositions were to small. The model parameters are not valid for the total catalyst lifetime because the deactivation process is fast in the beginning and slower after some time. Data from a larger period of time is needed to estimate a model that is valid over the total catalyst lifetime. It has been demonstrated that the historic process data contains enough information to estimate a catalyst deactivation model that describes the effect of temperature, but too little information to estimate the effect of the reaction mixture composition. 1. I N T R O D U C T I O N In the methanol synthesis, synthesis gas is converted to methanol over a lyst in a fixed-bed reactor. The following exothermic reactions occur [ 1]:
Cu/Zn/Al203 cata-
CO2-I- 3H2 ~
CH3OH+H20
(1)
CO+H20
CO2 + H 2
(2)
~
A flow sheet of the process is shown in Figure 1. The loop contains two reactors in parallel, but only one is shown in the flow sheet. The catalyst deactivates over time, and lasts for 3-4 years under normal operating conditions. Sintering is the main deactivation mechanism. The catalyst deactivates asymptotically, fast in the beginning and slower after some time. A good deactivation model is necessary to predict the methanol production and to optimize the operation of the reactor system. In previous work, the operation of the methanol synthesis has been optimized using a deactivation model from the literature [2]. Developing of a deactivation model for an industrial scale reactor involve several problems: Because of the asymptotic behavior, data for long times, preferably many years, are needed. Such time-consuming experiments can not be done in the laboratory. Planned experiments in process plants with large variations in input variables are expensive and can cause operation problems. Passive data from process plants contain less information, so it may not be possible to identify a model. The challenge in this
220
Fig. 1. Lurgi's methanol synthesis loop.
work has been to decide if the historic process data contains enough information, and if so, to estimate a mechanistic deactivation model. 2. DEACTIVATION MODEL FORM Few deactivation models are published, and they predict quite different deactivation rates. Most of the models are developed in laboratory scale and therefore predicts too fast deactivation [3-6]. There is agreement in the literature that sintering increases with increasing temperature and increasing fraction of water in the reacting mixture [7,8]. One model that considers the reaction gas composition is published [9], but the mechanism that is assumed is in conflict with other literature. This model also predicts too fast deactivation. A power law expression (PLE) with high order are normally used to model deactivation by sintering. A generalized power law expression (GPLE) describes the typical asymptotically deactivation better than a power law expression (PLE) [10]. The following deactivation model on the GPLE form is hereby proposed:
da dt
__ --kd (a
aeq) m
at=o -- ainit
[_~(1
kd :
k (1 + PH20 n) exp
1 )]
T
Tref
(3)
A limiting activity aeq is approached at infinite time. The term accounting for the effect of water is a first approach: if n is estimated to be different from zero water has an effect, and the term can be refined. Reactor temperature, partial pressure of water and activity are all functions of reactor position z and time, indexing is omitted for simplicity. 3. ESTIMATION The model parameters were estimated my minimizing the maximum likelihood objective function of the measurements. The simulation and estimation tool gEST/gPROMS were used in combination with Excel. The estimation problem consists of two parts: estimation of the initial activity profile ainit (Z) and estimation of the five model parameters in Equation 3. It was
221 more difficult to get a good estimation in the first part, because a continuous activity profile were estimated from only seven measurements.
3.1. Process data Historic process data from a limited period of about a year was used. This period is at the end of the catalyst lifetime, were the catalyst deactivation slows down. Data for a longer period was not available, because a complete variable set had not been recorded earlier. The following variables were used for estimation: 9 Input: Pressure, temperature, composition and flow in the reactor feed. 9 Control: Cooling water temperature 9 Output: Flow and water content of raw methanol, temperature at six axial measuring points in the reactor, and temperature at the reactor exit. Day mean values of all variables except water content were available. The water content was measured every week, and day values were obtained by linear interpolation. Several temperature measurements were taken at each point inside the reactors, and both the reactor temperature profile and reactor exit temperature were measured in the two parallel reactors. Mean values and variance were calculated from the temperature measurements. The variance in the raw methanol flow was estimated to 1%. The datas were smoothed by omitting unlikely data points and trip days.
3.2. Modeling The reactor was modeled by a two-dimensional, heterogeneous model with catalyst deactivation as the only dynamic effect. The model is presented earlier [11,2]. Because not all components were measured, two assumptions were made; (1) Other components than methanol and water in the raw methanol were neglected. (2) Methanol in the feed stream was neglected. The neglected components are present in very small concentrations, and the two assumptions have opposite effect on estimated methanol conversion. The reactors were operated without cooling in the lower section. This causes an after-reaction that raises the reactor exit temperature compared to the temperature measured in the reactor. To capture this effect in the model, cooling was omitted in the lower section of the reactor. The reactor model is discretized by backwards finite differences in the axial direction. The discretization points are not located exactly at the measuring points. Linear interpolation between the discretization points were used to calculate the temperature at the measuring points. To simplify the initial estimation problem, the initial activity profile was described by to parameters a)nit and a i2n i t . .
a init
~
a ~nit
a init
--
a ~nit -t-
forz (a2nit--a]nit)(Z--Zl)
=
0""Zl
for z =
Zl 9 9" z 2
Z2 - - Z l
a init
--
a2nit
forz
=
z2"'"
1
(4)
222 4. RESULTS 4.1.
Initial
condition
The two parameters in Equation 4 were successfully estimated with 10% standard deviation1, and gave a good fit of the temperature profile, see Figure 2. The deviation in the first measuring point is large. The temperature profile is steep around this point, so a small deviation in the location of the measuring point will cause a large deviation in the measured temperature. The standard deviation in this measuring point is also large, witch means that the contribution to the objective function from this measurement is small. Note that the deviation in the reactor exit temperature is small. The predicted initial production rate is 99.9% of the measured initial production rate.
9 Measured
Predicted
Predicted
Measured
1.1
40 3O
1.o
S 20
0.9
lO o o
!
i
0.5
1
Z
Fig. 2. Estimated and measured relative initial temperature profile.
4.2.
Model
0.8
,
0
100
'
,
,
200
300
Time[days]
Fig. 3. Estimated and measured relative methanol production rate.
parameters
The reactor exit temperature Tout caused problems in estimation of the deactivation model parameters. The estimation gave a large and systematic deviation in Tout that increased over time. The large deviation in Tout dominates the objective function and gives a poor fit of the other measurements. No systematic deviation in the reactor temperatures were seen. This suggest that there is an unknown effect on Tout that are not described by the model. Measuring error could be the cause of this effect. It was decided to exclude Tout from the estimation because if this reasons. Estimation without Tout gave a good fit of the production rate and reactor temperatures as can be seen in Figure 3 and 4. Some of the estimated model parameters are shown in Table 4.2. The model parameters k and E were estimated with 5 % standard deviation 1. Surprisingly, no measurable effect of water was found. The variations in feed compositions were probably not large enough to estimate the deactivation effect of water. Second order deactivation and a limiting activity aeq different from zero is consistent with the literature on GPLE models [10]. 1Values are confidential
223 Table 1 Estimated model parameters. Parameter
Value
m
2
n
0
aeq
ainit
1
Fig. 4. Estimated and measured relative reactor temperature at the 6 measuring points.
5. C O N C L U S I O N A deactivation model on the GPLE form was successfully fitted to historic process data from a limited period of time. The estimated model is of second order. No measurable effect of water was found, probably because the variations in the feed compositions were to small. The model parameters are not valid for the total catalyst lifetime because the deactivation process is fast in the beginning and slower after some time. Data from a longer period of time is needed to estimate a model that is valid over the total catalyst lifetime. It has been demonstrated that the historic process data contains enough information to estimate a catalyst deactivation model that includes the effect of time and temperature, but too little information to estimate the effect of reaction mixture composition.
224 6. NOTATION
E[J/mol]
Activation energy
aeq(Z)
ainit (Z)
FMeon[ton/day] Methanol production rate 1 ainit Gas constant a2nit R[J/molK] kd [day-1] T (z)[K] Reactor temperature Trey[K] Reference temperature k[day_l] Tout[K] Reactor exit temperature m
Tin[K]
Reactor inlet temperature
Tim[K] a(z)
Measurement i of T (z) Catalyst activity
n
Limiting catalyst activity Initial catalyst activity Parameter in eq.4 Parameter in eq.4 Deactivation rate Deactivation rate Deactivation order Parameter in eq.3
p142o(z)[bar]Partial pressure of water z[m/m] Axial reactor position
REFERENCES
1. K.M. Vanden Bussche and G.E Froment. A steady-state kinetic model for methanol synthesis and the water gas shift reaction on a commercial Cu/ZnO/Al203 catalyst. Journal of Catalysis, 161:1-10, 1996. 2. I. LCvik, M. Hillestad, and T Hertzberg. Modeling and optimization of a reactor system with deactivating catalyst. Computers and Chemical Engineering, 23:839, 1999. Supplement. 3. C. Kuechen and U. Hoffmann. Investigation of simultaneous reaction of carbon monoxide and carbon dioxide with hydrogen on a commercial copper/zinc oxide catalyst. Chemical Engineering Science, 48(22):3767-3776, 1993. 4. A. Cybulski. Liquid-phase methanol synthesis: Catalysts, mechanism, kinetics, chemical equilibrium, vapor-liquid equilibria, and modeling - a review. Catalysis review: science and engineering, 36(4):557-613, 1994. 5. G.W. Roberts, D.M. Brown, T.H. Hsiung, and J.J. Lewnard. Deactivation of methanol synthesis catalyst. Industrial & engineering chemistry, 32:1610-1621, 1993. 6. M. Sahibzada, D. Chadwick, and I.S. Metcalfe. Methanol synthesis from C02/H2 over Pd-promoted Cu//ZnO/A1203 catalyst: kinetics and deactivation. In M. el al. de Pontes, editor, Natural Gas Conversion IV, volume 107 of Studies in Surface Science and Catalysis. Elsevier Science B. V., 1997. 7. H.H. Kung. Deactivation of methanol synthesis catalyst - a review. Catalysis Today, 11:443-453, 1992. 8. J. Ladebeck. Improve methanol synthesis. Hydrocarbon Processing, pages 89-91, March 1993. 9. M.R. Rahimpour, J. Fathikalajahi, and A. Jahanmiri. Selective kinetic deactivation model for methanol synthesis from simultaneous reaction of C02 and CO with/-/2 on a commercial copper/zinc oxide catalyst. The Canadian Journal of Chemical Engineering, 76:153-761, August 1998. 10. C.H. Bartholomew. Sintering kinetics of supported metals: New perspectives from a unifying GPLE treatment. Applied Catalysis A: General, 107:1, 1993. 11. I. LOvik, M. Hillestad, and T Hertzberg. Long term dynamic optimization of a catalytic reactor system. Computers and Chemical Engineering, 22:707, 1998. Supplement.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
225
N o n l i n e a r analysis of an industrial a m m o n i a reactor with h e t e r o g e n e o u s model E. Mancusi a, P. L. Maffettone b, F. Gioiaa and S. Crescitelli a* a Dipartimento di Ingegneria Chimica Universit/t "Federico II" Piazzale Tecchio 80, 1-80125 Napoli, Italia b Dipartimento di Scienza dei Materiali ed Ingegneria Chimica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italia
A model for industrial ammonia reactors that includes intraparticle and interphase effects is studied with simulations and parameter continuation. A thorough description of the attractors both static and dynamics is given for typical operating conditions. 1. INTRODUCTION In a recent paper, Morud and Skogestad (1998) addressed the problem of an industrial incident in an ammonia production plant. They analyzed the reactor dynamics with a linear stability analysis, and showed that periodic solutions emerge from an Hopf bifurcation well within typical operating conditions. Subsequently, Mancusi et al. (2000) presented a dynamical analysis based on a continuation approach of the same pseudo-homogeneous model, and were able to explain the presence of periodic solutions far away from Hopf bifurcation points; different loss of stability scenarios were also detected and described. Although pseudo-homogeneous models are attracting for their simplicity, they cannot describe any intraparticle and interphase phenomena. In heterogeneously catalyzed reactions, mass and heat transfer between gas and catalyst surface, and the possible presence of gradients within the catalyst are often relevant. In this work the non-linear analysis proposed in Mancusi et al. (2000) is extended by considering a heterogeneous model.
2. THE MODEL The reactor consists of a series of three adiabatic beds with fresh feed between each bed, and feed preheating with the effluents. Figure 1 gives a sketch of the system. The heat exchanger and the mixers are assumed to have no dynamics, as in Morud and Skogestad (1998), since the reactant residence time within both is small compared with that in the fixed catalytic bed. The heat exchanger efficiency is considered as a parameter. The effect of the internal and external resistances on the reaction rate are accounted for by using global effectiveness factor rl defined as (Froment and Bischoff, 1990):
Corresponding author Email:
[email protected] (S. Crescitelli)
226 Rp
3 Ir ~ r~ (~o.~,r,.)dr q
__
0
(1)
w1
1 W2
2 [.t
W3
3
wh
Tf
Figure 1 A sketch of the ammonia reactor A pseudo-homogeneous model can be used when q is close to one. This is not the case for ammonia converters unless the catalyst pellets are very small, but this situation is usually avoided since it would cause exceedingly high pressure drops. Mass and energy balances are written for each catalytic bed and are reported in Table 1. Table 1. Model equations MASS BALANCE FOR GAS PHASE
0a~g --w Sgp~ at
c3o9g + p.,. ,; (~,~,r~),7, ~,~ (o,0 = ~o,~ az
(2:)
ENERGY BALANCE FOR GAS PHASE
a~
~ p~c~ at - - ~ - ~
a~
-h, av(r~-~.), T~(o,,)-~
(3)
227 MASS B A L A N C E
o=
~.
~
y2
F O R SOLID PHASE
a~ j + P ' ~
(O)s. .
, ~ ) ,,~,,]~=~ ~o, ~ I~o : o _
(4)
E N E R G Y B A L A N C E FOR SOLID PHASE
d~
(5)
The model assumptions are discussed in details elsewhere (Mancusi et al., 2001). Uniform initial conditions for temperature and concentration profiles in each reactor and in the catalyst pellet are assumed. It should be remarked that the pseudo-homogeneous model developed by Morud and Skogestad (1998) is recovered once the concentrations and the temperature in the catalyst pellets are constant and their values are equal to the corresponding concentration and temperature values in the gas phase. The model discretization is composite: A finite-difference scheme is used for eqs. (2-3), while the solid mass balance (eq. 4) is treated with an orthogonal collocation method (Villadsen and Michaelsen, 1978). The effectiveness factor defined in eq. (1) is calculated with an accurate Radau-Labotto quadrature. Time integration of the discretizated eqs (2, 3, 5) is performed with the VODE routines (Brown et al., 1989). 3. RESULTS AND COMMENTS The model is studied with two different approaches: parameter continuation and simulations. The continuation analysis is performed with the popular code AUTO97 (Doedel et al. 1997) and characterizes the model asymptotic behavior as one parameter value is varied. Simulations are used to elucidate transients behaviors. Both numerical approaches are based on the same discretization scheme mentioned in the previous section. All the results have been obtained for the parameter values reported in Mancusi et al. (2000). Figure 2 shows typical effectiveness factor profiles along the three beds for P=200 bar and s=0.628 at the steady state. It is worth noting that the effectiveness factor is always substantially smaller than unity. This fact enforces the use of heterogeneous modeling.
228 0 65
0.60
0 55
0.50
0 45
0.40 00
0.5
1.0
1.5 Reactor
2.0
25
30
length
Figure 2 Effectiveness factor profile under normal operating condition. Figure 3 shows a solution diagram obtained with the continuation approach. The outlet gas temperature is plotted versus the bifurcation parameter, i.e., the reactor pressure P. The exchanger efficiency is 0.628, and the other parameters are kept fixed to usual operating conditions (Mancusi et al., 2000). Periodic solutions are represented with the maximum value attained during the oscillation. It is apparent a qualitative similarity with the results obtained with the pseudo-homogeneous model (compare with Fig. 2 in Mancusi et al. (2000)). A high conversion periodic solution branch (see the inset in Fig. 3) is found. This branch emerges from the supercritical Hopf bifurcation (H1), and is initially stable. Then, after a fold bifurcation (F1), the periodic solution becomes unstable; a second fold bifurcation (F2) makes the solution stable again. This scenario is very similar to that found with the pseudohomogeneous model (Mancusi et al., 2000). As in that case, the presence of two folds (F 1 and F2) has an important effect. Indeed, although the Hopf bifurcation (H1) is supercritical, the presence of the bifurcations F1 and F2 makes the situation "catastrophic" since any small pressure reduction larger than ]API=IPH1-PFllresults in the jump from a stable static solution to the stable periodic solution with large amplitude oscillations (the stable branch between F2 and the third fold bifurcation F3). At P=PF3 the fold (F3) marks the onset of an unstable branch that eventually dies out in another supercritical Hopf bifurcation (H2). The fold bifurcation F3 plays an important role, as well. In fact, a decrease in the reactor pressure from its high conversion steady state set point is "recoverable" if P does not go below the leftmost fold bifurcation value (PF3). If, on the other hand, the fold is surpassed (P< PF3), then the model predicts the reactor shut down. A successive increase of the reactor pressure would result in a new "start-up" route along the low conversion branch up to the limit point at P=Ps2.
229
500
S l ~ ~
,-..,
?
,._.,
S 4o0
o
F3
~-~
~ ~
F2
..........
o o -) ~
~
~ o ~
~ ~-~
~
o o
o Oo
H2
"~
300
200 100
150
200
250
300
350
P [bar]
Figure 3 The solution diagram for e=0.628. Solid lines represent stable stationary solutions; dashed lines unstable stationary solutions; filled circles stable limit cycles; unfilled circles unstable limit cycles; filled squares Hopf bifurcations. The inset shows details of the zoomed rectangular region. A final comment is in order. As already mentioned, a substantial qualitative agreement with the pseudo-homogeneous results has been found. From quantitative point of view, however, there are significant differences. For example, the shutdown pressure PF3 is 146.57 bar while with the pseudo-homogeneous model it was around 138 bar. This discrepancy could be obviously relevant in process control.
NOTATIONS av Cpc Cpg Dcrl hf H P T Tf Ti To r
Rp rc
Specific surface area of catalyst pellet catalyst specific heat gas specific heat effective diffusivity heat transfer coefficient per unit particle surface area reactor length reactor pressure temperature, ~ Feed temperature Pre-heater inlet temperature Pre-heater outlet temperature radial position in spherical particle Radius of catalyst pellet reaction rate
t u w z AH
time superficial velocity gas mass flux reactor axial coordinate heat of reaction
/3
Prater number
e eg es q te~
exchanger efficiency void fraction of the bed void fraction of the catalyst pellet effectiveness factor effective thermal conductivity
pg Pcat
typical gas density catalyst density
(-AH)Def/(o,.Ir=gp )tea. T,.I1=%
230 COg COs
ammonia mass fraction in gas phase ammonia mass fraction in solid phase
Subscripts g in s
gas phase bed inlet conditions solid phase/catalyst
REFERENCES Brown P. N., G. D. Byme, and A. C. Hindmarsh, "VODE: a variable Coefficient ODE solver," SIAMJ. Sci. Stat. Comput., 10, (1989) 1038-1051. Doedel E. J., Champneys A. R., Fairgrieve T. F., Kuznetsov Y. A., Sanstede B., and Wang X., "AUTO97: continuation and bifurcation software for ordinary differential equations", July (1997) Froment G. F. and K. B. Bischoff, Chemical reactor analysis and design, 2nd ed., Wiley, New York, (1990). Mancusi E., Merola G., Crescitelli S. and Maffettone P.L., "Multistability and hysteresis in an industrial Ammonia reactor", ", AIChE J., 44, (2000) 824-828. Mancusi E., Maffettone P.L., Gioia F. and Crescitelli S., "A heterogeneous model for industrial ammonia reactors", submitted for the publication to AIChE J. (2001). Morud J. C., and Skogestad S., "Analysis of instability in an industrial ammonia reactor", AIChE J., 44, (1998) 888-895. Villadsen J. and Michelsen M.L., Solution of differential equation models by polynomial approximation, Prentice-Hall, Englewood Cliffs, (1978).
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rights reserved.
231
Mixed Mode Simulation- Adding Equation Oriented Convergence to a Sequential Modular Simulation Tool Marcelo Marchetti, Ashok Rao, David Vickery and the Aspen Plus Development Team Aspen Technology, Inc. 10 Canal Park, Cambridge MA, USA 02141 For years, a debate has raged over which method of solution for flowsheet simulations is b e s t - Sequential Modular (SM) or Equation-Oriented (EO). In truth, they both have strengths and they both have weaknesses and no one answer fits all problems. This paper describes some issues in designing a simulation environment that combines the strengths of both approaches and brings the power of both methods to all users - expert and novice alike. 1. I N T R O D U C T I O N Sequential Modular simulation of chemical processes has been used for decades and is the technique employed in most commercial simulation programs. The "modules" being solved may refer to material balance, energy balance, and equilibrium equations used to describe a single operation or individual unit operation models in a flowsheet. The variables that describe the state of the model or flowsheet are associated with specific equations describing the relationships among these variables. These variables and equations are then separated or "torn" into multiple groups that are solved in succession for the paired variables. An early advantage of the tearing or Sequential Modular methods was that the number of equations being solved at any given time was reduced. This made hand calculations or calculations on early computers which had limited amounts of available memory feasible. The tearing of the variables also means that initial estimates were required for only a subset of the variables and the methods are generally tolerant of poor initial guesses. In addition, these techniques can be tailored so that they are very good at solving specific types of problems. Finally, given the limited scope of the problem being solved at any given point, it is generally possible to give detailed diagnostics when problems in convergence are encountered. Thus SM simulation packages are generally very User friendly and easy for a novice to use. Some of the disadvantages of Sequential Modular or tearing methods are: 9 The solution algorithms are often too specific. One that works for distillation may not work for absorption- even though the governing equations are the same. 9 Convergence, while generally reliable when using an appropriate method, is often slow.
232 9 It can be difficult to find a pairing of equations and variables that always allow feasible solutions to the sub-models. Thus allowable specifications for a given algorithm may be limited. 9 Converging to non-standard sets of specifications involves adding more and more outer loop iterations - slowing convergence of the problem even more. While not new, Equation Oriented methods are gaining in popularity and availability due to increased performance of computing equipment and improved high-level computer languages. In essence, Equation Oriented techniques (also called Equation Based techniques) involve expressing all the governing relationships as residuals that have a value of zero at solution. In Sequential Modular form, the governing equations had been written as xj = gj(x)
(1)
In residual format, that same relationship could be expressed as fj(x) = xj - gj(x)
(2)
Some variant of Newton's method is typically used to drive all these residuals to a vlue of zero at the same time. These solution techniques are often quadratic in convergence- much faster than most SM techniques. Furthermore, all the equations describing the system are "open" and any reasonable set of specifications may be used to describe the system. Solving all the equations at once also means there are no nested convergence loops to handle - making these techniques well suited for systems with multiple and complex recycles. However, EO solution methods are not without weaknesses. Among the most common are: 9 It is generally necessary to be "near" the solution in order for EO methods to solve the equations, i.e., good initial estimates of the solution variables are required. 9 Derivatives of the governing equations are required for linearization of the residuals. 9 Solving large systems of equations is memory intensive. 9 Trouble-shooting convergence problems can be very difficult. If Sequential Modular and Equation Oriented techniques could be combined into a single framework, users have the potential to gain the strengths of both techniques while minimizing their weaknesses. 2. MIXED MODE CALCULATIONS When switching to Equation Oriented simulation, most users will be more comfortable if it is possible to migrate a flowsheet to EO solution in stages rather than being required to transition the entire flowsheet at once. Sections of the flowsheet that involve unusual specifications or complex recycles that benefit most from Equation Oriented solution can be converted first while the rest of the flowsheet convergence remains Sequential Modular. This allows users to tailor the level of conversion to create the best payback on their effort. This is possible when Equation Oriented solution is simply another convergence technique within a simulation tool.
233 Another way to mix the solution techniques to improve productivity would be to converge flowsheets that are slow or difficult to converge using SM techniques to a loose tolerance in the SM mode. The entire flowsheet can then be switched to EO convergence for final closure of residuals. This strategy can reduce time for converging highly recycled process flowsheets by an order of magnitude or more. In either case, it is necessary that the transitions between SM and EO simulation be straightforward, painless and work in both directions. 3. ENABLING EQUATION ORIENTED PROCESS SIMULATION TOOL
SIMULATION
IN
A
COMMERCIAL
Adding Equation Oriented solution techniques to an existing Sequential Modular flowsheet simulation tool requires the addition of many tools and features to guide Users through specifying, solving and interpreting results of EO simulation. Some of the key elements are described below. 3.1 The EO Variable Grid
The nature of EO techniques can lead to massive amounts of information being made available to Users. Variable specific information includes variable names, current values, units of measure, bounds, specifications, as well as other attributes. A tractable method of displaying this information is critical to understanding the Equation Oriented representation of the simulation problem. The nature of this information makes use of a grid, similar to that of a spreadsheet, ideal for displaying and manipulating the information. At a minimum, such a grid must be capable of presenting all current variable information to the user. Some simple functionality that will tremendously improve the usability of a variable grid include: 9 The ability to sort the entire grid based upon any variable attribute. Among these attributes are index, name, and specification. For example, it is often desirable to see variables that are held fixed for a given simulation at the top of the variable list. 9 Visual cues for specific actions, results or states of the variables. For example, backlighting a value which has been manually changed by the user or changing the color of variable which is at a process constraint (bound). 9 The ability to view the flowsheet variables in an hierarchical manner. For example, (1) the entire plant, (2) the purification section and (3) a specific distillation column in the purification section. 9 The ability to create customized variable grids in which certain criteria (queries) are used to limit the variables in the grid. For example, a grid that contains only the temperature variables of a distillation column or that always contains variables at a bound.
234
3.2 Specification Management Often, one of the reasons for switching to Equation Oriented mode is that the solution to the desired specifications is easier and/or faster using an Equation Oriented technique. In this case, the SM view of the problem may use one set of specifications that are chosen to provide reliable initialization of all variables. These specifications are then changed in the EO simulation to the desired specifications. (Re-)Specifying the problem in Equation Oriented mode involves two aspects. How do the Sequential Modular specifications translate into the EO view and how are specifications changed in the EO view? The answer to the first question is very simple: the default EO specifications should match those in the SM view. This makes it easier for the user to switch back and forth between SM simulation and EO simulation. A key aspect of the second question is the concept of Degrees of Freedom. For every variable in the EO formulation of the problem, there must be a direct specification of the value (via fixing the value or allowing the math engine to manipulate the value as a decision variable)
235 or a residual equation that allows its value to be determined. If these criteria are met, the problem is well posed. Furthermore, the EO simulation may be run in one of several modes; simulation, parameter estimation, data reconciliation or optimization; and it might be desirable to have a given variable behave differently in different modes. The specifications for each run mode must give a well-posed problem. The table below outlines a set of extended variable specifications that allow this change in behavior.
For the sake of maintaining a well-posed problem, it is desirable to change specifications in groups that preserve degrees of freedom. For example, one specification group may contain two variables whose original specifications were Constant and Calculated but are being changed to Measured and Parameterized. Defining a specification group involves listing the variables for which the specifications are to be changed and the new variable specifications. A result of changing the specifications in multiple groups is that the order in which the specification groups are processed can affect the behavior of the problem if a given variable appears in more than one specification group. Thus, it is necessary to be able to specify the order of specification group processing.
3.3 Connection Management Connection management for the EO view of the flowsheet can be viewed as having three components: port definition, connection definition and processing, and visualization in the process flow diagram. A port can be viewed as an arbitrary collection of variables. That collection may be a single variable and the "port" appellation is generally dropped, the variables representing a material stream, or it may be more abstract. Connections are generally added in the EO view to represent constraints that were either ignored or implicit in the SM view. One example is reaction kinetics data that is stored in a single location and used by all models in the SM view but app2ars as explicit variables in each reactor model in the EO view. Defining a connection is a simple matter of declaring a connection source (port) and a connection destination (port). The declaration of source or destination is used to help with specification management in the EO view. Every connection introduces a new equation to the system. In order to preserve the degrees of freedom, this means that some specification must move from Fixed to Free. When connecting a Constant to a Calculated variable, the Constant
236 will become Calculated and the declaration of source or destination is redundant. However, if two variables which are both Constant are to be connected, the source variable should remain Constant and the destination variable will become Calculated. As with the specification groups, the final problem definition may be dependent on the connection processing order. Finally, it is desirable to have an indication of this information flow on the process flow diagram (PFD).
3.4 Objective Functions Very often, a process simulation is used to determine optimal operating conditions for the process or a reconciled material balance for the plant. In these cases, the ability to define an objective function for the simulation is a necessity. Objective functions can be generally grouped as one of three types: (1) linear, (2) sum of squares, and (3) general. Many profit functions can be expressed as a linear objective function: Profit = Z(Product Rate * Value)- Z(Feed Rate * Value) - Z(Utility Rate * Value)
(3)
Most data reconciliation cases to match plant operating conditions have an objective function that can be expressed as the minimization of the error in plant measurements (or is the expected standard deviation for the value) : Error = ~{(Model Value-Measured value) 2 / o}
(4)
Customized objective functions can be used in situations where the above forms are not sufficient. In general, terms within the linear and general (custom) objective functions could be other objective functions.
3.5 Other Issues The issues listed here are only the beginning. Other areas of interest include online connectivity and the concept of measurement models, exploration of sensitivity of flowsheet outputs to operating conditions and inputs via derivative information and diagnosing convergence difficulties for the Newton based convergence methods. 4. CONCLUDING REMARKS This paper is intended to give readers a feel for some of the issues involved in designing and using a flowsheet simulation tool with Equation Oriented solution capabilities. The users will have varying degrees of expertise in both Sequential Modular and Equation Oriented flowsheet simulation skills. Ideally, the resulting simulation tool will allow the user to work comfortably in the realm in which they are most versed while making dabbling and translation to the other as easy as possible.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
237
Adaptive optimal operation of the Fenton's batch process for industrial wastewater treatment E. C. Martinez 82and G. D. L6pez Instituto INGAR (CONICET) and Universidad Tecnol6gica Nacional Avellaneda 3657, $3002 GJC Santa Fe, Argentina
This work presents a novel methodology for on-line adaptive optimization based on a lumped model of the Fenton's batch process for industrial wastewaters containing organic compounds. A simplified reaction pathway comprising only of complete and partial substrate oxidation is proposed to describe the very complex network of reactions occurring in the Fenton's chemistry. This model is integrated together with a two-step sequential adaptive estimation-optimization procedure. In the estimation step, the kinetic parameters of the lumped model are estimated on-line using as inferential measurements of waste degradation the oxidation-reduction potential (ORP) and carbon dioxide concentration. The optimization step uses the estimated oxidation rates to make a correction to the Fenton's reagent feeding strategy so as to maximize the overall rate of pollutant degradation. This estimationoptimization task is repeated at selected points in time until the desired ORP endpoint is achieved. Results are presented to illustrate the improvements that can be achieved in the Fenton's process automation strategy.
1. INTRODUCTION Despite the continuous efforts spent in developing more efficient/effective chemical manufacturing processes, successful wastewater treatment is still a major problem waiting to be solved. In fact, the number of low-cost alternatives available for dealing successfully with industrial wastewaters is still very narrow, and in some respects process treatment developments have not kept up with advances in manufacturing technology or in analytical methods. One reason for this lag is the wide variety of contaminants that might be part of the waste streams, their irregular flow rate patterns and their ever-changing compositions. Also, there has been an over-reliance on biological treatment to solve all waste problems. Even though a properly operated biological treatment is an inexpensive and relatively simple way to deal with many different types of industrial wastes, it cannot be used for wastes containing toxic, refractory or inhibitory organics 1. Many of these compounds do not really degrade biologically and pose a threat to biota and human populations. Such wastes are common in industries associated with chemicals, pharmaceuticals, insecticides, dyes and inks, explosives, petroleum processing, plastics and adhesives. Chemical oxidation is increasingly gaining attention to treat this type of industrial wastewaters. One cost-effective alternative is the socalled Fenton's reagent resulting from hydrogen peroxide with an iron catalyst2'4. The key for a widespread usage of the Fenton's process as a standalone solution or integrated with other technologies is achieving a breakthrough in the degree and type of automation used.
82Author to whom all the correspondence should be addressed. E-mail:
[email protected]; Fax: +54 342 4553439.
238 2. THE FENTON'S PROCESS
Today it is well known that the oxidation mechanism discovered by H. J. H Fenton is due to the extremely reactive hydroxyl radical generated by the catalytic decomposition of hydrogen peroxide in an acidic medium. Thus, many organic substrates are attacked and destroyed by the hydroxyl radical tl-41. The key piece in the Fenton's batch process is a semibatch reactor where, through the oxidation process, the toxic wastewater is reacted with inexpensive ferrous sulfate and hydrogen peroxide to yield oxidized intermediates and eventually carbon dioxide and water. Complete oxidation of the pollutant load is dependent on the ratio of hydrogen peroxide to organics, whilst the effective rate of oxidation is heavily dependent on the iron concentration and temperature evolution. The amount of iron needed is low, but the reaction is highly exothermic, so that to prevent excessive heating, the peroxide addition should be carefully controlled. Some form of heat exchange may be called for to diminish excessively long treatment duration. The pH should be adjusted to the region 3-6 since if the pH is too high the iron precipitates as Fe(OH)3, which might serve as a catalyst to decompose the H202 to oxygen thus creating a hazardous condition. The core of the process is made up of a stirred reaction vessel, which normally works nonpressurized and an operating temperature between 30 and 40 ~ Metering pumps are used to feed a ferrous sulfate catalyst solution and industrial strength (35-40%) hydrogen peroxide in a (w/w) 1:5 ratio, while the pH is maintained between 3 and 4 by sensing it and adding acid o base accordingly. To control the reagent addition rate sensors for ORP, carbon dioxide, oxygen and temperature are required. Other necessary equipment include level measurement and control, a venting system for volatile organic compounds, means for adding a flocculant and an air sparger system tied to an organics detector I. 3. LUMPED KINETIC MODELING The detailed mechanism for the Fenton's oxidation process of organic compounds is very complex 3. Even for a pure compound such as Chlorobenzene the exact reaction pathway is unknown 4 and many intermediate oxidation products are formed. In some cases, the oxidized intermediates formed are recalcitrant or refractory such as quinones, hydroxylated aromatic and dimeric compounds. The build-up of these intermediates might constitute a "bottleneck" in the series of partial oxidation reactions. The reactivity of a given substrate will thus depend on its constituents and the oxidized intermediates they give rise upon treatment. To model this extremely complicated reaction mechanisms and relate it to readily available measurements such as ORP we can only resort to lumped reactions and lumped concentrations such as the Total Organic Carbon (TOC) content involving all organic pollutants and their oxidized intermediates. The overall mechanism by which hydroxyl radicals oxidize an organic substrate can be represented as follows S + O H " --~ I + C O 2 + H 2 0
(1)
where the overall load of organic compounds in the waste stream is lumped as an aggregate compound S, and the overall pollutant concentration will be represented as the total organic carbon (TOC). Similarly, oxidized intermediates formed whilst adding the Fenton's reagent are all lumped together in the aggregate compound L The following pair of parallel lumped reaction pathways are used here to describe the Fenton's process.
239 Complete substrate oxidation S + O H " ---> CO 2 4- H 2 0 *;
Partial substrate oxidation S +OH" - - + I + H 2 0
(2)
Previous research on Fenton's oxidation mechanisms for different types of organic substrates emphasizes the multi-step first-order kinetic behavior regarding both IOn'] and organic compounds 4. Assuming a first-order kinetic behavior for both s and I in the lumped reactions (2) and using pseudo-kinetic parameters kl and k2 that are dependent not only on the reactor temperature but also on the p H and reagent' s addition policy. The overall dynamics of the oxidation process can be expressed as
a[I__]]=
_ d[S___J = (k 1 + k2 ) [ S ] ;
dt
k2 [S]
dt
(3)
Integrating this pair of equations from the initial condition [s]0 = Voc]0 and [I]o = 0 for t= 0 and bearing in mind that [s]+[I]=Voc] at any time t, it only takes elementary calculus to derive [TOC] [TOC]o
k2
k1
kl + k 2
kl + k 2
= ~
+
x exp[-(k
I +
k2).t ]
(4)
Equation (4) provides a compact representation of the time dependent TOC concentration in terms of parameters that indirectly account for reagent addition policy along with temperature and pH evolutions. The kinetic parameters can be easily estimated by fitting experimental data (TOC) to the model (4) using any nonlinear regression technique such as a neural network. For optimal operation of the Fenton's process, the key issue to be solved is how to adapt on-line the reagent feeding strategy upon estimation of kl and k2. 4. ADAPTIVE OPTIMAL OPERATION Variable amounts and types of organic pollutants in the waste stream imply not only batchto-batch variations but also time-varying rates of oxidation within the same batch. Consequently, on-line estimation of rates kl and k2, and subsequent adjustments to the reagent's feeding strategy through dynamic optimization are necessary to minimize treatment time 5. The only problem is that direct TOC measurements are both expensive and delayed for on-line optimization. In this work, we propose to use instead two inferential measurements that can be readily available on-line at low cost: oxidation-reduction potential (ORP) and the production of carbon dioxide in the reactor. The ORP reading of the reactor content for a given operating condition is directly related to how much refractory to oxidation by the hydroxyl radicals are the organic pollutants present. In terms of the kinetic parameters of the lumped model, this can be simply stated as:
1
O R P oc ~ k1+ k2
(5)
This inverse proportionality relationship explains why ORP steadily increases whilst oxidation is progressing as only the more refractory compounds are left in the reactor. Thus, ORP has a strong correlation with the overall rate of oxidation. To differentiate between partial and total oxidation another measurement is needed. From equation (2) it is clear that kl is strongly correlated with the [CO2] in the outlet stream.
240 Since the kinetic constants kl and k2 are affected by the Fenton's addition policy, temperature and pH, the estimation is based on a rather short time series of these highly relevant variables, named inputs hereafter for short. A functional relationship ,p(.) that relates kl and k2 to the chosen inputs is learned off-line using prediction errors of infrequent and sparse measurements of TOC as a fitting criterion. To compactly represent the relationship q~(.), a feed-forward neural network is used. An initial set of training examples is generated off-line and later enlarged incrementally as more batches are treated. Fig. 1 summarizes the key points of the proposed on-line estimation block. The optimization problem can then simply stated as: "finding the minimum feed rate of the Fenton's reagent that maximizes the rate of pollutant degradation" k1 M i n Max. F
• exp[-(k 1+ k 2) .At]
k1+ k2
At
(6)
Subject to: {k,,k2} = qo(F,-) O <- F <- Fmax
where Fm~x is imposed to prevent runaway conditions and dangerous accumulation of unreacted hydrogen peroxide. It is noteworthy that knowledge about the initial TOC content is not needed to solve the optimization problem (6) on-line, assuming that q~(o) is available. However, it is convenient to consider taking two or three samples in each treatment to improve the fitting of q~(o) for next treatments. Anyone might be tempted to think that maximizing the rate of degradation in (6) does necessarily produce the trivial solution 1:- G,a~. However, this will depend on the reactivity of organic compounds and the rate of production of radical hydroxyls. If the degradation is effectively controlled by the intrinsic reactivity of the organic substrate there is no point in accumulating Fenton's reagent in the reactor. Very often, the local rate of pollutant degradation in (6) varies with F according to a saturation-like function. The very point of seeking on-line a solution to (6) is to determine how far away the current addition policy is from the "optimum." Sometimes, it may require increasing the reagent feed so as to speed up oxidation reactions. Conversely, it may be advisable to lower addition rate because the same oxidation rates can be achieved with less addition. Once a change to the feed rate is decided and implemented, a new estimation step is initiated to re-estimate the lumped model kinetic parameters followed by another optimization of the addition rate. Even though it is less common, the ratio catalyst to hydrogen peroxide can also be used as an optimization variable. In some cases, a minor increase in this ratio has produced a dramatic improvement in the rate of oxidation when combined with a tighter pH control around a selected value. The proposed approach can easily incorporate this and other extensions.
4.1. Batch-to-batch learning The key requirement to solve the on-line the optimization problem (6) is to model (data-driven) the function {kl,k2}= q0(.) from batch-to-batch. Thus, measurements, decisions about reagent additions and outcomes (both intermediates and final) from previous treatments are carried over to next batches through a predictive capability that helps speed up pollutant oxidation via Fenton's reagent. To achieve this, a neural network approximation is used here based on its versatility and the capability for adaptation as more training examples are added. Choosing the right structure for the network is not so difficult because we already know what are the inputs that influence most on the oxidation rates. There are only two structural issues to be solved: i) the amount of 'memory' (previous readings) to
inductively
241 incorporate in the set of inputs, ii) and the number of weights. Regarding the first one, due to the time-varying nature of oxidation rates is not recommended to use an excessively long time frame of previous data. Using measurements that cover a time interval between 15 and 20 minutes is a good choice. Always make sure that each data subset is obtained according to the reagent addition strategy that produces it. Whenever possible, it is advisable to use averages obtained from conveniently filtered data and, if automatic control is well done, for pH and temperature is practical to use simply the chosen set-points as inputs. With regards to the number of parameters in the neural network approximation is important to account for the well-know 'bias-variance' dilemma. This issue is related to finding a proper balance between the size of the training set and the number of weights (or hidden nodes) in the network. Initially the number of training examples is necessarily small, but its size will increase as more batches are treated and data gathered is used to generate new training examples. The very point of adding new examples is to provide the network with information that is not already present in the training data set. The addition of similar examples creates the problem of 'overfitting" that diminishes the generalization capabilities of the model. On the other hand, adding new examples requires extra degrees of freedom (weights) to allow learning the additional features associated with the relationship q~(*). Thus, to avoid 'underfitting' the hidden layer should incorporate new nodes as additional examples are introduced in the training data set. Considering the network architecture in Fig. 2, where there exist nine inputs and one bias, enlarging the hidden layer by one node after 10 new training examples are incorporated, guarantees a good trade off between fitting error and generalization. It is worth pointing out that the neural network should be re-trained whenever its structure is changed. 4.2. Results
To illustrate how adaptive optimal operation of the Fenton's process can be achieved with the proposed approach, let's consider now a very simple laboratory scale study carried out using a l-liter reactor. The batch reactor set up is a very conventional one allowing strict temperature control through a thermostatic bath at 38 ~ and efficient pH control in the range 3 to 4. ORP measurements in the liquid bulk and carbon dioxide concentration readings in the outgoing stream are the only special features that are required. The aqueous mixture to be treated is a made up of 50 mg/L of phenol which is quite reactive to the Fenton's reagent, along with 50 mg/L of 2-4-6-Trichlorophenol, a much more refractory compound 2. The reagent addition policy used for this synthetic waste is of multi-shoot type where a given portion m of the total amount M (stoichiometric plus 15% excess) of reagent required is added at each step. Each reagent addition policy is only constrained by the fact that Z m = M at the end of each treatment. To allow time for oxidation reactions to occur a fixed addition cycle of 15 minutes was set. ORP and carbon dioxide readings were averaged over five minute intervals. Thus, three different aggregates providing a trend for these key variables will be available between additions. Samples for TOC assessment were taken at the beginning, and after three new additions have been made (i.e., 45 minutes). Since each treatment takes an average of 3 hours, each run provides between 3 and 4 new training examples for approximating q~(-). The neural network model used for estimating the rate parameters k l and k2 is shown in Fig. 2. The model has 9 inputs, a bias and two outputs. After 10 treatment runs have been completed and with 35 training examples, the prediction error when the "optimal" addition strategy was used for the treatment run #11 is shown in Fig. 3. The corresponding addition profile (normalized) is depicted in Fig. 4. The well-known Matlab | "Neural Networks" and "Optimization" packages were used to implement the adaptive optimization approach.
242 5. R E M A R K S
A novel two-step sequential adaptive optimization for the Fenton batch reactor based on a lumped kinetic model has been proposed. The rate of degradation is estimated on-line using the oxidation-reduction potential (ORP) and carbon dioxide concentration as inferential measurements. The optimization step uses the estimated oxidation rates to decide on-line a correction to the Fenton's reagent feeding strategy. Information gained from successive batches is accumulated in a predictive model using a batch-to-batch learning scheme. REFERENCES [ 1] R. J. Bigda, Chem. Eng. Progress, 91 (1995) 62. [2] F. J. Benitez, et al., Ind. Eng. Chem. Res., 38 (1999) 1341. [3] D. L. Sedlak and A. W. Andren, Environ. Sci. Technol., 25 (1991) 777. [4] A. R., Bowers, et al., Wat. Sci. Tech, 21 (1989) 477. [5] D. Ruppen, D. Bonvin and D. W. T. Rippin, Computers and Chemical Engineering 22 (1998) 185.
Fig. 1. On-line estimation of Oxidation rates using the lumped model.
Fig. 3. TOC-decayprediction compared for run #11.
Fig. 2. Neural Network predictive model
Fig. 4. Fenton's reagent addition profile for run #11
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
243
Generalized Reactor Model: An O b j e c t Oriented Approach to Reactor Modeling Artm S. Moharir a, Sunil S. Shah a, Ravindra D. Gudi a, Brian M. Devereux b, Kurt Vanden Bussche b, and Ganesh Venimadhavanb aIndian Institute of Technology, Bombay, India bUOP
LLC. USA
Industrial reactors differ from each other in many aspects such as geometry, hydrodynamics and reaction kinetics. The paper focuses on the development of a generalized modeling tool that can accommodate the above variations and help in the analysis, design, and synthesis of more novel reactor configurations. An object-oriented framework has been used for the reactor modeling and analysis. In the proposed framework, the design was accomplished by deconstructing the reactor model into simpler objects (such as reactor geometry, reaction network, reaction kinetics, etc.) and then defining the state and behavior of these objects as well as their interrelationships. The implementation of the generalized model has been done in a Microsoft Component Object Model (COM) framework which allows the model to be used by any application or programming language that can function as a COM client. Case studies, involving industrially significant applications, have demonstrated the potential of the framework to solve reactor design and synthesis problems effectively. 1. INTRODUCTION The need for accurate process models in the design and analysis of process plants cannot be overemphasized. However, the development and deployment of accurate process models of specialized and complex process operations has hitherto not been possible. This is due to a lack of appropriate modeling environments that promote rapid development and aid the use of these models in various routine tasks of process engineering. The limitation of flowsheeting technology in enabling fast and flexible assembly of detailed and accurate process models has long been recognized. In a recent paper, Bieszczad et al. (1999) have introduced the paradigm of phenomena based modeling and propose it as a flexible alternative to unit-operation based models that are currently offered by most flowsheet simulators. Different applications in process engineering require varying rigors of modeling and a quick way to develop rigorous models would be to build upon simpler models and add the required degree of complexity in them. Towards this objective, Marquardt and co-workers (1996) have proposed proper structuring of the process models and a formalization of their representation in an overall object-oriented framework. This object-oriented representation with structuring was shown to greatly facilitate the task of model development through reuse and adaptation of existing models. A number of object oriented modeling languages such as ASCEND (Piela et al. (1992), OMOLA (Nilsson, 1993) ,MODEL.LA (Stephanopoulos et al. (1990)) and gPROMS (Oh and Pantelides,
244 (1996)) have also been proposed and allow for the declarative and structured knowledge representation; this permits models of varying degrees of rigor to be represented and used in a flexible way depending on the requirement of the application at hand. The focus of this paper is on the development of a generalized environment for modeling and analysis of chemical reactors. Flowsheet simulation technology, as it exists today, has rigorous unit modules, and has attempted generalization to some extent, for almost all the major unit operations such as heat exchange, compression, single or multi-stage fluid contacting devices (flash, distillation, absorption, extraction etc.). However, for the reactor, which is typically at the heart of any flowsheet, most simulators provide either simple yield based reactor modules or a set of idealized models like CSTR, plug flow, etc. with limited capabilities for defining reaction kinetics. Perhaps, one of the reasons for the lack of generalization of reactors is that they occur in a broad variety in terms of their geometric configurations, hydrodynamics, reaction types involved, reaction kinetics, and the phases involved in them. Thus, building reactor models quickly, correctly, and economically still remains a challenging goal. The object-oriented programming (OOP) and component-based programming paradigms offer many advantages for reactor modeling over conventional structured programming techniques. OOP techniques allow for more extensible and maintainable models. In addition, a proper object-oriented design makes model development more intuitive since the underlying objects are faithful abstractions of familiar physical concepts. A modular, component-based implementation allows the model to be consistently used in a variety of development environments. In this paper, the design and development of an object-oriented framework for reactor modeling and analysis is discussed. This design was accomplished by deconstructing the reactor model into simpler objects such as reactor geometry, reaction network and reaction kinetics, since these are the features that contribute to the diversity in reactors. It has been shown that these simpler objects can be generalized relatively easily. By appropriate definition of the state and behavior of these objects as well as their interrelationships, it is shown that the overall reactor structure can be reconstructed. Various issues related to the design and implementation of the generalized model in an object-oriented framework, for deployment on commercial user platforms, are discussed and illustrated with a case study. 2. GENERALIZED REACTOR MODEL As stated earlier, at a broad level, reactors differ from each other in terms of their geometry, hydrodynamics, reaction network, and reaction mechanisms. Each of these aspects presents itself with a large variety, the combinatorial nature of which is perhaps quite difficult to generalize at the reactor level. However, a closer look at these features helps to reduce the dimensionality of the problem and to identify and generalize the commonality in the variety that these features offer. Figure 1 shows the skeletal interaction of these features in building up a reactor model. 2.1 Approach to Generalization The object-oriented framework has been designed to incorporate the generalizations at various hierarchical levels to support efficient model building through inheritance and code reuse. At the highest level, the model structure has the following three classes: a) Boundary: This class defines the control volume across which the conservation equations are written. There is a US patent pending on this methodology. The reactor equations
245 thus generated can be solved only after information regarding the reactions and phase interactions, which take place are specified. This is the objective of the next two classes, viz. the reaction and the inter-phase class. b) Reaction: Quite aptly named as ARIS (A Reaction Information Semantic) in our generalized modeling environment, this class defines the nature of the reactions that occur in the reactor and is broadly defined in terms of the following statements: (i) Stoichiometry: This statement is designed to facilitate the synthesis of the reaction terms that need to be incorporated in the model equations. The stoichiometric statement is made up of species that take part in a reaction. (ii) Kinetic Statement: This statement is designed to accommodate various commonly occurring simple as well as complex rate expression forms such as power law and LHHW forms. Additionally, a template class for user defined rate kinetics is also included in the kinetic statement. (iii) Thermodynamic Statement: This statement is concerned with the specification of the heats of reaction for the various reactions and equilibrium parameters for the reversible reactions. The thermodynamic statements corresponding to phase equilibrium are considered in the third class viz. Inter-phase. Figure 2 summarizes the structure of the ARIS class. I GeneralizedReactorModel: GRM [
I ~ o"oo
I
r
I
I
4,
ARIS
I
'nut
I ReactorModel
Inter-Phase ~
I
Output
I Solver
I
i
"1
Para. Est.
I
--I~
Figure 1" Data Flow Diagram: GRM A REACTION INFORMATION SEMANTIC: ARIS
I
COMPONENT
]
THERMO I REACTION
STOICHIOMETRY [
IRR/REV REACTIONS RATE CHOICE: 9 ELEMENTARY 9 POWERLAW 9 LHHW
RATEMODEL
--....
RATEMODEL PARAMETERS
Figure 2. Class Structure for Reactions
PHAs
I
246 c) Inter-phase: This class defines the phase equilibrium thermodynamics and interphase heat and mass transfer kinetics for multi-phase reactors. Like the reaction class, the phase class structure is also designed to facilitate the calculation of depletions from, or additions into, each phase due to mass transfer, in a generalized fashion. For a given number of phases, the class structure features the automatic generation of interfaces and their templates for characterization of the mass transfer resistances across each interface through a phase interaction matrix. In addition to the above broad class structure that defines the overall rigor of the generalized model, other classes that are used to customize the generalized model are those that define the component, solver and parameter estimation attributes. The component class is designed to connect to any component data bank for the component properties or accept data from a user interface. Specialized solvers for solving the resultant set of model equations (algebraic, differential, differential-algebraic, partial-differential) and procedures for their customization to the problem at hand are defined in the solver class. The parameter estimation class defines various procedures to set up the generalized reactor model for estimating kinetic/hydrodynamic parameters from experimental data. Again, a specialized optimizer has been deployed to perform the task of parameter estimation. The solvers and the optimizer have been object-enabled using their earlier, rigorously tested, legacy code forms.
2.2 Implementation Issues Implementation design of GRM closely follows the client-server architecture currently prevalent in the Information Technology industry. Different GRM classes discussed in the previous sections form the modelling/simulation engine or the server. The server has been developed using the Microsoft Component Object Model (COM) technology. Each of the simple GRM objects, is provided with a COM based interface, thereby permitting quick instantiating of different object models. COM structure also permits rapid development of customized user interface for entering reactor/reaction parameters. COM based implementation allows the server to be used by any application or programming language that can function as a COM client. This allowed rapid development of a spreadsheet-based tool for kinetic parameter estimations and the successful integration with a commercial process simulator. GRM server has been connected to Microsoft Excel for carrying out stand-alone modelling and simulation of reactors and for parameter estimation. GRM server has also been connected to a commercial simulator. A rigorous reactor model can be developed on the stand-alone package. Subsequently the same model may be integrated into any flowsheet developed by the commercial simulator. This permits consistent use of reactor models across different application environments. In addition to increasing GRM's technical capabilities, work is also in progress to enable the server to support multiple clients simultaneously, enable remote machine access and allow the server access through inter and intra nets. A significant number of case studies involving systems with adiabatic/ isothermal, multiphase, axial or radial, membrane reactors, reactors with complex geometries have been tested for simulation and kinetic parameter estimation. A simple illustrative example is considered below.
247 3. ILLUSTRATIVE CASE-STUDY
The methodology was applied to a transalkylation reaction system of industrial importance. The reaction scheme is as follows:
A+C<=>2B B->D where B is the main product and D is an undesirable by-product. The reactions were assumed to be elementary and the heats of reaction were obtained from the literature. The reaction rate constants were written in the form:
E k = A' * A * e x p ( - ~ -
RT
E'
)
RT
where A and E are the pre-exponential factor and the activation energy and A' and E' are the parameters that are estimated using the optimization program. To illustrate the flexibility of the object oriented approach, the same kinetic object ARIS was used in two different reactor configurations, namely, a PFR, and a CSTR. Robustness is of prime importance for the wide range of applications in which models are used. To ensure this, data is collected over a wide range of process conditions, using state of the art equipment. Some of the more advanced tools that are routinely applied include TAP, TSR and TEOM. TAP, or Temporal Analysis of Products, is used to probe the reaction mechanism in detail, by sampling a catalyst surface using high precision pulse valves and looking at the effluent using millisecond time resolution. This allows the scientist to formulate a set of rate equations that accurately describe the chemistry that is taking place on the catalyst. This technology has received its share of attention in the literature (e.g., Ebner and Gleaves (1986), Gleaves et al. (1988), and Svoboda (1993)) and has successfully been applied at UOP for various technologies. Once the mechanism is established, the TSR or Temperature Scanning Reactor is used to collect data over a wide range of temperatures and space velocities, linear velocities, and feed variations. This allows weighing of the importance of various reaction pathways and building of a model that is robust over a wide range of conditions and compositions. Papers on the TSR methodology include Rice and Wojciechowski (1997), Wojciechowski (1997), and Wojciechowski and Asprey (2000). Finally TEOM, or Tapered Element Oscillating Mass analyzer (e.g., Chen et al. (1996); Rebo et al. (1998); and Lodeng et al. (2000)), allows one to study the deactivation of catalyst systems while simultaneously monitoring yields and selectivities at various conditions. A detailed description of the effect of changes in feed slate and process conditions on catalyst life allows the refiner or petrochemical producer to make more informed decisions in planning, scheduling and load optimization. All of these data typically fit into the generalized reactor model through the ARIS routine, often using the solver routine for in situ parameter estimation. 4. CONCLUSIONS An object oriented framework for generalized reactor modeling has been proposed in this paper. The framework structure is designed to support axial or radial multiphase flow configurations and most typical forms of kinetic expressions. The model can be programmatically extended to support arbitrary kinetics. The Microsoft Component Object Model (COM) based implementation allows the model to be used by any application or programming language that can function as a COM client. This allows the rapid
248 development of a spreadsheet-based tool for kinetic parameter estimations and the successful integration with a commercial process simulator. Case studies, involving industrially significant applications, have demonstrated the potential of the framework to solve reactor design and synthesis problems effectively. Acknowledgements: The authors would like to thank Mr. Parag C. Patil and Mr. Ashish Y. Joshi from liT-Bombay and Dr. Hemant W. Dandekar from Eastman Chemicals, Kingsport, TN for their contributions in developing this methodology. REFERENCES
1. Bieszczad, J., Koulouris, A. and Stephanopoulos, G., MODEL.LA: "A phenomena based modeling environment for computer aided process design," AIChE Symp. Ser, 323, p438-441 (1999). 2. Bogusch, R. and Marquardt, W., " A formal representation of process model equations," Computers and Chemical Engineering, 19, $211-$216 (1995). 3. Chen, De, Gronvold, A., Rebo, H.P., Moljord, K., Holmen, A., " Catalyst deactivation studied by conventional and oscillating micro-balance reactors," Appl. Catal., A 177(1), L l-L8 (1996). 4. Ebner, J.R., and Gleaves, J.T., U.S. Pat. US 4,626,412 (2.12.1986). 5. Gleaves, J.T., Ebner, J.R., and Kuechler, T.C., "Temporal analysis of products (TAP)- a unique catalyst evaluation system with sub-millisecond time resolution," Catal. R e v . Sci. Eng., 30(1), 49-116 (1988). 6. Lodeng, R., Chen, De., Jacobsen, C.K., and Holmen, A., "A study of coke formation kinetics by a conventional and an oscillating micro-balance on steam reforming catalysts," Stud. Surf. Sci. Catal., 130D (International Congress on Catalysis, 2000), 3639-36 (2000). 7. Marquardt, W., " Trends in Computer Aided Process Modeling," Computers and Chemical Engineering, 20, (6/7), 591-609 (1996). 8. Nilsson, B., " Object Oriented Modeling of Chemical Processes," Doctoral Dissertation, Department of Automatic Control, Lund Institute of Technology, Sweden (1993). 9. Piela, P.C., McKelvey, R.D. and Westerberg, A.W., " An Introduction to Ascend: its language and interactive environment," Journal of Management and Information Sciences, 9, 91 - 121 (1992). 10. Rebo, H.P., Chen, D., Blekkan, E.A. and Holmen A., "Application of the oscillating microbalance catalytic reactor: kinetics and coke formation over Pt-Sn/A1203 in propane dehydrogenation," Stud. Surf. Sci. Catal., 119 (Natural Gas Conversion V), 617-622 (1998). 11. Rice, N.M., and Wojciechowski, B.W., "The temperature scanning reactor. II: Theory of operation," Catal. Today, 36(2), 191-207 (1997). 12. Stephanopoulos, G., Henning G. and Leone, H., " Model.LA: A Language for Process Engineering- part I and II," Computers and Chemical Engineering, 14, 813-869 (1990). 13. Svoboda, G.D., "Fundamental transport-kinetics models for interpretation of temporal analysis of products (TAP) reactor transient response data with application to reactive systems," Ph.D. Thesis, Washington University, St. Louis (1993). 14. Wojciechowski, B.W., "The temperature scanning reactor. I: Reactor types and modes of operation," Catal. Today, 36(2), 167-190 (1997). 15. Wojciechowski, B.W., and Asprey S.P., "Kinetic studies using temperature scanning: the oxidation of carbon monoxide," Appl. Catal., A 190(1,2), 1-24 (2000).
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
249
Step Restriction for a Bounded Newton's Method w. Morton, L. Kozel, P.P.S. Lim, D. Douglas University of Edinburgh, School of Chemical Engineering Edinburgh, Scotland EH9 3JL
The robustness of Newton's method is improved by modifications which consist of a partial solver for singular equations, use of variable bounds and optional step size bounds through a trust region, step restriction strategies and optional insistence on norm reduction in the NLAEs. The singular equation solver reliably allows escape from singular points. Most problems in a test set could then be solved with hard bounds but no trust region. Of the restriction methods, a partial LP reduction of the linearised residuals was the most effective. Use of a trust region usually helps convergence, but not always. Insisting on norm reduction was a hindrance, except in terminating runs trapped away from a solution. We recommend a methodology for solving NLAE systems. At present this is a procedure to be applied in stages by the user rather than a fully automated algorithm. 1. I N T R O D U C T I O N Systems of nonlinear algebraic equations (NLAEs) h(x) = 0 require solution in process simulation and design. Newton's method offers second-order local convergence but can fail to converge from an arbitrary starting guess. Newton linearises the NLAEs at an estimate xi giving a linear equation system gi =_ Jis~i+ l + hi -- O
(1)
where hi - h(xi) and Ji is the Jacobian matrix with elements Ji,pq -- ~ h p / ~ X q evaluated at Xi. Eqs. (1) are solved for the Newton step si+ aIn [ 1], strategies to improve robustness of Newton-like methods are reviewed. The approach in [ 1] was to convert equations (1) into a linear programming (LP) problem. Differential homotopy methods can also be used to improve robustness [2] but we do not include them here, though they may work better on some of our more difficult test problems. The measures we propose to promote convergence are all simply related to Newton's method, and allow the Newton step (s N) to be taken when it is deemed safe to do so. We consider (a) bounds on the variables to prevent calculation errors; (b) an optional trust region (TR) to limit the step; (c) a 'singular equation solver' (SES) which allows partial solution of eqs. (1) when J is singular; (d,e) two classes of step restriction (SR) strategies, needed when the Newton step violates step bounds; and (f) optional enforcement of norm reduction (NR) in ]]h II. These methods were tried on many of the problems in [ 1] and on a non-ideal dewpoint problem [3].
250
2. ROBUSTNESS STRATEGIES (a) ' H a r d ' upper and lower bounds are applied to all variables to prevent calculation errors like division by zero. They can also be used to isolate physical solutions, e.g. keeping mole fractions in [0, 1]. In the tests, we classified bounds as 'BB' as in [ 1]" 'PH'" physical bounds if different from [ 1]; 'LP'" large positive bounds (0 _~ x _< L); or 'LPN'" large positive or negative bounds ( - L _< x _< +L). L is a large value (e.g. 1010 or 1012), and does not restrict steps in convergent runs. Smaller 'large' bounds (+100) were used for arguments of 'exp' to prevent overflow. To avoid calculation errors, some variables are kept strictly positive even in the 'LPN' case. (b) A square trust region can optionally be used to restrict the step, si, at iteration i. For a TR of size Pi and upper and lower hard bounds x U and x L, the step bounds are Smin ~_ Si ~ Smax where Smi n = max(x L - - X i - 1 , - p i ) and Snag - - min(x U - - X i _ l , +p/). The initial TR size, Pl, must be given as a control parameter. If the nonlinear residuals h are close to the linearised residuals g after a step we increase the TR, and vice versa. Optimisation texts (e.g. [4], p.96) compare the actual and predicted reduction of the residual norm. Not all our step restriction methods guarantee reduction in the norm even of the linearised equations, so we generalise the idea. Let L1 norms of the equation residuals at the start and end of the (i + 1)th step (s) be: f0 = Ilhill - Zj Ihj(xi) l
gl --liCe§
Ej Ihj(xi) + ~kJjkSkl
fl - - [ I h i + l l [ - ~jlhj(xi+s)l where j and k run from 1 to N, the number of equations and variables. The linear norm change for the step is defined by ~, - f0 - g l and is positive if the linear norm is reduced. A nonlinear contribution (NLC) to the norm at the end of the step is defined by v - fl - g l and gives the nonlinearity across the step. The basic TR adjustment formula is p - min(qmax, max(qmin, I~/Ivl)) Ilslloo where Ilsll.o is the infinity norm of the previous step. L/v is the multiple of the present step which would give zero norm reduction (fl - f0) assuming that the NLC is quadratic and ~,, v > 0. This formula reduces the TR if there is NL norm increase and linear norm decrease, which is necessary with NR enforcement. If the NL norm decreases (0 < v < L), the TR is increased. qmax and qmin prevent rapid change in the TR. We set them to 2.0 and 0.5 respectively. (c) A Singular Equation Solver (SES) has been written to deal with points where J is singular. This often occurs at a starting guess. Eqs. (1) have no (unique) solution for singular J but the SES allows a partial solution for the step, which permits a well-posed problem to escape from the singularity. In the SES, all components of the step (s) are initialised with a default value. During Gaussian elimination we simply skip any columns which have no pivot greater than a small threshold value. After elimination, these 'singular' columns are (logically) moved to the right of the matrix, the right side vector is modified and backsubstitution proceeds as normal. The default step vector is usually zero. This often moves the solution away from a singular point, but sometimes singularity remains. We term these cases 'severe singularities'. When this occurs the singular variables must be given an arbitrary non-zero step. We used a value of 0.001 for components of the step at severe singularities: a small default step may lead to a large subsequent step as we are still close to the singularity. (d) Newton step restriction strategies are needed, since the solution of eqs. (1) does not
251 always satisfy step bounds. Two classes of methods which can give quite different restricted steps were compared. In the first class, the Newton step, s N, is solved and then restricted as follows. 1. For the jth variable (j = 1...N), let ~j = m a x ( a j E [0, 1]) such that: ~ J 4 ~> Smin,j if sffj < 0, or (zjs~j~ Smax, j if s~j > 0. ~j is the largest fraction of s u which satisfies the step bounds for sj. 2. Let [3 -- min{~j, j = 1 . . . N I ~j > A} where A E [0, 1] is a parameter defining the particular method. 3. Calculate a provisional step: s = [3su. 4. If there are any variables with ~j < A then s still violates a step bound. For those variables set sj - stain,j if s~j < 0, or sj -- Smax,j if S~j > 0. Setting A = 0 gives the pure 'fractional Newton' (FN) step, which is the largest fraction of the Newton step satisfying the step bounds. FN guarantees linear norm reduction (gl < f0) and ensures that linear subsets remain satisfied. However, FN easily becomes trapped on a hard bound when the Newton step crosses the bound, and solution terminates. We found that FN often failed in this way, and did not consider it further. If A = 1 the algorithm independently forces all variables which cross a step bound back on to the bounds. We call this the 'Variable Chopping' (VC) method. Solution estimates keep moving much better than in FN but there are theoretical drawbacks. Norm reduction is not guaranteed even for I[g[I. This complicates the TR adjustment procedure ((b) above), and precludes a convergence proof since reduction of [[h[I cannot be guaranteed for small steps. All connection with the Newton step direction may be lost, which can cause failure. However, VC often works well, e.g. on utility problems [5]. If A is set to an arbitrary small value, say 10 -1~ the FN step is taken except when a variable already on a bound tries to cross it. This variable is held on the bound but the others take a fraction of their Newton step values. We call this approach 'Hybrid Fractional Newton' (HFN). HFN has the good properties of FN when it takes a FN step but less readily becomes trapped by a hard bound. However, when any variable is trapped against a hard bound there is again no guarantee of linear norm reduction. (e) LP-based methods are another approach to SR. Whereas HFN and VC modify the Newton step after it has been calculated by the SES, we can solve an LP (as in [ 1]), to minimise the L1 norm [[g(s)[]l and find s. Rigorous minimisation is not necessary to reduce this norm. We have written a 'Partial LP' (PLP) solver which moves variables one at a time from s = 0 to give a descent step in I le(s)lll. PEP progressively satisfies as many as possible of the linearised equations subject to step bounds. Once an equation is satisfied it remains in the basis. PLP is illustrated in Figure 1, which also shows how PLP often has a well-defined solution when the Jacobian is singular, unlike the Newton step. At the start of NLAE solution by either SES or PLP, we perform a sparse reordering based on the structure of J. In SES, this cannot affect the solution unless J is singular. In PLP the reordering fixes the order in which variables are moved and this can affect which equations and variables are unpivoted. Figure 1 shows how a different PLP solution is obtained if s2 is moved before Sl. We tested two initial reorderings: (1) the SPK1 algorithm (our default) which uses minimum row count [6]; and (2) a 'minimum column / minimum row' (MCMR) reordering which uses minimum column count first. Although SPK1 and M C M R may give different restricted steps, descent directions and solutions are affected much more by equation
252 scaling both in PLP and in the rigorous LP method used in [ 1]. If the non-zero residual values in Figure 1 (10 and 5) were interchanged, the PLP solution would be at the 'north-west' comer of the TR.
Fig. 1. PLP solution steps for singular linearised equations
Trust Region
1
i=10 2
se of
current step
s
1
first
I
j2
s first II
~
Linearised equations I=0 1 I=0 2 have no solution PLP solution steps denoted by
,-\/\\ I1= 5 \
~"\ \
....
or
PLP solutions
(f) N o r m reduction in the nonlinear equations can optionally be enforced. Acceptance of a step then requires f 0 - fl > E ( f 0 - gl), where ~ is a small number which we set to the equations convergence tolerance, to ensure finite norm reduction. Even if this option is active, step rejection is considered only if there is strict linear norm reduction: gl < f0. Hence in VC, and to a lesser extent HFN, NR is applied partially, but PLP gives linear norm decrease allowing consistent enforcement of NR. 3. TEST P R O B L E M S Most tests were performed on problems described in [ 1]. We tested their problems 1 - 12, 15 - 24 and 26 - 33 but did not try pipe network (13 - 14) or distillation problems (34) or a contrived algebra problem (25). The problems are small, having at most 14 variables, but some are relevant to reaction engineering or equilibrium separations. Our approach has also proved effective on substantial flowsheeting problems involving distillation, heat exchanger networks and utility cogeneration systems. For this study we added a non-ideal dewpoint problem for acetone and water on which, unusually, the VC method performed poorly. This has 10 variables and uses Antoine and Wilson models for vapour pressure and activity coefficients. We gave three starting guesses: one good (the solution with ideal VLE), one with ballpark unscaled values and one poor, with all variables guessed as 1.0. (Pressures were in mm Hg). Bounds were essentially 'LP', but variables in log arguments had small positive lower bounds and Celsius temperature had a negative lower bound large enough to prevent division by zero in the Antoine equations. Upper bounds were large, except for mole fraction variables (1.0). 4. RESULTS Our test set comprised 33 problems, separately counting three versions of a catalysis problem with different parameter sets (problem 29 in [1]). Guesses were taken from [1], and other
253
guesses were tried. For many problems and guesses we gave alternative bounds, usually by trying the bounds in [1] and also the less constraining 'LP' and 'LPN' cases. There were 99 guesses and 159 guess & bound cases. Full results are given in [3]. First we ran the guess and bound cases with hard bounds but no TR. 71 of the cases converged with no step restriction, having an average iteration count of 13.4. Another 54 cases converged without a TR for all SR methods but encountered hard bounds. From these we can compare the efficiency of the SR methods: SR method HFN VC PLP - SPK1 PLP - MCMR
Average iters 12.5 12.2 16.4 16.5
SD of difference - from HFN
- from SPK 1
1.6 10.7 2.5
6 of these cases had singular guesses but the resulting steps were not affected by bounds. No method shows a significant advantage, though there is a hint that PLP is slightly slower. Another 6 cases with singular or severely singular guesses converged for all SR methods, but they were not counted because, without a TR, iteration count depended strongly on the default step size. Both SES and PLP easily coped with all singularities, provided SES was given a non-zero default step at severely singular points. In 16 cases without a TR, at least one SR method failed but at least one converged. HFN failed on 11 of the 16, VC on 7, and PLP on 3 with MCMR or 1 with SPK1 reordering. PLP is apparently the most robust method. The remaining 12 cases could not be solved by any SR method without a TR. However, 8 of these converged if PH or LP bounds were relaxed to LPN, albeit to an unphysical solution in some cases, e.g. with negative concentrations. We also tried trust regions on all cases. The TR often reduced iteration count, especially from poor guesses when no hard bounds constrained the steps. However, the opposite could also happen and in a few cases, introducing a TR destroyed convergence by trapping the path close to a local minimum of Ilhll. In all the cases where no SR method converged without a TR, we always found a TR of some initial size and at least one SR method which converged to a solution within physical or BB bounds. A successful TR may need to be small, especially when without a TR the path terminates on bounds. The TR appears to play a similar role to the bound-shifting recovery procedure in [ 1], though it works differently and is not guaranteed to succeed. It can be important in some cases to prevent the path reaching hard bounds early on. Especially with HFN, it is easy to estimate from a failed run without TR the maximum 91 which avoids an initial step to bounds from an interior guess. In difficult problems, a value smaller than this was usually successful. HFN and PLP are the most robust step methods with a TR, each working better on different problems. VC is seldom superior to both and is often worse, because decoupling of variables from the equations and the Newton step is harmful. NR enforcement seldom improved iteration count or robustness despite its theoretical importance. Iteration counts tended to increase, most markedly when nonlinear equations had large residual scales and the TR was small. Once the solution approached such an equation, TR increase was prevented by norm increase (fl > f0) even after quite modest steps. This effect was most noticeable with PLP step restriction, which does the most effective job at reducing residual
254 norms early on. It was also seen for poorly scaled problems without NR enforcement, albeit less severely. All our test runs were unscaled: scaling is discussed in [3]. The only real use we found for NR enforcement was to terminate runs with a TR which otherwise cycled endlessly near a minimum of llh I1. 5. PROPOSED METHODOLOGY
Based on the test results in [3], we propose a solution strategy given an NLAE problem and a starting guess. 1. Apply bounds to prevent calculation errors. If desired, let the bounds be tight enough to isolate physical solutions. Try Newton with no TR using HFN or PLP with SPK1 reordering, which is likely to be the most robust SR method. 2. At a severe singularity, apply a non-zero default step when using SES. 3. If 1 and 2 are unsuccessful, and the solution path is trapped on bounds, try relaxing the bounds. Try with PLP and HFN step restriction. 4. If 3 fails or gives an unphysical solution, try a TR. The HFN run from 1 or 2 provides a maximum Pl which keeps the first step away from bounds [3]. Try with a smaller TR using HFN. Repeat with smaller TRs to prevent trapping on bounds, or to find other solutions. If HFN fails try PLP. 5. If the TR gives endless fairly small steps use NR to terminate the path, probably near a minimum of[lh II. In this case rescaling of the equations may help, as proposed in [1 ]. This approach at first omits the TR, to avoid trapping the path near a minimum of I lhll. However, we saw this in only a few cases. In many others, a TR reduced the iteration count by keeping the solution path to reasonable values. REFERENCES
[1] Bullard, L.G. and Biegler, L.T., Iterative Linear Programming Strategies for Constrained Simulation, Computers and Chemical Engineering, 15 (1991) 239-251. [2] Wayburn, T.L. and Seader, J.D., Homotopy Continuation Methods for Computer Aided Process Design, Computers and Chemical Engineering, 11 (1987) 7-25. [3] Morton, W. and Douglas, D., A NLAE Solver incorporating Bounds, Partial Singular Equation Solution and Trust Regions, in preparation (2001). [4] Fletcher, R., Practical Optimization (2nd ed), Wiley, New York (1987). [5] Rodriguez-Toral, M.A., Morton, W. and Mitchell, D.R., Using New Packages for Modelling, Equation Oriented Simulation and Optimization of a Cogeneration Plant, Computers and Chemical Engineering, 24 (2000) 2667-2685. [6] Chen H_S. and Stadtherr M.A., Sparse Matrix Methods for Equation-based Chemical Engineering Flowsheeting, I - Reordering phase, Computers and Chemical Engineering, 8 (1984) 9-18.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
255
Improving Robustness using Homotopy as an Open Solver in a Dynamic Simulation Package Jorge R. Paloschi AspenTech Ltd. Sheraton House, Castle Park, Cambridge, United Kingdom Email:
[email protected] We present a homotopy implementation for improving the robustness of nonlinear equation solving in the commercial dynamic simulation package Aspen Custom Modeler TM. The implementation is realized by making use of open solver interfaces available in the product. The code implementing the homotopy solver is presented as a Dynamic Load Library and communicates with the simulator only via the interfaces. The implementation is tested on an industrial distillation column, containing 50 plates and about 4500 equations. The internal nonlinear equation solver fails to converge when a poor initial estimate point is used, while the homotopy solver achieves convergence. 1. INTRODUCTION Robustness in terms of solving problems from bad initial points is a desirable feature in a process simulator. This feature is particularly important when an entirely new simulation is attempted, where the true location of the solution may be completely unknown (or even the fact of whether there is a solution). In the context of steady state or dynamic simulation of chemical processes, robustness is mainly related to the capacity of solving a set of nonlinear algebraic equations (NLAE's) which arises either from the steady state equations or the initialization of the set of differential algebraic equations (DAEs) in the dynamic case. In the dynamic case, robustness is also needed while solving the DAEs after the initialization stage, but since in this case we have the solution at the previous time as an initial guess, the robustness of the nonlinear solver is less important. Traditionally, solvers based on Newton's method have been used to solve NLAEs. Convergence properties for this class of methods are only local, requiring the initial point to be in a neighborhood of the solution. However, in practical terms, convergence can be achieved from points not too close to the solution by using heuristic algorithmic details devised to enlarge the convergence domain (see Dennis and Schnabel, 1983). But, there is no guarantee that convergence will be achieved.
256 There have been several attempts to devise global methods, that is, methods which ensure convergence from initial guesses far from the solution. Ideally, global methods will guarantee convergence from any point in the domain under consideration, but for NLAEs satisfying this condition is sometimes difficult (or impossible) to ensure. The earliest attempt to develop a global solution method can be traced back to Lahaye's (1934) use of homotopies or continuation methods for single nonlinear equations. This technique consists of replacing a system of NLAEs f(x) = 0
(1)
h(x,t) = 0
(2)
by a homotopy function
where f : ~tt" --+ 9t" and h : ~n+l ~ ~n. The homotopy function h is such that h(x,O) = 0 has a known solution (or it is easy to find), while the solution of h(x,1) = 0 coincides with that for equation (1). Equation (2) defines a solution path x(t), and because of the conditions on h, we can, in theory, start with x(O) and follow the continuation path until x(1) is reached, which is the solution to (1) we are seeking. The main idea behind using (2) instead of (1) is that we can attempt solving h(x,t)=O from a previous value of the parameter t, close enough to ensure convergence. The actual implementation of these methods is not as simple as it looks, since a robust implementation implies transforming this problem further into a set of ordinary differential equations, and not solving it as stated here. We refer the reader to the excellent book by Allgower and Georg (1990) for details on how this is achieved. Alternatives to continuation do exist and they share the same aim, that is, enhancing the domain of convergence. One of them is the interval method, where instead of working with numbers, we work with intervals. Hence, if we know the solution belongs to a bounded domain, we could, theoretically, use an interval method based on the domain to guarantee convergence. See the book by Neumaier (1990) for details on these methods. Another alternative, is to use global optimization methods, which have received particular attention in recent years and look very promising. See the discussion by Floudas(1999) for an overview of these methods in the context of Chemical Engineering. Each global methodology has its own advantages and drawbacks, and there is scope for all of them in the wide range of problems to be solved. All of them have a common factor, they are very expensive in execution time (because of long homotopy paths and the combinatorial nature of the other approaches), which is the price to pay for global convergence. An advantage for the homotopy methods is that they can be implemented using no more than the residual and Jacobian information that is already available for an implementation of Newton type methods. They can even be implemented without the availability of the Jacobian (using efficient sparse finite difference perturbations). Interval and global optimization methods, on the other hand, require the availability of the analytical residuals, but they provide much better global convergence properties.
257 In this paper, we will show how homotopy methods can be implemented in a commercial simulation package, using open solver interfaces available in Aspen Custom Modeler. These interfaces will be briefly discussed in Section 2, while Section 3 will be devoted to presenting the algorithmic details of the homotopies implemented. Section 4 will provide an example of use, followed by conclusions in Section 5. 2. OPEN SOLVER INTERFACES IN ASPEN CUSTOM M O D E L E R T M The Aspen Open Solver set of interfaces (AOS) have been introduced in the commercial dynamic simulator Aspen Custom Modeler (ACM) in order to allow plugging external solvers to solve NLAEs. They are available in ACM 10.2, which is the release used for the results in this paper. Once an external solver is plugged-in, it can be used as an alternative to the internal solvers available. We have chosen to implement homotopies in ACM using these interfaces, which allows the homotopies to become completely independent of the product itself. An advantage of this approach is that as AOS interfaces become implemented in other products, a solver implementation based on the interfaces can then be plugged-in without needing any adaptation or modification. The AOS set of interfaces basically consists of an Equation System Object (ESO) interface and a Services interface, both implemented and exposed by ACM. The ESO interface allows the solver to fetch properties of the equation set including residuals, Jacobian, variable names, etc. A third interface is the open solver interface, which is used to implement the homotopies. This interface receives a pointer to an AOS ESO interface on creation, which provides the problem data needed for the solution process. ACM provides a "plug" which accepts a solver implementing the AOS solver interface. The interfaces are presented in Paloschi et al (2000). Similar interfaces were proposed by the Cape Open initiative, with the main difference being that they were designed for COM or Corba implementations, while the AOS set of interfaces is based on pure C++ abstract classes for increased performance. 3. IMPLEMENTED HOMOTOPIES
The homotopy we have implemented is the sparse bounded homotopy proposed by Paloschi(1996):
h~.h (x,t) = P ( x ) h ( x , t ) + v(x) - v(x h )
(3)
which is based on a standard homotopy h(x, t). See the original paper for a detailed description of P(x) and x b. In this paper, we have used h(x,t)=f(x)-(1-t)f(x ~ while v(x)=F ~ x, where F ~ is the diagonal of F'(x~ the Jacobian off(x) evaluated at x ~ The roles of P(x) and x b are to keep the homotopy path within a bounded domain Y~. P ( x ) - i within Y~ and 0 < P(x) < 1 outside, while xb=x inside ~ . Therefore, inside ~ , h.,.h - h .
All the involved functions are smooth
(hence differentiable), plus their derivatives are easily defined in terms of those for h(x, t) and
f(x). This homotopy can be implemented based only on the residuals and the Jacobian, as shown in the original paper. The main advantage of (3) is that it can be ensured that, if x is within a
258 bounded domain f2, the homotopy path x(t) will be bounded within ~ . This is particularly important when solving chemical engineering problems involving physical property calculations because the properties package will refuse to evaluate outside the valid domain. As the tool to track the homotopy path, we have used the code PITCON from Rehinboldt and Burkardt(1983). Because the use of homotopies is a very expensive procedure, the algorithm first tries to use a nonlinear solver to converge the problem, and only resorts to using the homotopy if it fails to converge. In this way, we can use the homotopy algorithm as a general nonlinear solver without incurring in additional cost, unless it is absolutely necessary. The way we do this is by having an AOS Solver plug in the homotopy solver itself, which enables the use of any solver which implements the AOS solver interface. The homotopy solver is implemented as a DLL exposing the AOS solver interface. The following diagram shows a sketch of how it looks:
Homotopy] Solver [
DLL ~ / ,......." \\
Pitcon
&
@OSSo,vow) ES ) @OSSo,e)
\
ACM
~
....A..OlSSolver implementation
DLL
The ovals denote the AOS interfaces. The one in the middle corresponds to the ESO interface, implemented by ACM and used by the homotopy solver. The one on the left corresponds to the AOS solver interface, implemented by the homotopy solver and used by ACM. The last on the right, is again an AOS solver interface, implemented by an external solver dll and used by the homotopy solver. In our particular case, we have used the internal ACM nonlinear solver, which is also implemented in a dll exposing an AOS solver interface. 4. EXAMPLE OF USE As an example showing the use of the solver implementing the homotopy, we have chosen a distillation column simulation. It consists of a dynamic industrial distillation model instantiating a column with 50 plates, with all the necessary controls. It provides a very detailed simulation of a real column posed in Aspen Plus TM and exported for use in ACM. Once instantiated in ACM, the problem has 5418 variables, 4495 equations and 14926 nonzeros. By using an internal tearing strategy, it is decomposed into 1653 blocks, many of which are nonlinear. The problem converges without difficulties using the internal Newton-type solver in ACM, if the initial point provided is used. If the initial point is changed to one where all the variables are set to their default values, the internal solver does not converge anymore. In this case, the
259 initial point is very bad since there are no profiles at all in the column and all variables for each plate in the column are set to default values. By using the homotopy solver described here, having plugged it into ACM using the AOS interface available, convergence is achieved. Of all the nonlinear blocks to be solved, 32 do not converge with the internal solver, but converge using the homotopy. Figure 1 presents the homotopy path for all 20 variables in the first block where the homotopy was used. All the equations involved in the block belong to the last stage immediately before the reboiler. Because the variables have a wide range of values, we have scaled them in the interval [ 1-10], so we can show the homotopy path for all of them in a single graph. The horizontal axis corresponds to the homotopy parameter t, while the vertical one is common for all the variables. As we can see from Figure 1, some variables have a simple homotopy path (almost a straight line), others start and come back to the same value, while others have a more complex path. As a reference, we also present in Table 1, the starting and final values for all variables in the homotopy path, for the first block. As you can see from Table 1, some variables start from initial guesses which are far from the solution. In Table 2, we show the starting values for some of the variable types. Table 1 9Start and end values of homotopy path for variables in first block t
0
xl
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11 x12 x13 x14
x15
x16
x17
x18
x19
x20
0.04 0.50 1.00 0.00 0.00 25.00 0.50 0.50 0.50 0.50 0.50 1.00 2.50 2.50 32.04 0.01 23.63 832.82 10.00 10.00
1 0.05 0.97 1.48 0.00 0.00 86.84 0.03 0.50 0.50 0.66 0.34 0.05 3.39 0.09 32.02 0.01 24.03 769.45 0.14
1.77
Table 2 9Initial values for some variable types
Iv.rT, ool
L'01
Initvalue I 5971.15
o' a01 ow assJ 'ow o' I ,owVo, J Fracl He,0u0Mol 14330.8
I
1000
I
10
116.66710.51
2.5
[olOI:YJllil,,,I~I:l~J
Although the use of the homotopy is expensive, as compared with a standard Newton solver, it is not prohibitively expensive. For this problem, it takes about 50 seconds of cpu time to solve it, while it takes about 5 seconds to be solved from a good initial guess (i.e., not needing the homotopy). Hence, the price we pay is not really that much considering the alternative with the internal solver (which is not being able to solve the problem). The cpu times given above were obtained using a Pentium I1266Mhz PC. 5. C O N C L U S I O N S We have presented an implementation of a homotopy solver in the commercial simulation package, Aspen Custom Modeler. The implementation has been done using open solver interfaces available in ACM, which are the only way in which the homotopy solver communicates with ACM. The homotopy solver itself uses the interfaces to plug in an external nonlinear solver. The path following code PITCON is used as an internal tool for path tracking. The homotopy solver is implemented as a DLL and plugged into ACM to be used as an alternative to the internal Newton-type solvers.
260 Its use in the package is illustrated with a large distillation column example. Using a poor initial point, the homotopy solver obtains convergence, while the internal ACM nonlinear solver fails to converge, This shows how we can use homotopy techniques to improve the domain of convergence and robustness of a dynamic simulator. The methodology used in the implementation shown in this paper requires no more information than is already available for a Newton type method, that is, residuals and derivatives. No analytical information on the residuals is needed.
REFERENCES
Allgower E.L. and Georg K., Numerical continuation methods-An introduction, Springer Verlag,(1990) Dennis, J.D. and Schnabel R.B., Numerical methods for unconstrained optimization and nonlinear equations. Prentice Hall, (1983) Floudas C., Recent advances in global optimization for Process Synthesis, Design and Control: Enclosure of all solutions, Computers Chem.Engng., ppS963-$973, (1999) Lahaye E., Une methode de resolution d'une categorie d'equations transcendantes. C.R.Acad.Sci.Paris 198, pp1840-1842,(1934) Neumaier A., Interval methods for system of equations. Cambridge University Press, (1990) Paloschi J.R., Bounded homotopies to solve systems of sparse algebraic nonlinear equations.
Computers Chem.Engng.,21,5,pp531-541,(1996) Paloschi J.R., Laing D.M., Zitney S.E, Open Solvers Interfaces for Process Simulation, AIChE Annual Meeting, Los Angeles, November 12-17,(2000) Rheinboldt W. and Burkardt J., A locally parameterized continuation process. A CM Trans.Math.Software 9,pp215,235,(1983)
.
.
.
.
Figure 1 :Homotopy paths for variables in first block
1.2
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
261
Using wavelets in process identification: A new link to the state space Heinz A. Preisig a aTechnical University Eindhoven, PO Box 513, 5600 MB Eindhoven, The Netherlands Spline-type modulating functions were derived based on state-space considerations: they are the result of solving an observer problem. Spline wavelets are derived based on their properties by the signal processing community. That the two are the same has not been recognised by either of the two communities. In fact, the spline-type modulating functions are multi-wavelets covering a wider range of functions than this is the case with ordinary wavelets. The resulting link gives new and interesting insights. 1. I N T R O D U C T I O N The project, reported in the three papers [3-5], was on the identification of models for an industrial reactor using the spline-type modulating functions introduced by Maletinsky [ 1]. The first of the three papers [3] reviews the history, discusses structural issues and the basics of the methodology as it was known in 1990, respectively. The second paper [4] describes the implementation for on-line and off-line applications and the third [5] is devoted to the application. Spline-type identification uses B-Splines as state variable filters, which are known to counter the bias problem in estimators. At the time this work was done (1978-84) [2], "wavelets" were not yet in fashion. Whilst the wavelets boomed shortly thereafter, the modulating function method never really made it into the ranks of the primary identification methods as the small and scattered literature body gives clear testimony [3]. The two literature bodies of modulating functions and wavelets remained essentially separated though even a simple visual inspection of the modulating functions suggests that B-spline modulation functions are indeed wavelets. The reason for this separation lies in the fact that the two theories come from two different scientific communities, which, in earlier years were connected, but lost each other out of view on the way. This contribution uncovers some of the links, namely the view of the systems-oriented group and the signal-processing group. 2. B-SPLINE M O D U L A T I N G FUNCTIONS The B-spline modulating functions, as they were first derived by Maletinsky [1 ], are obtained by solving an observer problem for unknown initial conditions in SISO systems. The elimination of these initial signal derivatives yields an integral transformation, which compensates for the unknown quantities by a filtered piece of signal history. Maletinsky [1] used the oberver canonical form to derive this result, whilst in [4] parts of the derivation are done in the Laplace
262 domain. Given the SISO model in transfer function form
~aiDiy
:-- ~ i
i
biDiu,
(1)
elimination of the unknown initial conditions of the n integrators, except the last one which is directly measured by the output, transforms the model into
n E
s ai < (~n,i,y >
i
:--
bi < t~n,i,u >
(2)
9
i
The functions (~n,0 are the modulating functions, B-splines of the order n, the first index, and 0-times differentiated, the second index. Some of the important characteristics of the B-spline functions are: (P1) Compact base
t~a,(~(t) =
(P2) Derivative I Integral
Oa,a-1 (t) =
(P3) Stem function of order n
0
for r [0,nT];
(3)
Oa,a(t)dt;
(4)
,n,n(t)-- T-n ~ (-1)i (;) 8(t-iT);
(5)
i:--0
(P4) Zero Boundaries
d~a,(~(O)-- d~a,a(nT) --0.
(6)
{~n,jlJ "-
The family of modulating functions ~n,0 form together with their derivatives an upper triangular function matrix shown in Box 1.
---
Box 1: Generating the function matrix
The upper triangle of functions is generated stepwise. The first row are the Cholesky factors, the rows of the Pascal triangle with alternating signs. The Pascal triangle can be generated by consecutive convolution: ~a,a*~b,b = ~a+b,a+b thus convoluting ~1,1 n-times with itself ((~l,1)*n) generates the first row. The other elements of the triangle can be generated by successive integration (property P2) or, equally well, through convolution. It is easy to show with property P2 that the following relation holds: ~a,o~rf~b,~
:="
~)a+b,t~+~-
(7)
----,--
, ' -
~-,,
r~
~
t
|
,
I
" l
V1
1,1 2,2 3,3 4,4 5,5 6,6 1,0 2,13.2 4,3 5,4 6,5 2,03,14,25,36,4 3,04,15,26,3 4.05,16,2 5,0 6,1 order,derivative
I
i
[-1 rn~
V
w
~ _
0, ..,n}
@
~
263 3. M O D U L A T I N G FUNCTIONS VS WAVELETS: T H E LINK Strang [6], p250 writes: "Splines are piecewise polynomials, with a smooth fit between the pieces. They are older than wavelets. The 'two-scale equation' or dilation equation was at first not particularly noticed." Indeed, the relation has not been noticed and neither has Maletinsky's derivation. With the latter derivation being based on a signal generating system, Strang's statement "Almost every formula in this book comes out neatly and explicitly for splines." (p250) does not come much of a surprise. The two-scale relation is readily found in the above triangle of spline functions. The "straight" wavelet theory of B-splines uses triples of functions from the function triangle, high pass filter (~n,n, scaling function (~n,0 and wavelet (~2n,n. The respective wavelet is then f~2n,n
--- (~n,n *On,O ,
(8)
which is a direct use of Equation (7).
The wavelet community derives its functions based on requesting certain properties such as symmetry, orthogonality and norming. The analysis and development is driven by properties of the signals. The required tools are heavy gear and it is by no means trivial to enter the field and work your way to the front. The results are impressive and the information is very complete. In contrast: the derivation of the spline-type modulating function is driven by a viewpoint of the system (plant) generating the signal from a base (input) signal. The approach is simple and elegant, uses a minimum of the mathematical tools and yields surprisingly much of information [21, [1], [3]. Benefit can be drawn from a cross-fertilization. (i) The modulating function approach yields the view that the filters in a column yield average consecutive derivatives. The modulating function apply a weighted average, whereby the weight is the modulating function and thus corresponds to the convolution with the scaling function in one time scale of the wavelets. "Normal" spline wavelets use thus filtered zero-order derivative information (scaling) and n th derivative of the 2 n-order filtered signal, where, when looking at a single time scale, the wavelets use the same compact base. (ii) The wavelets pick three elements in the function matrix. (iii) The main filter characteristics is the time between two impulses of the high-pass filters, the characteristic time T of the modulating function theory [1], [3]. (iv) The modulating functions are derived from a state-space representation of a signal transforming system: There is a state. (v) The modulating functions can also be multi-scaled and implemented on-line [4]. The remaining part of this paper is devoted to some of the lesser-documented points of the above list. We shall have a look at what appears to be most spectacular, namely the combination of the multi-resolution wavelet thought and the thought of differentiation in the modulation sphere. 4. T I M E - S C A L E ANALYSIS USING B-SPLINE MODULATION The above thoughts triggered a project in which we attempt to seek time-scale information of the signal generating system - our plant of interest - in identification, using multi-scale modulation. The thought is naturally also motivated by the singular-perturbation approach as it is used in time-scale-based model reduction. This type of problem is well known and overlaps with our research on modelling theory [8].
264 As an illustration we use a third-order model, composed of three first-order systems in series, each with a different time constant (1, 0.1, 0.01). As an input a periodic pulse function was used, which starts after a delay used to initialise the modulation filters. In view of the limited extent of this paper, we present only some results of two analysis: (i) continuous wavelet analysis using B-spline wavelets and (ii) filtering of the input and output signals using a multi-scale implementation of the B-splines [4] generating all the signals {< d~n,j,U >, < ~n,j,Y > [j :-O, 1 , . . . , n - 1}.
Fig. 1. modulated input and output signals
The top row in Figure 1 shows the original signal and the 3-rd-order-modulated input signal, thus the modulated signal and the first and second derivative of it. The bottom row shows the corresponding output signal. Time is shown pointing to the right and different-length (resolution) modulation of order 3 towards the back with a scale indicating the filter constant T. Modulation is done "very hard", that is, a lot of the original signal is filtered out. Most of the tipples, caused by the high-frequent input signal, are ironed out, and as the filtering increases, the remaining ridge gets smaller. This hard filtering was done because we sought the dominating time constant, which indeed can be extracted easily by assuming a first-order model (equation (1)) and simply solving for the only parameter at every point. The result (not shown) is a pretty-much fiat plane at the dominating time constant. The general procedure is described in [5]. Also the effects one can expect when undermodelling the dynamics [4]. The B-spline modulating functions can be readily introduced into the Matlab wavelet toolbox. Applying the B-splines of order 1,2 & 3 yields the three graphs shown in Figure 2. The left graph shows that more of the additional terms are used to reconstruct the signal in the first phase. In the middle graph, this is significantly reduced and in the fight is essentially nonexistent. The difference between the middle and the fight, is very small such that one could conclude that the signal could live in the B-spline wavelet space of order 2 just as well as in the one of order 3. Clearly it is not as well represented in the space of order 1. The wavelet analysis can thus be used to determine an appropriate order.
265
Fig. 2. modulated input and output signals
5. A FEW THOUGHTS ON THE PARAMETER IDENTIFICATION TASK The high-pass filter is not very suitable as a modulating function, because it cuts out all pieces of the signal between the impulses. One thus uses a modulating function that is at least an order bigger than the minimal. How many of the derivatives one needs, depends on the number of parameters in the model and precisely on how one would like to perform the identification task. In a transfer function model, the maximum number of parameters, related to the eigenvalues of the state-space realisation and the gain, is 2n + 1. The complete state space representation, {A,b_,c_T,d}, has n 2 + 2n + 1 parameters, which is n 2 more than for the transfer function representation, n being the dimension of the state-space model. The n 2 parameters are for the similarity transformation matrix transforming the canonical representation back to the "meaningful" state-space representation. The number of parameters thus depends on the order of the model and the representation. Having determined on how many parameter need to be identified, the user has still a number of choices on how to extract these parameters from the filtered signals {< ~n,j,Y > (k), < ~n,j,u > (k) I J "= 0, . . . , n - 1,k "= 1,2, ..... ,number of samples}, which in general are available as discrete signals sampled with some rate determined by the application and realisation of the filter. In our current implementation the minimal shift time is equal to the sampling time of the original signal. Thereafter any multiple can be set as one of the method parameters. For constant model order one may choose to allow the parameters to change over the length of the experiment and use higher-order derivative information by simply differentiating the model equations sufficiently many time as to generate enough equations to solve for the parameter vector.
a+, a T [ ~0 y2~
9
]
-
-
Ua+
1
9
]
,
(9)
where the vector ~q collects the derivatives p , . . . , q of the respective modulated signal. Obviously, the order of the modulation must be chosen accordingly. Alternatively, if the parameters do not change quickly, one may choose another approach and select consecutive filtered values along the signal. In this latter approach, one needs to be aware of the fact that the modulations overlap, thus extract correspondingly identical pieces of information from the signal. In [4] it is argued that Shannon's sampling theorem suggests that the modulations should be at least T/2 shifted, which is equivalent to the discrete wavelet shift. In practice, one should though use excess information and augment the above procedure with a method such as minimal-sumof-squares, to minimize the effect of remaining noise components in the modulated signals. In the past, we used the same signal interval as one would obtain when shifting only T/2 but
266 overlapped at least 5 times more and then used a minimisation to determine the parameters. The latter also provides a measure for the remaining variance mapped into the estimation of the parameters. 6. CONCLUSIONS (i) B-splines modulating functions are multiwavelets. (ii) "straight" B-spline wavelets are defined by the triplet high pass filter ~n,n, scaling function ~n,0 and wavelet q~2n,n. (iii) The 2-scale relation operates along the horizontal axis of the B-spline function matrix. (iv) The modulating function approach operates along the column of the same matrix. (v) The use of differentiated signals in identification has become acceptable with the use of wavelets in signal processing, without being notice, though. (vi) On the sideline: the detection of events using wavelets, as being advertised in the discrete event dynamic community, can also be interpreted as finding discontinuities in higher-order derivatives on different time scales. (vii) Identification is still very much an art. The discussed relation, though, enriches the "bag of tricks" with an interesting new one. An extension to fractional B-splines has been published by [7] yet another variation of the theme. REFERENCES
1. V. Maletinsky, 1-i-P Identifikation kontinuierlicher dynamischer Prozesse. PhD Thesis, ETH-Ztirich No 6206, 1978, 143 p. 2. H A Preisig, On the Identifcation of Structurally Simple Models of Energy Distribution in Industrial-Scale Reactor Equipment. PhD Thesis, ETH-Ztirich No 7616, 1984. 3. H A Preisig, Theory and Application of the Modulating Function Method-I. Review and Theory of the Method and Theory of the Spline-Type Modulating Functions. Comp & Chem Engl7, No 1(1993), 1-16. 4. ibid.-II. Algebraic Representation of Maletinsky's Spline-Type Modulating Functions. Comp & Chem Engl7, No 1(1993), 17-28. 5. ibid.-III. Application to Industrial Process, A well-Stirred Tank Reactor. Comp & Chem Engl7, No 1(1993), 29-38. 6. G Strang, T Q Nguyen, Wavelets and Filter Banks. Wellesley-Cambridge Press, 0-96140887-11996490, 1996. 7. Unser M, Blu, Fractional splines and wavelets. SIAM Review 42, No 1(2000), 43-67. 8. Westerweele M R, Preisig H A, Weiss M, Concept and design of Modeller, a computeraided modelling tool. ESCAPE-99 Budapest Hungary Comp & Chem Eng. (1999).
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
267
Evaluation of coupled reactive distillation performances by means of a rigorous simulation procedure D. Rouzineau, M. Meyer, M. Prevost INP/ENSIACET, LGC Equipe S6paration Gaz Liquide 18 Chemin de la Loge, 31078 Toulouse Cedex 4, France A non equilibrium model (NEQ) for multi component separation processes, including liquid and/or gas reactions, is developed in this paper. This model includes the finite mass transfer rate described by Maxwell Stefan equations. It is assumed that the bulk of both the vapour and liquid are perfectly mixed and that the resistance to mass and heat transfer is located in two films at the liquid/vapour interface (film theory). There are no restrictive hypotheses as to the nature and the localisation of the chemical reactions. The phase non ideally and the diffusion coefficient variation are taking into account. The original formulation of mass transfer is described and solved in order to avoid the bootstrap problem. The resulting DAE system is then solved. We describe the resolution and focus on two keys points : the coherent initial state and the reduction of differentiation index. With several examples, a comparison between equilibrium stage model (EQ) and the NEQ model is effected and a sensitivity analysis is done to highlight the importance of the film thickness. 1. I N T R O D U C T I O N Reactive distillation is a unit operation of great industrial interest enabling active process intensification. It can reduce capital and production costs by combining two units into one, and this unit can improve selectivity, heat integration, conversion, etc... Simulation and design of multi component reactive distillation is usually carried out using the equilibrium stage model. The limitation of conventional equilibrium stage efficiency calculations is discussed by many authors [1-5]. These authors assume that the generalised nonequilibrium model should be preferred for the simulation of a column for reactive distillation to the equilibrium model, because an accurate prediction of individual Murphee tray efficiencies (or HEPT for packing) is very difficult in the case of simultaneous multi component separation and reaction. The non equilibrium model seems to be better because the model takes into account the technological characteristics of the column (type of plate, packing...), nearer to reality. So, the NEQ model is developed by numerous authors. But this author include the Maxwell Stephan approach which is delicate to solve. Indeed, this formulation involve the bootstrap problem mentioned by Taylor & Krishna [4]. Moreover, the non equilibrium model requires parameters which can be as difficult to find as an efficiency. In this paper, the first part deals with of NEQ model; we focus on the diffusional layer near the interface, where a complete multi component reactive mass and heat transfer model is described. The numeric resolution, avoiding the bootstrap problem, is discussed in the second part. Finally, after the model requirement list, some examples will be discussed to illustrate the non equilibrium simulation in practice.
268 2. NONEQUILIBRIUM MODEL THEORY A schematic representation of the non equilibrium model (NEQ) is shown in Fig.1. This NEQ stage may represent a tray or a section of packing. It is assumed that the bulk of both the vapour and liquid are perfectly mixed and that the resistance to mass and heat transfer are located in two films at the liquid/vapour interface according film theory [6,7].
Figure 1 : The nonequilibrium model representation of stage j 2.1. Mass and heat transfer model A novel model is used to compute heat and mass transfer through the diffusion layer. Indeed, the fluid is considered as an NC component reactive non ideal mixture. The balance equations for simultaneous heat and mass transfer are written in steady state, taking into account the reactions. Fick formulation is limited to two components mixtures because it does not take the interactions between the different components into account; moreover the non ideality for the driving force is not considered. So, for mass transfer, the Maxwell Stefan diffusion law is used, in a novel formulation. Neither the diffusion coefficients, nor the molar flux due to the reaction, are considered to be constant. No assumption is made on the type or the number of reactions, thus they can be controlled by kinetics or equilibrium. The complete formulation for mass transfer, for NC components in non ideal mixture, is :
t)Ni 3z
M a s s continuity.
NRC @
EagijRj j=l
(
NRE "1- E '[)ij'~j j=l
t}Inyi
Diffusion law"
~
Equilibrium equation-
Kj : H a ?
8iJ "~"Xi
0X.1 T,P
-0
)~Xi= )/)z - ~ ( x iN j-xiN,) }=1
c-tDil
for/= 1 toNC
(1)
for i=ltoNC
(2)
for j= 1 to NRE
(3)
NC i=1
The dimension of system (I) formed by equations (1),(2) and (3)is 2NC+NRE. In a traditional model, only NC-1 equations of (2) are considered because of equation dependency. Indeed, equation (4) is obtained by summation of the NC terms in equation (2) : Olny; i=l l-l-Xi
0xt ~
" + 2 1+
~X--~T,p) ~z'~ i=1 ""
i=l
__~z= _ ~ - ' ~ gi ~Xn T,P
i=l j=l
___ ctDij
(4)
269
Taking into account the Gibbs Duhem relation
xi i=l
Dij-Dji,
equation (4) becomes:
~)ln~i
= 0 and the coefficient symmetry
~X j
(5)
' =0 i=l ~Z
In our model, equation (5) is never used also NC equations of type (2) are usable. Consequently, no additional equation is needed, as proposed by Taylor & Krishna [4], to obtain directly the molar fluxes Ni. For the heat transfer, the Dufour and Soret effects are neglected and the heat diffusion rate is evaluated by Fourier's law. The whole system is called HMT model and involves differential algebraic equations (DAE).
2.2. Phase equations The phase equations are classically the mass balances and energy balance in the bulk phase for each stage [4]. These equations take into account reactions, and there are no restrictive hypotheses as to the nature and the localisation of the chemical reactions. The temperature of the vapor and the liquid phases are not assumed to be equal. The entire column is taken as a sequence of a number of such stages. We consider a N stages column where stage 1 can be a total or partial condenser and stage N a reboiler. The modelling leads to a system of differential and algebraic equations (DAE).
2.3. Interface equation Physical equilibrium is assumed at the vapor liquid interface for each component. Moreover, mass and energy continuity is described by mass and energy balances. 3. N U M E R I C R E S O L U T I O N
3.1 DAE integration of HMT model Firstly, the HMT model is solved. No analytic solution is available, so, the DAE system is solved by a Gear method. To be efficient, a such integrator has need of a coherent initial state and a differentiation index as small as possible.
Initial conditions A robust procedure, leading to a coherent initial state ( ie. all algebraic equations must be satisfied) before starting the integration, has to be used. The following equations are solved by Newton method" Mass balance
9
Fox~ - Fxi + Z o ~ j - o NC
D,,3
Equilibrium equation 9
Kj
Summation equation:
1 - ~ xj = 0
= H ai' i=1
for i : 1 to n
(6)
forj = 1 to NRE
(7)
(8)
j=l
With this resolution, we obtained the molar fraction at the interface in agreement with the equilibrium reaction constraint.
270
Differentiation index An automatic substitution procedure is used to reduce the number of mass balances in order to take into account the chemical equilibrium constraints and to eliminate the ~ variables. Matrix formulation of equation (1) is"
-1
~ -~Z dN1
O1,NRC R1
~)1,1 ...................
~)I,I
...............
~)I,NRE
"
+
~
dN. -1 ~
=0
+
~.I).,,..................V,,,NRC
NRC
E
With a pivot strategy, canonic form of equation (1) is derived: 0 0 9
dz
[
"~
9
i 1
a 1
.
=(a"
+(B) 9
i
9
~
i 6
,~~
dN,,
So, the enhancement variables ~ become explicit, and we can substitute their value in the other equations. So, the 2n+NRE dimension system (I) is reduced to a 2n dimension system. Moreover, the differentiation index of the DAE system is reduced from 3 to 1.
3.2. Integration of phase equations Secondly, the general balances in both phases and at the interface leads to a system of differential and algebraic equations. Limit conditions are given at both extremities of the column, so a discretisation of DAE system has to be done and the resulting algebraic system is solved by a traditional Newton's method 9 4. SIMULATIONS RESULTS Two examples will be discussed to illustrate the use of NEQ models 9The first example is a reactive distillation where Acetic Anhydride reactes with Water to obtain Acetic Acid. The reaction rate is given by Marek [8].The column is composed of 15 sieve trays with a feed at the level 6 ( composition : 0.161 acetic anhydre, 0.484 water, 0.355 acetic acid and floxrate of 27.6 mol s-1).The first simulation is done by NEQ model where the vapor film thickness is fixed a t 10-4 meters and the liquid film thickness at 10.5 meters 9 The interfacial area is obtained by the Zuiderweg correlation. The results from NEQ model are compared to the EQ model results obtained successively with an efficienciy equal 1, predicted by Mac Farland [9] or calculated by NEQ model. The correlation of MacFarland predicts efficiencies ranging from 0.69 to 0.72 for Acetic Anhydride, from 0.67 to0.83 for Water and from 0.69 to 0.99 for Acetic Acid. So, as we can specify only one efficiency for a plate, mean value of 0.7 is chosen. In figure 2 are presented the Murphee efficiencies along the column calculated plate to plate from the results of the
271
NEQ model. The variation between 0.54 to 1.14 indicate that the conventional prediction of Murphee efficiencies from an empirical correlation may not be reliable for multicomponent reactive mixture. In Fig. 3, the Acetic Acid molar fraction profile is shown for different simulations. From these profiles we can conclude (the conclusion is the same for the other components) that, with a good prediction of efficiencies (here 0.7 from Macfarland), the two models give similar results.
........~...,.~, ~ AddAcetique
4-
-e-. Water
[
s-
--.-
6-
k;
'
~7
~.
..o-.E -~,- EQ M o d e l Elf = 0,7 ( M a c
lO11 12 14 ,, 0,5
"~..~'~.. ~,,.
Farland)
- - * - - N E Q M o d e l el =10-5 ev = 10,4 . . . .
.
0,6
.
0,7
.
.
0,8
.
0,9
MurpheeEffielencles
1
1,1
,
15
1,2
0
0,1
0,2
0,3
.
0,4 0,5 0,6 Liquid molar fraction
',~,
.
i~
0,7
0,8
0,9
1
Figure 3 "Liquid Acetic Acid profile
Figure 2 9Murphee efficiency profile
The last example is the case of a packing distillation column in which the esterification reaction of ethanol is done. Two simulations are realised with different film thickness. In the first simulation, the vapour film thickness is fixed at 10 -4 meters and the liquid film thickness at 10 -5 meters (reference values found in literature). In the second simulation, the two thickness are estimated by correlations. Indeed, the film thickness is calculated by the ratio diffusion coefficient / transfer coefficient. By using Onda correlation for transfer coefficients, Wilke and Chang correlation for the liquid diffusion coefficient and Fuller for the gas phase, the liquid and gas film thickness are respectively equal to 1.30.10 -4 and 2.47.104m. 1,2-
0,8
Height of packing 0,6(meter)
0,4
0,2
84
0 0
0,1
0,2
0,3
0,4
0,5
0,6
Molar fraction
Figure 4 : Liquid composition profile The profiles of liquid molar fraction are shown in figure 4. The results are very different and illustrate the fact that the thickness is a very sensitive parameter. Experimental values are needed to discriminate these simulations and to fit these parameters. It seems to be sure that accurate results of NEQ models are obtained only if film thickness is predicted with a good accuracy.
272 5. CONCLUSION We have developed a non equilibrium model for multi component reactive separation techniques. This model is solved numerically and then included in the ProSimTM environment. The originality of this model is the Maxwell Stefan formulation which is solved in its complete formulation. Moreover, the reaction can be controlled by kinetics or equilibrium. The non equilibrium model is tested and compared with the classical equilibrium model. Our model takes into account the phase non ideality via activity coefficient models, the interaction between components in mass transfer via Maxwell Stefan law, the chemical reaction without restrictions concerning kinetics or location. But a very poor representation of hydrodynamic phenomena is used: the double film model introduces the film thickness as a sensitive parameter. So the prediction of this parameter is the main difficulty of a NEQ model. In order to overcome this problem, we are now developing an experimental pilot plant to validate the simulation results. If experimental values are not required in the future, the development of a more realistic approach of the hydrodynamics must be considered. This may be done by using a local approach of the three transfer phenomena. Rj Rate of reaction j (mol/m2/s) NOMENCLATURE xi Molar fraction component i a i Activity component i yi Activity coefficient component i c,
Total concentration total (mol/m 3)
Dii
Maxwell Stephan diffusion
coefficient binaries i-j (m2/s) F ,Fo Molar flow (mol/s) Kj Equilibrium constant for equilibrium N i
N,
NC NRE NRC
reaction j Molar flux component i (mol/m2/s) Molar flux molar total (mol/mZ/s) Number of components Number of equilibrium reactions Number of kinetic controlled reactions
vo
Stcechiometric coefficients of
0( 0
component i for kinetic controlled reaction j Component order i equilibrium
z)ij
reaction j Stoechiometric coefficients of
~:j
component i for equilibrium reaction j Enhancement of equilibrium reaction j (mol/m2/s)
ACKNOWLEDGEMENT We are grateful for research support provided by ProSim S.A. (France). Helpful discussions with Dr Sere Pereygain (ProSim Technical Manager) are acknowledged. REFERENCES 1. R. Baur, A.P. Higler, R. Taylor and R. Krishna, CEJ 76, 33-47, 2000. 2. A. Higler, R. Krishna, Taylor, AIChE J, Vol.45, 11, 1999. 3. Lee Jin-Ho, M.P. Dudukovic, Computers and Chemical Engineering 23, 159-172, 1998. 4. R. Taylor and R. Krishna, Multicomponent Mass Transfer, Wiley Series in Chemical Engineering, (New York), 1993. 5. J.A. Wesselingh, Distillation and Absorption, Vol. 1, 1-21, 1997. 6. R. Krishna, Chem. Eng. Sci., 32, 659-667, 1977. 7. R. Krishna and G. Standart, AIChE J., 22, 383-389, 1976. 8. J. Marek, Coll. Czech. Chem. Commun, 21,1561, 1956. 9. S. A. MacFarland, Hydrcarbon Processing, 111-114, July 1972.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
273
An Adjoint-Based Algorithm for Mixed-Integer Dynamic Optimization V. Sakizlis a, V. Bansal b, R.Ross b, J.D. Perkins a and E.N. Pistikopoulos a * aDepartment of Chemical Engineering, Centre for Process SystemsEngineering, Imperial College, London SW7 2BY, U.K. bprocess Systems Enterprise Ltd. 107aHammersmithBridge Road London W6 9DA, U.K. This paper presents an adjoint-based algorithm for the solution of mixed-integer dynamic optimization problems. The algorithm is based on Generalized Benders' Decomposition, where the solution alternates between a series of primal dynamic optimization problems (which give upper bounds on the solution) and master, mixed-integer linear programming problems (which give lower bounds). In contrast to earlier adjoint - based works [ 1,2], the algorithm is independent of the type of method used for integrating the differential-algebraic equation system in the primal problems and allows easier construction of the master problems. The application of the algorithm is illustrated with a process example taken from the literature. 1. INTRODUCTION In the last few years, dynamic modelling and simulation have become popular tools in academia and industry for improving process design and operation. This is usually accomplished by conducting a large number of simulations for different values of manually selected process parameters and choosing the set that gives the most desirable process performance (e.g. measured in terms of cost) while satisfying operating, safety and environmental constraints. This approach is time-consuming since guaranteeing feasibility in a system with more than one or two constraints is often very difficult. Even if a feasible solution can be found, this approach does not guarantee an optimal solution [3]. The shortcomings of exhaustive enumeration can be overcome by applying a systematic dynamic optimization approach. Here, a mathematical algorithm is employed that automatically manipulates a selected set of process parameters in order to optimize (minimize or maximize) a performance index, while guaranteeing satisfaction of time-dependent constraints. The efficiency and reliability of methods for this purpose have developed to the extent that realistic engineering dynamic optimization problems, involving thousands of variables, can be readily solved using commercial codes such as gPROMS/gOPT [4]. In practice, engineering problems do not involve only continuous decisions, such as the determination of the optimal distillation column diameter; they also involve discrete decisions such as the determination of the optimal feed tray location. These discrete decisions are usually modelled via integer (most commonly binary) variables; thus, in the context of optimizing dynamic systems, this gives rise to mixedinteger dynamic optimization (MIDO) problems, which are very difficult to solve. A number of algorithms have recently been proposed in the literature for solving MIDO problems (for a complete review, see [5]). The most efficient and reliable approaches directly *To whom correspondence should be addressed. Tel.: (44) (0) 20 7594 6620, Fax: (44) (0) 20 7594 6606 E-mail: e. pist ikopoulos @ ic. ac. uk
274 decompose the MIDO problem into a series of primal problems (upper bounds on the solution) and master problems (lower bounds on the solution). The primal problems correspond to continuous dynamic optimization problems where the values of the binary variables are fixed. These are commonly solved using control vector parameterization (CVP) techniques, where only the time-varying control variables are discretized. CVP methods offer significant benefits when applied to complex, large-scale dynamic systems, particularly because they exploit the progress made in DAE numerical integration technology and solve the optimization problem in the reduced space of the control variables. The MIDO algorithms that employ CVP for the primal problems mainly differ in how they construct the master problems, where the latter correspond to mixed-integer linear programs (MILPs) whose solutions give new sets of binary values for subsequent primal problems. Variant2 Generalized Benders' Decomposition-based (GBD-based) approaches [ 1,2,6], Outer Approximation-based (OA-based) approaches [7] and approaches based on "screening models" [8] have been developed. These MIDO algorithms tend to depend on a particular type of method for integrating the DAE system in the primal problems, require the solution of a complex intermediate problem in order to construct the master problem, or are case study-specific. The purpose of this paper is to outline a new algorithm for solving MIDO problems that generalizes the GBD-based approach [1,2] so that it is independent of the DAE integration method and allows easier construction of the master problem. 2. THEORETICAL DEVELOPMENTS
Consider a general MIDO formulation" min
x(t),z(t),u,d,y
d?(Yc(tf),x(tf),z(tf),u(tf),d,y, tf)
s.t.
0000 >
f(Yc(t),x(t),z(t),u(t),d,y,t) c(x(t),z(t),u(t),d,y,t) r(x(to),Yc(to),z(to),u(to),d,y, to) q(Yc(tf),x(tf),z(tf),u(tf),d,y, tf) to
(1)
Here, x E 9~nx,z E ~r~nz are the vectors of the differential states and the algebraic variables respectively. The vectors u E ~t~nu,d E 9~nd represent the control and the time - invariant design variables, whereas y E {0, 1}ny is the vector of the discrete binary variables. The functions f, c and r represent the differential equations, the algebraic equations and their initial conditions respectively. The objective function is denoted by ~ and the constraints by q. The binary variables y participate only in a linear form in the objective function, the differential system and the constraints, since this is a necessary condition for applying the variant-2 of GBD [9] to a mixed integer optimization problem. The primal problem is constructed simply by fixing the binaries to a specific value y - yk. Then problem (1) becomes an optimal control problem that is solved with control vector parameterization. The control variables u are discretized to time-invariant parameters. From now on, the new total set of optimization variables will be denoted as v and includes the design and the parameterized controls v = {Ul,U2,..,UNu,d}, v E ~nu'Nu+nd. In GBD-based approaches the master problem is constructed using the dual information of the primal at the optimum solution. The dual information is embedded in the Lagrange multipliers
275 /1 of the constraints q and the adjoint time-dependent variables )~(t), p(t) that are associated with the differential system of equations, i.e. f,c. The master problem formulation is then: min rl y,rl
s.t. r I > d~+(pk)T.q+(O~) T. --
[:1
f
k= l,K k c K
[l
+(03~) T f
C
+(pk)T'r+
o
E
I:].,
[(~J~)T(pk)T]. C ,
(2)
p, COyand COoare multipliers that are evaluated from the first order optimality conditions of the optimal control primal problem [10]. In the master problem, the continuous variables (x(t), z(t), v) are fixed to the values obtained from the solution of the primal problem. The Lagrange multipliers p are also fixed in this way. The only variables that are left to be calculated in order to proceed with the master solution are the adjoint variables )~,p that are not provided directly at the optimal solution of the dynamic optimization problem. For their evaluation, it is necessary to perform an extra integration of a differential algebraic equation system, the so-called adjoint differential system, that is derived from the original DAE [10]: --
d{[~]T')~(t)} ~X OCT = -[ ]7".)~(t)_[~x] "p(t)"
Of lr
[
dt
30
(tz) - -{( yx) +
3q
O-
-[
~Z T
Oc]T
] "~(t) - [:3z
"P(t)
(3)
Of r 3c
(-Yxx)[: + [(yx)
Solution of equation (3) involves a backwards integration and can be computationally expensive. Nevertheless, the extra integration can be eliminated by adapting an adjoint-based approach for evaluating the gradients of the constraints and the objective function of the primal problem. This provides at the optimal solution of the primal, a set of vectors of adjoint variables that are associated with the constraints and the objective function, denoted as [)~(/) p~(/)], [)~q(t) pq(t)] respectively. Those adjoint functions are given by the same linear DAE system as the adjoint functions that are necessary for the master problem construction. However, [~(/) p~(t)], [)~q(t) pq(t)] are given by different final conditions. Their final conditions are:
Of lr
k,(t:) -
O~ar +
Of) r 3c r ]:.(co:),]
0fir Oc [0~t~.kq(rl) = _r(Oq)r L ~ : + [( ~xf ) r (~)r]e-(cOl)q]
(4)
The linear properties of the adjoint differential system and its boundary conditions, allow the evaluation of the adjoint variables required for the master problem ~,, p as a function of [;k~p,], [)t,qpq] from the equations: L(t) -- ~,(t) +/Ir)~q(t); p(t) -- po~(t) + l.trpq(t)
(5)
Equation (5) can be proved using transition matrix theory [11,12]. Consequently, rigorous adjoint integration for master problem construction is not required after the primal has terminated and the derivation of dual information isreduced exclusively to equation (5). Even if the now easily obtained time-dependent functions ~(t), p(t) are supplied to the master problem many calculations are required due to the presence of the time integral in equation (2) and the usually complicated non-linear functions involved in the DAE system. In order to
276 simplify further the master problem, equations f,c are decomposed in terms of the binary and continuous variables:
f = f'(Yc, x,z,v,t)+ fY(k,x,z,v,t)'y; c=c'(x,z,v,t)+cY(x,z,v,t)'y r = r'(.fCo,Xo,Zo,V, to) + rY(Yco,Xo,Zo,V, to) "y
(6)
fY, cy, ry are matrices of dimensions nx
x ny, nz x ny, nx x ny respectively. This separation is allowed, since the binaries participate in the DAE in a linear form. At the primal solution fk = 0 r162(f,)k = _(fy)k. yk Similarly for c, c', cy, r, r', rY. Finally we have:
f = (fy)k. (y_yk); c = (cY)k. ( y _ yk); r = (rY) k. ( y _ yk)
(7)
Once equation (7) is substituted in equation (2) the modified master problem becomes: min 11 y, rl
s.t. r I >__q~+ ( / ~ ) r q + {o3~
Cy
cY f
+
rY +
o
[)~rpr]
fY cY
d t } k ( y _ yk) (8) t
In this manner, the size of the master problem formulation is reduced. The multiplier of the binary terms y - yk corresponds to a vector whose evaluation requires the integration of an equivalent ODE system of size equal to the dimension of the binaries, ny. Alternatively, if formulation (2) was retained, every equation that contains a binary term would have to be integrated generating an ODE system of order of magnitude O(nx + nz)> > ny. From an implementation point of view, problem (8) is more convenient to construct, because matrices fY,cY, rY are the Jacobians of the DAE with respect to the binaries that are generated numerically or analytically using well-established commercial codes [13]. In the master optimization problem (8) the only variables that are allowed to vary are the binaries and the continuous objective. Since these optimization variables participate in a linear form, the problem is an MILP which can be solved with current well-established methodologies. 3. M I D O A L G O R I T H M STEPS
The steps of the algorithm are summarized as follows: 1. Fix the values of the binary variables, y = yk, and solve a standard dynamic optimization problem (kth primal problem). An upper bound, UB, on the solution to the MIDO problem is obtained from the minimum of all the primal solutions obtained so far. 2. At the solution of the primal problem, using equation (5) obtain the adjoint functions
)@), p(t). 3. Use the problem variables x(t),z(t), v, the adjoint functions )~(t), p(t) and the Lagrange multipliers of the constraints/1 to construct the kth relaxed master problem (8) from the kth primal solution. The Master problem solution provides the lower bound, LB, on the MIDO solution. If UB-LB is less than a specified tolerance e, or the master problem is infeasible, the algorithm terminates and the solution to the MIDO problem is given by UB. Otherwise, set k = k + 1 and yk+l equal to the integer solution of the master problem and return to step 1.
277 4. REMARKS ON THE MIDO ALGORITHM
The MIDO algorithm presented above has the advantage that it does not depend on a particular type of method for the integrations associated with the DAE model of the process and the gradient evaluations for constructing the master problem. Two other significant features are that the dual-adjoint evaluation in (3) is avoided and that the master problem is more easily constructed than in previous approaches [ 1,2,6,7]. However, it relies on using an adjoint-based approach for evaluating the gradients, unlike the algorithm described in [5], which does not depend on how the primal problem is solved. It should also be noted that, as with all other MIDO algorithms proposed in the literature, the algorithm in this article does not guarantee global optimality, although it does guarantee convergence to a local optimum.
5. MIDO EXAMPLE
The use of the MIDO algorithm presented in the previous section is now demonstrated with a process example from literature [5,14]. The process consists of distillation column that separates a binary mixture. The full details of the model can be found in [5]. The dynamic behaviour of the system is caused by a step-change in the feed composition. The number of trays is constant while the objective is to find the optimal feed tray location, vapor boil up B and reflux flow rate R that minimize the integral square error (ISE) of the bottom and top compositions with respect to their set-point values. In addition to that, there are two end point constraints, one for each of the two compositions (XN+I ~_ 0.98, Xb < 0.02). The feed flow is constrained so that it can only enter the column above the 10th tray. The time horizon considered is fixed to 400 min and initially, the system is assumed to operate at steady state. Figure 1, (taken from [5]) shows the column superstructure and the values of certain parameters. The proposed MIDO algorithm was applied to this example. The primal optimal control problem was solved using Matlab (1999). The optimal control code was constructed with Matlab components. Namely, an NDF integrator was employed for the solution of the state DAE system. NDF methods for the integration of DAE systems are an extension of the well - established BDF methods. In addition, the NLP solverfminopt was utilized that implements an SQP - quasi Newton - line search method for tackling the optimization problem. The MILP master problem was solved using GAMS/CPLEX [ 15]. The results are summarized in Table 1. The optimal feed location is tray 25, giving an ISE=0.1817. This corresponds to the solution reported in literature [5,14]. The feed location tends to approach the top of the column in order to result in a quick response of the distillate composition (smaller ISE). However, the feed cannot be located too high because this would cause a very slow response to the bottoms composition (i.e. ISE25 < ISE26) and also deteriorate the separation quality (constraints violation). An exhaustive enumeration of all the possible feed tray locations would require the solution of 20 dynamic optimization problems. However, the proposed method finds the solution solving only 4 optimal control problems implying 80% computational savings.
6. CONCLUSIONS This paper has outlined an adjoint-based algorithm for solving mixed-integer dynamic optimization problems. The proposed method generalizes earlier works [1,2], since it does not depend on a particular type of numerical integration method and allows easier construction of the master problem. The application of the algorithm has been successfully demonstrated via a process example from the literature.
278 V, YN L ~ Vap~
-- D, xN+t xNI LiquidV
v!,~., ~-~ ~!x,-~qui~ V.ly~_, L,,.,,~x,,., F, zf Liquid
.
.
.
.
v!y,
:--<:L:,
.
.
.
.
.
.
.
Value No. of trays, N 30 Relative volatility, ~ 2.5 Tray liquid hold-up, m 0.175 kmol Feed flow rate, F I kmol/min. Distillate set-point, x~+t 0.98 Bottoms set-point, x~ 0.02 Parameter
,
l
LiquidJ~t,x t V'Y~ ( 0 ~ Vapour ~
B'Xb Liquid
Table 1. Results on Illustrative Example N ~ Iterations Primal Feed Loc. V(kmol/min) R (kmol/min) ISE UB Master Feed Loc. LB
1
2
3
4
20 1.223 0.684 0.1991 0.1991
26 1.783 1.243 0.1827 0.1827
24 1.411 0.871 0.1833 0.1827
25 1.543 1.002 0.1817 0.1817
26 24 25 23 0.1577 0.1813 0.1815 0.1847
Fig. 1. Illustrative Example
REFERENCES
1. M.J. Mohideen, J.D. Perkins and E.N. Pistikopoulos, Comput. Chem. Eng. 21 (1997) $457. 2. R. Ross, V. Bansal, J.D. Perkins and E.N. Pistikopoulos, In: AIChE Annual Meeting. American Institute of Chemical Engineering. Miami Beach Florida, US. 1998. 3. C.C. Pantelides, Process Modelling in the 1990s. In: 5th Worm Congress of Chemical Engineering. San Diego, California. 1996. 4. Process Systems Enterprise Ltd., gPROMS Advanced User's Guide. London 2000. 5. V. Bansal, PhD Thesis. Imperial College, University of London 2000. 6. C.A. Schweiger and C.A. Floudas, In: Optimal Control: Theory, Algorithms and Applications (W.W. Hager and P.M. Pardalos, Eds.). Kluwer Academic Publishers B.V. (1997) 388. 7. M. Sharif, N. Shah and C.C. Pantelides, Comput. Chem. Eng. 22 (1998) $69. 8. R.J. Allgor and P.I. Barton, Comput. Chem. Eng. 23 (1999) 567. 9. C.A. Floudas, Nonlinear and mixed-integer optimization. Oxford University Press, New York, 1995. 10. V. Vassiliadis, PhD Thesis. Imperial College, University of London 1993. 11. W.L. Brogan, Modern control theory. Second Edition, Prentice Hall, Inc. New Jersey, 1985. 12. S. Trahanas, Differential equations, Vol. I. University editions. Irakleio, Greece, 1989. 13. Maple V Release 5.1. Waterloo Maple Inc. (1998). 14. C.A. Schweiger, A.Rojnuckarin and C.A. Floudas, MINOPT: User's Guide. Version 2.0 Ed., Princeton University, 1997. 15. A. Brooke, D. Kendrick and A. Meeraus, GAMS Release 2.25: A User's Guide. The Scientific Press, San Francisco, 1992.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jargensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
279
A Hybrid Mathematical Model for a Three-Phase Industrial Hydrogenation Reactor P. L. Santana; E. C. Vasco de Toledo; L. A. C. Meleiro; R. Scheffer, B. B. Freitas Jr., M. R. W. Maciel, and R. Maciel Filho* Laboratory of Optimization, Design and Advanced Control. Department of Chemical Processes. School of Chemical Engineering. State University of C a m p i n a s - UNICAMP. Campinas SP (Brazil) - CP 6066 - CEP 13081-970, Fax. +55-19-37887840 This paper presents a hybrid mathematical model for describe a three-phase reactor behavior, which combines a neural network architecture (as a predictor block for the liquidsolid mass transfer coefficient) and phenomenological equations describing the mass conservation principle. The optimization procedure used in the network training was based on the Fletcher-Powell algorithm. Results of the network training and validation showed the predictive capacity of the proposed model and its great potential to be used as a support for process modeling and control. 1. INTRODUCTION Processes based on multiphase reactions occur in a broad range of application areas and form the basis for manufacture of a large variety of intermediate and end products. Catalytic processing, is widespread in the production of fuels, commodity chemicals, specialty chemical and pharmaceuticals, pollution abatement, as well as in production of food where both conventional and enzyme catalysis is employed. More information on these reactors can be found in the following references, Shah (1979), Ramachandran and Chaudari (1983), Gianetto and Silveston (1986), Deckwer (1982), Biardi and Baldi (1999), Dudukovic (1999) Dudukovic et al. (1999) and Santana (1999). A realistic description of catalytic three-phase reactors should be based on the real mass an heat transfer characteristic of the system, i.e., on kinetic models, mass and heat transfer models for both the bulk phases of gas and liquid as well as for the catalyst particles, (Salmi et al., 2000). However, the development of efficient and reliable models for these multiphase reactors is still a difficult task because it involves many aspects including hydrodynamics, gas-liquid and liquid-solid mass transfer, heat transfer, pore diffusion, reaction kinetics and deactivation. Model assessment has mostly been reported for a single reaction, or reactions obeying simplified kinetic laws, under isothermal conditions. Nevertheless, exothermic reactions with a multistep reaction, complex kinetics and rigorous comparison of the performances in several multiphase reactors, is a subject of great interest in industrial applications (Bergault et. al., 1997). This paper presents a comparison between two models. The first model consists of balance equations for the catalyst particles as well as for the bulk phases of gas and liquid, and a *Correspondingauthor (
[email protected]).
280 energy balance for the coolant fluid, which is not usually considered by the works found in the literature (Santana, 1999). The second model is a hybrid mathematical model, which combines a neural network architecture (which works as a predictor block of the liquid-solid mass transfer coefficient) with the phenomenological equations describing the mass conservation principle of three-phase process. The underlying system is an industrial three-phase catalytic slurry reactor, with tubular geometry and coolant system, operating in steady state and continuous regime in which a highly exothermic industrial hydrogenation reaction takes place. The mathematical model assumes plug flow regime and considers mass and energy balances for fluid phase and catalyst particle. A comparative study is carried out between real and simulated concentration and temperature profiles. To explore the potentialities of the ANN, two different approaches were tested in this work: /) the Standard ANN Modeling (also called black-box), where ANN was used to represent the whole process behavior by mapping its input to output process data (Figure 1) and; ii) the Hybrid ANN Modeling, where ANN was used to predict the liquid-solid mass transfer coefficient that is a parameter for the determinist model (composed by the first principles equations). Figure 2 shows the hybrid model approach.
I Figure 1 - Standard ANN Modeling.
1
Deterministic M o d e l
o,,,.. " " "-~
Figure 2 - Hybrid ANN Modeling.
2. MODELING OF A CATALYTIC THREE-PHASE SLURRY REACTOR The mathematical model consists of the mass and energy balance equations. The formulation was made considering as case study a hydrogenation reaction, but the same approach can be adapted with great simplicity to any other three-phase system. Although it is usual to neglected equations for the particle (due to the reduced dimensions typically found in the practice) in the modeling of slurry catalytic reactors, in this work these equations were considered explicitly. The following hypotheses were adopted in the development of this three-phase reactor dynamic model (Santana, 1999): a) steady state; b) plug-flow for the reactant fluid and for the thermal fluid; c) suspension (liquid + solid) homogeneous, considered as a pseudo-fluid; d) negligible pressure variations; e) axial dispersion was neglected; J) reaction of the type vAA(g) + VBB(I)-->Vc C(1) happening in the catalyst and with a kinetics depending on the concentrations of the A and B; g) gas phase hold up negligible; h) it doesn't happen phase change in the system. 2.1. Fluid phase
Mass balance for the reactant A in the gas phase: 0Ag
-Ug 0z = K L a g ( A * - A I )
(1)
281 Mass balance for the reactant A in the liquid phase: 0AI
-u,-~z
+ KLag (a* - a , )
s
= (Ksap) A(AI - As)
(2)
Mass balance for the reactant A in the solid phase: (Ksap) A(A, - A~) = w tiC a A(A~, BSs,Ts~)
(3)
Mass balance for the reactant B in the liquid phase: 0BI
- u, - ~ z : (Ksap)B
(B i - B S) s
(4)
Mass balance for the reactant B in the solid phase: (Ksap)B(B , - B ~ ) = v w n ~ RA (A~,BSs,T#)
(5)
Energy balance for the reactants in the fluid phase: c3T 4U (UlCplPl -t'- UgCpgpg)--~- Z -- (-AH) w rio RA (ASs, B ~ , T # ) - ~-.- (T - T R)
(6)
Energy balance for the reactants in the solid phase: (hsap)(Ts ~ - T ) = w rio (-AH) R A(A~,BSs,T~)
(7)
Energy balance for the coolant fluid: p Cp r ~ _ ~ 0_Y4r U
-u r
r
0Z
(y r -
T)
(8)
Dt
with the following initial conditions (in z = 0): Ag --- Agi, A l = Ali , B 1 = Bli , Y = Ti, Yr = Tri
(9)
2.2. S o l i d P h a s e
Mass balance for the reactant A in the solid phase: / N dA s DeA d [r 2 ] (A B s Ts) r 2 dr\ dr J = P p R A s, ,
(10)
The equations for the reactant B in the solid phase may be written in a similar way. For the modeling of solid catalytic reactors where heterogeneous approach is assumed, it always arises the need to calculate the effectiveness factor, r/c. which may be defined as:
>p_ 3DeA (deAl RA
P
A
r=Rp
282 The formulation of the mathematical model for three-phase reactors leads to a system of algebraic-differentiate equations strongly non-linear. Equations 1 to 9 form an initial value problem that to be properly integrated requires, in each integration point, the resolution of the equations of the solid phase. The equations that describe the particle behavior constitute a boundary value problem, which is solved using the orthogonal collocation method. The application of this method generates a system of non-linear algebraic equations. For the resolution of the initial value problem the software LSODAR was used, and for the solution of the system of non-linear algebraic equations the Newton-Raphson method was applied. To simplify the solution of the reaction-diffusion problem that occurs in the particle, it can also be used the Bischoff approach to calculate the effectiveness factor, which generates accurate results of r/c. 3. THE HYBRID M O D E L
The hybrid neural network model is composed of two blocks (Figure 2). The ANN block estimates a process parameter (the liquid-solid mass transfer coefficient), which is used as input to the second block, represented by the deterministic equations of the process (mass and energy balance equations). For this process, optimization methods that consider the whole system were adopted. The training process is then reduced to an optimization problem from a global cost function that includes the ANN and the deterministic equation blocks. The weights are adjusted in such a way to minimize the following cost function:
NT Fobj --Z(Yhyb,p--dp)
2
(12)
p=l
where NT is the total number of training data; Yhyb is the output vector predicted by the hybrid model; d is the target vector (real values); and p is the pth pair of the training data set. In this work, the Fletcher-Powell algorithm was used to minimize the cost function (12) associated to the hybrid model. 4. RESULTS Here are presented some simulations from the use of the deterministic and hybrid models to describe the three-phase slurry reactor behavior. Deterministic model was used to simulate the real process providing the training data set used by the ANN. Especial attention was given to component B concentration in the liquid and solid phases to exemplify the prediction capacity of the studied models. Figure 4 depicts the temperature effect on the concentration profiles of the B component in the liquid phase and in the particle surface. It can also be noticed a reduction of the concentration in both, liquid and solid phases. This effect is due to the progress of the reaction along the reactor, which is much more intense for the highest temperatures. Deterministic model is compared to real industrial operational data (Figure 4) showing its capacity to predict the reactor behavior. As the deterministic model showed high performance and accuracy when compared to real process data, its outputs were used as training and test data sets to the ANN models. To illustrate the use of the ANN-based models, it was considered the feed temperature, Ti, as input, and the concentration of the component B (o-cresol) in the liquid phase (bt) as the
283
output of the ANN. This choice is due to the fact that in real three-phase slurry reactors, conversion of the liquid component is the most important variable of the process. Temperature was chosen as the input because its major influence on the o-cresol concentration in the liquid phase. 1.1 -I
i
-
&
-&.
o.8 P 07
~o~ E
o5 i
OA O3
~~ 1 oo
9
00
i
9
0.1
i
,
|
0.2
,
0.3
i
,
0.4
i
9
0.5
,
,
06
i
9
0.7
i
-
0.,
t
09
,
!
i:5 O2
10
0.0
0.1
i
0.2
0.0
Dimensionless length (z)
Figure 3" TemperatureEffect in B l and BSs 1,00 0,95 0,90 0,85 0,80 0,75 0,70 0,65 0,60 0,55 0,50 0,45 0,40 0,35 0,30 0.25
i
9
|
9
w
0.4
0.6
Dimensionless
9
|
'
9
Axial
0.8
1.0
Distance
Figure 4: Profile of Three-Phase Reactor Temperature
_
1,1 . 1,o-
.
.
.
.
.
.
i
o.,: 0,5-
T=610 K
.~ == E
.._
0,4o,30,2-
~ - ..... .-
Process
Standard ANN Model Hybrid ANN Model
',,
o13
-.. 1 .......... i L
o.10,2
0.4
0,6
0.8
1.0
Dimensionl~s Icnght (z)
Figure 5 - Results of the Hybrid ANN-based model.
o,o
i .
. . . .
,
.
.
.
o14 .
Dimensionless
o:,
o; lenght
9
o~ t"
9
oi,
9
o:....
9
-~
(z)
Figure 6 - Results of the Hybrid and Standard ANN-based models.
The same training and test data set were used to compare the performance of the both, the standard and the hybrid models (described in the figures 1 and 2). This procedure allows a better evaluation of the performance of the proposed approaches. According to the results obtained, the hybrid approach showed the best performance. This result is probably due to the fact that hybrid models incorporate some knowledge about the process by means of the deterministic equations coupled to the model. Figure 5 shows the performance of the hybrid ANN model and figure 6 depicts the comparison between the results of the hybrid and standard ANN models. The hybrid A N N model tends to be more accurate since it provides a prediction of the liquid-solid mass transfer coefficient, which is a difficult parameter to be obtained from the deterministic model. 5. C O N C L U D I N G R E M A R K S Industrial multiphase reactors present very complex behavior, which is difficult to be predicted by deterministic mathematical models. To be able to do that sophisticated models
284
have to be derived which contain parameters that are difficult and expensive to be determined. Through the hybrid model approach, coupling deterministic and ANN models is possible to develop models with very good prediction capabilities from industrial concentration and temperature profiles this makes this procedure to have a great potential to be used in the model development of industrial systems. A further work will tune the Hybrid Model with real data and it will be compared with deterministic model with constant transfer coefficient. NOTATION ag and ap = gas-liquid and liquid-solid interfacial areas respectively, ml; A = concentration of the component A, kmol/m3; A* = solubility of the component A, kmol/m3; B = concentration of the component B, kmol/m 3; Cjo = concentration of the component j in the particle centre, kmol/m3; Cjs = concentration of the component j in the particle surface, kmol/m3; Cp = heat capacity, kj/kg.K; De = effective diffusivity, m2/s; Dt = reactor diameter, m; AH = heat of reaction, kj/kmol; rio = catalytic effectiveness factor; hs = heat transfer coefficient, kj/m2.s.K; KL and Ks= mass transfer coefficients gasliquid and liquid-solid respectively, cm/s, L = reactor length, m; ~ef = effective thermal conductivity, kj/m.s.K; v = stoichiometric coefficient; r = particle radial position, m;
Rp = radius particle, m; 9 = fluid density, kg/m3; T = temperature, K; u = linear velocity, m/s; U = reactor to wall heat transfer coefficient, kj/m2.s.K; z = reactor axial position, m; w = catalyst concentration, kgcat/m 3. Subscripts A = component A B = component B g = gas phase; l = liquid phase; i = initial value (reactor inlet); p = particle; r = coolant fluid; s = solid. Superscripts s = particle surface.
REFERENCES Bergault, I., Rajashekharam, M. V., Chaudhari, R. V., Schweich, D. and Delmas, H., (1997) Chemical Engineering Science, Vol. 52, No. 21/22, pp. 4033-4043. Biardi, G. and Baldi, G., (1999) Three-phase catalytic reactors, Catalysis Today, Vol. 52. Deckwer, W. D., (1992) "Bubble Column Reactors", John Wiley and Sons, New York. Dudukovic, M. P., Larachi, F. and Mills, P. L., (1999), Chemical Engineering Science, Vol. 54, pp. 1975-1995. Dudukovic, M. P. (1999), Catalysis Today, Vol. 48, pp. 5-15. Gianetto, A. and Silveston, P. L., (1986) "Multiphase Chemical Reactors: Theory, Design, Scale-up", Hemisphere Publishing Corporation, Washington. Ramachandran, P. A. and Chaudhari, R. V., (1983) "Three-Phase Catalytic Reactors", Gordon and Breach Science Publishers, New York. Salmi, T., W~irna, J., Toppinen, S., R6nnholm, M. and Mikkola, J.-P. (2000), Brazilian Journal of Chemical Engineering, Vol. 17, No. 04-07, pp. 1023-1034. Santana, P. L. (1999), "Mathematics Modeling for Three-Phase Reactor: Deterministic, Neural and Hybrids Models ", Ph. D. Thesis (in Portuguese), UNICAMP. Campinas, SP, Brazil. Shah, Y. T., (1979) "Gas-Liquid-Solid Reactor Design", McGraw-Hill Inc., New York.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
285
Complex Reactive Absorption Processes: Model Optimisation and Dynamic Column Simulation R. Schneider a, E. Y. Kenig b and A. G6rak b aBayer AG, Corporate Technology, ZT-TE 7 Overall Process Engineering, D-51368 Leverkusen, Germany bDortmund University, Chemical Engineering Department, D-44221 Dortmund, Germany
The optimal design of reactive separations is impossible without reliable process models. Especially for the dynamic simulation and the model-based control of complex reactive absorption processes the model development leads to a contradiction between the required model accuracy to reflect the process complexity and the feasibility of process simulations regarding the computation time. In this respect, we have developed a new rigorous dynamic two-phase model based on the two-film theory as a first step, which takes into account the influence of chemical reactions and additional driving forces in electrolyte systems on mass transfer considering thermodynamic non-idealities as well as the impact of column internals on the process hydrodynamics. Based on results of sensitivity studies, we have accomplished different model modifications leading to a stabilisation of the numerical solution and an appropriate model complexity without affecting the good agreement between simulation results and the experimental data. 1. INTRODUCTION Reactive absorption processes represent a combination of complex mass transfer and chemical reactions in two-phase systems. In general, modelling and design of this operation is based on the theoretical description of the reaction and mass transport in multicomponent fluid systems covering the superposition of many phenomena like multicomponent diffusion, chemical interactions, convective flows, multicomponent thermodynamic interplay, etc. Current industrial reactive absorption processes usually operate far from their optimum because a reliable design and model-based control require very detailed steady-state and dynamic models. Dynamic results are also crucial for the start-up and shut-down phases, process safety as well as for the on-line optimisation. Optimal reactive absorption models have to be both rigorous enough in order to reflect the process complexity and simple enough in order to ensure feasibility of dynamic process simulations. 2. MODEL DEVELOPMENT AND OPTIMISATION A comparison of different model approaches revealed that traditional equilibrium stage models and efficiency approaches are inadequate for reactive absorption processes [ 1].
286 Therefore, a rigorous rate-based model based on the two-film theory has been developed [2,3]. This model takes into account the direct influence of chemical conversion (film reaction) and additional driving forces in electrolyte systems (Nernst-Planck transport equation) on the mass transfer, as well as the thermodynamic non-idealities and the impact of structured packings and liquid distributors on the process hydrodynamics. Dynamic differential mass and energy balances with the simultaneous calculation of accumulation terms such as the liquid hold-up on each column segment reflect the continuous and dynamic character of the process. In the dynamic component material balances for the liquid bulk phase, changes of both, the specific molar component and the total molar hold-up, are considered. These balances are expressed by the following partial differential equations (1)
i-- 1.... , N C
a-7
where xi is the component mole fraction, Ut is the specific molar liquid holdup, L is the liquid flow rate, ni is the molar flux entering the liquid bulk phase, a is the specific interfacial area, Ri is the reaction rate, ~bis the volumetric liquid holdup, A is the column cross section and N C is the number of components. Due to the chemical conversion in the film, the values of the molar fluxes at the interface and at the boundary between the film and the bulk phase differ, and the changing mass transfer rates along the film coordinate have to be considered. Therefore, we have taken into account the chemical reaction kinetics and mass action laws in differential equations describing the liquid film region and resulting in non-linear concentration profiles l dn/ --~5 drl
R/ = 0 ;
(2)
i = l ..... N C
where r/ is the dimensionless film co-ordinate and 6 is the film thickness. For the determination of the film thicknesses, empirical mass transfer coefficient correlations are used which allow for the influence of column internals and hydraulics. The Nernst-Planck equation has been implemented as constitutive relation taking into account the gradient of the electrical potential as additional driving force in systems containing electrolytes [4]: n[ =
_ c f D~e# ( d X [ + x f z~ Fd ---s
Rr
~/+
i x, , , , ,
1,. :
NC
(3)
..,
where m is the solvent index, zi is the ionic charge and ~b is the electrical potential. The consideration of the electrical potential requires an additional condition of electroneutrality that has to be satisfied everywhere in the liquid phase. The proposed rigorous dynamic rate-based model serves as a reference description and leads to a system of partial differential, differential and algebraic equations, which have to be solved numerically. For a model-based control and dynamic on-line simulation a reasonable model complexity has to be determined. Therefore, in this work several feasible model reductions concerning both physical and numerical parameters have been investigated.
287 Different film and packing section discretisations, several mass transfer and hydrodynamic correlations, and different driving forces and diffusion models have been thoroughly tested. 3. APPLICATION AND MODEL VALIDATION As an application example, the reactive absorption of sour gases in an air purification process with packed columns is simulated. The aim of this process is the selective removal of HzS, NH3 and HCN by suppressing competing reactions of the major impurity CO2. The system of reactions comprises 14 components and 8 parallel reactions, 3 of them are kinetically controlled [5]. The numerical solution of the model equations requires a discretisation with regard to the axial (column height) and normal (film thickness) coordinates. A sensitivity analysis regarding numerical parameters leads to the required number of grid points (5 film segments and 5 axial segments per meter of packing are sufficient). The steady-state simulations are validated by experiments performed at the Technical University of Berlin in a DN 100 pilot plant absorber. The column is equipped with Sulzer MellaPak | 350Y structured packing and three liquid distributors [6]. The results of the ratebased simulations show a good agreement with the experimental data (Fig. 1), whereas the equilibrium stage model overestimates the CO2 absorption rate leading to a totally wrong absorber performance. This can be explained by the importance of mass and heat transport in reactive absorption processes since in practice mass and heat transfer are actually rate processes that are driven by gradients of chemical potential and temperature.
Fig. 1. Liquid phase axial concentration profiles for the H2S absorber; comparison between experimental and simulation results based on different model approaches Single stage simulations including the Maxwell-Stefan approach reveal that intermolecular friction forces do not lead to reverse diffusion effects and thus can be neglected. The impact
288 of electrical forces enhances the absorption of the strong electrolytes H2S and HCN by 3-5 %, while the CO2 absorption rate is dominated by the reaction in the film. Significant changes in the concentration profiles and the component absorption rates due to the film reaction have been observed (Fig. 2). As a model simplification, a linearisation of the film concentration profiles has been studied. This causes no significant changes in the simulation results and at the same time reduces the total number of equations by half and stabilises the numerical solution. The assumption of chemical equilibrium in the liquid bulk phase does not change the absorption rates significantly which indicates fast conversion. Therefore, neglecting the film reaction unrealistically reduces the absorption rates. On the other hand, neglecting the reaction kinetics within the film results in completely different orders of magnitude for the calculated removals. As a consequence, the reactions of carbon dioxide should not be regarded instantaneous although the corresponding Hatta number of about 7 characterises the reaction as very fast [7].
Fig. 2. Simulated column absorption rates obtained with different model assumptions The most sensitive components appeared to be those involved in kinetically controlled reactions, especially CO2. In this respect, the process is mostly influenced by the reaction kinetics of the carbamate formation reaction and by the value of the interfacial area. These two parameters determine the reactive absorption selectivity. 4. DYNAMIC SIMULATION The optimised model allows for a dynamic real-time simulation of the entire absorption process. As the dynamic behaviour is mainly determined by process hydraulics, it is necessary to consider those elements of the column periphery which lead to larger time constants as the column itself. Therefore, major elements of the column periphery such as distributors, stirred tanks and pipelines have been additionally implemented into the dynamic model. With this
289 extension of the model, the process dynamics is investigated by local perturbations of the gas load and its composition. A significant dynamic parameter is represented by the liquid hold-up. Fig. 3 demonstrates the changes of the solvent composition after a decrease of the gas flow rate from 67 m3/h to 36.4 m3/h and a simultaneous small increase of the liquid flow rate.
Fig. 3. Change of solvent composition after a sudden significant decrease of the gas flow rate and a simultaneous small increase of the liquid flow rate The liquid hold-up of the packing section decreases which leads to a lower conversion of the kinetically controlled reactions of CO2 and a reduction of the CO2 absorption rate. As a consequence, the solvent mole fractions of HCO3- and carbamate decreases whereas the relative fraction of HS ~ increases. The selectivity of the absorption process towards the HzS and HCN reduction is enhanced by minimising the liquid hold-up of the column. At the same time, a larger interfacial area improves the performance of the plant. Therefore, modem industrial sour gas scrubbers should be equipped with structured packings. Fig. 4 illustrates the response after a sudden increase of the gas flow by 20 % and its HzS load by 100 %. As expected, the HzS load increases everywhere along the column height in the gas phase. The change is more significant in the lower part of the absorber than at the top because some additional hydrogen sulfide is absorbed. The new steady state is already achieved after 30 minutes which justifies the implementation of dynamic models for the column periphery. The simulation results agree well with the experimentally measured concentration profiles. 5. CONCLUSIONS Several conclusions can be drawn from this application which are of industrial significance and generally valid for reactive absorption processes. Due to the very fast process dynamics,
290
Fig. 4. Dynamic axial H2S column concentration profile on-line simulation and model based control is rather complicated and requires a mathematical description with an optimised complexity. The results demonstrate that the process is dominated by the chemical reaction and its influence on the diffusional mass transfer whereas an implementation of the Maxwell-Stefan approach is usually not required. The most significant parameters are the interfacial area and the reaction kinetics which have to be determined accurately. A linearised description of the film reaction leads to an optimised model which, as a result, can be considered superior as compared to previous approaches and is well suited to the dynamic modelling of entire reactive absorption columns.
REFERENCES
1. J. C. Charpentier, Trans IchemE, 60 (1982) 131-156. 2. E. Y. Kenig and A. G6rak, 1995, Comp. Chem. Eng., 19 (1995) $287-$292. 3. R. Schneider, E. Y. Kenig and A. G6rak, Trans IchemE A, 77 (1999) 633-638. 4. R. Taylor and R. Krishna, Multicomponent Mass Transfer, John Wiley, New York, 1993. 5. E. Y. Kenig, R. Schneider and A. G6rak, Chem. Eng. Sci., 54 (1999) 5195-5203. 6. J. Mayer, R. Schneider, E. Y. Kenig, A. G6rak and G. Wozny, Comp. Chem. Eng., 23 (1999) $843-$846. 7. R. Zarzycki and A. Chacuk, Absorption: Fundamentals & Applications, Pergamon Press, Oxford, 1993.
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rightsreserved.
291
A Web-based Library for Testing Performance of Numerical Software for Solving Nonlinear Algebraic Equations M. Shacham a, N. Brauner b and M. B. Cutlip c aDept, of Chemical Engineering, Ben Gurion University, Beer-Sheva 84105, Israel* bSchool of Engineering, Tel-Aviv University, Tel-Aviv 699 78, Israel; CDept. of Chemical Engineering, University of Connecticut, Storrs, CT 06269, USA The desirable structure for a web-based test problem library of nonlinear algebraic equations is investigated. Limitations of existing test problem collections--regarding the type of information included and the form in which this information is stored and presented--are demonstrated. It is concluded that a web-based test problem library with beneficial aspects for potential users should contain the problem definition with the following information: the model equations in the same form as the input for the numerical solver, the explicit definitions of constraints on the variables, the initial estimates and function values at the initial estimates, and the variable values and function values at the solution. All variables and function values should be reported with the same precision as that of the numerical solution. 1. INTRODUCTION Steady state and dynamic simulations of chemical processes require numerical solution of large systems of nonlinear algebraic, ordinary differential, and differential algebraic equations. It is too often taken for granted that the numerical solver used can find the correct solution for a particular problem if it exists and that warning messages will be issued in case of doubts regarding the accuracy and/or correctness of the solution. Unfortunately, the solution provided by the numerical solver cannot always be trusted as some of our previous work [1][2][31 ' ' , for example, has demonstrated. Testing the software's reliability requires that it be challenged by a large set of benchmark problems, which are known to be difficult to solve. The need to validate the software is common to both the numerical and statistical software. For statistical software, a group of statisticians in NIST (National Institute of Standards and Technology of the U.S.) took the initiative and placed a large set of test problems on the Web (http://www.itl.nist.gov/div898/strd/index.html). This library contains problems related to: analysis of variance, linear and nonlinear regression, and univariate summary statistics. For the case of nonlinear regression, for example, the data set contains the data points, the correlation model equations, and "certified" values of the calculated parameters including their standard deviation and the resultant variance. The problems are of various size and difficulty level and can be downloaded as ASCII files for testing software packages. We have been working on the development of a similar test problem library for systems of nonlinear algebraic equations (NLEs). As a basis for the library, we are using collections of test Author to whom correspondenceshould be addressed, e-mail:
[email protected]
292 problems published in the literature (see for example references [2] to [6]) and problems obtained by personal communications from individuals who needed help in solving particular types of equations. While preparing the library, attempts to reproduce the results obtained form various sources in the literature revealed many of the limitations of those test problem collections. These limitations helped to identify the type of information that should be included in a problem definition, the desired form in which this information should be stored and displayed, and the general structure of the library. Due to space limitations in this paper, only the considerations related to the single problem definition will be discussed herein. The computations related to this article were carried out with the NLE solver program of the POLYMATH 5.0 package [copyrighted by M. Shacham, M. B. Cutlip and M. Elly (http://www.polymath-software.com/)]. The NLE library was implemented with Excel [Excel is trademark of Microsoft Corp. (http://www.microsoft.com)]. 1. LIMITATIONS OF THE EXISTING TEST PROBLEM C O L L E C T I O N S The limitations of the existing test problem collections will be demonstrated with reference to the problem of "Combustion of Propane in Air", which was used as a test problem, for example, by Hiebert [s], Shacham TM,Bullard and Biegler [61, and Meintjes and Morgan [71. Table 1. Hiebert's [51 Version of the "Combustion of Propane in Air" Test Problem No.
Equations
Variable
Xo
1 2 3 4 5 6 7 8 9 10 11 12
f l = x1+x4-3 = 0 f2 = 2*x1+x2+x4+x7+x8+x9+2*x10-R = 0 f3 = 2*x2+2*x5+x6+x7-8 = 0 f4 = 2*x3+x5-4*R = 0 f5 = x1*x5-0.193*x2*x4 = 0 f6 = x6*sqrt(x2)-0.002597*sqrt(x2*x4*xs) = 0 f7 = x7*sqrt(x4)-0.003448*sqrt(x1*x4*xs) = 0 f8 = x8*x4-1.799e-5*x2*xs = 0 f9 = x9*x4-0.0002155*x1*sqrt(x3*xs) = 0 f l 0 = x10*x4^2-3.846e-5*xs*x4^2 = 0 R=40 xs = x l + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 + x 8 + x 9 + x l 0
xl x2 x3 x4 x5 x6 x7 x8 x9 xl0
2 5 40 1 0 0 0 0 0 5
X *a
f(x *a)
X *b
2.995 3.967 79.999 0.005 0.001028 0.001916 0.0622 1.553 12.043 8.19
0 2.9976 2.00E-04 3.9664 1.72E-04 80 -9.72E-04 0.0023645 -7.50E-04 6.04E-04 7.90E-07 0.0013659 -3.28E-06 0.064573 -8.80E-07 3.5308 -4.21E-06 26.432 2.05E-04 0.0044998
108.81714
117.00021
f(x *b) -3.55E-05 3.37E-04 -5.34E-05 6.04E-04 -1.99E-08 -1.00E-07 2.57E-08 -3.59E-08 1.45E-06 -1.56E-13
aSOlution obtained by Bullard and Biegler[61, bSolution obtained by ShachamTM The equations as presented by Hiebert [5] are shown in Table 1. It should be pointed out from the outset, that Meintjes and Morgan Iv] had found that this model is chemically incorrect and does not represent a physical system. This illustrates a very important point that many of the published test problems contain typographical errors. The only way to avoid such errors in the library version is to use electronic transfer of the problem's equations from the numerical solver to the library and vice versa. Following this principle, the equations in Table 1 are shown in the form as they were copied from the input data set of the numerical solver. This set of equations is very difficult to solve because the system may have several solutions, some of them are physically infeasible (xi represents number of moles of various components thus all xi must be >0). This brings up an additional issue, that constraints are an integral part of the mathematical model and, as such, they must be explicitly displayed in the library. In this particular example, the solution algorithm is challenged by the need to calculate the square root of some of the variables, which may attain negative values along the solution path, and especially if the solution itself is very close to zero.
293 Hiebert [5], Shacham TM and Bullard and Biegler t61 solved the set of equations of Table 1 for several different values of the parameter R and from several different starting points. Some of the data and the results for R=40 and one set of initial estimates (x0) are also shown in Table 1. The initial estimates have been reported in these studies, but the function values at the initial estimate were not reported. The information regarding f(x0) is essential for a user who wants to reproduce the results, since differences in function values at the initial estimate signal errors in the problem setup. Such errors can be detected irrespective as to whether convergence to a solution is achieved by the software that is being tested. The function values at the initial estimate can also provide some information regarding the order of the magnitude of the various terms comprising a particular function. The order of magnitude of the various terms is important for determining the attainable accuracy as indicated by the function value at the solution point. The information that can be derived from f(x0) is easier to interpret if different initial estimates are used for the different variables. Thus, using an initial estimate of zero value for five of the ten variables, as shown in Table 1, can hide some important information that can be deduced from f(x0) Hiebert t51 attempted to solve this system of equations using 9 different software packages and reported the relative performance of the various packages. The values of the variables at the solution were not reported. This makes the reproduction of the results rather difficult and uncertain, as the user can never be sure whether the same solution is found or even whether the same problem is solved. Bullard and Biegler TM have found two solutions to this system using an iterative linear programming strategy. The first solution, as they reported it, is shown in the column marked with x *a in Table 1. Shacham TM found one solution to this problem using the CONLES t81 algorithm. The latter is shown in the column marked with x *b. The three solutions are completely different. For example, x10=8.19 in x *a, x10=0.0044998 in x *b, whereas x10=6.465 in the additional solution reported by Bullard and Biegler E61(not shown in Table 1). The existence of three different solutions to this problem raises several options: 1. The problem has multiple solutions, and all the solutions obtained are valid. 2. There are errors in the problem set-up in one or more of the cases. 3. The solution method converged to a local minimum instead of the zero of the system of equations. In order to find out which of the reported solutions are valid solutions of the problem, the function values should be checked. Neither Bullard and Biegler [6] nor Shacham TM reported function values at the solution. The function values were calculated in this work by introducing the values shown under x *a and x *b into the equations. The resultant function values are shown in Table 1 (in the columns marked with f(x *a) and f(x *b) ). It can be seen that in both cases the largest function values are of the order of 10-4, a number that can be considered as a non-zero value. One reason for the large function values obtained at the solution point is the low precision used for reporting the xi* values. Bullard and Biegler [6] report x*, in most cases, with a precision of four decimal digits. For numbers greater than one (in absolute value), it is understandable that even in a linear equation, the function value can exceed 10.4 due to error introduced by rounding the numbers to four digits. Thus, in order to verify that x* is indeed a solution of the system of equations, the results should be reported with a much higher precision, preferably with the working precision of the computer program (most NLE solver programs work with double precision, approximately 15 significant decimal digits). In this particular case, our work verified the solution provided by Shacham TM. The utilized computer program (the "constrained" option of the POLYMATH 5.0 program) converged to the same solution as shown in the column x *b of Table 1, and the resulting solution was obtained
294 with 15 decimal digits of accuracy. These yield a highest function value of the order of 10"14, which can be safely considered as zero in a double precision computation. The solutions reported by Bullard and Biegler t6] are suspected to be incorrect. This can be seen, for example, by considering the values of the two terms comprising fl0 at the solution reported by them. The first (positive) term value is: 8.19'0.0052=2.05"10 4. The value of the second (negative) term is -3.846* 10-5* 108.817*0.0052= - 1.046* 10-7. Thus, the function value is equal to the value of the positive term, the negative term is insignificant and the solution shown is not a zero of fl0. The conclusions that can be illustrated by this example regarding the structure and the information that has to be included in a test problem library are the following: 1. The model equations must be stored in the same form as the input for the numerical solver, to prevent introduction of typographical and other errors. 2. Constraints on the variables are an integral part of the mathematical model and should be explicitly included in the library. 3. Different values should be used as initial estimates for the different variables, and function values at the initial estimate must be reported in order to enable detection of errors in the problem set-up. 4. The variable values at the solution must be reported with the same precision that the solution was obtained (15 significant decimal digits for double precision). 5. Function values at the solution must be included. In case the function values are suspected to be too high, the order of magnitude of the various terms comprising the function must be compared to validate that the reported solution is a zero of the function. 2. S T R U C T U R E OF THE TEST P R O B L E M LIBRARY Many of the problems associated with the example presented in the previous section could have been easily detected if the physical nature of the model and the various variables was known. Although the inclusion of the description of the physical basis of the model in the library is not a must, it can be rather helpful in many cases. Meintjes and Morgan [7] have traced the "Combustion of Propane in Air" problem back to a paper from 1943 by Damkohler and Edse [9]. The stoichiometric equation of the reaction is CaH8+R(Oz+4NE)/2--~Products, where R is a parameter expressing the relative amounts of air and fuel. The list of products is shown in Table 2. Given this list of species, R must be greater than 3 for a physically feasible solution to exist; if R
10 it is "lean". Table 2 shows the equations (as provided by Meintjes and Morgan [7]) in the form they are stored in the test problem library. The unknowns nl, n2, ... nl0 represent the number of moles of product i formed per mole of propane consumed. An output variable (which appears on the lefthand side) is designated to each equation. In explicit equations, the output variable is assigned according to the calculation order, while in implicit equations, the assignment is arbitrary in order to allow the presentation of the variables, function definitions and values in a concise and compact format. The description of the variables and the equations are included as comments (separated from the equation by the # sign). Constrains on the variables are an integral part of the problem definition and they appear in a separate column of Table 2. All the variables represent moles of product formed; thus they all must be non-negative at the solution point. A constraint that must be satisfied all the way to the solution is marked by (a). This is necessary in order to avoid a negative value inside the square root function, in this particular example.
295 Table 2. Meintjes and Morgan [7] Version of the "Combustion of Propane in Air" Test Problem No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 aAn
Equations =
Constrains b
f(nl) = n1+n4-3 #Mol of Carbon Dioxide - Carbon Balance f(n2) = 2*n1+n2+n4+n7+n8+n9+2*n10-R #Mol of Water-Oxygen Balance f(n3) = 2*n2+2*n5+n6+n7-8 #Mol of Nitrogen - Hydrogen Balance f(n4) = 2*n3+n9-4*R #Mol of Carbon Monoxide - Nitrogen Balance f(n5) = K5*n2*n4-nl*n5 #Mol of Hydrogen - Equilibrium Expression f(n6) = K6*sqrt(n2*n4)-sqrt(nl)*n6*sqrt(p/nt) #Hydrogen atom - Equilibrium Expression f(n7) = K7*sqrt(nl*n2)-sqrt(n4)*n7*sqrt(p/nt) #Hydroxyl Radical- Equilibrium Expression f(n8) = K8*nl-n4*n8*(p/nt) #Oxygen Atom - Equilibrium Expression f(n9) = K9*nl*sqrt(n3)-n4*n9*sqrt(p/nt) #Mol Nitric Oxide - Equilibrium Expression f(nl0) = K10*nl^2-n4^2*nl0*(p/nt) #Mol of Oxygen - Equilibrium Expression nt = nl+n2+n3+n4+n5+n6+n7+n8+n9+nl0 #Total Number of Moles of Combustion Products K5 = 0.193 #Equilibrium Constant at 2200 K K6 = 2.597e-3 #Equilibrium Constant at 2200 K K7 = 3.448e-3 #Equilibrium Constant at 2200 K K8 = 1.799e-5 #Equilibrium Constant at 2200 K K9 = 2.155e-4 #Equilibrium Constant at 2200 K K10 = 3.846e-5 #Equilibrium Constant at 2200 K R = 10 #Air to Fuel Ratio p = 40 #Pressure (atm.) implicit equation is indicated by f(..)=. Output variable,~ assigned arbitrarily for implicit eqns.
>--0 (a) >_0 (a) __.0(a) >0 (a) ~0 >0 zO >-0 >-0 ->0
b Constraint on the output variable. An (a) indicates that the constraint must be always satisfied
Table 3. Initial Estimates and Solution for the "Combustion of Propane in Air" Test Problem Function and variable number 1 2 3 4 5 6 7 8 9 10 nt
Initial value
f0
n1(0)=1.5 n2(0)=2 n3(0)=35 n4(0)=0.5 n5(0)= 0.05 n6(0)= 0.005 n7(0)=0.04 n8(0)= 0.003 n9(0)=0.02 n10(0)=5 44.118
-1 5.563 -3.855 30.02 0.118 -0.0032339 -0.0209598 -0.0013330 -0.0076095 -1.1332377
n* 2.915725423895220 3.960942810808880 19.986291646551500 0.084274576104777 0.022095601769893 0.000722766590884 0.033200408251574 0.000421099693392 0.027416706896918 0.031146775227006 27.062238
f(n*) -3.11E-15 -7.11E-15 3.55E-15 -8.53E-14 1.94E-15 3.61E-16 1.16E-16 -2.98E-17 -3.25E-17 -7.59E-19
The introduction to the problem (as presented in the previous paragraph) and the data in Table 2 represent a complete definition of the problem. This includes the mathematical model and the physical basis. The equations as they appear in the second column of Table 2 can be directly copied into the POLYMATH 5.0 program for solution. If other programs are used for solution (such as MATLAB or MATHEMATICA), some modifications may be required. The required editing can be easily performed with Excel. The initial estimates, function values at the initial estimates, the solution and the function values at the solution are shown in Table 3. It can be seen that when 15 decimal digits are used for n*, the largest absolute value off(n*) obtained is of the order of 10"14, raising no doubt with
296 regard to the solution validity. All mole numbers are positive and their values make physical sense. This is an indication that the model is correct in contrast to the formulation in Table 1, where the nitrogen balance, for example, is grossly off because of an error in equation (4). 3. DISCUSSION AND CONCLUSIONS A web-based test problem library for NLEs that is most beneficial for potential users needs to include more information and in a different form than that found in existing test problem collections. Several important aspects of the proposed library have been demonstrated by the previous example. The equations should be stored in the same form as the input for the numerical solver, constraints on the variables should be defined explicitly, and initial estimates and function values at the initial estimates should be included. Variable values at the solution should be reported with the same precision that of the numerical solution, and function values at the solution must be given. While the inclusion of the physical basis of the model represented by the system of equations is not essential, it can be often helpful in verifying the physical validity of the solution. Additional aspects of the test problem library include: 1. Categorization of the problems according to size, difficulty level, number of solutions, and type of physical model. 2. Modification of the equations for alleviating the solution process, and 3. Initial estimate selection for various levels of difficulty. These have not been discussed here due to space limitation. REFERENCES 1. N. Brauner, M. Shacham and M. B. Cutlip, Chem. Eng. Educ., 30 (1996) 20. 2. M. Shacham, N. Brauner and M. Pozin, Computers chem. Engng., 22 (1998) 321. 3. M. Shacham, pp.891-924 in Westerberg, A.W. and Chien, H. H. (Eds) Proceedings of the 2nd International Conference FOCAPD, CACHE Corp., 1984. 4. M. Shacham, Computers chem. Engng., 14 (1990) 621. 5. K.L. Hiebert, ACM Trans. Math. Software 8 (1982) 5. 6. L.G. Bullard and L.T. Biegler, Computers chem. Engng., 15 (1991) 239. 7. K. Meintjes and A.P. Morgan, ACM Trans. Math. Software, 16 (1990) 143. 8. M. Shacham, Intern. Journal of Numerical Methods in Engineering, 23 (1986) 1455. 9. G. Damkohler and R.Z. Edse, Elektochem., 49(1943) 178.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
297
Analysis and Consistency of Process Models with Application to Ammonia Production v. Siepmann a, T. Haug-Warberg b and K. W. Mathisen a aNorsk Hydro ASA, Research Centre, P.O.Box 2560 N-3907 Porsgrunn, Norway bDepartment of Technology, HCgskolen i Telemark, N-3914 Porsgrunn, Norway Process models are key factors for continuous improvement in the chemical industries. This work shows that it is possible to combine (i) energy and exergy efficiency analysis with (ii) process simulation, and (iii) sensitivity analyses and optimization tasks. One consistent model is used instead of three incongruous approaches. Data and calculations of the primary reformer section of an ammonia plant illustrate the ideas. 1. INTRODUCTION Planning and reporting tasks, e.g. energy performance reporting, are generally decoupled from process simulation. Less suited tools based on different sets of thermodynamic data and spreadsheets are often used. This makes the calculations inconsistent. Several authors have suggested to shift enthalpies in the reference state [1,2] to obtain a reasonable enthalpy and exergy level reflecting the available energy in process streams. As recently shown by Haug-Warberg [3,4], the thermodynamic reference state can be shifted to define a new heat of formation Af/) with Af]-I i -- 0 for an independent set of recipient components. Equality of enthalpy and the sum of heating value and sensible heat allows energy efficiency analysis using ordinary process simulators. Methods to calculate relative exergy efficiencies are, among others, suggested by Sorin et al. [5]. This work presents a generic way to define unit based efficiencies. The approach is applied on different levels of granularity, and the results are compared to stream based efficiencies. The final part of the paper shows a way to obtain the sensitivity of the calculated efficiencies with respect to parameters of process units, such as heat exchangers.
2. ENERGY ANALYSIS Most process simulators allow user added stream properties. Redefining the heat of formation Af]-/provides a variable, which directly expresses the sum of the heating value and the sensible heat component of a stream. Using e.g. H20 and CO2 as reference components, Table 1 shows the shifted enthalpies at standard conditions, which fit the heating values found in the literature [6]. The shifted plane AfT/is used for exergy analysis (described in section 3). Figure 1 gives a general view of the observed process section. The energy flows are obtained by choosing Af]'/as the enthalpy basis. Combustion air has no energy contribution and is therefore left out. The cold purge gas leaving
298 Table 1 Standard and shifted heat of formation in kJ/mol. Standard data Afh* from [6-8]
H20 Afh* Afh,v Afh
Afh* Af~/ Afh
CO2(ig)
CO(ig)
02(ig )
H2(ig )
A/'(ig)
-285.83(1) -393.68 0.00(1) 0.00 64.82(ig) 80.70
-110.61 283.07 332.32
0.00 0.00 62.88
0.00 285.83 257.37
0.00 0.00 55.78
CH4(ig) C2H6(ig) C3H8(ig) n-C4Hlo(ig) -74.90 -84.74 -103.92 -126.23 890.44 1560.12 2220.44 2877.64 887.34 1564.38 2232.07 2896.63
n-C5H12(ig) -146.54 3536.84 3563.19
N2(ig) 0.00 0.00 55.81
the plant is taken to be a loss stream. These figures can easily be used to determine the overall absolute loss to 78.2 MW and the relative energy efficiency to 1] = 87.1%. The purge gas cool combustion air reforming steam
cold purge gas 22.2 MW 10.6 MW
feed and~][ Feed evaporation heat and mixing 403.2 MW I 389.0 MW I
fuel and _[ Fuel preheating ..__ heat ] and mixing 183.0 MW 174.3 MW
~ ,
403.2 MW MW vapourized feed 388.2 preheated air 15.5 MW 8.7 MW
I], Purge gas cool down [
preheated feed 448.1 MW 409.9 MW,
heat for high pressure steam and process air"21.3 MW 12.7 MW hot purge 69.8 MW gas
45.2 MW
hot fuel 198.5 MW I and air mix 173.9 MW ~ [ Primary reformer Ii
effluent 508.7 MW 450. 0 MW
Fig. 1. Simplified primary reformer section of an ammonia plant and enthalpy and exergy streams. Dashed lines represent streams unused in energy efficiency analysis.
down section loses 10.3 MW and has a relative energy efficiency of 97.9 %. Within the scope of the simulation model and the set of data, the absolute efficiencies are non-ambiguous, but the relative efficiencies are not. Haug-Warberg [3,4] suggests methods to define a reference state for process engineering, which guarantee positive values and thus a relative efficiency r I C [0, 1]. However, the process environment is never in thermodynamic equilibrium (e.g. cooling water at 4~ and air at 20~ ), so there is no unique definition of relative efficiency - as long as the system is treated as a black box. 3. E X E R G Y ANALYSIS
Disregarding chemical reactions, exergy is often defined as the maximum obtainable work output of a process, which converts the substance to a dead state just by exchanging heat with the environment [6,9]. But, as explained later in [6], and also used in [3,4], exergy is here
299 defined as the maximum obtainable work of a process described above, including chemical reactions. Haug-Warberg [4] exemplifies the problem of choosing proper sets of dead state species and discusses the influence on the exergy efficiency analysis of industrial processes. The exergy E is defined by E def H - HO - To ( S - So). Taking chemical conversions into account, H 0 - To So is the dead state Gibbs energy Go of the recipient stream:
E :
- G o + H - Tog = - G o + H res + HId + H~d - To Is]d -~- 82d - S~d + S res]
I
-- t~.. ni -tzo,i+h res i + A f h i*+
1-
, Cp,idT-To
T*
/
s ,i - R l n ~ j n jnip p,
)]
+s~eS
(1)
Euler's theorem of homogeneous functions [9] applied to E yields E = 0 by solving the system OE/Oni = 0 for a convenient set of Affl i -- Afh* -lUo,i by use of a proper ambient system of recipient components (To, po,no,i). Due to the ideal gas behaviour and occurrence of all required recipient components, ambient air is the most suitable dead state for the ammonia plant. Table 1 shows the new enthalpy plane based on the following equation for T = To:
Afhi
-
, ( Po no,i ) To si - R To In )--; + In ~,,jno,j
(2)
Introducing E as a stream variable in the flow-sheet simulation yields the exergy values printed in italics in Figure 1. The overall exergy loss is 111.2 MW and the relative exergy efficiency 80.6 %. The purge gas cool down section loses 12.7 MW exergy and therefore has an efficiency of 97.1%. 4. DETAILS OF E X E R G Y ANALYSIS The exergy analysis treats the observed system as a black box. This method is convenient, whenever no model exists and the input output data is based on real plant measurement. When a simulation is given, loss streams and streams at ambient conditions are all known exactly in terms of the accuracy of the used model. This permits a more intuitive definition of efficiency, which only considers converted parts of the observed measure.
4.1. Unit operation exergy efficiencies For instance, the heat exchanger efficiency is not influenced by the heating value of the hydrocarbons, and the pump efficiency does not depend on the temperature of the transported medium, but is defined as TI de__f('v'Ap)/Peg. Some publications [10,11] suggest a classification of different energy types. The concept of transiting exergy is used in [12,13] to describe the part of unaffected exergy. In order to exclude this part from efficiency, exergy can be subdivided into a chemical and sensible part E c = H(To) - ToS(To) and E s = [ H - H(T0)] - To[S- S(T0)]. The input and output is compared for each material independent system and each type of exergy. The resulting exergy vector AE = Eout- Ein contains zeros for transiting parts, and positive and negative numbers for supply and delivery. The definition of efficiency of units is 1]
def =
AE-b + AE.b-
with
b+ -
/ 1 0 %,
for ~ i else
> 0
and
b~- - l - 1 0
for LkEi < 0 else.
(3)
%,
The key idea is to define system boundaries of minimal exergy throughput, which still include all relevant process information in the described unit.
300
EcCld,in EcSld,in E~St,in EhC~
,, 1 ~ i i.
.
.
.
.
EcCold,out E~old,out E~St,out EhCt,out S
r
v
~ .
.
.
.
.
.
.
.
.
.
Fig. 2. Converted and transiting exergy parts in a heat exchanger
4.2. Mixer efficiency Eq. (3) is directly applicable for a mixer. In this case, there is only one material system, but exergy can be converted from chemical to sensible form: AE
--
EoSut-~
EiS,j
E~,Ct-~ EiC,j
j=l
N: number of mixer inlets
(4)
j=l
The chemical contribution will always be negative and appear in the denominator of Ti. The sensible part can be zero, positive or negative, and a nonzero efficiency occurs if AE s > 0. An example of this is nitric acid neutralization by ammonia in fertilizer plants, provided that the heat of mixing is used. 4.3. Heat exchanger efficiency Heat exchangers contain two separate material systems. Both have sensible and chemical exergy, so that:
Egot ]
E
-
level, hence AE = \E~Sld/
E~~176 E sh~ \EcSld,out
=:ff TI -- EcSld'~
EcSld'in
(5)
E(~ot,in -- E~St,out
0
E~Sld,in/
4.4. Exergy efficiency of the purge gas cool down section Figure 3 gives a more detailed view of the purge gas cooling section. The indicated relative efficiencies give suitable information about the units, but they are not sufficient to calculate the overall efficiency of the process section unless the transiting contribution is also known. In that case, a modified flow-sheet can be drawn as shown in Figure 2 for a single heat exchanger. Transiting parts are led around the unit block. Using this technique, a unit-based efficiency can also be calculated for the observed process section. For instance, the chemical exergy of the fuel is - apart from the m i x i n g - not affected, so it is led around the flow-sheet. The resulting value of r I = 71.2 % is therefore more suitable to compare two similar process sections.
5. SENSITIVITY ANALYSIS The model of the primary reformer section (Figure 1) is implemented in C+ +. It is made use of a new developed object oriented data type called Dfloat, which evaluates all partial derivatives with respect to other variables runtime. In contrast to automatic differentiation, this approach using operator overloading and lazy evaluation techniques - handles nested optimization loops
301
cold comb. air
cold ee x cold proc. air
I
HP steam
c~ 829c~
/
@
H2
88.8% H4
@
hot purge gas ]
] [
t
t
88.3% ~ (Sll] ] ~ f
0%~ ~~~-
~_~ ( ~
.~ H6 I 74.6%I @
, S
0 / r
[H7 154.3%
t-~" hOt cOmb alr ~ HP steam hot feed mix hot proc. air Fig. 3. Detailed model of the purge gas cool down section and relative exergy efficiency of unit operations ref. steam
'
80.4%
~
I
and allows specification of the required derivatives runtime. In this work, Dfloat is used inside the unit operation algorithms to perform second order optimization. Furthermore, it is applied to obtain sensitivity of exergy efficiency with respect to heat exchanger specifications. The dependency of the (UA)-product of H4 is included into the flow-sheet calculation. It can be found that for instance
~TIt~
--
5.5 x 10 .4 KkW -1
~(UA)IJ4
and
OTIH1
O(UA)H4
=
- 3 . 3 x 10 .4 KkW -1
(6) "
Similar derivative information can easily be used to perform an outer optimization loop.
6. CONCLUSION The primary reformer section of an ammonia plant has been modelled as a system of four black box models. Based on a suitable reference state, both energy and exergy efficiency calculations are integrated into the simulation. Afterwards, one part of the process section is simulated in detail, and additional information is used to obtain exergy efficiencies of single operation units defined in an intuitive but generic way. Extending this concept to agglomerated sets of units, suitable exergy efficiencies are obtained. The capabilities of the process model can be extended to cope with sensitivity analysis tasks. With very little programming effort, derivative information is obtained automaticly and gives a useful insight into a complex network of interacting units.
302 SYMBOLS
A b Cp E G H h
heat exchanger surface input - output vector heat capacity exergy Gibbs energy enthalpy molar enthalpy
[m 2] [-] [J/(molK)] [J] [J] [J] [J / mol]
n p S s T U
mole-number [mol] pressure [Pa] entropy [J] molar entropy [J/(molK)] temperature [K] heat transfer coefficient [J/(m2K)]
rI
efficiency
[-]
/~
chemical potential
^ * id s f
regarding energy analysis standard state (298.15 K, 105 Pa) ideal sensible formation
0 res c
regarding exergy analysis reference state residual (non-ideal) chemical
[J/moll
REFERENCES
1. G. Wall, Exergy flows in industrial processes, Tech. Rep. 83-11, Physical Resource Theory Group, Chalmers Univ. of Technology and Univ. of Grteborg (Jul. 1986). 2. D.R. Morris, J. Szargut, Standard chemical exergy of some elements and compounds on the planet Earth, Energy (Oxford) 11 (8) (1986) 733-755. 3. T. Haug-Warberg, Exergy analysis of industrial processes. I Basic theory, submitted to Energy Convers. Mgmt. 4. T. Haug-Warberg, Exergy analysis of industrial processes. II Calculated results, submitted to Energy Convers. Mgmt. 5. M. Sorin, J. Lambert, J. Paris, Exergy flows analysis in chemical reactors, Chem. Eng. Res. Des. 76 (A3) (1998) 389-395. 6. K. Wark, Advanced Thermodynamics for Engineers, McGraw-Hill. Inc., 1995. 7. M.W. Chase, C. A. Davies, J. R. Downey, D. J. Frurip, R. A. McDonald, A. N. Syverud, JANAF thermochemical tables. Third edition. Part I, A1-Co, J. Phys. Chem. Ref. Data, Suppl. 14 (1) (1985) 1-926. 8. M.W. Chase, C. A. Davies, J. R. Downey, D. J. Frurip, R. A. McDonald, A. N. Syverud, JANAF thermochemical tables. Third edition. Part II, Cr-Zr, J. Phys. Chem. Ref. Data, Suppl. 14 (1) (1985) 927-1856. 9. J.W. Tester, M. Modell, Thermodynamics and Its Applications, 3rd Edition, Int. Series in the Physical and Chemical Engineering Sciences, Prentice Hall PTR, 1997. 10. A. W. Culp, Principles of Energy Conversion, 2nd Edition, Series in Mechanical Engineering, McGraw-Hill. Inc., 1991. 11. W. R. Dunbar, N. Lior, R. A. Gaggioli, The component equations of energy and exergy, J. Energy Resour. Technol. 114 (1992) 75-83. 12. M. Sorin, J. Paris, Integrated exergy load distribution method and pinch analysis, Comput. Chem. Eng. 23 (1999) 479-507. 13. M. Sorin, A. Hammache, O. Diallo, Exergy based approach for process synthesis, Energy 25 (2000) 105-129.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
303
Dynamic Modelling of Chromatographic Processes: A Systematic Procedure for Isotherms Determination H.K. Teoh a, E. SCrensen a*, M. Turnerb and N. Titchener-Hookerb aDept, of Chemical Engineering, University College London, Torrington Place, London, WC1E 7JE, United Kingdom bDept, of Biochemical Engineering University College London, Torrington Place, London, WC1E 7JE, United Kingdom Due to the non-linear and dynamic nature of large-scale chromatographic processes, these processes are difficult to design. The accuracy of chromatography models is particularly dependent on the description of the relevant component isotherms. Identifying the correct isotherms, and determining the corresponding parameters, is a major obstacle. In this paper, we present a simple but efficient method for isotherm parameter estimation based upon the individual elution profiles of the components in the mixture. The approach requires minimal experimentation and yields a significant reduction both in terms of the time and the effort involved in finding the most suitable isotherm. The approach is demonstrated through a case study involving a binary mixture. 1. I N T R O D U C T I O N
Successful application of process-scale chromatographic separation processes is widespread in the pharmaceutical and biotechnology industries. Traditionally, the design and realisation of a particular process-scale chromatography process is based on detailed experimental work at analytical or preparative-scales with significant efforts involved in isotherm determination. This procedure is both tedious and time consuming. Hence, a better understanding of the process dynamics and equilibria of chromatographic separations is needed in order to reduce the time and effort from process conception to actual process realisation. The main objectives of this paper are: 1) to develop a systematic procedure for isotherm determination, 2) to determine isotherm parameters for the candidate isotherms by estimation methods based on a minimal set of experimentally obtained elution profiles and 3) to identify the most appropriate isotherm model which can accommodate the displacement and competitive effects that characterise the non-linear chromatography of realistic feed stocks. When modelling chromatographic processes, a significant number of parameters must be determined experimentally a priori, as the transport and equilibria relationships are too complex to model directly from first principles. Model accuracy is particularly dependent on the isotherm description, which relates the solute concentrations in the mobile and stationary phases [5]. In *Authorto whomcorrespondenceshouldbe addressed: Fax: +44 20 7383 2348; Phone: +44 20 7679 3802; email: [email protected]
304 the case of multi-component mixtures, an additional complexity results from the competition between the different components as they interact with the stationary phase. The amount of a component adsorbed at equilibrium is a function of the concentration of this component, as for single component isotherms, but also of the concentration of all the other components present in the solution which are adsorbed by the stationary phase [2].
2. ISOTHERM DETERMINATION Traditionally, isotherms are determined experimentally by static or dynamic methods. Static methods are time consuming batch methods, in which the adsorbate concentration in the fluid phase is monitored by gravimetry or infrared adsorption. Dynamic methods can describe the fast mass transfer kinetics and near equilibrium behaviour of the phase system utilising a chromatographic column, eg. frontal analysis (FA), frontal analysis by characteristic points (FACP), elution by characteristic points (ECP) etc. [ 1]. For the dynamic methods, the isotherm is determined by tedious fraction analysis of the elution profile from the column. It has been demonstrated that of all the methods only FA supplies accurate single component and competitive isotherm data, and only as long as the mass transfer kinetics is not too slow [2]. Unfortunately, FA methods are time consuming and require a significant amount of relatively expensive pure chemicals. A faster and simpler way of determinating such parameters utilising a numerical parameter estimation technique, coupled with minimal experimentation, is proposed in this study. This procedure can then be employed to identify more efficiently an appropriate isotherm with which to model the process equilibria. The proposed systematic procedure for isotherm determination is as follows: Step 1: The elution profiles of the individual components of a mixture as well as for the whole mixture must be generated experimentally (The individual elution profiles are for parameter estimation whilst the components elution profiles are for isotherm validation). In order to capture the competitive and displacement effects of the mixture under consideration, the experimental conditions must be such that sufficient degree of overlapping between the components is achieved. Step 2: The number of theoretical plates, Ne,i, the height equivalent to a theoretical plate, Hp,i and the apparent dispersion coefficient, Dap,i, of component i can be calculated directly from the experimental elution profile according to the following equations [2] [5]:
Np,i= 5.54(tg'i) 2
(1)
L Hp,i = Np,i
(2)
ui.L Dap,i = 2 "Np,i
(3)
where tn,i is the retention time for component i, Ai is the peak width of component i at half height, L is the column length and ui is the mobile phase velocity for component i.
305 Step 3: An UV calibration curve which relates the column outlet concentration, Gout, i to the UV absorbance, UV~ for component i must be constructed based on the Beer-Lambert Law [6]:
(4)
UVi :- Ei " Cout,i
where Ei is the Beer-Lambert Law's coefficient for component i. Provided there is no interaction between the various components, the Beer-Lambert's Law is still valid for a solution containing more than one adsorbing species in the following form [6]: n
UVtotal --"
u g i = E E i "Cout,i i=1
(5)
i=1
where Ugtota l is the total UV absorbance. This relationship holds in the linear region of the UV absorbance which is the normal operational region of chromatographic separation. Step 4: An appropriate isotherm model must be selected. Examples include: Langmuir isotherm, competitive Langmuir isotherm, bi-Langmuir isotherm, Fowler isotherm, Freundlich isotherm etc. [1] [2]. Step 5: Parameter estimation is conducted to determine the parameters of the selected isotherms. Step 6: The determined parameters can then be employed to validate the elution profiles of the mixture as a function of the operating conditions using the components elution profiles obtained in Step 1. 3. CASE STUDY A case study is presented to verify the approach outline above. The separation of a binary mixture of naphthalene and fluorene was considered. The mixture was separated in an HPLC column (4.6 X 150 mm) at different flowrates (0.8, 1.0, 1.2 and 1.5 mL/min). Different sample volumes of 10, 20, 30 and 50 lzL with 0.1 g/L feed concentration for each of the aromatic compounds were injected into the column. The mobile phase was 90 % acetonitrile and 10 % water. The stationary phase consisted of Jupiter 15 t~m C18 particles (Phenomenex, Macclesfield, Cheshire, United Kingdom). The separation was carried out under isocratic conditions. The response of the UV detector was found to be linear in the range of experiments undertaken. 4. PARAMETER ESTIMATION Both Langmuir (Equation 6) and competitive Langmuir (Equation 7) isotherms are considered for this separation:
aiCi qi = 1 + b/G
(6)
aiCi qi -- 1 +
ETbjC j
(7)
where qi and Ci are the solute concentrations of component i in the mobile phase and stationary phase respectively, ai and bi are the isotherm parameters for both the competitive Langmuir isotherm and the single-component Langmuir isotherm of that component.
306 (a)
(b)
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(~o 0 . 4 t'M
ur~
30.3
1.2 1
~ o.8 N 0.6 o
-8 0 . 4 ~
0.1 0
~:~ 0 . 2
0
-
1
2 Time,
3 min
4
--
5
0
1
1.5
I
II 2 Time,
21,5 min
i
3
j_
3.5
Fig. 1. Simulated and real elution profiles for naphthalene (a) F = 0.8 mL/min & Vinj - 20 pL (b) F = 1 mL/min & Vinj - - 5 0 / / L (Continuous line = simulated elution profile and 'x' = exp. elution profile)
The chromatographic column section was modelled using the well-known equilibrium-dispersive model, coupled with appropriate boundary and initial conditions [2] [5]. The single isotherms parameters, ai and bi, for both the components were estimated using gEST within the gPROMS package [4]. Although a large number of experimental data sets were available, only the experimental data for a flowrate (F) of 1 mL/min with sample volumes (Vinj) equal to 10 and 50111-, were employed for the parameter estimations. An orthogonal collocation finite element method was applied to solve the dynamic model. The computational load required for the parameter estimations was 11179 and 12781 CPUs for naphthalene and fluorene, respectively. Running on an IBM RISC System 6000 (43P-140), each parameter estimation took around 3 to 4 hours which is insignificant compared to the time and effort required to construct the detailed experimental work for isotherm determination. Further saving can be incurred in terms of reduced operating cost, raw material, overhead etc. The parameters were then used to predict the single elution profiles for both the individual component separately at different flowrates (0.8 and 1.0 mL/min) and different sample volumes (10, 20 and 30 pL). This was done to assess the goodness-of-fit of the parameter estimations. Excellent agreement was obtained between the simulated and experimental elution profiles at different operating conditions. Some of the examples are shown in Figures 1 and 2. 5. RESULTS AND DISCUSSION The isotherms parameters found for the individual components were used in predicting the elution profiles for a 1:1 binary mixture (both naphthalene and fluorene) at different flowrates (0.8 and 1.0 mL/min) with 10 #L sample volume utilising either the Langmuir or the competitive Langmuir isotherms. This was done to determine the most appropriate isotherm to describe the process equilibria of the case study investigated. Figures 3 and 4 show the simulated and experimental elution profiles for a 10 !11-, binary mixture (1:1) at 0.8 and 1.0 mL/min assuming a Langmuir isotherm and a competitive Langmuir isotherm, respectively. For both the isotherms, good agreement in terms of peak positions was obtained for both the flowrates considered. When the flowrate was 0.8 ml_,/min, slight
307 (a)
(b)
0.5 E 0.4 E i.o (.o cM
E t.,.o 0 . 4 r t-M 0.3 t.-
~ o.e o .-o "~ o . 1
-eL 0 . 2 o -.o ~ o.1 0
2
.......
3
Time,
min
illl
4
5
2
Time,
min
3
Fig. 2. Simulated and real elution profiles for fluorene (a) F = 0.8 mL/min & Vinj = 10 pL and (b) F = 1.0 mL/min & V~,,j = 10/A_, (Continuous line = simulated elution profile and 'x' = exp. elution profile)
(a)
(b)
0.7
0.7
0.6
~
0.5
@ o.5
8 o.4 r-
8o.4 e-
E r-
~
E
0.6
0.3
~
0.3
0.2
~
0.2
::3 0 . 1 0
~0.1 0
1
2 Time,
3 min
4
I
." : 5
0
0
.......
1
III
Time,
I 2 min
3
Fig. 3. Simulated and real elution profiles for the binary mixture assuming a Langmuir isotherm with a sample volume of 10 pL; (a) F = 0.8 mL/min (b) F - 1 mL/min (Continuous line = simulated elution profile and 'x' = exp. elution profile)
differences in terms of peak heights were observed whilst when the flowrate was 1 mL/min, good agreement in terms of peak heights was obtained. This is as expected due to the fact that the experimental data sets when the flowrate was 1 mL/min were used for the parameter estimation. Both the Langmuir and competitive Langmuir isotherms predicted the elution profiles reasonably well in these cases. However, from a computational load point of view, the Langmuir isotherm will be preferred due to its simple mathematical nature. Some oscillation in the numerical solutions was observed for both the isotherms considered due to the inherent characteristic of the orthogonal collocation finite element method employed. This can be reduced by using a higher number of finite elements though this is at the expense of longer computational times, q
308 (b)
(a) E
t--
0.7
~
E
0.6
0.7 0.6
o.~
~ o.5
80.4 I:::
8r-- o.e
~
~
0.3
~
0.2
-nO.1 0
0
......
1
N
2 3 T i m e , rain
4
I
I-
5
~
o.a
~
0.2
~
o.1 0
0
..........
1
2 Time, min
3
Fig. 4. Simulated and real elution profiles for the binary mixture assuming a competitive Langmuir isotherm with a sample volume of 10/zL; (a) F - 0.8 mL/min (b) F = 1 mL/min (Continuous line = simulated elution profile and 'x' = exp. elution profile)
6. CONCLUSION A systematic procedure for determinating isotherm parameters utilising numerical parameter estimation and requiring minimal experimentation effort has been developed. The most appropriate isotherm which can describe the equilibria relationship for a particular purification process can easily be identify using this method. Significant reduction both in terms of time and effort can be then realised. For the case study considered, good agreement between the experimental data and the simulated elution profiles was obtained under different operating conditions. Both the Langmuir and the competitive Langmuir isotherms captured successfully the displacement and competitive effects in non-linear chromatography for the case considered. Future work will consider the application and verification of this isotherm determination procedure at a larger scale of operation. REFERENCES
1. Bellot J.C. and J.S. Condoret, Selection of competitive adsorption model for modelling displacement chromatography, J. of Chrom. A, 657, 305-326, 1993. 2. Guiochon G.,S. Golshan-Shirazi and A.M. Katti, Fundamentals of Preparative and Nonlinear Chromatography, Academic Press, Boston, 1994. 3. James E, M. Sepulveda, E Charton, I. Quinones and G. Guiochon, Chem. Engng. Sci., 54, 1677-1696, 1999. 4. Process Systems Enterprise Ltd., gPROMS Advance User's Guide: Release 1.8, London, 2000. 5. Teoh H.K., M. Turner, N. Titchener-Hooker and E. S0rensen, ESCAPE 10, 8, 193-198, 2000. 6. Thomas, M., Analytical Chemistry by Open Learning: Ultraviolet and Visible Spectroscopy, 2nd Edition, John Wiley & Sons, 1997.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
309
Process Simulation and Analysis with Heterogeneous Models John E. Tolsma a and Paul I. Barton a aDepartment of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA This paper describes how symbolic techniques can be applied to general Fortran code in order to perform automatically many tedious, time-consuming, and error prone tasks required when using state-of-the-art algorithms for numerical calculations. Using this new approach, external models written in Fortran can be properly incorporated into an equation-oriented modeling environment. This allows the modeler to create heterogeneous models using both Fortran and the high level input language of the process simulator, depending on which is more suitable for the particular facet of the overall model. 1. INTRODUCTION AND MOTIVATION For several decades, modeling and simulation has played an important role in the design and analysis of chemical processes. The development of increasingly sophisticated processes, tighter environmental constraints, and the necessity to become more competitive will certainly make this impact even larger in the future. One reason for the widespread use of modeling and simulation in the chemical process industries is the availability of advanced process modeling environments, such as modular simulators (e.g., Aspen Plus, HYSYS, and Pro/II) and equationoriented simulators (Aspen Custom Modeler, gPROMS, and ABACUSS II). These tools provide an environment where the modeler can concentrate on constructing a correct model and not have to worry about the myriad of additional details associated with performing efficient and correct calculations. Unfortunately, the features that make these modem modeling environments convenient and user-friendly also make them inflexible. For example, modem modular simulators provide a library of unit operation models that can be assembled together (usually with a user-friendly graphical environment) to construct the flowsheet of interest. However, the modeler is effectively limited by the relatively small number of unit operation models provided in the library. Proprietary models developed in-house can be incorporated into the library, however, this is often a very difficult task, requiring a great deal of expertise in the modular simulator employed. In contrast, most modem equation-oriented simulators provide a high level, declarative input language with which the modeler constructs the mathematical model of the process. Although custom or proprietary models can be coded with this input language, problems occur when attempting to use existing legacy models, usually in the form of Fortran subroutines. (In this paper, the Fortran programming language is emphasized due to its popular and historical use for mathematical modeling. However, the ideas presented are directly applicable to any programming language employed.) One option is to re-write the existing Fortran model in the input language of the process simulator. However, this is time-consuming, error
310 prone, and sometimes not possible due to the limitations of the input language. Alternatively, most equation-oriented process simulators provide the option for linking external models as "black-boxes" into an overall flowsheet. This is problematic for the following reason. Modern numerical algorithms require substantially more information other than simply the numerical values of the equation residuals. Accurate partial derivatives, obtained efficiently, can often dramatically improve the performance of many numerical algorithms (in particular, parametric sensitivity analysis). If the model is sparse, significant improvements can be realized by exploiting sparsity for memory savings and speed increases. In addition, discontinuities within the external model must be handled explicitly to ensure efficient numerical integration [2] and correct parametric sensitivity analysis [1,4]. These discontinuities may result from nonsmooth intrinsic functions such as MIN and MAX in addition to the more obvious IF statements. By coding the model in the input language of the equation-oriented simulator, this additional information is readily extracted and exploited. In contrast, if the model is linked as a "black-box" then derivatives are computed using finite differences, sparsity is not exploited, and discontinuities are hidden, resulting in a degradation of performance and possibly incorrect results. In the following section, we comment on the advantages of process modeling using low level programming languages. We then present an approach for automatically extracting the additional information required for proper and efficient numerical calculation and demonstrate how external models can be readily and properly incorporated into a modern equation-oriented modeling environment. This allows the modeler to combine the power and flexibility of low level languages such as Fortran (as well as leveraging the large amount of existing legacy code) with the convenience of a modern equation-oriented environment.
2. HETEROGENEOUS MODELING
Even with the emergence of modern process modeling environments, the importance of Fortran models should not be underestimated. The obvious reason is that large amounts of proprietary or classified legacy code currently exists and embodies a great deal of knowledge and understanding of the process. In addition, complex physical property models are often only available as Fortran subroutine libraries. However, the importance of Fortran code is not limited to the exploitation of existing models. For example, if the modeler intends to perform a dynamic simulation with a flowsheet containing a new or proprietary model, the input language of the equation-oriented simulator may not be adequate to represent the model. Using Fortran not only offers the modeler more flexibility, but also allows the modeler to embed custom-tailored solution algorithms for solving the new model. As stated in the previous section, modern modeling environments allow the user to incorporate external models as "black-boxes" into an overall flowsheet. We present an alternative approach where external Fortran code is incorporated into the modeling environment in a manner identical to native models (i.e., models provided with the modeling environment or written in the input language of the environment). Using symbolic and code transformation techniques, the external Fortran code is analyzed and new Fortran code is generated providing all of the additional information required when performing numerical calculations using state-of-the-art algorithms. The ideas described above are implemented in a software library called DAEPACK [3]. The DAEPACK library is divided into two main parts: components for analyzing code and generating new information, and numerical components that exploit the automatically gen-
311 crated information. The symbolic library is described in this paper. The symbolic operations consist of two phases: the translation phase and code generation phase. During the translation phase, the source code for the external model is read and a symbolic representation is constructed in computer memory. During the code generation phase, the symbolic representation of the model is analyzed and new code is constructed which can be compiled and linked to the application to provide the additional information, including analytical derivatives, sparsity patterns, and discontinuity information (this is described in more detail in [3]). Figure 1 illustrates this process. In this figure, the user provides the source code for the external model and a specification file which describes what code is to be generated, the independent and dependent variables, etc. DAEPACK then generates automatically new code for computing analytical derivatives, sparsity pattern, and discontinuity-locking information. However, the code generated is not limited to just these examples. Other codes that may be generated can return the interval extension or convex estimator of a given model. Also, what is important to emphasize is that the external model does not have to be a simple set of assignments but may embed sophisticated solution algorithms, for example a root finding procedure for an equation of state. Not only does the approach described above allow legacy models to be properly uti-
User Supplied
DAEPA CK Generated l1
A SUBROUTINE KVAL_AD (NC, K, X, Y, . . .'I
DKDX,NE, IROW,JCOL)
/ IMPLICITNONE User specification file defining independent and dependent variables and code generation options. ~
. . . . . . . . . . .
- ~ _
~ ~ "7 n A i : D A ~ . i ( .......
f S~BRO~INE KVA~,NC,K,X,Y.... , ~ / / IMP~.ICI~NONE /'/
INTEGER NC
/
END
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~
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/
|
/ J
/,
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hi I ~ r X C X T NONE "1 INTEGE~we
~
~
I I
J
~
_
1
NSTATES, NDISCONS, DISCONS)
Fig. 1. Automatic generation of additional code using DAEPACK.
lized, it also allows c o n s i s t e n t heterogeneous models to be formulated. The best language (e.g., Fortran, C, or the input language of the modeling environment) can be selected for different portions of the model based on the expressive power of the language, what is being modeled, and the skill of the modeler. This approach has been implemented in the equation-oriented process simulator ABACUSS II (http://yoric.mit.edu/abacuss2/abacuss2.html). ABACUSS II, like other equation-oriented environments, provides a high level declarative input language. However, ABACUSS II also employs DAEPACK for properly incorporating portions of an overall flowsheet model represented as Fortran code. ABACUSS II translates the portions of model in the form of ABACUSS II input files and constructs data structures that may be analyzed and
312 manipulated to provide the necessary symbolic information for performing a numerical calculation. In addition, ABACUSS II calls DAEPACK to construct subroutines automatically that return the same symbolic information from portions of the model represented by Fortran code. Rather than calling this external code as a "black-box", all portions of the model are treated in a correct, consistent manner. Figure 2 demonstrates how a heterogeneous model evaluation is performed. The left side of Figure 2 shows the symbolic representation of the equations con-
Fig. 2. Evaluating the residuals of a heterogeneous model and incorporating discontinuity information for proper hybrid discrete/continuous simulation.
structed from the ABACUSS II input files. Residual values are obtained by interpreting these equations and assigning the values to the appropriate position of the residual vector. In contrast, the right side of this figure, shows the portions of the model that are computed with external Fortran code. During a residual evaluation, the external code is called to compute values of the dependent variables given the current values of the independent variables, and these values are inserted into the appropriate location of the overall residual vector. This process is the same as other equation-oriented simulators that allow the user to link in "black-box" external code. What distinguishes ABACUSS II is that DAEPACK is used to construct new code from these external models which extract the hidden discontinuities so that they may be exploited during hybrid discrete/continuous calculations. This is shown at the bottom of Figure 2. Figure 3 shows how a heterogeneous Jacobian evaluation is performed. The symbolic equations constructed from the ABACUSS II input files are shown on the left side of this figure. The gradients of these equations (rows of the Jacobian matrix) are accumulated from symbolic representation and inserted into the overall Jacobian matrix. The rows of the Jacobian matrix corresponding to the external portions of the model are evaluated by calling the derivative code
313 generated automatically by DAEPACK. This has the following advantages: the derivatives are exact (up to round-off error), they are evaluated efficiently, and any sparsity of the external model is preserved in the overall Jacobian matrix. In contrast, if the external code is called as a "black-box" then the corresponding portions of the Jacobian matrix would be evaluated using finite differences and any sparsity would be lost.
Fig. 3. Evaluating the Jacobian matrix of a heterogeneous model.
The discussion above describes how external Fortran code can be incorporated into an equationoriented simulator. However, there may be situations where the reverse may be desirable, that is, creating Fortran code from ABACUSS II input files. ABACUSS II provides the option for generating Fortran code from the portions of the model described by ABACUSS II input files and integrating this output with other portions of the model originally available as Fortran code. The resulting Fortran model can then be processed by DAEPACK to generate all of the additional information described above (e.g., analytical derivatives, sparsity patterns, and hidden discontinuities). This process is desirable when the model is to be used in applications where speed is crucial (e.g., if the model is embedded within another application), or when the model is to be used with a custom or proprietary numerical algorithm. ABACUSS II can be used to construct, debug, and validate the model, and then Fortran code can be generated, compiled, and linked into an application requiting fast model and derivative evaluations. Thus, the features described in this paper provide the modeler with full inter-operability between Fortran and ABACUSS II. 3. COMMENTS ON OTHER INITIATIVES The CAPE Open committee has developed a set of interface specifications based on CORBA and COM for unit operation models, physical property models, numerical solvers, and graph analysis tools. By using components adhering to this standardization, a modeler can assem-
314 ble a collection of components, developed in-house and/or purchased from several third-party vendors, to solve a wide variety of problems. Unfortunately, substantial effort is required by the modeler in order to make the model CAPE Open compliant. This is particularly true when the model is used for dynamic simulation, where the current CAPE Open standard requires the modeler to provide not only residual values, but also sparsity patterns, analytical derivatives, and the state task network corresponding to the discontinuous equations in the model. Fortunately, the ideas described in this paper can also be used to generate automatically the additional information required by the CAPE Open standard, simplifying the creation of CAPE Open compliant components. It should be noted that the CAPE Open committee has recognized the need for assisting the user in component development and has started AD-CAPE, an initiative for incorporating automatic differentiation into the standardization. However, as described in this paper, derivatives are only a piece of the puzzle when performing numerical calculations properly. By creating standard interfaces, the CAPE Open compliant process modeling environment will be able to solve heterogeneous models, provided the model components also adhere to the standard. Although these and similar interface standards will increase the flexibility of process modeling environments, the communication overheads associated with the component architectures may impair performance. These approaches may be better suited for non-communication intensive applications distributed across multiple platforms. These additional communication overheads are not present when binary-compatible compiled code is linked directly into an application. Thus, the ideas presented in this paper provide both an enabling and alternative technology to the heterogeneous modeling approach advocated by CAPE Open. 4. CONCLUSION Using the symbolic and code generation components of DAEPACK, external Fortran code can be properly incorporated into a modem process modeling environment. Like modem equation-oriented environments that can generate all of the information necessary for robust, efficient, and correct numerical calculation from a model written in the simulator's input language, DAEPACK can generate new code which computes the same information from the original Fortran code. This allows heterogeneous models to be developed where the modeler selects different languages for different portions of the flowsheet model depending on the expressive power of the language, what is being modeled, and the skill of the modeler. Acknowledgments - This research was supported by the EPA Center for Airborne Organics at MIT and ABB Research Limited.
REFERENCES 1. 2. 3. 4.
Santos Gal~in, Willian E Feehery, and Paul I. Barton. Parametric sensitivity functions for hybrid discrete/continuous systems. Applied Numerical Mathematics, 31:17--47, 1999. TaeshinPark and Paul I. Barton. State event location in differential algebraic models. ACM Transactions on Modeling and Computer Simulation, 6(2):137-165, 1996. John E. Tolsma and Paul I. Barton. DAEPACK: An open modeling environment for legacy models. Industrial and Engineering Chemistry Research, 39(6): 1826-1839, 2000. John E. Tolsma and Paul I. Barton. Hidden discontinuities and parametric sensitivity calculations. in preparation, 2000.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
315
A Structured and Selective Framework for Hybrid Mechanistic-Empirical Model Building Pedro Vale Lima and Pedro M. Saraivaa a Department of Chemical Engineering, University of Coimbra Pinhal de Marrocos, 3030-290 Coimbra, Portugal phone: 351-239798700, fax: 351-239798703 e-mail: eq3pvl @eq.uc.pt, eq lpas @eq.uc.pt
In order to address issues related with process operation, diagnosis, optimization, improvement and control, among other tasks, several kinds of models have been used. They usually fall under the scope of two distinct paradigms: mechanistic first-principles and empirical approaches. Both have been adopted but very few frameworks were developed so far in order to combine and integrate features from each one of them into hybrid models, which share mechanistic and empirical components. In this article we describe a methodology for overcoming this lack of integration efforts, through an algorithm that leads to the construction of process models that contain mechanistic and localized empirical elements, achieved by exploring symbolic manipulation of the first-principles model equations. This new framework was tested and evaluated by application to a simulated CSTR case study. 1. INTRODUCTION The construction of mathematical models to forecast and understand the behavior of chemical processes forms the basis for a countless number of tasks (planning, optimization, improvement, fault diagnosis, control, etc). Depending on the nature of the specific process and its desired goals, several kinds of models have been developed, differing namely in scope, level of detail, and underlying structures. However, for many practical situations two separate schools of thought have emerged: on one hand, we have fully mechanistic approaches, where models are built based upon first-principles equations; on the other hand, and specially for operation analysis and improvement at existing plants with complex transformations, fully empirical techniques have also been suggested and applied, relying in operators knowledge extraction, data analysis based upon machine learning [1 ] or statistical tools. But very few efforts have been done in the past to combine and integrate both of the above paradigms, although they are conceptually believed to be complementary to each other: empirical components will in general lead to better local prediction capabilities through the full exploration of all information that is available, while mechanistic elements make it possible for one to get a better understanding of the underlying physico-chemical phenomena, predict the values for unmeasured state variables, provide additional trust, reliability and extrapolation characteristics. In the limited previous work that has been done in order to build hybrid models [2,3], one may find different perspectives: choice among alternative first-principles models followed by some parameter fitting to existing data;
316 combination of the available mechanistic equations with an empirical model adjusted to the residuals associated with first-principles predictions. However, none of them do take into account the detailed fine structure of the available underlying mechanistic model, neither do they allow for partial localized and selective introduction of empirical elements. In this article we present a framework for building hybrid models that provides these capabilities.
2. ALGORITHM FOR HYBRID MODELING Our approach for selectively building localized hybrid models covers six basic steps: 1) 2) 3) 4) 5) 6)
Choose an initial mechanistic model. Perform symbolic reformulation of the mechanistic set of equations. Solve parameter estimation problem for the initial mechanistic model. Perform best equation structure change analysis (BESCA). With the resulting structure, calculate prediction error over testing data set. Change the model structure and go back to step 4 until stopping criteria based upon overall data adjustment quality are met.
In the forthcoming paragraphs we will describe some of the above steps in more detail.
2.1. Symbolic Reformulation We conduct a symbolic reformulation of the original set of equations, provided by the user, which correspond to the available initial process mechanistic model. This symbolic reformulation is intended to transform a generic equation, of arbitrary complexity, into a set of linear equations and special algebraic structures, such as the bilinear type xy, power xy or other atomic functional forms f ( x ) . For instance, if we start with equation 2 x e 4/y - z : 0, the introduction of additional variables wi originates the equivalent formulation: 2Wl -- Z ----0 W3 -- 4W4
Wl = XW2
W2 ~ e w3
W4 - - y
-1
(1)
All the symbolic decomposition and reformulation steps mentioned are performed automatically, by means of storing in the computer memory the initial set of equations, expressed as nested lists [4]. As a result of this symbolic reformulation, we obtain a set of linearized initial equations and additional ones, related with the wi auxiliary variables definition. Both groups taken together are equivalent to the original mechanistic model.
2.2. Parameter Estimation Within the scope of this paper a model has the form of a set of algebraic equations f ( 0 , z) - - 0
(2)
where 0 stands for p unknown parameters, z represents n process variables, with k output variables, and f covers q equations. The error variables ej are defined by ej - ~ j - y j
j -
1,... ,m
(3)
where 33j represents the set of predicted values for the process output variables and y j* stands for the j case from a set of m training records of data.
317 Considering maximum-likelihood estimators, by solving the optimization problem rn
minL-- ~ 0
rn
ejTV-lej- ~_,Lj
j=l
(4)
j=l
where L is the likelihood function and V the covariance matrix for the output variables, we obtain the desired adjusted 0 values.
2.3. Best Equation Structure Change Analysis We now consider h, the subset of both linear equations (with length s) and the extended set of variables v - - z U w that includes z and w (set of t additional variables resulting from the symbolic reformulation performed). For each variable v, in each equation h, a new model structure is generated through the addition of a Tv term, where y is a new free parameter added to 0 (table 1).
Table 1 Terms used to generate new structures
fl
Zl
...
Zn
W1
...
Wt
'YZl
...
~Zn
"~1
...
~4~t
~Zl
...
'YZn 'yw1
...
'~t
. o .
fs
Then, optimization problem (4) is solved for each case, resulting in a matrix with values of the merit function L for each alternative structural change, and the model structural change introduced corresponds to the best value found.
2.4. Stopping Criteria The model resulting from a particular BESCA iteration is used and evaluated over a testing data set with r records, leading to a prediction performance provided by: etest = ~ e j T v - l e j j=l
(5)
When no significant performance improvements over the previous iteration are being achieved, we stop the algorithm and use the corresponding structure as our final model set of equations. 3. CASE STUDY To evaluate the above framework we will now describe its application to a known case study [5]. The plant model represents, with some simplifications, a steady state adiabatic CSTR with irreversible first order reaction (A ~ B) and a bypass stream. Simulated operating data for this unit were obtained using different input values for inlet temperature, To, and concentration of A, A0, and obtaining the associated outputs, outlet temperature, T, concentration of A, and B, from the following equations (that we assume to provide a perfect fit to real plant data and use to generate simulated operating values):
318 (1 - 5)Ao - A s - kl'CAs = O
kl -- O1 eO2(1-800/r)
- B s + kl'CAs = 0
A = 8,40 + (1 - 8)As
T,(--Z~r)
To- T + ~ k l A s - O
(6)
B--(1-8)Bs
pcp
where "c stands for the residence time in the reactor (100s), ~ r is the heat of reaction p the mixture density (1 g / l ) , Cp its heat capacity of the mixture ( 4 . 1 8 J / g K ) , 5 the bypass ratio (0.05), 01 and 02 are kinetic parameters (0.017719s -1, 12.483). Three data sets were built this way: the first (training) has 30 points distributed over the input space A0 E [0.8, 1.01], To E [425,544]; the second (testing) has 64 points forming a complete grid over the same space; the third (for extrapolation evaluation) has 120 points distributed in a grid over the space A0 E [0.74, 1.07], To C [391,544]. Since the previous model (6) was used just for the sake of generating simulated operating data, we must now consider an initial mechanistic approximate model for the plant, whose complete detailed first-principles equations are assumed not to be known. For that purpose, we took the basic CSTR model equations (without bypass) as our initial mechanistic plant model, which, after symbolic reformulation, corresponds to: (-4180J/mol),
Ao - A - 10001w4 = 0
w2 + 80002Wl - 02 = 0
- B + 10001w4 = 0
w3 = e wE
Wl - 1 / T
To - T + 106 01w4 ~- 0
w4 = Aw3
(7)
where the values of 01 and 02 were adjusted to the training data. Then, our selective structural hybrid modeling approach was employed in order to improve the underlying initial model set of equations fitting performance (for A, B and T). The first iteration BESCA matrix suggested the addition of T1 w4 to the first equation, while the second iteration BESCA matrix suggested the addition of 72 w3 to the fourth equation (table 2), thus leading to the following final model structure: Ao - A - (100 01 - T1) w4 = 0
W4 ~-- A e wE
- B + 10001 W4 -- 0
w2 - 02(1 - 800/T) - 72 e w2
(8)
TO-- T + 10601 w4 = 0
Table 2 Likelihood values and parameters for different hybrid model stages Stage 0 1 2
L
01
02
~1
~2
0.3534 8.459• 10 -4 1.871• -8
0.01888 0.01785 0.01867
12.530 12.296 12.486
0.08907 0.09330
0.09653
We compared the performance of this model also with the one obtained with a purely empir-
319
ical approach, consisting of an artificial neural network (ANN) trained and tested over the same datasets (table 3). As an illustrative example, the prediction error for the output temperature is plotted as function of the distance from the center of the input variables scaled space (figure 1). It can thus be seen that our hybrid model not only provides excellent fitting capabilities but clearly outperforms both of the altemative approaches considered (mechanistic with adjusted parameters and empirical).
Table 3 Prediction mean square error for the (M)echanistic (with adjusted parameters), (H)ybrid and (A)NN models as predictors of T, A adn B. Interpolation Extrapolation T A B T A B M 1.77 x 10 -1 1.14 • 10 -4 1.78 • 10-7 2.26 • 10-1 1.43 • 10 -4 2.25 • 10 -7 H 1.90 • 10 -7 1.71 • 10 -13 1.89 • 10 -13 7.92 • 10-6 7.92 • 10 -12 7.19 x 10 -12 A 1.88 x 10 -5 6.74 • 10 -3 2.88 • 10 -4 1.47 • 10-1 9.98 • 10 -4 1.04 x 10 -4
1
e.
J
interpolation :1'
9
|
I
I 0.5
i
n
o
B
,.,. a8
[]
I
I
0
0.1
0.2
|
|
: t"o2 B a ..B d
I
!,a o
8 8o ~ o8
I
I
0.3
0.4
o
~d.
,
~
.
Oo
o
a
a8
[]
o~ a;~
i
|
,~ 8 a m ~
[]
I
-0.5
0
extrapolation
o
.. 0"[] .
,
og
0 8
o
-
.
o
o~
"
.,o
, o
I
I
I
0.5
0.6
0.7
Fig. 1. Temperature prediction error with distance from center of the (D M e c h a n i s t i c , / k Hybrid, o ANN).
(Ao,To) grid
One should point out the small number of structural changes that were needed here, as opposed to the other formula building techniques, such as genetic programming [7,6]. Here, we did conduct our search over a full space of model atomic structural changes. In more complex problems, the efficiency of the algorithm can be improved by excluding some elements of the search space, based upon available process knowledge.
320 A careful analysis of the generated final model structure could be used to discover process features that were not included in the initial mechanistic model, thus enlarging our firstprinciples knowledge, although in this particular case study that was not possible, namely because the model structure used for generating simulated data falls outside from the search space considered for hybrid model construction. 4. CONCLUSIONS We have introduced and developed a framework that is supported by the symbolic reformulation of a set of first-principles equations, in order to derive hybrid models, which opens new possibilities regarding the integration of empirical data-driven with mechanistic knowledge components for model building. The use of this model reformulation strategy results in a set of atomic equations that allow for empirical elements to be added selectively and locally. A simple case study shows that such an approach may lead to quite interesting and competitive results. Since this sort of approach is able to identify and deal with simpler substructures that result from a decomposition of the original set of equations, it is likely to provide more and more added value as the complexity of the original first-principles model increases, because it is able to guide selectively the incorporation of empirical elements to specific zones of its structure. At present we are examining extensions of the approach to address noise, large scale problems and performance comparisons with a number of alternative methodologies. ACKNOWLEDGMENTS The authors acknowledge financial support provided by research grants FCT POCTI/1999/EQU/32647 and PRAXISXXI/BD/15518/97. REFERENCES
1. P. Saraiva, chapter in Stephanopoulos and Han (Eds.), Intelligent Systems in Process Engineering, Academic Press (1996) 377. 2. M. Thompson, M. Kramer, AIChE J. 40 (1994) 1328. 3. R. Oliveira, PhD Thesis, Martin-Luther Uni., Halle-Wittenberg, 1998. 4. E. Smith, PhD Thesis, Imperial College, London (1996). 5. I. Kim, M. Liebman, T. Edgar, AIChE J. 36(7) (1990) 985. 6. K. Bettenhausen, P. Marenbach, GALESIA'95 (1995). 7. J. Koza, Genetic Programming, MIT Press, Cambridge, MA (1992).
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rights reserved.
321
Validations of the nonequilibrium stage model and of a new efficiency correlation for non ideal distillation process through simulated and experimental data M. R. Wolf-Maciel a*, C. Soares a and A. A. C. Barros b
-
a Laboratory of Separation Process Development - Chemical Engineering School - State University of Campinas (UNICAMP) - P.O. Box: 6066 - Zip Code: 13081-970 Campinas - SP Brazil - E-mail: [email protected] and [email protected] b Chemical Engineering Department- Regional University of Blumenau (FURB) B l u m e n a u SC - B r a z i l - E-mail: [email protected] The objective of this work is to use a robust software developed for distillation process involving the nonequilibrium stage model, the equilibrium stage model with Barros & Wolf correlation for efficiency calculation, the mass and heat transfer calculations and the automatic procedure for comparison of profiles calculated and obtained from experimental data. The case study is the system ethanol/water separation The intention is to compare the performance of the equilibrium stage model with the new correlation for efficiency. 1. I N T R O D U C T I O N The equilibrium stage model for distillation process is commonly used, since it does not need a complete description of the equipment. It is supposed that the streams leaving the stage are in equilibrium to each other. The material balance, the equilibrium relationships, the summation equations and the heat balance for each stage are solved using available algorithms, obtaining the liquid and vapour flows, as well as, the temperature and composition profiles along the distillation column [1]. However, it is necessary the introduction of the efficiency concept to correct the deviations of the fact that the stages, in the practice, rarely provide the equilibrium among the phases that leave them. However, the limitations presented in the several efficiency correlations, stimulated the development of a new correlation that considers parameters of mass and heat transfers as fundamental in its evaluation. An empirical efficiency equation developed by the authors of this work will be used to evaluate the behaviour of temperature and composition profiles in a distillation column. Equation (1) was obtained based on techniques of factorial design and it is dependent on the properties of the mixture, for the calculation of plate and component efficiency. ]]i,j
"--
I-- 1-004516
~38.5309 k-L~ P-L~,j ~2
*To whom all correspondence should be addressed
(1)
322 For the determination of the component efficiencies, the same correlation is used, but as function of the pure component parameters. A software involving the equilibrium stage model with the new efficiency correlation was developed using subroutines for the calculation of all parameters present in equation (1). 2. N O N E Q U I L I B R I U M STAGE M O D E L The nonequilibrium stage model, described with details by [1,2,3] is characterised by the absence of efficiency values. Besides, the calculations are based on the simultaneous solution of the mass and heat transfer equations, written independently for each phase. The plate is assumed to be at mechanical equilibrium, the mass and energy balances are written for each phase separately and the thermodynamic equilibrium is assumed only at the interface. It is established that transfers from the vapour phase to the liquid phase are positives. The vapour and liquid mass transfer rates are calculated using the Krishna and Standart method [4]. The equations that describe a separation process [ 1, 2, 3] in the steady state in the nonequilibrium stage model are: Liquid Phase Component Material Balances, Vapour Phase Component Material Balances, Interface Component Material Balances, Liquid Phase Energy Balance, Vapour Phase Energy Balance, Interface Energy Balance, Interface Summation Equations and Interface Equilibrium Relationships. The Interface Component Material Balances are represented by the mass transfer flow relationships: RFv = Nj - NjV -- 0
where: j=l to c-1
(2)
RFL = Nj - NjL = 0
where j=l to c-1
(3)
The Krishna and Standart method based on the exact solution of Maxwell Stefan's equations is used for the calculations of the mass transfer rates. The equations in the matrix form are given by: (4)
(NV)c-1,1= [K v] c-1,c-1.a. (yV-yI)c-1,1 + N T . (yV)c - 1,1 (NL)c - 1,1= [KL] c-1,c-I .a-(XI-XL)c-1,1 +NT. (XL)c- 1,1
where N T = ~ N j j=l
(5)
The interfacial area is combined with the transfer coefficients and a correlation that gives the product of these parameters is used. The binary mass transfer coefficients on the trays are calculated using the AIChE correlation. The Interface Energy Balances is given below by the equation: BE I = e v - e L = 0
(6)
The energy transfer rate equations are obtained by the sum of a conductive heat flow and a convective contribution due to the transport of enthalpy in the interface. The energy transfer
323 coefficients are calculated for the vapour phase using the Chilton-Colbum analogy and for the liquid phase through the penetration theory: v
=h
v
.a-
-T
+
Nj .Hj
(7)
L = hL -a- (TI - TL) + 2 NjL -Hi -L j=l
(8)
j=l
The nonequilibrium stage model is described by (5c+3) equations and variables for each stage. The condenser and the reboiler are described by the thermodynamic equilibrium relationships. The equations are simultaneously solved according to the method described by Naphtali and Sandholm [5] and by Newton-Raphson method. The mass and energy transfer rates together with the total and partial enthalpies and the equilibrium constant which are calculated using the model UNIQUAC for the activity coefficient are determined separately. 3. METHODOLOGY Using the equilibrium stage model with the efficiency correlation of Barros & Wolf and the nonequilibrium stage model described, a program was developed for the simulation of distillation columns in steady state, enabling, also, that results from simulations be compared with experimental data. The developed program uses Fortran language. All necessary physical properties (binary diffusion coefficients, viscosity, density, heat capacity, thermal conductivity) for pure components and mixtures, the enthalpies of the vapour and liquid streams entering and leaving a stage, the partial molar enthalpies of the bulk vapour and liquid phases are calculated rigorously using proper subroutines. Both concepts, the equilibrium and nonequilibrium, have the same number of degrees of freedom and, therefore, the initial estimates and the specifications are the same ones. To the program are supplied all the critical properties of the components, the location and the conditions of the feed, type of condenser, number of plates of the column, among other data. For the nonequilibrium stage model are necessary, besides the mentioned parameters, the column design parameters. It was observed that the computing time involved in the simulation of the nonequilibrium stage model is larger than the equilibrium stage model; that is evident, since the number of equations in the nonequilibrium stage model (5c+3) is higher than in the equilibrium stage model (2c+ 1) [6]. For validating the comparison of the obtained results using the equilibrium and nonequilibrium stage models, experimental data obtained in a distillation column made of stainless steel was used. The column with 140 mm diameter has eight sieve plates and in all the test runs made in this investigation, the column was operated with the ethanol-water system at total reflux. At the steady state condition, the liquid samples and the temperature were taken along the distillation column and the required data were recorded. 4. SIMULATION AND RESULTS Tanking into account that the computing time of the nonequilibrium stage model is quite large for on line control application, it is very important to carry out the evaluation of the equilibrium stage model using the new efficiency correlation in relation to the more realistic
324 nonequilibrium stage model and also with experimental data. Ethanol-water system was used and the temperature and composition profiles were compared. The temperature were taken along the distillation column, except in the condenser and in the reboiler. In the figures, stage 1 corresponds to the reboiler and stage 10 to the condenser. For the simulations, the specifications used were the reboiler duty (0.194E8 J/s) and the mole fraction of the ethanol on the top of the column (77.9%). The ethanol mole fraction in feed was 7%. 100 362
o Experimental ,~ Equilibrium N o n e-qou-i l i b r i u m 6 ~ __~
~,~ 360 358
o
60
~ 40 ,~
~ 356 [,.
0
i
0
354 1
2
3
4
5
6
7
8
9
~"(~-~- O ~ . x ~ e r
~/
~ 2o
l
1 2
i
i
i
~
~
i
3
4
5
6
7
8
t
i
9 10 1
Stage N u m b e r
Stage Number
o Experimental Lx Equilibrium o Nonequilibrium
Fig. 1. Comparison of experimental predicted temperature profiles
and Fig.2. Comparison of experimental predicted composition profiles
and
In Figure 1, the temperature profiles of the equilibrium (using Barros & Wolf correlation for efficiency) and nonequilibrium stage model and experimental are compared. The temperature of the liquid phase for the equilibrium and nonequilibrium stage model coincide in all stages, except for stage 2, a stage very close to the reboiler. A comparison of the predicted mole fraction of the ethanol and water with the measured mole fraction along the distillation column is shown in Figure 2. In the figures, it can be observed that the mole fractions are practically coincident in the upper part of the column, and that in the lower part, the nonequilibrium is more coincident with the experimental data. Based on results obtained in Figures 1 and 2, the behaviours of the heat and mass transfer coefficients along the distillation column were calculated by the software (Figures 3, 4, 5 and 6). It was observed that the mass transfer coefficients for the liquid phase increase in direction to the bottom of the column, and that the values for both pairs are very close (Figure 3). It can be said that the coefficients tend to increase, increasing the gradients of concentration among the phases, which happen from the top to the bottom of the column. The results obtained for the liquid phase were used in the simulation, which allowed to get the behaviour of the parameters in the vapour phase. It is observed that the binary mass transfer coefficients in the liquid phase are larger than the ones in the vapour phase (Figure 4). This behaviour shows that the resistance to the mass transfer is larger in the vapour phase. Furthermore, these coefficients increase in an opposite way.
325
200
35
=o ~
o Ethanol-Water rn Water-Ethanol
.N
34
o Ethanol-Water 190 [] Water-Ethanol
o
O O
[]
8 ~ [-
~
32
k
~
18o
[] [] []
31
~
8 8 ~ 8 8 ~
i 1
i
2
3
t
i
i
i
i
i
4
5
6
7
8
9
I0
~
Stage Number
16o 1
2
3
4 5 6 7 8 Stage N u m b e r
9
10
Fig. 3. Liquid phase mass transfer coefficients Fig.4. Vapour phase mass transfer coefficients profiles profiles ~' r~ a~. ~ ~ ~ ~ ~ ~
19000 18000 17000 16000 150OO
~
14000
~
75
a ~
73
<>
<>
O
<> <> <>
<> <> O
1
2
3
4
5
6
<>
7
Stage N u m b e r
O t
<> i
8
9
~e~o
69
~
67
<>
65 10
1
i
i
t
i
i
i
i
i
2
3
4
5
6
7
8
9
10
Stage N u m b e r
Fig. 5. Liquid phase heat transfer coefficients Fig. 6. Vapour phase heat transfer coefficients profiles profiles For expressing the heat transfer coefficients in the liquid phase, the penetration theory was used, described as a function of the mass transfer coefficients in the liquid phase, the heat capacity and the mass diffusivity. For the vapour phase, the Chilton-Colbum analogy was used, which relates the average of the mass transfer coefficients in the vapour phase, the heat capacity and the Lewis number, all present in the developed software. The behaviours of the heat transfer coefficients follows the ones for the mass transfer. 5. C O N C L U D I N G R E M A R K S The software developed by the authors of this work, which includes the equilibrium and nonequilibrium stage models, were used to simulate the distillation process. Results show the prediction of the models to represent a real non-ideal process. It can be said that, using the Barros & Wolf correlation for efficiency, both stage models present profiles practically coincident and are validated with experimental data. This is an important result since one can
326 choose the most appropriate model for a particular application. Moreover, liquid and vapour phase mass and heat transfer coefficients can be calculated and analysed along the distillation column. NOTATION a BE c Cp D h
N PM RF T X
y
interfacial area (m2) energy balance function number of components average heat capacity (J/mol.K) mass diffusivity (cm2/s) partial molar enthalpy (J/mol) heat transfer coefficient (J/(cm2.s.K) multicomponent mass transfer coefficient (mol/(cm2.s) average thermal conductivity (W/cm.K) interface mass transfer rate (mol/s) average molecular weight (g/mol) mass rate relation functions temperature (K) liquid mole fraction vapour mole fraction
Subscripts i stage j component Superscripts I interface L liquid phase V vapour phase Greek Letters e interface energy transfer rate (J/s) rI Barros & Wolf efficiency P average density (g/cm3)
la
average viscosity (cP)
AKNOWLEDGEMENTS
The authors are grateful to FAPESP (Fundag~.o de Amparo 5. Pesquisa do Estado de Sgo Paulo) for the financial support for this project. REFERENCES
[ 1] R. Krishnamurthy and R. Taylor, AIChE Journal, Vol. 31, No. 3 (1985a) 449. [2] R. Krishnamurthy and R. Taylor, AIChE Journal, Vol. 31, No. 3 (1985b) 456. [3] R. Krishnamurthy and R. Taylor, AIChE Journal, Vol. 31, No. 12 (1985c) 1973. [4] R. Krishna and G. L Standart, AIChE Journal, Vol. 22, No. 2, (1976) 383. [5] L. M. Naphtali and Sandholm, AIChE Journal. Vol. 17, No. 1 (1971) 148. [6] M. H. Pescarini, A. A. C. Barros and M. R. Wolf-Maciel, Computers Chem. Engng., Vol. 20, Suppl. (1996) $279.
European Symposium on Computer Aided Process Engineering - | l R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
327
Simulation of an industrial olefin polymerization FBR operating under condensed mode A. Yiagopoulos a, H. Yiannoulakis a, J. Morris b and C. Kiparissides a a Department of Chemical Engineering and Chemical Process Engineering Research Institute Aristotle University of Thessaloniki, P.O. Box 472, Thessaloniki, Greece 540 06
b Department of Chemical and Process Engineering, University of Newcastle-upon Tyne, NE1 7RU, U.K. In the present study a comprehensive mathematical model is developed to describe the behavior of an industrial gas-phase olefin polymerization FBR operating under condensed mode conditions. Assuming complete mixing and instant vaporization of the liquid feed in the bed, detailed simulations were carried to investigate the effect of the amount of liquid in the ethylene feed and the composition of the recycle stream on the reactor temperature, spacetime yield and heat removal rate. It was shown that the introduction of a liquid feed stream in the reactor substantially broadens the safe operation window of the FBR with concomitant the increase of the space-time yield and the heat removal rate from the reactor. 1. INTRODUCTION Fluidized bed solid catalysed olefin polymerization has long been recognized as one of the main manufacturing processes for the production of polyolefins. In such a process, catalyst particles are continuously fed into a fluidized bed and react with the incoming monomers to form a particulate polymer product that is withdrawn from the bed at a point above the distributor. Heat is removed by cooling the recycle gas stream via a heat exchanger. It has been reported that cooling of the recycle stream below its dew point and the subsequent injection into the reactor of a mixed liquid-gas feed, can further improve the heat removal rate from the fluidized bed [ 1]. This is accomplished due to the higher heat capacity of the liquid fraction, lower inlet temperature of the feed stream and latent heat of vaporization of the liquid phase. Replacing noncondensable components such as nitrogen, with condensable isopentane [2] can lower the dew point of the recylce stream mixture. When the recycle stream is cooled below its dew point the fluidized bed reactor is said to operate under condensed mode. The typical level of condensation in the recycle stream is between 5-20 wt. %, although values of up to 50 wt. % have been reported [ 1-4]. The liquid is fully or partially vaporized in the bed, upon contact with the hot active polymer particles, thus removing excess heat via its latent heat of vaporization [5]. The obvious benefit of this technique is the broadening of the safe operation window of the FBR, meaning that the catalyst feed rate, and consequently the production rate, can be increased without the risk of temperature runaway. As a result, the reacrtor space-time yield can be significantly increased without increasing the size of the fluidized bed [2].
328
'~
Recycle Stream
C yc,o e
.................. Cooler
Reaction Zone
Compressor ..............
Gas Make-up Feed
>
1-Hexene Make-up Feed
Product Prepolymer "1 ~1111 Feed i,, , ,i II I I',1 i
i
~
i
i
i
i
Gas Feed
..
Flash Drum
i
i
"".,.,, Condenser
Ir Liquid Feed . j
Fig. 1. Schematic representation of an FBR operating under condensed mode conditions. In the present study a comprehensive mathematical model for a gas-phase olefin polymerization fluidized bed reactor operating under condensed mode is developed. Extensive simulations are carried out to investigate the effect of degree of condensation of the recycled stream on the dynamic and steady-state fluidized bed reactor operation. 2. F L U I D I Z E D BED R E A C T O R M O D E L
A schematic representation of an olefin polymerization fluidized bed reactor operating under condensed mode is presented in Figure 1. Liquid is injected in the bed in the form of dispersed droplets through a series of nozzles. Injection of the liquid occurs at a point above the distributor where reactor temperature is high enough to ensure adequate vaporization upon entrance in the bed. Chinh et al. [4] reported that the fluidized bed of figure 1 can be approximated by a well-mixed vessel, while 98% of the liquid entering the reactor is completely vaporized. According to Chinh et al. [4] the weight percentage of liquid that can be safelly introduced in the bed varies from 6 to 20%. Higher levels of condensation will probably lead to liquid 'weeping' in the bed and the development of a distinct three-phase gas-liquid-solid fluidization regime near the reactor' s gas distributor [7]. Based on the above observations the fluidized bed reactor can be approximated by a continuous pseudo-homogeneous phase stirred tank reactor [6]. Moreover, the liquid feed is assumed to be completely vaporized upon entrance in the bed. Since no bubble phase is included in the model, the bed voidage, abed, accounts for the overall gas volume fraction in the bed. A detailed kinetic model is used to describe ethylene and 1-hexene copolymerization [6]. Assuming uniform temperature and monomer concentration throughout the bed, the unsteady-state material and energy conservation equations can be derived. Thus, the dynamic molar balance for the "i" monomer in the reactor can be written as:
329
d[Mi]
_
FI
dt
P m l A H l~bed
[M ]in,!
U0 ([Mi Hgbe d
]in,g _ [Mi ])_ [Mi ]Q0
0 - gbed)RMi
(1)
g bed
where F1 is the liquid feed rate, Q0 is the product removal rate and u0 is the superficial gas velocity. RMi denotes the consumption rate of monomer 'T'. The dynamic molar balances for hydrogen, nitrogen, ethane and the moments of number chain length distributions as well as the mass balance for the polymer in the bed can be found elsewhere [6]. Accordingly, the dynamic energy balance equation can be written as follows:
H accum dT/dt = I2Igas,in "+-III liquid,in + fl gen -- fl gas,out
-- flprod,out
-- I[Ivap
(2)
Nm
Haccum = ( Z N i ] C p M i W [ H 2 ] C p H 2
CppolOp,- ]. +[N2]CpN2 +[C2]Cpc2 + (1 - gbed) --
(3)
gbed
i=l Nm
IiIgas,in =
U0 ( Z N i ] i n , g Hgbed
CpMi q-[H2]in CpH 2 + [N2lin CpN 2 + [C2]in,g Cpc2)(Tin - Tref)
(4)
i=l Nm
liquid,in
--
FI
(Z[Mi]in,lCpMi + [C2]in,lCpc2)(Tlin- Tref)
PmlAHgbed
(5)
i=l
Nm
(1 - ~bed) gen
--
~3bed
(Z RMiMWi)AHrxn
(6)
i=l Nm
I2Igas,out
_-- U0 ( Z N i ] C p M He bed i=l
i -I-[H2]CpH 2 +[N2]CpN 2 +[C2]Cpc2)(T-Tref)
(7)
Nm
Q0 ( Z H9prod,out - g"kl 717_1 I
N i ]CpM i at_[H2]CpH 2 +[N2]CpN 2 +[C2]Cpc 2 4-
(8)
i=l
(1 - e bed)
- g bed
yap = FI AH v a p / A H e
C ppolP p )(T - rre f ) bed
(9)
Notice that, in the derivation of the energy balance equation, the enthalpy associated with the prepolymer feed stream is considered to be negligible. In order to make a quantitative comparison between the heat removal rates from a conventional and a condensed mode FBR, the following dimensionless quantity is introduced:
Heat.emova. Factor-Hen//Has.ou.+ Hpo.ou.+ Hva )
(10)
330 To calculate the weight percentage of liquid that enters the reactor one must determine the vapor-liquid equilibrium in the flash drum. In the present study, the Soave-Redlich-Kwong (SRK) equation of state [8] was employed to calculate the amount of liquid entering the bed, based on the operating conditions in the flash drum and the composition of the recycle stream. 3. SIMULATIONS RESULTS AND DISCUSSION
To demonstrate the predictive capabilities of the present model, an ethylene-hexene-1 copolymerization FBR operating under condensed mode was considered. Numerical simulations were carried out to investigate the effect of the partial condensation of the recycle stream on the safe operation window of the FBR. In particular, the effect of the weight fraction of liquid in the feed stream and the recycle stream composition on the reactor temperature, space-time yield and heat removal rate was thoroughly examined. The reactor and flash drum operating conditions are reported in Table 1. The kinetic rate constants for ethylene-hexene-1 copolymerization can be found elsewhere [9]. Table 1: Typical reactor and flash drum operating conditions [3,4] Variable Bed dimensions, HxD (m 2) Bed voide fraction, ~;bed Condenser inlet, The Temperatures (~
Flash drum, Td Reactor liquid feed stream, Tlin Reactor gas feed stream, Tin Pressure, P (bar) Mole fractions of flash feed stream: (H2, N2, C2H4,C2H6,C6H12) Prepolymer feed rate, Fpre(g/s) Inlet active site concentration, P0,in(mol/L)
Value 16x4.85 0.5 75 55 55 55 20 (3.8, 55.4, 30.2, 1.5, 9.1) 30 0.0284
The increase in the heat removal rate from the FBR operating under condensed mode is expected to broaden the safe operation window of the reactor in comparison with the operation of conventional FBRs (e.g., 0% liquid in the bed). Figure 2 depicts the effect of the amount of liquid in the feed stream (e.g., 0, 6.4 and 12.2 %) on the reactor temperature for various prepolymer feed rates. The amount of liquid that enters the reactor can be controlled by adjusting the operating temperature of the flash (e.g., 75, 66 and 55 ~ respectively). The dew point of the feed stream was calculated to be 75 ~ In all cases, the reactor temperature increases as the prepolymer feed rate increases. Moreover, the safe operation window broadens significantly as the amount of liquid fed to the reactor increases. As an example, for a prepolymer feed rate of 20 g/s and when no liquid is fed to the reactor (see solid line) the reactor temperature is equal to 80 ~ When the liquid weight fraction in the total feed stream is 12.2 % (see dotted line), the reactor temperature is less than 40 ~ for the same prepolymer feed rate of 20 g/s. This means that the prepolymer feed rate can significantly be increased without the risk of temperature runaway. Similar behavior is observed when the amount of inert and/or active condensables in the recirculation feed stream increases. Figure 3 illustrates the effect of the hexene-1 concentration in the recycle stream on the safe operation window of the reactor. Notice that for a given prepolymer feed rate, as the amount of hexene-1 increases (e.g., from 9.12% to 11.12 %) the reactor temperature decreases. This can be explained by the
331 140
120
c..) %.-' 12o
%-11oo
~1oo 80 ~D
t..,
o
60
.ca
C6H12 mole fraction
80 ~D
iquidinFeed ~ 0
o
20
"9
r
60
O
. . . . . . . . . 6.4 -........... 1 2 . 2
0 ' 2'0' 4'0' 6'0' 8'0 '100'1 89
Prepolymer Feed Rate
//~
/I
(D k.4
iii . . iiii!1 ...... .
~176
(g/s)
Fig. 2. Effect of the amount of liquid in the feed on reactor temperature,
*~ 4o20
0
....
20 40 60
, , , , , , , .
80 100 120 140 160
P r e p o l y m e r F e e d R a t e (g/s) Fig. 3. Effect of the hexene-1 concentration in the recirculation stream on reactor temperature.
decrease of the dew point temperature of the recirculating feed stream. Thus, as the amount of condensables increases, the dew point of the mixture decreases, while the liquid weight fraction in the feed stream increases. The main benefit of the condensed mode operation in olefin polymerization FBRs is presented in Figure 4. As can be seen, for a constant reactor temperature (e.g., 85 ~ the space time yield increases linearly with the amount of liquid in the reactor feed. This can be explained in conjunction with the results of Figure 2. As the amount of liquid increases, the allowable prepolymer feed rate increases which in turn leads to an increase of the overall polymerization rate. According to the results of Figure 3, when the amount of liquid is increased from 0 to 6 %, the space-time yield increases as much as 200 %. Finally, Figure 5 shows the effect of the amount of liquid in the feed stream (e.g., 0, 6.4 and 12.2 %) on the heat removal factor given by eq. (10) for various prepolymer feed rates. As can be seen, in the case of a conventional FBR (solid line) the heat removal factor is always above unity, even at low prepolymer feed rates. On the other hand, when a mixed liquid gas feed is introduced into the bed (e.g., 6.4%, 12.2%) the heat removal factor reduces significantly (e.g., heat removal increases due to the latent heat of vaporization). Thus the FBR can operate at a higher prepolymer feed rate, resulting in a higher space-time yield.
4. C O N C L U S I O N S In the present study, a model for the simulation of an ethylene copolymerization FBR operating under condensed mode has been developed. Under the assumptions of complete mixing and instant vaporization of the liquid feed in the bed, detailed simulations were carried out in order to investigate the effect of the liquid weight fraction in the feed stream and the composition of hexene-1 in the recycle stream on the reactor temperature, space-time yield and heat removal rate. It was shown that the introduction of a liquid feed in the reactor substantially broadens the safe operation window of the FBR with concomitant the increase of
332 lOO
10t/-r------r
~" 9o
r~
~8
80
,,"
,,'
........ 6.4
,.~ 70
,,"
~6
"~ 60 ~
Liquid in Feed (% w)
~0
.
N 5o ~ ~.~
40
~
20
~4
30-
~-~10 14
Liquid in F e e d (% w) Fig. 4. Effect of the amount of liquid in feed on the reactor's space time yield (Tr = 85 ~
0
0
20
40
60
80
100
120
P r e p o l y m e r F e e d Rate (g/s) Fig. 5. Effect of the amount of liquid in the feed on the heat removal factor
the space-time yield and the heat removal rate from the reactor. These results are in qualitative agreement with findings reported in the patent literature [ 1,3-5]. REFERENCES 1. J.M. Jenkings III, R.L. Jones, T.M. Jones and S. Beret, Method for Fluidized Bed Polymerization, US Patent No. 4 588 790 (1986). 2. Y. Jiang, K.B. McAuley and J.C.C. Hsu, Ind. Eng. Chem. Res., 36 (1997) 1176. 3. M.J. DeChellis, J.R. Griffin and M.E. Muhle, Process for Polymerizing Monomers in Fluidized Beds, US Patent No. 5 405 922 (1995). 4. J.-C. Chinh, M.C.H. Filippelli, D. Newton and M.B. Power, Polymerization Process, US Patent No. 5 541 270 (1996). 5. R.J.N Bemier, R.L. Boysen, R.C. Brown, L.S. Scarola and G.H. Williams, Gas Phase Polymerization Process, US Patent No. 5 453 471 (1995). 6. H. Hatzantonis, H. Yiannoulakis, A. Yiagopoulos and C. Kiparissides, Chem. Engng Sci., 55 (2000) 3237. 7. L.-S. Fan, Gas-Liquid-Solid Fluidization Engineering, Butterworths, U.S.A., 1989. 8. R.C. Reid, J.M. Prausnitz and B.E. Poling, The Properties of Gases and Liquids, McGrawHill, U.S.A., 1988. 9. A.Yiagopoulos, Ph.D. Dissertation, Aristotle University, Thessaloniki, Greece, 2000.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
333
An Extended Self-Organizing Map with Application to the Modeling of Pulse Jet Fabric Filters Hualiang Zhuang and Min-Sen Chiu* Department of Chemical and Environmental Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 Pulse jet fabric filters (PJFF) have become an attractive option of particulate collection utilities due to the feature that they can meet the stringent particulate emission limits regardless of variation in the operating conditions. The dynamics of the filter has complex nonlinear characteristics as reported in [1]. In this paper, the framework of local model networks (LMN) is employed to approximate the process dynamics of the pulse jet fabric filter that is subject to multiple operating regimes. To do so, an extended self-organizing map is employed to partition the PJFF's operating range and construct the LMN automatically. Simulation results illustrate the proposed approach and a comparison with the conventional approach is made. 1. INTRODUCTION Pulse jet fabric filters (PJFF), sometimes called bag house filters, have become an attractive option of particulate collection utilities due to the feature that they can meet the stringent particulate emission limits regardless of variation in the operating conditions. Other merits of pulse jet fabric filters are high collection efficiency, on-line cleaning applications and outside collection which allows the bag maintenance in a clean and safe environment. Despite their wide applications, investigations in the modeling and control of the pulse jet fabric filters can only be described as rudimentary. To the best knowledge of the authors, two models were reported in the literature [1,2]. However, these first principle models are too complicated to be incorporated in the controller design. Therefore there is a need to develop a simple model with reasonable accuracy for the PJFF. In this paper, a modeling methodology based on the framework of local model networks (LMN) [3] is developed to construct an accurate and simplified dynamic model of a PJFF. Despite the recent advances of LMN, a prior knowledge of the processes has to be exploited for determination of the LMN structure and the weighting functions. In this paper, an extended self-organizing map (ESOM) network, which can overcome aforementioned difficulties, is developed to construct the LMN using the input-output data. This ESOM network is a refinement of that proposed by Ge et al [4] with the improvement procedures of employing competitive learning algorithm for data clustering and using the partial least square algorithm for computing the parameters of the local models.
* To whom all correspondences should be addressed. Phone: (65) 8742223, Fax: (65) 7791936, Email: [email protected]
334 2. L M N F O R PULSE JET FABRIC FILTERS The operation of the pulse jet fabric filter (Figure 1) can be briefly described as follows: during the filtration cycle T, influent gas is passed through a filter bag and dust cake is built up at the upstream side of the bag surface. At time T, the bag is subject to a pulse jet of air of high pressure, removing a certain fraction of the cake. The average flow-rate Q of the exhausted gas during cycle time T, i.e. the filter's filtration capacity, is the key output of the process. In this paper, Q and T are chosen to be the output and input of PJFF. Owing to the nonlinearity of the filtration process, a LMN framework is employed to tackle this problem. LMN is constructed as the weighted sum of a set of simple local models across the specified operating range. The properly that it can approximate nonlinear systems has been investigated in a great detail recently [3]. In the LMN structure, a weighting function is assigned to each local model to indicate its respective degree of validity in the operating space. To apply LMN to PJFF, the operating space 9 is decomposed into n . operating regimes ~ j . Define fl(t-1) dynamics as given by fl(t -
1) =
[Q(t -
1),..., Q ( t
as an operating point vector that characterizes the process
- nQ ) , T ( t - n d ) . . . . . T ( t - n d - n v +
where nQ and n T are integers related to the system's order; weighting function is denoted by n~ Q ( t ) : ~_~ p } ( f l ( t j=l
1))~ r ( t
pj (fl).
1)]v
na
(1)
is the process time delay. The
The LMN of PJFF can then be presented as follows, (2)
- 1)Oj
where ~(t - 1) is the regression vector and Oj is the local parameter vector as given by gt(t- 1) =
[Q(t-1),...,
Q(t-
nQ ) , T ( t -
9
n d ) . . . . . T ( t - n d - n~ +
O.j : [O.j,1 .... , Oj,nQ , Oj,no +], .., Oj,tiO +tiT' O.j,tiQ +nT +1
]r
1), 1]v
(3) (4)
In the following section, an extended self-organizing map approach is proposed to construct the LMN automatically using the input-output data. 3. E S O M A L G O R I T H M
Ge et al [4] developed an extended self-organizing map (ESOM) network that can partition the operating space of nonlinear processes automatically using the input-output data. However, their method suffers two shortcomings: (1) the computation time increases dramatically as the number of local models increases, (2) the least square algorithm employed is not reliable when the matrix to be inversed is ill-conditioned. To overcome these problems, a competitive learning method for cluster-center searching is used in the proposed ESOM network and a partial least square algorithm is also employed to compute the parameters of the local models. As shown in Figure 2, the ESOM consists of three layers" an input layer, a Kohonen layer and an output layer. The functions of these three layers are described in below.
335
Input layer: Given the input-output training data v and y, the input of this layer includes y as well as v, i.e. ~ = (v, y ) , where v ( t ) = [Q(t - 1)..... Q ( t - n Q ) , T ( t - n d ) .... , T ( t - n d - n T + 1)] T
(5)
y(t)=Q(t)
(6)
K o h n o e n Layer: The weight vector of the nodes is formatted as Wi = ( W / ' , W , Y ) , where W/' is called the master weight vector and Wiy is the slave weight vector. The following gives the self-organizing algorithm to determine a set of cluster centers f~j, j = 1,...,n o , which characterize the dynamics of the nonlinear process. S t e p 1. Initialize Wi, i = 1, 2,--., K 2. S t e p 2. At each learning step l, only the master weight vector W,", i = 1, 2,..-, K2, is used to
determine the winner node whose weight vector W,~' best matches the input v, i.e.,
where H]I denotes the Euclidean norm. S t e p 3. Update every weight vector W~ = ( W [ , W f ) in the Kohonen layer as follows:
w~ (l + 1) = w, (l) + y ( i , l ) ( ~ ( l ) - w, (l)),
1
where y ( i , l ) =
(8)
, Pi and Ps are the positions of the node i and winner node.
/(1 + l)e [pi-p''[[2 S t e p 4. Check for the convergence of W~. If not, go to step 2. Otherwise, reset learning step
l= 1 and go to Step 5. From step 5 to step 7, a competitive learning rule [5] is employed to determine a fixed number of neurons f~j, j = 1,2,-..,n~,, which are considered as the centers of the clusters of the nodes with W/', i = 1, 2,..-, K 2 in the Kohonen layer. S t e p 5. Initialize f2j, j = 1,2,..., n~,. S t e p 6. If l > K2, l = 1. Determine the neuron whose weight vector f2,. (l) best matches the
weight vector of Wtv (l), i.e., (9) Update the weight vector of the neuron f2 s as follows: ~ s (l + 1) = ~ , (l) + Z ( 0 ( g ~ (Z) -
~,
(Z))
(10)
336 where Z(I)=
, Pl and Ps are the positions of the node l and neuron ff2s. ~/(1 + l)e IIp'-p'IIz
Step 7. Check for the convergence of f2j (l) j = 1,2,...,n.. If all of them converge, stop. Otherwise, l = l + 1 and go to step 6. In relation to LMN, these cluster centers form the local models and the weighting functions Ps are chosen to be the normalized gaussian functions as follows,
(11)
Ps(~) = .. j=l
where o-s2 is a constant variance.
Output Layer Set ~3= [v, 1], we then obtain no
y(t) = ~ p j v
^T
Oj --/30
(12)
j=l
where/3 = [pl~3r
,-..,
pn.~3r ]r ~ 0 = [Or
,...,
Onr ]r , and Oj is defined in (4)
Given the input-output data {v(t), y(t), t = 1, 2,...,
K t
}, /3
is fixed after the self-organizing
process such that the output .~ is actually a linear combination of the elements of O. In this work, a partial least square method [6] is chosen to find the solution of O, which minimizes P0 -
Yll, where
P = [/3(1) ..... /3(K,)]r, y = [y(1),..., y(K, )]r.
4. S I M U L A T I O N RESULTS For the operating space under consideration, the input signal, i.e. the cleaning cycle time T, ranges from 30 seconds to 150 seconds. The corresponding steady state output Q varies from 408.3 x 10 -4 m/s to 304.1 x 10 -4 m/s. In what follows, the proposed ESOM algorithm will be performed to construct the LMN for the PJFF based on the data obtained from the nonlinear simulations of the first principle model given in [1 ]. The input-output data is generated as follows: 500 independent random signals with uniform distribution ranging from 30s to 150s are used as inputs. The corresponding plant outputs in time sequence are generated. These input-output data can be grouped in the format of (5) and (6) so that it is suitable for the ESOM learning algorithm. During the ESOM learning, a trial and error procedure is required to determine the suitable order and number of local model. In this work, a third-order is used as the order of all local models. The effect of the number of local models, n o , is also studied by selecting four
337 different values, 3, 5, 7 and 9 for comparisons. For the purpose of comparison, a single global model and conventional LMN approach are also constructed. By conventional LMN approach, we means that the operating space is uniformly partitioned with respect to the value of output variable Q. Similarly, four cases with 3, 5, 7 and 9 local models are obtained. To obtain local model at each operating point, 200 points of input-output data are generated at each operating point. To compare the predictive performances of these models, a series of step changes of input T is performed as show in Figure 3. The mean absolute errors for all nine cases are summarized in Table 1. Due to the space constraint, only the simulation results of the single model, the LMN with n o = 7 based on the conventional approach and the proposed ESOM network are illustrated in Figure 4 to Figure 6 respectively. Evidently, the ESOM method outperforms both single model and the conventional LMN. For ESOM method, the accuracy of the model increases as the number of local models increases (see Table 1). 5. CONCLUSION This paper refines the previously proposed extended self-organizing map for the LMN based modeling approach. By using the input-output data, the proposed method automatically partitions the operating space and constructs the LMN. Simulation results of a pulse jet fabric filter example show that the proposed method has better modeling accuracy as compared to both single global model and conventional LMN approach. ACKNOWLEDGMENT
The authors appreciate the grant of National University of Singapore (RP279000043112). REFERENCES
1. J. Ju, M. S. Chiu and C. Tien, J. Air & Waste Management Assoc., No. 50 (2000) 600. 2. M. D. Ravin and W. Humphries, Filtration & Separation, May/June (1988) 201. 3. T.A. Johansen and R.M. Smith, Multiple Model Approaches to Modeling and Control, Taylor & Francis, NewYork, 1997. 4. M. Ge, M. S. Chiu and Q. G. Wang, Ind. & Eng. Chem. Res., No. 39 (2000) 3778. 5. T. Kohonen, Self-Organizing and Associative Memory, Springer-Verlag, 1987. 6. P. Geladi and B. R. Kowalski, Anal. Chim. Acta., No. 185 (1986) 1.
Table 1. Mean absolute errors of validation results for three modeling methods ,,
no = 3
no = 5
no = 7
no = 9
Conventional LMN
5.23x10 -4
4.91x10 -4
4.73x10 -4
4.71x10 -4
ESOM based LMN
4.97 x 10 -4
2.03 • 10 -4
1.77 x 10 -4
1.69 x 10 -4
Single Model
9.22 x 10 -4
338
Figure 1. Schematic of a pulse jet bag filter.
Figure 3. Input sequence for model validation.
Figure 5. Validation of conventional LMN with n . = 7. ~ : m o d e l ; ---:plant.
Figure 2. ESOM architecture.
Figure 4. Validation of single model. :single model; :~t,
Figure 6. Validation of ESOM method with n o = 7. ~ : model; - - - : plant.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
339
Feed Pretreatment for Binary Distillation Efficiency Improvement Rakesh Agrawal and D. Michael Herron Air Products and Chemicals, Inc., Allentown, Pennsylvania 18195 In this paper we discuss thermodynamic efficiency for cases where two feeds of the same composition but different enthalpy are fed to a binary distillation column. We present a quick method to determine for each of the feeds whether it should be a saturated liquid or vapor or two-phase. This study provides a process designer with a quick and reliable method for selecting thermodynamically efficient configurations through preconditioning of the feed streams and addition/removal of heat from intermediate locations of a binary distillation column. 1. I N T R O D U C T I O N
A
q><
IC
FCON Feed A,B FREB
>
,R,-~B
Fig. 1" Different options for adding and removing heat from intermediate locations,
Quite often, the feed to a binary distillation column originates from another distillation column or from a unit operation within a process. Thus, there is an opportunity to obtain this feed as a saturated liquid, a saturated vapor, a two-phase stream, or as multiple streams with different enthalpies. Moreover, sometimes lowpressure steam may be available as a waste heat source to allow pre-adjustment of feed enthalpy. To a process designer, if the impact of proper feed conditions on thermodynamic efficiency were known a priori, the opportunities could then be exploited to create an optimum solution through manipulation of feed conditions. Generally, preconditioning of feed is an integral part of the overall strategy to add/remove heat from an intermediate location of a distillation column [1 ]. Several options to accomplish this task are illustrated in Figure 1. The intermediate reboiler is shown as IR and intermediate condenser as IC. The feed heat exchanger FHX is optional. FHX could be a feed condenser for a vapor feed or alternatively it could be a reboiler for a liquid feed. FCON is
340 a feed condenser and FREB is a feed reboiler. When FHX is used in conjunction with FCON or FREB then the quality of feeds exiting the two heat exchangers are different. A process designer is faced with evaluating the merit of each of these five heat exchangers. In earlier work, we developed a model to systematically study the thermodynamic efficiency of binary distillation [1-3]. Through extensive computation and model development, a framework was developed that provides quick and reliable answers to the questions: (1) Is the optimum feed condition to the column saturated liquid or vapor, or twophase? (2) When does the use of an intermediate reboiler or an intermediate condenser provide a meaningful improvement in thermodynamic efficiency? (3) Which of an intermediate reboiler or condenser is more effective? (4) Where is the optimum location for an intermediate reboiler or condenser? Sometimes two feeds that are of the same composition but different enthalpy are used to improve the performance of distillation [4-5]. There are no available methods to quickly assess the proper thermodynamic state of each feed for this case. In this article, we provide a method to quickly decide for each feed whether the optimal state is all liquid, all vapor or two-phase. However, before this issue can be addressed, it is essential to briefly review the model and the optimum location for intermediate heat addition or removal. 2. BRIEF M O D E L DESCRIPTION
The analysis in this and previous work is made with the assumptions that imply that the mixture to be separated is an ideal binary mixture of constant relative volatility. The performance was expressed in terms of thermodynamic efficiency r/"
minimum work of separation total exergy loss + minimum work of separation References 1-3 provide the derivation of efficiency equations and the extensive computation through which the answers to some of the earlier posed questions were obtained. This analysis is based on the distillation of a binary mixture AB into pure products. The parameters needed to answer the questions are simplified to: mole fraction of the light component A in the feed ( ZA ), relative volatility of A with respect to B ( a ) and the quality of the feed q (liquid fraction in the feed stream). Surprisingly, the reboiler, condenser and feed temperatures are not needed to solve the final thermodynamic efficiency equations. 3. O P T I M A L LOCATION FOR INTERMEDIATE HEAT A D D I T I O N / R E M O V A L
Once it is determined that an intermediate reboiler or condenser can lead to a meaningful improvement in efficiency, the next hurdle is to find its optimum configuration and location [2-3]. The guidelines FOR selecting locations were developed based on the following two parameters [1 ]" I+Z~ aiR
"
-
~
z8
I+Z A .
alC
=
~
ZA
341 For a saturated liquid feed, the optimal location for intermediate heat addition is determined by comparing the actual relative volatility a with the value of aiR. 1. a < aiR, the highest efficiency is achieved by withdrawing a side liquid draw, subjecting it to total vaporization, then returning as shown for IR in Figure 1. As a increases, the side liquid draw location moves up the stripping section of the column. 2. a = aiR, a Z A fraction of the feed is totally vaporized and fed to the column (FREB).
3. a > aiR, FREB becomes a partial vaporizer. It is worth noting that as the value of a increases above aiR, the transition of the stream exiting FREB heat exchanger from total vapor to significantly two-phase is found to be quite gradual. Therefore, unless a is significantly greater than aiR, one would not see a significant decline in efficiency when using FREB as a total vaporizer. Similarly, for a saturated vapor feed, the relative volatility a can be compared with alC. For a < Ctic a side vapor draw with IC as a total condenser is used. When ct = ctlc, a ( Z B) fraction of the feed is totally condensed in FCON. partially condensed in FCON.
For ct >_Ct~c, a fraction of the feed is
4. TWO FEEDS OF SAME COMPOSITION BUT DIFFERENT ENTHALPY We have shown that feed pretreatment is an integral part of the strategy to add or remove heat from an intermediate location of a distillation column. These findings are now extended to deduce the optimal state for each feed when there are two feeds of the same composition and no intermediate reboiler nor condenser is to be used. Additionally, the effectiveness of an intermediate reboiler or condenser for the special case when one feed is a saturated liquid and the other a saturated vapor can also be deduced. These guidelines are shown graphically in Figure 2 and depend on relative values of a , air and Ctic. When neither intermediate reboiler nor condenser is used, the optimal feed conditions are determined by combining results of two thought experiments. In the first, the top (upper) feed is assumed to be a saturated liquid and the optimal feed condition for the bottom (lower) feed is deduced. Similarly, the second thought experiment is conducted by assuming the bottom feed is a saturated vapor. For the first thought experiment the top feed is saturated liquid. (a) a < air : Since the use of an IR is preferred over an all-vapor bottom feed, but no IR is to be used, it follows that the bottom feed to the distillation column be a vapor stream. (b)ct = aiR: Taking a Z A portion of the assumed liquid feed and completely vaporizing is preferred. Therefore the bottom feed must be saturated vapor. (c) tz > air : The bottom feed to the distillation column must be a two-phase stream. For the second thought experiment the bottom feed is saturated vapor. (d) cr < aic : The more effective method of removing heat is through the use of an IC. Since no IC is to be used, the top feed to the distillation column should be a liquid stream. (e) a = Ctic: A ( Z 8 ) portion of the assumed vapor feed should be totally condensed and fed
to the distillation column. In this case, the top feed must be saturated liquid.
342 (f) a > alC : The top feed to the distillation column must be a two-phase stream.
(~IC
~IRI
%op--2~
\t ~ 2,C=declines
~
FBOT=2 ~
iR77
/
10 ~FTop=L ~ FBOT-2(i) ~ IC=yes ~ IR=no
y
~ FTOp=2(D FBOT =v
IIR=yes C=no F op-L
FBoT=V
ICpreferred !
0.2
|
IRpreferred !
0.4 ZA 0.6
!
0.8
Fig. 2: Two-feed map for selection of optimum configuration. IC and IR recommendations are for saturated liquid and vapor feed case. 2~ = Two phase.
By combining the rules set forth by items (a) through (f) the selection of optimal feeds states can be further refined. First consider the region in Figure 2 where the feed is A-lean, i.e., Z A < 0.5. In this region air a z c , from items (c) and (f), both the feed streams should be two-phase. Similarly, optimum feed conditions for the mirror region in Figure 2, when Z~ > 0.5, can also be easily derived. For Z A = 0 . 5 , as a is increased to much higher values than 3 (where air
=arc),the
quality of the two two-phase streams approach one another. This implies that when Z A is in the neighborhood of 0.5 and relative volatilities are quite large, it will be sufficient to have one two-phase stream and the quality of this feed will be around 0.5. It is worth noting that transitions from saturated liquid or saturated vapor to two-phase across air and arc curves in Figure 2 are not sharp. This is especially true for higher values of a , i.e., for an A-rich phase (Z A > 0.5 ) when a is in the neighborhood of a~R and for an A-lean phase (Z A < 0.5) when a is in the neighborhood of a~,c . The primary reason lies in
343 the fact for an A-rich stream, as a exceeds aiR, the top feed is a two-phase feed and not a saturated liquid. As a result, its temperature would be warmer than that of the saturated liquid of the same composition. Therefore, if the optimum temperature for intermediate heat addition with a saturated liquid feed is at the saturated vapor feed temperature (for a = air ) then with a much warmer two-phase feed one would expect this optimum temperature to also shift to a temperature that is warmer than the saturated vapor feed temperature. This will require that for the A-rich feed, as a exceeds air but is in the neighborhood of c•/R, the bottom feed be saturated vapor feed and not a two-phase feed. Only when a sufficiently exceeds %R and the optimum location for intermediate heat addition has moved above the temperature of the saturated vapor feed would the bottom feed become two-phase. Therefore, in Figure 2, when c~ is in the neighborhood of a~R but exceeds aiR, it is better to use the bottom feed as a saturated vapor feed. Similar reasoning can be applied for an A-lean feed when a is in the neighborhood of a~<: but is greater than aK., in this case, the top feed should be saturated liquid. It is important to note that such reasoning is not applicable for an A-rich feed when a just exceeds a~<. or for an A-lean feed when a just exceeds a~R. In these regions one would expect a somewhat sharper transition from one-phase to two-phase. As stated earlier, an advantage of the exercise above is that one can also deduce whether the use of an intermediate reboiler or condenser would lead to meaningful improvement in efficiency (> 3%) when the top feed is a saturated liquid and the bottom feed is a saturated vapor. The preferred option of an intermediate reboiler or condenser for the different zones is shown in Figure 2. These choices can be readily explained. For example, consider the region for an A-lean feed. First, earlier work [3] has demonstrated that an intermediate reboiler is ineffective since the bottom feed is A-lean vapor. Whenever a is less than a~c, for a single vapor feed the optimum method of heat removal is to remove heat from a temperature below the saturated liquid feed temperature. Therefore, with the saturated liquid feed (along with the saturated vapor feed) use of an intermediate condenser IC will still be attractive. On the other hand, for a saturated vapor feed, when a is sufficiently larger than ale, optimum location for heat removal is at a temperature that is warmer than the saturated liquid feed temperature. Therefore, use of a saturated liquid feed is analogous to removing heat from a temperature that is lower than the optimum temperature. For such cases, as the value of c~ increases beyond aIc, the effectiveness of an intermediate condenser declines. Similarly, all the choices of intermediate reboiler and condenser in other regions of Figure 2 can be deduced. 4.1 An Example Detailed computer simulations were performed with a commercial simulation package using available thermodynamic data to determine the optimum feed state for the distillation of ethane/propane feed mixtures. In the table, results are summarized for three different feed compositions. The value of a was varied by changing the pressure of the distillation. For the two feeds a values are not identical, therefore, the tabulated values are averages. A large number of separation stages were used in the distillation column to simulate a pinched condition. For a given pressure in the distillation column, the flow of each feed stream and its quality was optimized to maximize the thermodynamic efficiency. Overall good agreement is obtained between the data in Table 1 and the predictions from Figure 2.
344 T A B L E 1: Efficient Feed Conditions for Ethane/Propane Distillation
Basis: Total Feed Flow = 1.0; FT = Top Feed Flow ETHANE - LEAN
5 0 - 50 M I X T U R E
ETHANE-
Z A = 0.25,azR = 2.33,a~c = 5.0
Z A = 0.5,a~R =arc =3
Z A = 0.75,air = 5.0,a~c = 2.33
c~
FT
q~op
qsoT
a
FT
qToP
qsoT
a
FT
qvoe
q~ov
2.3 3.7 5.4 8.4
0.79 0.77 0.76 0.73
1 1 1 0.99
0.0 0.09 0.15 0.24
2.7 3 3.8 6.0
0.59 0.59 0.58 0.57
1 1 1 0.97
0 0 0.05 0.07
2.3 3.6 5.5 9.2
0.35 0.37 0.38 0.42
1.0 0.92 0.89 0.75
0 0 0 0.01
RICH
. .
For an ethane-lean feed, at lower values of a , the top feed is saturated liquid and the bottom feed is saturated vapor. As the value of a is increased, the first transition for the bottom feed occurs in the neighborhood of air and its optimal state becomes two-phase. As expected from the earlier discussion when a exceeds alc, the second transition for the top feed from saturated liquid to a two-phase feed is delayed until a is sufficiently greater in value. Similar agreement is found for the other two feed compositions. 5. C O N C L U S I O N In an earlier work it was shown that optimal policy for adding heat or removing heat from an intermediate location of a distillation column requires that preconditioning of the feed be part of the overall decision-making process. In this paper we extend the work to analyze cases where two feeds of the same composition but different enthalpy are fed to a binary distillation column. We present a quick method to determine for each of the feeds whether it should be a saturated liquid or vapor or two-phase. For the special case when one of the feeds is saturated liquid and the other is saturated vapor, the method also helps in ascertaining the relative effectiveness of an intermediate reboiler or condenser. The easy-to-use results are graphically presented in Figure 2. In spite of the simplifying assumptions used in the model, good agreement is obtained with real systems. 6. R E F E R E N C E S
1. 2. 3. 4. 5.
R. R. R. P. Z.
Agrawal and D. M. Herron, AIChE J., 44 (1998) 1316. Agrawal and D. M. Herron, AIChE J., 43 (1997) 2984. Agrawal and D. M. Herron, AIChE J., 44 (1998) 1303. C. Wankat and D. P. Kessler, Ind. Eng. Chem. Res., 32 (1993) 306. T. Fidkowski and R. Agrawal, Ind. Eng. Chem. Res., 34 (1995) 1287.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
345
A Flowsheet-Centered Architecture for Conceptual Design B. Bayer
a,
K. Weidenhaupt
b, M .
Jarke b, W. Marquardt
a,,
a Lehrstuhl ftir Prozesstechnik, RWTH Aachen, D-52056 Aachen, Germany b Lehrstuhl fiir Informatik V, RWTH Aachen, D-52056 Aachen, Germany The current state of design support in chemical engineering is characterized by numerous software tools developed for specific purposes. For an improvement of process design, an integration of software systems into a coherent, but not monolithic environment is needed. In this paper, a flowsheet-centered integration architecture is proposed, t h a t - in contrast to commercial solutions - takes into account the central role of flowsheets in chemical engineering design. A novel flowsheet editor has been realized as the center of such an architecture, which allows the creation and management of hierarchical flowsheets for various design alternatives in different versions on different levels of detail for a variety of applications in the design process. It has an open architecture for data exchange with application tools and is integrated with a coordinating workflow component. 1. INTRODUCTION In the process industries there is a growing demand for an improvement of the design process in order to obtain better plants in shorter development cycles. Existing software tools support only isolated parts of the design process. A sustainable improvement can only be achieved by a tight integration of application tools into a design environment combined with the support of work processes [1]. For the integration architecture, the structure of work processes and the information used during process design needs to be taken into consideration, without hard-coding detailed assumptions into the tools [2]. Fig. 1 shows schematically some classes of tools used during conceptual process design together with major documents and data they are working on. Information about flowsheet elements and their structure is needed for the development of mathematical models in some modeling tool and for the specification of steady-state and dynamic simulations in the respective simulators. Results obtained from simulations may implicate changes in the design and thus in the flowsheets. Mass and energy balances as well as cost calculations that are performed in spreadsheet programs or specific application tools also depend on flowsheet information, even though those tools do not handle this information directly. Within the Global CAPE OPEN project [3], a workflow model has been developed describing the overall lifecycle of a chemical plant together with the information flows occurring. Among all documents created during the lifecycle, flowsheets play the major role: Block flow diagrams (BFD) are used in preliminary studies and conceptual design to sketch the process by means of process steps. They are refined to process flow diagrams (PFD) *Correspondence should be addressed to W. Marquardt, [email protected]
346
Fig. 1. Tools and documents within process design.
during conceptual design, where equipment is specified to realize process steps and unit operations. Based on the PFD, piping and instrumentation diagrams (P&ID) are developed during basic and engineering design. The P&ID forms the basis for plant construction and serves as a reference for plant operation and maintenance. During all lifecycle stages, flowsheets are reference documents for the exchange of information about the process and the status of its design in the design team. Due to this central position of flowsheets, flowsheet information is stored and handled in different tools (see also Fig. 1), leading to redundancies and inconsistencies of data. An integration of tools into a coherent design environment can overcome these problems. For an efficient support of chemical process design, a dedicated tool for defining and retrieving flowsheets is suggested to form a central and prominent component of such an environment. In order to provide a single user interface and to avoid inconsistencies, there should be no flowsheet functionality in any of the other applications. Interfaces should be provided for exchanging information between the flowsheet tool and application tools to enable the use of the best tool for solving specific design problems without reentering information and as the basis for an access to all results through the flowsheet. During the last years, several design environments were developed that support the early phases of the plant lifecycle, like Aspen Zyqad or COMOS PT by innotec. They support design starting from the simulation of flowsheets as it is performed for example in Aspen Plus or Hysys. There is no support for flowsheet design and editing independently from other design activities like modeling and simulation. Instead of having a flowsheet as the central document, to which all other information is associated, the flowsheet depends on other documents like simulation specifications. Furthermore, mainly tools of one vendor are tightly linked together within those commercial systems. This makes extensions difficult and leads to monolithic and hardly maintainable systems, which can be adapted to the peculiarities of the design process in a specific company only within tight constraints.
347 In order to support the central role of flowsheets during design, a novel flowsheet editor has been developed to serve as the center of an open and fexible design environment. In the next section, requirements on the flowsheet tool itself and on tool integration are identified. Section 3 presents the flowsheet editor and its integration with existing application tools. 2. R E Q U I R E M E N T S F O R A F L O W S H E E T - C E N T E R E D A R C H I T E C T U R E 2.1. Requirements for a flowsheet tool A flowsheet tool must support the creation and change of all kinds of flow diagrams. Predefined elements for the different flowsheets must be available and the definition of new flowsheet elements should be possible. During process design, the process is successively refined and detailed. An initial BFD representing the overall functionality of the process can be decomposed into a BFD where single process units are given. This can be further refined to a PFD by specifying items of equipment. More details are added regarding equipment, piping and instrumentation until the final P&ID is reached. This series of flowsheets can be seen as a hierarchy, where alternative refinements and decompositions exist leading to a complex tree of flowsheets on different levels of detail. A flowsheet tool should support that hierarchical approach both in a top-down and a bottom-up manner: decomposition of entire flowsheets and flowsheet elements should be allowed as well as the grouping of several elements into one process step. Alternatives and versions need to be managed so that browsing within the flowsheet tree is possible. A documentation of the different flowsheets and of the design history is needed in order to provide support for changes in design and to give guidance for the design process with its multiple and creative activities. 2.2.
Integration requirements
All data and documents that can be associated to the flowsheet should be accessible through the flowsheet tool. This includes information that can be edited within the flowsheet tool itself like process parameters or equipment specifications, but also information that is generated within other tools. Examples are simulation results including stream flow rate and compositions or cost information about the equipment. For all designers involved in a design project, the flowsheet tool should serve as an access portal to information available about the design project just like the flowsheet itself is the central document within process design. Therefore, data integration between the flowsheet tool and other application tools is needed. This can be done either by the definition of interfaces between the tools that allow the exchange of flowsheet related information or by the introduction of a central data storage for all tools using a file system or a database management system [1]. For the respective implementation, a well defined data model is needed that covers the relevant data in order to reduce the number and complexity of data transformations between tools; examples for such data models are CLiP [4], MDOOM [5] or those parts of STEP related to chemical engineering (e.g. [6]). 3. A N O V E L F L O W S H E E T EDITOR We introduce a flowsheet editor, which has been developed within the Collaborative Research Center IMPROVE as the center of an open software environment supporting conceptual process design [1]. This editor builds the main interface between the technical
348 designer and the support environment, where several existing, primarily commercial tools are integrated that are commonly employed in industrial practice. The flowsheet editor was realized on the basis of Visio [7], a commercial drawing tool that is used in chemical engineering and in other technical domains. Visio was chosen because of its extensive and adaptable library of drawing elements and its open interfaces that give access to all internal programming elements of Visio which is necessary for the implementation of tool integration. 3.1.
Features and functionalities Some functionalities were added to the general drawing features of Visio, that allow the creation of hierarchical layers, the browsing within a hierarchy, or the in-screen expansion of refined process steps. An adjustment of the object model of Visio was needed: objects typical for process design such as process steps or unit operations were added to the data model with their characterizing attributes. Also, the concept of hierarchical and alternative refinement was introduced into the data model. Finally, Visio was connected to a database which is used for storing the flowsheets together with associated data instead of the Visio file format [8]. The flowsheet editor is integrated with a workflow component [2]. Single activities that are performed are traced in order to catch design knowledge. These traces are documented and filtered; depending on the actual design context, they are offered to the user as a very specific and flexible work process support. In Fig. 2, a BFD of a polyamide6 process - the design of this process serves as a reference scenario within the CRC I M P R O V E - drawn with the flowsheet editor is shown. The process consists of three process steps: reaction, separation and compounding. The shaded area around those three process steps indicates, that there exists an aggregation: the process with
Fig. 2. Browsing between hierarchical layers.
349 in- and out-flowing streams. The dashed lines around process steps show the user, that there are alternative refinements. In the example in Fig. 2, the user wants to browse the refinements for the reaction. He selects the reaction block and the menu item Choose alternative refinement. A popup menu appears with a list of the existing refinement alternatives. The user can select one and the respective refinement will be shown as part of the entire flowsheet.
3.2.
Integration with application tools
Within the database of the flowsheet editor, only major information about the flowsheet elements is stored, including characteristic parameters of process steps and equipment items as well as major stream and piping information. All additional data, like simulation results or cost calculations, are kept in those tools where they are used and generated. In order to make that data accessible through the flowsheet, integration with application tools is needed. Fig. 3 shows the status of the actual implementation which integrates the flowsheet editor with a general purpose flowsheet simulator and a dedicated extruder simulator. For the exchange of information between the flowsheet editor and other tools, integration tools were implemented. These allow the exchange of data between different tools and ensure consistency between data storages and documents [9]. The integration tool between the flowsheet editor and Aspen Plus creates incomplete Aspen input files using all information needed for a simulation that exists within the flowsheet. This includes flowsheet elements and their connecting streams, which are translated to the respective modeling units. Since there is often no non-ambiguous mapping between flowsheet elements and model units, user interaction is needed. The generated input files can be edited in Aspen Plus; simulation results are transferred back into the flowsheet editor. In a similar manner, an integration between the flowsheet editor and the simulation tool MOREX [10] is implemented that is used for the design of compounding extruders. Due to these integrations, the flowsheet functionality of the simulation tools has become redundant. A process data warehouse [11] provides information about the tools, the type and the structure of the information they are working on and their specific data storages like data
Fig. 3 Integration architecture.
350 bases and files. It guides the user from the flowsheet editor to all associated information and the tools where it is available. The aforementioned workflow component coordinates the user's work processes within the flowsheet tool and across tool boundaries. This linkage between the flowsheet tool and application software through integration tools, a data warehouse, and a workflow component is very flexible in comparison to monolithic commercial systems. 4. CONCLUSIONS AND FUTURE W O R K
Since the flowsheet is the central document during chemical process design, integrated environments for design support should be centered around a dedicated flowsheet tool which is separated from individual application tools. A flowsheet editor has been presented that can serve as the center of a design environment due to its open and flexible interfaces. It allows the creation and editing of hierarchical flowsheets and thus supports the work with flowsheets during conceptual design where the process is successively refined and detailed. Integrations between the flowsheet editor and simulators exist that allow the generation of (incomplete) simulation specifications and the transfer of simulation results back to the flowsheet. Integration with additional application tools are planned. Furthermore, a coupling of the flowsheet editor with administrative support systems for managing design processes in collaborative teams [12] will be investigated. Another focus will be laid on the visualization of data from different tools in the flowsheet for a better support of decisions. ACKNOWLEDGEMENTS
This work is supported by the DFG, Deutsche Forschungsgemeinschaft, in the CRC 476 'IMPROVE'. The authors thank M. Nagl and R. Schneider for many fruitful discussions. REFERENCES
[1] W. Marquardt, and M. Nagl, DECHEMA-Monographie 135 (1998) 95-126. [2] K. Pohl, K. Weidenhaupt, R. D6mges, P. Haumer, M. Jarke, and R. Klamma, ACM Trans. Software Engineering and Methodology 8 (1999) 343-410. [3] B.L. Braunschweig, C.C. Pantelides, H.I. Britt and S. Sama, Chemical Engineering Progress, September (2000) 65-76. [4] B. Bayer, R. Schneider and W. Marquardt, Comp. Chem. Engng. 24 (2000) 599-605. [5] M.L. Lu, R. Batres, H.S. Li, and Y. Naka, Comp. Chem. Engng. 21 (1997) S 11-S 16. [6] ISO TC 184/SC4/WG3 N600. Product data representation and exchange: Application Protocol: Functional Data and their schematic representationfor process plants, 1997. [7] Microsoft Corporation, Visio Home. http://www.microsoft.com/office/visio/, 2000. [8] K. Weidenhaupt and B. Bayer, In: Informatik "99- Informatik ~berwindet Grenzen, K. Beiersd6rfer, G. Engels and W. Sch~fer (eds.), Springer, Berlin (1999) 305-313. [9] M. Nagl (ed.), Building Tightly Integrated Development Environments: The IPSEN Approach. Springer, Berlin, 1996. [10]E. Haberstroh and M. Schltiter, In: Tagungshandbuch 20. Kunststoffiechnisches Kolloquium des IKV, Verlag Kunststoffinformation, Bad Homburg (2000) Part 7, 4-7. [11] M. Jarke, T. List and J. K611er, Proc. 26th Intl. Conf. Very Large Data Bases (VLDB 2000), Cairo, September 2000, 473-483. [12] B. Westfechtel, Graph-Based Models and Tools for Managing Development Processes. Springer, Berlin, 1999.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
351
Systematic Generation of the Optimal and Alternative Flowsheets for Azeotropic-Distillation Systems B. Bertok a, F. Friedler a, G. Feng b, and L. T. Fan b aDepartment of Computer Science, University of Veszprem Egyetem u. 10, Veszprem, H-8200 Hungary bDepartment of Chemical Engineering, Kansas State University Manhattan, KS 66506, U.S.A. A systematic and rigorous method for synthesizing azeotropic-distillation systems, which is of utmost practical importance, is yet to be fully established. The available methods are based mainly on heuristics and graphical procedures. Our experience indicates that even in synthesizing a simple separation network, the structure of the optimal solution may be counterintuitive. For synthesizing a complex network structure necessary for an azeotropicdistillation system, therefore, the probability is very high that the solution generated would be far from the optimal one unless the method is systematic and rigorous. The proposed method is capable of algorithmically synthesizing optimal, near optimal, and other feasible structures for an azeotropic-distillation system from a set of candidate operating units. Initially, the residue curve map of the system is transformed to a unique multidimensional representation to facilitate the systematic partitioning of the feasible regions into lumped materials bounded by the thermodynamic boundaries and pinches. This renders it possible to derive analytical expressions of the resultant materials in terms of the coordinates of these boundaries and to automatically identify with dispatch the candidate operating units, such as separators, mixers, and decanters, for possible inclusion in the system. The processgraph (P-graph) representation of these operating units serves as a basis for the synthesis procedure including combinatorial algorithms. The method is equally applicable to various other complex processes with phase transition and/or phase separation with any number of components. Crystallization, extraction, reactive distillation, and their combinations are examples of such processes. A case study in which ethanol is separated from its aqueous solution with toluene as the entrainer demonstrates amply the efficacy of the method. 1. INTRODUCTION The current contribution is concerned with the synthesis of feasible alternative flowsheets of azeotropic-distillation systems. Specifically, it aims at developing an algorithmic and systematic method for synthesizing azeotropic-distillation systems from an extensive set of candidate operating units, i.e., functional units. Azeotropic distillation is ubiquitous in chemical and allied industries. The majority of existing azeotropic-distillation processes were developed and designed through extensive tryand-error based on past experiences. Consequently, questions largely remain unanswered as to
352 what potential benefit there might be for improving existing processes as well as what methodologies are to be adopted for devising new processes. In contrast to the rapid progress that has been made on the analysis of azeotropicdistillation systems since the mid-1980's (see, e.g., Van Dongen and Doherty, 1985; Siirola, 1996; Widagdo and Seider, 1996), success achieved for the synthesis of azeotropic-distillation systems has been rather modest. The majority of the available approaches, often termed analysis-driven synthesis, are essentially based on the first principles and/or heuristic rules derived from the analysis of the residue curve map (RCM) of the system of interest (see, e.g., Siirola, 1996; Westerberg and Wahnschafft, 1996). In spite of the progress made to date, much remains to be resolved for establishing a systematic and comprehensive methodology for synthesizing azeotropic-distillation systems. In fact, some critical issues are yet to be resolved: For example, how can be feasible alternative flowsheets systematically and inclusively generated for the analysis-driven synthesis approach (see, e.g., Siirola, 1996). The difficulty of synthesizing an azotropic-distillation system is attributable to its physical/chemical intricacy, which inevitably leads to enormous combinatorial complexity in synthesis (Feng et al., 2000). In other words, it may result in an inordinately large number of plausible or candidate operating units, the possibility of which being included in a feasible structure must de determined in synthesis. The magnitude of the solution space renders it extremely difficult, if not impossible, to adopt a conventional MINLP method. Hence, the development of a combinatorially effective method is deemed highly desirable. Such a method is proposed here; it is based on the process graph (P-graph), an innovative mathematical system, which has been conceived for process synthesis by incorporating the specificities of process systems (see, e.g., Friedler et aL, 1992; Friedler et al., 1993; Friedler et al., 1995). 2. DEFINITION OF THE SYNTHESIS PROBLEM
The thermodynamic pinches or boundaries, e.g., azeotropes, distillation boundaries, and the boundaries of liquid-liquid equilibrium envelopes, are of critical importance for azeotropic distillation. Moreover, the composition of the feed- and product-streams must be specified to define the synthesis problem for an azeotropic-distillation system. Such information can be represented by RCM's. For illustration, the RCM of the ethanol-watertoluene system is depicted in Figure 1. The points F, E, W, and T, represent the feed, product, byproduct, and the entrainer, respectively.
Fig. 1. RCM of the ethanol(E)water(W)-toluene(T) system.
353 The materials and operating units are the essential building blocks of a chemical process system. In what follows, the materials and operating units will be defined first. Naturally, the materials concomitant with plausible or candidate operating units, i.e., the input and output materials to each of them, are also simultaneously identified as required by the P-graph approach. A countless number of plausible materials or mixtures may be identified for a systems of three or more components. For exhaustive inclusion of every plausible alternative, therefore, a need exists to partition all the materials within each of the areas defined by Fig. 2. Lumped materials (L1, L2, .. L13) various boundaries in the RCM. "' " Any RCM occupying an area or space comprises an infinite number of points; as a result, the number of plausible operations is also infinite. Feng et al. (2000) have proposed that the RCM be partitioned into a finite number of lumped materials covering the entire are as illustrated in what follows. In Figure 2, the whole RCM is partitioned into materials occupying the points, i.e., E, W, T, H, and F; those occupying the areas, i.e., L1 through L6; and those occupying the lines, i.e., L7 through L19. According to the topology of the RCM, the plausible operating units are the distillation columns producing ethanol (operating units 1 through 6 in Table 1); water (operating units 7 through 10); toluene (operating units 11, 12, and 13); ternary azeotrop H (operating units 14 through 23), decantors (operating units 24 through 30); and mixers. The set of products, raw materials, and operating units serve as the input to the process-networkF,k L , ~ ' - - ~ synthesis algorithms. Mixer.~
\ f
\
3. AZEOTROPIC DISTILLATION DESCRIBED BY P - G R A P H S
A set of operating units can be represented in a Pgraph, where the operating units are denoted by horizontal bars, and their input and output materials by solid circles. The P-graph is a directed graph; the direction of the arcs representing a process network is the direction of the material flows in the network; it is directed to an operating unit from its input materials and from an operating unit to its output materials. For example, distillation columns 6 and 9, decantor 29, and two mixers are represented by P-graph in Figure 3. A Pgraph is said to be a combinatorially feasible process structure or solution structure if it satisfies the five
Separat~L ~ 6 W ]~ . ~
1
Llsepar..9~t~~,~ El, Decantor Fig. 3. P-graph representation of a process structure.
354 axioms (S1) through ($5) of the Table 1 combinatorially feasible process structures (see, Operating units for production of pure e.g., Friedler et al., 1992 and 1993). Axiom ethanol from its aqueous solution with (S 1) implies that each product is produced by at toluene as the entrainer. least one of the operating units of the system; axiom ($2), a material is not produced by any # Type Input Outputs operating unit of the system if and only if this E, L7 1 distillation L1 material is a raw material; axiom ($3), only the 2 distillation L1 E, L8 plausible operating units of the problem are 3 distillation L1 E, L10 taken into account in the synthesis; axiom ($4), 4 distillation L1 E, L11 any operating unit of the system has a series of 5 distillation L5 E, Ll0 connections eventually leading to the operating 6 distillation L5 E, LlI unit generating at least one of the products; and 7 distillation L4 W, L10 axiom ($5), each material appearing in the 8 distillation L4 W, L9 system is an input to or an output from at least 9 distillation L6 W, L7 one operating unit of the system. W, L7 10 distillation F The union of all combinatorially feasible 11 distillation L2 T, L8 process structures is defined to be the maximal T, L9 12 distillation L3 structure. The maximal structure of a synthesis 13 distillation L3 T, Lll problem comprises all the feasible structures 14 distillation L1 H, L 18 capable of yielding the specified products from 15 distillation L1 H, L19 the specified raw materials. Naturally, the 16 distillation L5 H, L18 optimal network or structure is among the 17 distillation L5 H, L 19 feasible structures generated from the H, L14 18 distillation L6 maximum structure, which is the complete and 19 distillation L4 H, L14 yet simplest super-structure rigorously defined 20 distillation L4 H, L15 mathematically. The maximal structure is 21 distillation L3 H, L16 constructed via algorithm MSG (Friedler et al., 22 distillation L3 H, L 17 1993). The complete set of the combinatorially 23 distillation L2 H, L17 feasible process structures or solution structures 24 decanting L3 L 12, L 13 can be generated by algorithm SSG (Friedler et L4 L12, L13 25 decanting al., 1995). 26 decanting L5 L12, L13 To facilitate algorithmic synthesis, the L9 L12, L13 27 decanting mathematical models are derived for the 28 decanting L10 L12, L13 various operating units involved, i.e., 29 decanting Lll L12, L13 distillation columns, mixers, and decantors, 30 decanting H Ll2, L13 based on RCM's and analytical geometry. The boundaries on RCM's, i.e., the distillation boundaries and the boundaries of liquidliquid equilibrium envelopes, are often non-linear. The non-linearity of these boundaries leads to the non-linearity of the constraints involved in the mathematical programming problem that need be solved for optimal synthesis. This non-linearity usually gives rise to inordinate difficulty in solution. To circumvent such difficulty, the boundaries are linearized or sectionally-linearized. Note that distillation boundaries and the boundary of liquid-liquid equilibrium envelope in Figure 2 are linearized or sectionally linearized, thereby resulting in Figure 4. The linearized models serve for the structure generation but not for the analysis of
355 individual structures which need to be evaluated on the basis of the rigorous models to ensure optimality. The linearized RCM and flow-rate-based representation conceived for separationnetwork synthesis (Kovacs et al., 2000) render the mathematical models of mixers and separators linear. Moreover, the mathematical models of decantors can be made sufficiently linear. In the flow-rate-based representation, the materials are represented by the flow rates of their components, e.g., M = (Ml, M2, M3), where M1, M2, and M3 are the flow rates of the first, second and third components, respectively. The total flow rate is the sum of Fig. 4. Linearized RCM. the flow rates of the components. The massbalance constraints for the input and output materials of Table 2 operating units can be written in the form of a simple sum of Mathematical model of the vectors representing the materials, e.g., A = B + C or (A1, distillation column 9. A2, A3) = (B1, B2, B3) + (C1, C2, C3). If any material M is inside a convex partitioned region, M can be written as a nonnegative L7 = v~q + v2x linear combination of the vectors of the concentrations of the L6 = v3w + v4q + vsx points, which are at the intersections of the boundaries of the W = v6w region. For example, L6 = vlw + v2q + v3x, where vl, v2, and v3 v1, v2, ..., v6 >- 0 are the nonnegative variables (vl, v2, v3-> 0); and w, q, and x are W+L7=L6 the constant vectors of the concentrations of points W, Q, and X in Figure 3, respectively. The mathematical model for any operating unit involves the massbalance constraint and the constraints for its input and output materials. For example, the mathematical model of distillation column 9 consuming L6 and producing W and L7 is given in Table 2. The mathematical programming model of a process network includes the constraints for the operating units, e.g., the mathematical models of the operating units, and those for the materials, e.g., the mass-balance constraints or constraints for the products or raw materials. The mathematical models of the operating units defined in the preceding section, i.e., mixers, separators, and decantors are linear, and the flow-rate-based representations of the massbalance constraints are also linear. Thus, the mathematical-programming model gives rise to a MILP problem which can be solved effectively by the algorithms developed for the P-graph approach.
4. SOLUTION OF THE SYNTHESIS PROBLEM OF PRODUCTION OF PURE ETHANOL FROM ITS AQUEOUS SOLUTION WITH TOLUENE AS THE ENTRAINER To demonstrate the efficacy of the proposed method, the systematic synthesis of the feasible flowsheets for the 3 component system of ethanol-water-toluene given in Figure 1 has
356 been solved at the level of linearization given in Figure 3. The resultant synthesis problem is defined on the basis of 24 partitioned regions denoted in Figure 2; it involves 23 separators, 7 decantors, and 593 mixers, thereby yielding altogether 623 operating units. The resultant synthesis problem, however, is extremely complex because of this huge number of candidate operating units. Under the constraint that each process structure contains at most 7 operating units, the implementation of the algorithm has resulted in 15 feasible flowsheets in 130 minutes on a PC (Pentium II. Celeron 366 MHz). One of the feasible structures is represented on RCM in Figure 5 and by Pgraph in Figure 3.
Fig. 5. Feasible structure.
5. CONCLUSION A systematic method based on the first principles and minimal heuristics has been developed for the algorithmic synthesis of azeotropic-distillation systems. The method is capable of generating the complete set of feasible flowsheets for an azeotropic-distillation system, from which optimal and near-optimal flowsheets emerge once the cost of operating units are appropriately assigned. REFERENCES
1. Feng, G., L. T. Fan, F. Friedler, and P. A. Seib, Identifying Operating Units for the Design and Synthesis of Azeotropic-Distillation Systems, Ind. Eng. Chem. Res., 39, 175-184 (2000). 2. Friedler, F., K. Tarjan, Y. W. Huang, and L. T. Fan, Graph-Theoretic Approach to Process Synthesis: Axioms and Theorems, Chem. Eng. Sci., 47, 1973-1988 (1992). 3. Friedler, F., K. Tarjan, Y. W. Huang, and L. T. Fan, Graph-Theoretic Approach to Process Synthesis: Polynomial Algorithm for Maximum Structure Generation, Comp. Chem. Eng., 17, 929-942 (1993). 4. Friedler, F., J. B. Varga, and L. T. Fan, Decision Mapping: A Tool for Consistent and Complete Decisions in Process Synthesis, Chem. Eng. Sci., 50, 1755-1768 (1995). 5. Kovacs, Z., Z. Ercsey, F. Friedler, and L. T. Fan, Separation-Network Synthesis: Global Optimum through Rigorous Super-Structure, Comp. Chem. Eng., 24, 1881-1900 (2000). 6. Siirola, J. J., Strategic Process Synthesis: Andvances in the Hierarchical Approach, Comp. Chem. Eng., 20, 1637-1643 (1996) 7. Van Dongen, D. B. and M. F. Doherty, Design and Synthesis of Homogeneuous Azeotropic Distillations.1. Problem Formulation for a Single Column, Ind. Chen. Eng. Fundam., 24, 454-463 (1985). 8. Westerberg, A. W. and O. Wahnschafft, The Synthesis of Distillation-Based Separation Systems, Adv. Chem. Eng., 23, 64-170 (1996). 9. Widagdo, S. and W. D. Seider, Azeotropic Distillation. AIChE J., 42, 96-130 (1996).
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
357
Reactor Selection and Design for Heterogeneous Reaction Systems Christian Btihner, Gerhard Schembecker Process Design Center B.V., Joseph-von-Fraunhofer-Str. 20, D-44227 Dortmund, Germany This paper presents a heuristic-numeric strategy for the selection and the design of technical reactors for multiphase reaction systems in particular. On basis of fundamental data of the reaction system and its structure essential decisions are made with regard to relevant reactor selection criteria and the reactor selection itself. 1. INTRODUCTION Selection and design of a reactor is a decisive step within the framework of process synthesis. Chemical processes consist of the reaction section and the separation section. The reaction section and its outlet streams have strong influence on the structure of the separation section whereas recycle streams which origin from the separation section affect the conditions of the reaction section. Apart from this, the selection and design of a reactor strongly influence the conversion of reactants to desired products and the selectivity respectively. This short description shows that the reactor type, its characteristics and the reaction conditions within this reactor are important questions which should be answered at an early stage of process synthesis. The fact that many parameters to be considered during the selection procedure depend on one another makes the reactor choice a difficult task. In addition only fundamental data from initial experiments will be present at an early stage of process synthesis. This means that selection methods which are based on mathematical models and algorithms [1] are not suitable for the selection process since the necessary data for the utilization of such tools are usually not available. This paper presents a strategy for the selection of suitable reactors for multiphase reaction systems which uses a heuristic-numeric approach for the selection procedure. On the basis of fundamental data of the reaction system (e.g. kind of side reactions, qualitative data of kinetics) the strategy applies selection levels which are based on one another (see Fig. 1). Each selection level determines relevant reactor selection criteria to be used in the subsequent selection level and for the final selection of suitable technical reactors respectively. The strategy especially takes into consideration that many of the criteria investigated are based on one another or depend on each other. Therefore in contrast to publications of other authors (e.g. [2-3]) the strategy presented is not a simple straight forward procedure to determine selection criteria. The strategy has been implemented in the heuristic-numeric consulting system READPERT (_Reactor E_valuation and Design Expert System). The following chapters describe the major steps of the decision making process.
358 2. INVESTIGATION OF THE REACTOR PHASE
Beside homogeneous reaction systems the reactor selection and design strategy especially focuses on the investigation of multiphase reaction systems. The following phase systems can be handled:
Fig. 1: Structure of the strategy 9 homogeneous: gas, liquid 9 heterogeneous: gas-liquid, liquid-liquid, gas-catalyst, gas-liquid-catalyst The first step to investigate multiphase reaction systems is to analyze the reaction phase. E.g. for a gas-liquid reaction system with the reaction taking place in the liquid phase the liquid phase will be investigated. The gas phase is necessary for the supply of additional reactants. This strategy step proposes basic reaction conditions to be realized in the reaction phase by using fundamental data of the reaction system. These are mainly: 9 structure o f the reaction system: parallel reaction, consecutive reactions etc. 9 qualitative data o f kinetics: irreversible, reversible, autocatalytic, inhibited etc. 9 ratio o f reaction orders 9 temperature dependency of reactions (ratio of activation energies)
Based on this information READPERT proposes qualitatively favorable conditions with regard to the concentration levels of the reactants (by comparison of reaction orders), the residence time distribution and the temperature profile (by comparison of activation energies, structure of the reaction system). The goal for the investigations is the maximization of the reactor performance (i.e. conversion, selectivity or yield). A multitude of heuristic rules is applied which for homogeneous reactions systems have been created by Westhaus [4], now extended for the reaction phase of multiphase reaction systems. 3. DETERMINATION OF PHASE AND MIXING CONDITIONS
This step of the strategy takes into account the effects and characteristics typical for a multiphase reaction system. The following important criteria are part of this selection level: 9 backmixing characteristic of the phases present (total, partial, no backmixing) 9 contacting of the phases (co-, counter-current) 9 interracial area (area between the fluid phases, catalyst structure) 9 holdup of reaction phase
359 The investigation of the backmixing characteristic of the phases utilizes the results of the preceding selection level with regard to concentration levels and residence time distribution. In case two fluid phases exist (e.g. gas-liquid), the backmixing characteristic of the non-reaction phase (gaseous) only depends on the proposed concentration levels of the reactants transferred from the gas to the liquid phase whereas the backmixing characteristic of the reactionphase also depends on the residence time distribution. An example for the determination of the reaction phase backmixing characteristic is given for the following reaction system: (1) Ag + B!--~ CI (2) Ag + Bl-~ D!
(CI: desired product)
2
r1 - k 1.c A .c a 2
rE = k2 "CA "Ca
(3) Cl --~ El r3 = k3 "Cc On account of the reaction orders, the presence of a consecutive reaction and the ratio of the reaction rates of the side reactions the following heuristic is applied for the reaction phase (liquid phase): Rule: The backmixing characteristic of the reaction phase is, total backmixing ', if 1) the concentration level of the reactant supplied by the reaction phase(Bt) has to be kept ,as low as possible' 2) a narrow residence time distribution is necessary, 3) the reaction rate o f the parallel reaction is faster than the reaction rate o f the consecutive reaction.
Due to the ratio of the reaction rates of the side reactions the residence time distribution is not as important as the desired concentration level. As a result, total backmixing' of the reaction phase is proposed. The investigation of criteria such as e.g. interracial area requires knowledge about mass transfer limitations between the phases existing. It is sufficient to have knowledge about the ratio of the intrinsic reaction rate and the mass transport, called ,,kinetic regime": 9 mainly controlled by intrinsic kinetics 9 controlled by kinetics and mass transfer 9 mainly controlled by mass transfer In order to decide which kinetic regime is present READPERT needs information about possible conversion changes checked by simple laboratory experiments such as conversion change by variing the agitator speed of a laboratory reactor for gas-liquid or liquid-liquid reactions. Concerning gas-catalyst reaction systems the gas velocity and catalyst size are possible features for the determination of the kinetic regimes. The interracial area between the phases is an important aspect as it is possible to influence the conversion and selectivity. For reaction systems with two fluids (gas-liquid, liquid-liquid) one can affect the concentration level of the transferred components by changing the interfacial area [5-8]. The following rule is a simple example for the approach: Rule: The interracial area for a liquid-liquid system has to be as low as possible with regard to goal,selectivity'if 1) the reaction is controlled by kinetics and mass transfer, 2) the concentration level of the transferred components has to be as low as possible.
In case a heterogeneous catalyst is present (i.e. gas-catalyst, gas-liquid-catalyst) the determination of the interracial area refers to favorable catalyst sizes and pore structures [2]. Not only the concentration level is influenced by the catalyst size and pore structure but also the residence time in the catalyst pores as the heuristic given below shows:
360 Rule: The catalyst size should be ,as small as possible' and the mean pore size should be a 'wide pore structure' with regard to the goal selectivity, if 1) the reaction is controlled by kinetics and mass transfer (pore diffusion resistance), 2) parallel reactions and consecutive reactions occur as side reactions, 3) the reaction rate of the consecutive reaction is faster than the rate of the parallel reaction.
The consecutive reaction has more influence than the parallel reaction and therefore the reduction of the residence time is more important than the consideration of the concentration levels of the reactants. In addition to the catalyst investigation concerning concentration levels, the strategy also offers the option to determine the catalyst with respect to heat transfer from and to the catalyst as this also may have strong effects on conversion and selectivity. Another criterion dependent on the kinetic regime is the holdup o f the reaction phase. First it is checked whether explosive or poisonous phases are present and for which the holdup has to be kept low. For two fluid systems the holdup of the reaction phase should be as high as possible if the reaction is ,kinetically' or ,kinetically and mass transfer' controlled. In this case the reaction takes place in the bulk of the phase. If only mass transfer limitation occurs the holdup is not a decisive aspect, the interfacial area is much more important. Proposals for the catalyst holdup are made on the consideration of the proposed catalyst size and the kinetic regime as both give hints how much reaction zone is available. Within this selection level also hints are given for counter- or co-current phase contacting of two fluid phase systems. First of all it is revised whether a runaway of the reactor is possible (---~ no co-current operation) or if some reactants have to be supplied in excess (---~ phase contacting is without influence). In a second step the structure of the reaction system is regarded. If conversion is the goal for reactor selection a counter-current phase contacting is obviously preferable, especially for equilibrium and inhibited reactions. For the goal selectivity sensitivity studies have been carried out. The results show that the phase contacting does not have significant effect on the selectivity for parallel reactions whereas in the presence of consecutive reactions a counter-current operation improves selectivity. 4. OPERATING CONDITIONS This third step of the strategy provides recommendations for significant operating conditions of the reactor which are listed in figure 2. The figure shows that many of these operating conditions are coupled and cannot be analyzed separately. Dependent on the complexity of the reaction system it is possible that some of these criteria may have already been derived in the two preceding selection levels before which indicates that the strategy has to consider the dependencies between the criteria. This clarifies that the strategy is not a simple straight forward procedure of determining selection criteria. The proposals for the different operating conditions such as e.g. component removal or ratio of reactants are based on the investigation of the structure of the given reaction system. A product removal for example is recommended if consecutive reactions occur. A removal of side products may be advantageous in case the side products react with one reactant or product. The heuristics applied in this strategy level are meant to give hints and to generate ideas. The question if it is possible to transfer the recommendations to technical practice is not the aim here.
361
Fig. 2" Operating conditions and their dependencies 5. TECHNICAL R E A C T O R AND HEAT TRANSFER EQUIPMENT The proposals of the preceding steps for different reactor selection criteria are used on the final selection level for the selection of suitable technical reactors. Additional constraints and further selection criteria will enter the selection process of suitable technical reactors. First all reactors are collected from a database in READPERT which can handle the reaction phase system considered (e.g. liquid-liquid). In the succeeding step so-called knock out criteria as for example deposit forming or foaming caused in the reactor by the reaction mixture are applied. In case a reactor cannot handle such reaction mixtures it is excluded from the selection process. Having checked the knock out criteria the subsequent selection steps utilize so-called ranking criteria to find favorable reactors. The list of ranking criteria consists of the reactor selection criteria determined in the preceding selection levels (e.g. backmixing and contacting of phases, interfacial area, kinetic regime etc.). Additional criteria such as phase dispersion or corrosive phase are also part of the list of ranking criteria. Applying the ranking criteria- for each step the user of READPERT can select one ranking criteria out of the list the number of favorable technical reactors will be reduced step by step, so that at the end of the selection a list of most favorable reactors will be left. Moreover, READPERT offers the option to change the input of a selection step in order to create alternative solution branches. Having finished the reactor selection procedure it is possible to evaluate heat transfer options. At first the reactors are analyzed by shortcut methods concerning a feasible adiabatic operation. In case an adiabatic operation is not possible different heat transfer options will be regarded for each reactor such as boiling cooling, external heat exchange equipment, internals for heat exchange and heat exchange using the reactor wall. First general constraints are checked which prevent the usage of a heat transfer option for the reactor (e.g. boiling cooling cannot be applied in case strong foaming occurs). In a second step short cut methods for heat transfer options are applied in order to roughly estimate whether the heat production can be controlled by the heat transfer option. As a results one for example gets the necessary heat transfer area for external heat exchange equipment. An extra advantage of the presented heuristic-numeric strategy is the fact that it can provide useful hints with regard to additional laboratory experiments which may help to improve the investigation of the selection criteria and the reactor selection itself.
362 6. EXAMPLE: ALKYLATION OF 1-BUTENE WITH ISOBUTANE PRODUCING CsALKYLATE Reaction system
1. Isobutane + 1-Butene --->C8-Alkylate 2. Cs-Alkylate + 1-Butene --->C12-Alkylate Investigation of reaction phase 9 concentration level of isobutane: without influence 9 concentration level of 1-butene: without influence 9 residence time distribution: narrow 9 temperature profile: constant (as low as possible) Operating no special reactants supply ratio of reactants" excess of isobutane recycle streams: recycle of isobutane
9 9 9 9 9
Data of reaction system irreversible, catalyst: H2SO4, exothermic phase system: liquid-liquid exothermic reaction orders: first order ratio of activation energies: EA~< Em
Phase and mixing conditions 9 no backmixing characteristic of both phases favorable 9 interfacial area: as high as possible 9 dispersion of reaction phase 9 contacting of phases: counter-current 9 holdup of reaction phase: without influence conditions 9 C8-Alkylate removal recommended 9 incomplete conversion of isobutane 9 safety" no hint concerning necessity of adding inerts
Technical reactor 9 checked criteria: deposit forming, stable emulsion, backmixing characteristic, interracial area, kinetic regime, contacting, volume flow ratio 9 favorable: Rotating Disc Contactor, Kuehni or Scheibel extraction column 9 heat transfer: adiabatic operation not possible, employment of external heat exchangers
7. CONCLUSION The presented selection procedure has been implemented in the heuristic-numeric consulting system READPERT which is part of PROSYN | a tool for computer-aided process synthesis. The application of READPERT in numerous industrial process synthesis projects proofed that READPERT is a useful tool for improving the process structure and the reactor performance in special. REFERENCES V. Metha and A. Kokossis, Computers chem. Engng., Vol. 22 (1998), S 119 [1] R. Krishna and S. Sie, Chem. Eng. Sci., Vol. 49 (1994), No. 24A, 4029 [2] R. Jacobs and W. Jansweijer, Comp. & Chem. Eng., Vol. 24 (2000), 1781 [31 U. Westhaus, Beitrag zur Auswahl chemischer Reaktoren mittels heuristisch-nume[4] rischer Verfahren, Ph.D. Thesis, Universit~it Dortmund, 1995, Shaker-Verlag O. Nagel, B. Hegner and H. Ktirten, Chem.-Ing.-Tech., Vol. 50 (1978), 934 [5] K. Schiigerl and R. H~insel, Chem.-Ing.-Tech., Vol. 58 (1986), 308 [6] R. Doraiswami and K. Yesuda, Chem. Eng. Comm., Vol. 147 (1996), 119 [7] K. Samant, AIChE J., Vol. 44 (1998), No. 10, 2212 [8]
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
363
Generalized Disjunctive Programming Model For The Synthesis Of Thermally Linked Distillation Systems J.A Caballero a and I.E. Grossmann
b
a Department of Chemical Engineering. University of Alicante. Ap Correos 99. 03080 Alicante Spain b Department of Chemical Engineering. Carnegie Mellon University. Pittsburgh, Pennsylvania 15213.
This paper presents a systematic procedure for synthesizing distillation column configurations to separate a non-azeotropic mixture containing N components. It is shown that for the separation of an N component mixture it is possible to develop a superstructure that takes into account all the possibilities, from structures containing only conventional columns to fully thermally linked structures using only one reboiler and one condenser. All the partially thermally linked structures are included. The State Task Network (STN) formalism is used for generating that superstructure. A set of logical relationships between tasks is proposed in order to allow only feasible configurations that use the minimum number of column sections, taking into account that the number of column sections and the number of heat exchangers are not independent. The superstructure can be modeled using Generalized Disjunctive Programming (GDP) independently of the equations that represent each of the tasks. 1. INTRODUCTION Distillation continues being the most important separation system in the chemical process industry, even though it is one of the most energy intensive systems. The objective of a distillation based synthesis problem is to find the arrangement of columns that will provide the best scheme in terms of investment and operational costs. For a mixture of N components, N-1 conventional distillation columns are needed, but each one of those distillation columns has one reboiler and one condenser with their own contribution to the total investment cost and energy (utilities) consumption. Reduction in energy consumption is usually performed through energy integration. Either by synthesizing multieffect columns or by exchanging heat between different separation tasks. However, it is also possible to reduce the energy consumption and often also the total cost using fully (or partially) thermally linked distillation sequences. In this work we first present a systematic procedure for generating a general superstructure that varies from thermally linked distillation systems to conventional distillation systems, based on the STN formalism of Yeomans and Grossmann 1. We include a set of logic relationships among separation tasks that are enough to completely specify a feasible separation system (some of them previously proposed by Agrawal 2). Next, we show how this superstructure can be modeled using a disjunctive representation. An extended version of this paper can be found in Caballero and Grossmann 3.
364 2. SUPERSTRUCTURE FOR DISTILLATION SEQUENCES. The first aspect to take into account when generating superstructures for thermally linked columns is that there is a relationship between the number of heat exchangers and the minimum number of columns (or better column sections) of a given separation. While for conventional columns this relation is fixed: N-1 columns and 2(N-l) heat exchangers, this is not true for thermally linked distillation systems. A column (separation) section is defined to be a portion of a distillation column which is not interrupted by entering or exiting streams or heat flows 4. For the separation of a mixture of N components N-1 conventional columns are needed, or equivalently 2(N-l) separations sections. This is also the minimum number of column sections required to perform that separation. Using only conventional columns 2(N-1) heat exchangers are needed (one for each section). In thermally linked columns (side strippers or side rectifiers) the vapor and liquid flows from a given section in a column are shared with another column, i.e. the rectifying section of a column is shared with a side stripper, the enriching section of a columns is shared with a side rectifier. Therefore, the condenser is able to provide the condensing duty for the side stripping column and the reboiler is able to provide the boiling duty for the side rectifying column. Hence, the minimum number of reboilers and condensers that uses the minimum number of distillation sections is equal to the number of components in the feed mixture. However, it is possible to eliminate a reboiler or a condenser associated with a component of intermediate volatility by adding two more distillation sections. See for example Figure 1 in which the reboiler associated to the component B in scheme (1.b) is removed in the scheme (1.c). Note that the condenser associated with the lightest component and the reboiler associated with the heaviest one cannot be removed 2. Figure 1 shows graphically some of the aspects previously commented. A
A
ABC
ABC ABC
.~ B
A B/ B C
(a) 4 exchangers
(b) 3 exchangers
(c) 2 exchangers
Figure 1. Reduction of exchangers in ternary systems Summarizing, the minimum number of distillation sections for thermally linked systems is 2(N-l) and for this design, the minimum number of heat exchangers (reboilers and condensers) is N (See for example Figure l b). It is possible eliminate N-2 of those condensers, but this implies adding two new column sections for each of the exchangers that is removed. Thus, the minimum number of column sections in a thermal linked system is 2(N-l) + 2(N-E), where E is the number of heat exchangers (2 <E < N). If there are only
365 one
condenser
and
one
reboiler the
minimum
number
of column
sections
is
2(N-1)+2(N-2)=4N-6. Another aspect to take into account is that in a thermally linked (or partially linked) separation system it is not possible to pre-specify if a given separation task is going to be carried out by the main column or by a side stripper, a side rectifier, etc. It is possible to use configurations in which more that one column is integrated in the same column shell. Hence, we cannot pre-specify the equipment in which a given separation task will be performed. We only know which are possible separation tasks. Yeomans and Grossmann 1 (See also Sargent 5) proposed a general framework for systematically generating superstructures. The State Task Network (STN) formalism is especially well suited for this case in which the particular equipment used cannot be prespecified. We will illustrate the methodology with a mixture of four components, Assume that ABCD is a mixture of four compounds ranked according to their volatility, A is the most volatile and D is the least. The states are all the possible mixtures that can be produced after a separation. In our example the possible states are: ABCD; ABC; BCD; AB; BC; CD; A; B; C; D (defined in this case only as a function of their composition). Identification of tasks is not always trivial. The reason is that the simple enumeration of tasks can include tasks that cannot be performed by a thermally linked distillation system. However, this can be circumvented, if a set of logical relationships, between tasks and states are included to yield only correct separation sequences. Therefore, for a four components mixture, the tasks are: AIBCD; ABICD; ABIBCD; ABCIBCD; ABCICD; ABCID; A[BC; ABIBC, ABIC; BICD; BCICD; BCID; AIB; BIC; CID. Deriving the superstructure is now a simple task. We only need to join the different states with the tasks that the state gives rise. Figure 2 shows the resulting superstructure. oC ToD
Figure 2 Superstructure for separation of 4 components We must include some logical relations between tasks to avoid forbidden configurations or sequences with more than 4N-6 separation sections. The following 5 rules are enough to completely specify a feasible separation system
366 1. A given state can give rise to at most one task 9 2. A given state should only be produced at most by one task 9(except products) 3. For the products: the lightest and the heaviest should be produced only by one task. However, the intermediate products are produced by two sections. One by a stripping section of a task (considering the task as a pseudo column) and another by a rectifying section. 4. Connectivity: i.e. task ABIBCD implies tasks AIB and task ABIBCD implies task BICD or task BCICD or task BCID. 5. The number of tasks is equal to (4N-6)/2. Note that each task produces two separation sections. After the superstructure has been generated, the second step involves the modeling of the representation as a mathematical programming problem. Because there are conditional tasks that involve discrete decisions, it is necessary to use discrete mathematical programming models. In this case we will use Generalized Disjunctive Programming 6 (GDP) in which the conditional constraints are represented by disjunctions which have assigned a boolean variable that represents their existence. If we define the boolean variable Yt such that variable is True if task t is selected and False otherwise, conceptually the model can be written as follows 1' 7. min 9f ( x )
s.t. r(x) = 0
connectivity equations
V
1
ht(x)=O v B t x=O VteTASK
Lgt(x)-
(Pl)
f2(Yt )= True Yt ={True, False} XE~
n
where x is a vector of continuous variables, and f(x) is the objective function. The equations and constraints that are activated when task t is selected are given by ht(x)=O and g,(x)
367 3. The lightest and the heaviest products are produced only by one task, and the heat exchanger associated to these components will always appear in the superstructure. However, the intermediate products are produced by one or two contributions. 3.1. If the intermediate product is produced by two contributions, one must come from a stripping section of a task and the other from a rectifying section. There is no heat exchanger associated with this product. 3.2. If the intermediate product is produced by only one contribution, the heat exchanger associated to that product must be selected. 5. The number of tasks is equal to N-1 + N - E = 2N-1-E (E is the number of heat exchangers) Superstructures generated for thermally linked systems are also valid for systems that take into account all the combinations of non-conventional and conventional columns. Note that at the level of STN representation the only difference between thermally linked columns and conventional ones is the presence of intermediate condensers and reboilers. Again the thermal state of intermediate states is important. Note also that if we introduce the possibility of a heat exchanger associated to each one of the intermediate states, we obtain a superstructure in which all the possibilities are taken into account. Although at the level of superstructure there are no important changes, there are important modifications at the modeling level. In this case the connectivity equations cannot be treated as simple mixers and splitters, because the presence or absence of an intermediate heat exchanger changes the flows between columns. While in thermally linked systems separation tasks exchange vapor and liquid flows, if an intermediate heat exchanger is selected only liquid is fed to the tasks. To model this last case it is necessary to introduce a boolean variable to indicate the existence or not of these intermediate heat exchangers. It is important to point out that we differentiate between heat exchangers associated to pure products and intermediate heat exchangers (related to each one of the intermediate states). The reason is that the number of intermediate heat exchangers does not change the minimum number of column sections (or tasks) for performing the separation. They only break the thermal linkage between tasks, although the heat exchangers related to the final products are related to the minimum number of separation sections. EXAMPLES The procedure is illustrated with two examples, both for a 5 component mixture (nPentane, n-hexane, h-heptane, n-octane and n-nonane in the first case and i-butane, 1-butene, n-butane, trans 2-butene, cis 2-butene for the second example). A modified version of Underwood method proposed by Carlberg y Westerberg 8,9 is used to model the different tasks. In order to study the effects of the costs a conceptual objective function was used in which fixed cost coefficients were assigned to column sections, heat exchangers and utility systems. The cost of operation was assumed proportional to the total vapor flow through each column. The first case is a mixture that is relatively easy to separate while the second one it is much more difficult. The feed flow rate in both cases is 100 kmol/hr with compositions 10% A, 10% B, 20% C, 30% D, 30% E. Using in both cases the same cost coefficients the optimal configuration obtained are shown in Figure 3.
368 Notice that the configuration in Fig 3a there are 8 column sections, 4 condensers and 1 reboiler. In Fig 3b there are 14 column sections and only one condenser and one reboiler. The heat load ofthe reboiler in Fig 3a is 3275 MJ/h and in Fig 3b is 18812 MJ/h. It is important to also note that there are multiple representations for the optimal configurations in this problem, since many of them are thermodynamically equivalent. Therefore, the configurations in Figs. 3.a and 3.b are only one of the many solutions for the optimal sequence of tasks for N- 1columns.
r
.A
~B
AB
BCD
I
C
1)
AB~
BCD ~' ]
ABC
1
DE
(a) Easy
r-B
] ____.o
(b) Difficult Figure 3 Optimal configurations for easy and difficult separations.
REFERENCES
1. Yeomans, H. and Grossmann, I.E. A Systematic Modeling Framework of Superstructure Optimization in Process Synthesis. Comp. Chem. Eng. 1999 (23) 709. 2. Agrawal, R. Synthesis of Distillation Column Configurations for a Multicomponent Separation. lnd. Eng. Chem. Res. 1996, 35 1059-1071. 3. Caballero, J.A. and Grossmann I.E. Synthesis and Modeling of General Distillation Systems. Submitted for publication 2000 4. Hohmann, E.C.; Sander, M.T.; Dunford, H. A New Approach to the Synthesis of Multicomponent Separation Schemes. Chem. Eng. Commun. 1982 17 273-284. 5. Sargent, R.W.H. A Functional Approach to Process Synthesis and its Applications to Distillation Systems. Comp. Chem. Eng. 1998 (22) 31. 6. Turkay, M. and Grossmann I.E. A Logic Based Outer Approximation Algorithm for MINLP Optimization of Process Flowsheets. Comp. Chem. Eng. 1996, 20, 959. 7. Caballero, J.A.; and Grossmann I.E. Aggregated Models for Integrated Distillation systems. Ind. Eng. Chem. Res. 1999 38, 2330. 8. Carlberg, N.A.; Westerberg, A. Temperature Heat Diagrams for Complex Columns. 2. Underwood's Method for Side Stripers and Enrichers. lnd. Eng. Chem. Res. 1989a 28, 1379-1386. 9. Carlberg, N.A.; Westerberg, A. Temperature Heat Diagrams for Complex Columns. 3. Underwood's Method for the Petlyuk Configuration. lnd. Eng. Chem. Res. 1989b. 28, 1386-1397.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
369
Optimization Approach to the Design of Value-Added Soybean O i l Products
An
K. V. Camarda a, P. Sunderesan a, S. Siddhaye a, G. J. Suppes a, and J. Heppert b aDepartment of Chemical and Petroleum Engineering The University of Kansas 4006 Learned Hall Lawrence, KS 66045 bDepartment of Chemistry The University of Kansas 1023 Malott Hall Lawrence, KS 66045 This work focuses on the development of new products which can be synthesized from soybean oil and used in such applications as fuel additives, detergents and industrial solvents. Soybean oil derivatives have been found in previous studies to have excellent properties in terms of improving diesel fuel performance. However, a methodology which allows the determination of both the molecular structures of soybean oil-derived compounds and the relative amounts of these compounds within a mixture which will closely match a set of desired physical properties is needed. This work employs topological indices as descriptors of the molecules within a mixture in order to predict physical properties of interest. These indices take into account not only the types of atoms present within the molecules, but also the bonding environments between those atoms. New correlations have been generated based on these indices which are then used within an optimization framework to derive candidate mixtures which can be synthesized and tested for use in a given application. The optimization problem has been formulated as a mixed-integer linear program, which is then solved using standard techniques in order to generate both the molecular structures of the compounds in the mixture and the proportion of each compound within that mixture. The formulation is then tested using a number of industrial examples. 1. I N T R O D U C T I O N Currently, there is a significant need for new products in the fuel and consumer products industries which are created from renewable resources. Furthermore, such products tend to be environmentally benign and inexpensive to produce. The use of soybean oil derivatives to serve as improved fuel enhancers and detergents has been suggested as a example of the use of a renewable feedstock which also would greatly improve the potential market for this agricultural product. However, the challenge involved is to determine the optimal soybean oil derivative to choose for a given application. The use of optimization techniques coupled with molecular design, along with property estimation methods allows the determination of candidate molecules matching a set of target properties. This method can greatly reduce the
370 effort required to develop a new product, and is applicable to such varied products as polymers, solvents, and detergents. For example, it has now been reported (Hairston, 1998) that a computational algorithm has been successfully implemented in order to design a new pharmaceutical which fights cancer. Soybean oil is a mixture of fatty acids connected via glycerol molecules to form triglycerides. Slightly more than 80% of the fatty acid content is made up of unsaturated acids, including linoleic acid (51%), oleic acid (23%), and linolenic acid (7%) (Erickson et. al. 1992). These molecules also act as surfactants, with the carboxylic acid group serving as the head and the carbon chain as the tail. The formation of micelles above the critical micelle concentration (CMC) gives derivatives of these acids their excellent properties in terms of detergency and solubility. Many synthesis reactions are possible at the terminal carboxylic acid group, leading to a wide range of possible derivatives with varied properties. Thus this system forms an excellent basis for molecular design studies. In order for a molecular design algorithm to be successful, two main challenges must be overcome. The first is that physical properties of interest, such as detergency, density, solubility, or reactivity within a given system, must be estimated with a reasonable accuracy and using a very small amount of computational effort. Secondly, the optimization problem which results from the large set of possible molecules, which usually includes integer variables, must be solved to near-optimality in a reasonable amount of time. This work employs connectivity indices, which are numerical values which describe the electronic structure of a molecule, to characterize the molecule and to correlate its internal structure with physical properties of interest. Kier and Hall (1976) report correlations between connectivity indices and many key properties of detergents, such as density, solubility, and toxicity. Furthermore, these indices can be computed with a minimum of computational effort. The equations needed to compute the physical properties are then combined with structural constraints and reformulated into a mixed-integer linear program (MILP), which can then be solved using commercial software to find the optimal molecular structure. Further solutions can be generated using integer cuts, which results in a list of candidate molecules with a high probability of being effective for a specific application. These compounds can then be synthesized and tested for eventual development and introduction into the marketplace. Many of the applications of connectivity indices have been reviewed by Trinajstic (1975). Raman and Maranas (1998) first employed these indices within an optimization framework, and Camarda and Maranas (1999) used connectivity indices to design polymers which prespecified values of specific properties. In earlier molecular design work, group contribution methods were used to estimate the values of physical properties, as in Gani, et al. (1989), Venkatasubramanian et al. (1995), and Maranas (1996). The connectivity indices, however, have the advantage that they take into account the internal molecular structure of a compound. The property predictions generated from these indices are thus more accurate then those from group contributions, and furthermore, when a molecular design problem is solved using these indices, a complete molecular structure results, and no secondary problem must be solved to recover the final molecular structure. Connectivity indices also provide a method to compute the properties of mixtures, since the connectivity indices of individual compounds can be combined to estimate mixture properties in certain cases.
371
2. P R O P E R T Y P R E D I C T I O N VIA C O N N E C T I V I T Y INDICES The basis for many computational property estimation algorithms is a decomposition of a molecule into smaller units. Topological indices are defined over a set of basic groups, where a basic group is defined as a single non-hydrogen atom in a given valency state bonded to some number of hydrogen atoms. Table 1 gives the basic groups used in this work, along with the atomic connectivity indices for each type of group. In this table, the 8 values are the simple atomic connectivity indices for each basic group. They refer to the number of bonds which can be formed by a group with other groups. The 8v values are atomic valence connectivity indices, which describe the electronic structure of each basic group, including lone-pair electrons and electronegativity. The definitions of these indices are from Bicerano (1996). Note that atomic connectivity indices can be defined for any basic group, and the small table of groups used here is merely for illustrative purposes.
Table 1: Basic Groups and their Atomic Connectivity Indices 8 8v 8
-CH3 -CH2-
1 2 3 4 2 3
-CH< >C< -CH>C-
1 2 3 4 3 4
-OH -O-O -N< -NH-NH2
1 2 1 3 2 1
8v 5 6 6 5 4 3
Once a molecule is decomposed into its basic groups, and the atomic connectivity indices for those groups are known, then molecular connectivity indices can be computed for the entire molecule. The zeroth order molecular connectivity indices ~Z and ~ are sums over each basic group, and thus describe the identity of the groups in a given molecule: 1
~
ieG
0Z,. = ~--,~ 1 i~G
where G is the set of all basic groups in the molecule. The first order connectivity indices ~Z and 1zv are defined across the bonds of the molecule, and thus describe the internal molecular structure: iZ = ~--~~ 1 i,j~BX/-~,6,
1zV=~ i /~B
1 (~i (~j
where i and j are the groups participating in one of the bonds B of the molecule. Higher order connectivity indices can also be defined, and have been used to give a more precise description of molecular structure (Kier and Hall, 1986). However, most of the key properties of pharmaceuticals can be correlated with these four molecular connectivity indices.
372 Once the equations defining the (molecular) connectivity indices are in place, we can use these indices in empirical correlations to predict the physical properties of novel soybean-oil derived compounds. For example, the correlation derived in this work for the critical micelle concentration of a soybean-oil derived surfactant is given as 100 log (CMC) = - - ~ t[_ 1.063 ~Z +0.327 ~Z ~ +1.390 1Z - 0 . 4 3 6 1Z" + 0.828] /If
Data for this and other correlations was collected from Rosen (1978), Schick (1967), Shinoda and Friberg (1986), Shinoda et. al. (1963), and Van Os et. al. (1993). Other properties such as the hydrophilic-lipophilic balance (HLB) can also be computed using structural descriptors of this type. Since connectivity indices are defined in a very general way, they are capable of describing any molecule, and thus correlations based on them tend to be widely applicable and fairly accurate over a wide range of compounds. Correlations can also be created combining the connectivity indices of two or more different molecules, leading to equations describing the properties of mixtures. Using such correlations, an optimization problem has been formulated which has as its optimal solution a molecule derived from soybean oil which most closely matches a set of target property values for a fuel additive. 3. P R O B L E M FORMULATION The optimization problem which determines the best molecule for a given application uses an objective function which minimizes the difference between the target property values and the estimated values of the candidate molecule. This can be written as min s = ~
liP"
m~R p~cale
_ ptarget
,
I
where R is the set of all targeted properties, Pm is the estimated value of property m, Pmscale is a scale factor used to weight the importance of one property relative to another, and Pm target is the target value for property m. The molecule is represented mathematically using two sets of binary variables: a partitioned adjacency matrix with elements a(i,j,k) which are one if basic groups i andj are bonded with a k-multiplicity bond, and zero otherwise. This matrix is partitioned such that specific rows are preassigned to different basic groups, so that it can be determined a p r i o r i what 6i and 6iv values should be used for each basic group i in the molecule. Since we do not know how many of each type of group will occur in the final optimal molecule, the partitioned adjacency matrix will have many rows which do not correspond to a basic group. The binary variable wi is set to one if the ith group in the adjacency matrix exists in the molecule, and is zero otherwise. These two sets of variables provide sufficient information to compute the connectivity indices, and thus estimate molecular properties. The definition of the zeroth-order connectivity indices can be rewritten using these variables to become N
.=
ozv=
N
w
373 where N is the total number of rows in the adjacency matrix. The equations for the first order connectivity indices are also linear in this formulation, since the 8i and 6iv values are constants for a given i. Those equations become
~ ~a0~ "=
"=
]=i+l
j=i+l
v
v
j
Next, property correlations using the connectivity indices are included. Finally, structural feasibility constraints are needed to ensure that a chemically feasible molecule is derived. The valence balance for both single and double bonds can be written as i=l
N
Zaijk + ~auk = vikwi ,V i= l,...,N,k= l,2 j=l
j=i+l
where vtk refers to the number of kth multiplicity bonds which can be formed by the basic group at row i in the partitioned adjacency matrix.. Each bond which forms must also be of a single multiplicity:
~-'aok < I , V i = I , . . . , N ,
j=i+l,...,N
k=l,2
In order to guarantee that all the groups in the molecule are bonded together as a single unit, we include the constraints from a network flow problem into the formulation. These constraints require that a path through the bonds of the molecule exist between one group, called the source, and all other existent groups, called the sinks. A feasible solution to a network flow problem across the bonds of a molecule is a necessary and sufficient condition for connectedness, and the constraints required are linear and introduce no new integer variables. Other constraints include bounds on the variables and property values. The problem written in this form is an MILP, which for smaller examples can be solved directly using commercially available software. 4. E X A M P L E The example presented here produces a possible molecular structure for a fuel additive which could be synthesized from linoleic acid, the major component of soybean oil. Fixing the lipophilic portion of the molecule, the program determines the optimal structure of the surfactant head group which will give an HLB value near 8 and a low CMC, which is related to the amount of fuel additive required to give a detergent effect. An HLB value near 8 gives a sufficient lipophilic character to dissolve in hydrocarbon fuels, and yet is high enough to give the compound the capability to solubilize organic deposits. The possible basic groups in the molecule are those listed in Table 1, and the maximum number of basic groups allowed in the molecule is 35. The problem was solved using the solver CPLEX accessed through the GAMS modeling language (Brooke, et. al., 1988) on a Sun Ultra l0 workstation. The optimal structure determined is shown in Figure l, and has a predicted HLB of 7.93 and a CMC in water of 1.5xl 0-l~ . /(C H2)2~ C H-- O C H ~"-- (C H2)~--- [C H2 C H = C H]2---(C H 2)7-~ N \ O ~ (C H 2 ) 5 ~ O H
Figure 1: Structure of optimal soybean oil derivative
374 5. CONCLUSION This work has focused on the use of optimization techniques within a molecular design application to derive novel molecular structures for fuel additives derived from soybean oil. The use of connectivity indices to relate internal molecular structure to physical properties of interest provides an efficient way to both estimate property values and recover a complete description of the new molecule after an optimization problem is solved. The optimization problem has been formulated as an MILP and solution times are relatively low even for large problems. Further work will include a much larger set of physical properties and basic groups from which to build molecules, and will work toward the design of mixtures and the prediction of mixture properties via connectivity indices. REFERENCES
1. Bicerano, J. (1996). Prediction of Polymer Properties. Marcel Dekker, New York. 2. Brooke, A., Kendrick, D., and Meeraus, A. (1988). GAMS: A User's Guide. Scientific Press, Palo Alto, CA. 3. Camarda, K. V. and Maranas, C. D. (1999). Optimization in Polymer Design using Connectivity Indices. Ind. Eng. Chem. Res., 38, 5, 1884-1892. 4. Erickson, D. R., Pryde, E. H., Brekke, E., Ordean, L.. Mounts, T. L. and Falb, R. A. (1992). Handbook of Soy Oil Processing and Utilization. American Soybean Association, St. Louis, MO. 5. Gani, R., Tzouvars, N., Rasmussen, P. and Fredenslund, A. (1989). Prediction of Gas Solubility and Vapor-Liquid Equilibria by Group Contribution. Fluid Phase Equilib., 47, 2, 133. 6. Glover, F. (1975). Improved Linear Integer Programming Formulations of Nonlinear Integer Problems. Manage. Sci., 22, 4, 455. 7. Hairston, D. W. (1998) New Molecules get on the Fast Track. Chem. Eng., Sept., 30-33. 8. Kier, L. B. and Hall, L. H. (1976). Molecular Connectivity in Chemistry and Drug Research. Academic Press, New York. 9. Maranas, C. D. (1996). Optimal Computer-aided Molecular Design: A Polymer Design Case Study. lnd. Eng. Chem. Res., 35, 3403. 10. Raman, V. S. and Maranas, C. D. (1998). Optimization in Product Design with Properties Correlated with Topological Indices. Comput. Chem. Eng. 22, 6, 747. 11. Rosen, M. J. (1978). Surfactants and Interracial Phenomena. John Wiley and Sons, New York 12. Schick, M. J. (1967). Nonionic Surfactants. Marcel Dekker, New York. 13. Shinoda, K. and Friberg, S. (1986). Emulsions and Solubilization. John Wiley and Sons, New York. 14. Shinoda, K., Nakagawa, T., Tamanushi, B. and Isemura, T. (1963). Colloidal Surfactants - Some Physicochemical Properties. Academic Press, New York. 15. Trinajstic, M. (1975). Chemical Graph Theory. CRC Press, Boca Raton, FL. 16. Van Os, N. M., Haak, J. R. and Rupert, L. A. M. (1993). Physico-Chemical Properties of Selected Anionic, Cationic and Nonionic Surfactants, Elsevier Science, Amsterdam. 17. Venkatasubramanian, V., Chan K., and Caruthers, J. M. (1994). Computer-Aided Molecular Design using Genetic Algorithms. Comp. Chem. Engng., 18, 9, 833-844.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
375
E C O F A C - Computer Aided Solvent Design and Evaluation in Environmental Problems, Based on Group Contribution Methods with Association. M.Cismondi and E.A.Brignole. Planta Piloto de Ingenieria Quimica-PLAPIQUI (UNS-CONICET), Camino La Carrindanga Km 7, 8000, Bahia Blanca, Argentina Computer aided product design requires the generation of feasible compounds and the prediction of solution and pure component properties by group contribution methods. The stability of the generated chemical structures is based on the electronegativity of the group attachments. A new characterization of group combination properties and the corresponding feasibility criteria for computer aided generation of branched structures are presented. A synthesis procedure and a strategy for reducing the size of the combinatorial synthesis problem are presented. Group contribution methods for property predictions of aqueous systems in environmental problems are reported. 1. INTRODUCTION The application of group contribution methods, for prediction of pure component and mixture properties, opens the way for efficient computer aided evaluation of chemical compounds. The backward solvent selection problem was formulated as follows: "giving a set of property constraints and certain performance indexes, generate chemical structures with the desired set of physico - chemical and environmental properties"(Gani and Brignole [ 1]). For the computer aided generation and evaluation of molecular structures a combinatorial - partition strategy was proposed by Brignole et al. [2]. Several molecular design procedures based on optimization algorithms have also been proposed [3-6]. Applications have been reported for the design of polymers [5], refrigerants [3,6], product substitution [7], solvents [8,9], etc. Even though the generated molecules satisfy the neutrality condition, the chemical stability of the components is in many cases not guaranteed [5,6]. This is partly due to the way groups are defined in different group contribution methods and/or to the lack of proper combination rules for the groups. Some of the component properties of interest, from the point of view of environmental considerations, depend on mixture or solution properties. The first solvent design studies were based on solution properties derived from the UNIFAC group contribution method for computing activity coefficients [10]. Several revisions and extensions to electrolytes, polymers and equations of state, of the original UNIFAC predictive package have been presented [11]; a group contribution equation of state (GC-EOS) based on similar but more detailed group definitions, has been extended to new groups and gases [12-14]. Different group definitions have been proposed [15] for prediction of pure compound
376 properties, such as heat capacities, solubility parameters, formation energies, etc. In general, specific criteria [6] to assure the component valence neutrality and the fulfillment of the octet rule are required. These criteria are combined with optimization algorithms for the synthesis of molecules. However the generation of chemically unstable compounds is not avoided. The generation of feasible compounds can be achieved if the electronegativity of the group bonds is taken into account [2,8,9]. Three basic types of group valences, with increasing degree of electronegativity, were identified [8] in aliphatic compounds: J, L and K,. The methyl group (-CH3), even though it is a J type group, is identified as type M because of the different role it plays in the feasibility criteria analysis, with respect to the other J groups. Pretel et al. [8] proposed synthesis procedures based on intermediate and terminal groups for the generation of linear (not branched) molecules. An extension of the synthesis procedure for aliphatic branched molecules is developed in the present work. 2.
SYNTHESIS OF BRANCHED MOLECULES The extension of Pretel et al. feasibility criteria to branched structures would be:
Z i i Ki <_M + J
(final structure)
with
J = J2 + J3 + J4
(1)
where ~ or Ji are the number of K or J groups with 'T' attachments in the structure and M is the number of methyl groups. However this criteria leads in many cases to unfeasible structures like: (HCOO)(CH)(CH3)(OH), for a molecule with a tertiary carbon (CH). 2.1 New group combination property characterization It can be seen from the previous example, that after a combination of two J groups, there are residual free attachments whose combination properties may be modified when linking with K groups. Therefore the formulation of robust feasibility criteria for the synthesis of the branched structures requires not only the characterization of the group free attachments but also of its internal bonds, as in the case of groups having L attachments. When the internal and free bonds are taken into account, only two bond status: K (electronegative) and J (neutral) are required to define the combination properties. For example groups with L attachments are formed by a combination of two "pure" K and J groups, as is shown in Table 1. Therefore, a revision of the UNIFAC groups combination properties is presented in Table 2. With the new group characterization it is possible to formulate more general feasibility criteria. 2.2 General feasibility criteria for the synthesis of linear or branched structures Considering that there are pure "K" and "J" groups, the new synthesis concept is: each J group cannot be attached to more than one K group. In other words, the building of feasible molecules requires the existence of a J-J type bond for each K group incorporated
377 Table 1: Redefinition of group combination properties in terms of J and K bond status. UNIFAC Group Previous Decomposed New combination Group example Valence Characterization in Sub-groups properties (CHRC1) 1 (L,1) J2 + K1 (K,1) (J,2) (CHC1) 2 (L,2) J3 + K1 (K,1) (J,3) (CCL) 3 (L,3) J4 + K1 (K,1) (J,4) (CHzCO) 2 (K,1) (L,1) J2 + K2 (K,2) (J,2) (CHNH) 3 (K,1) (L,2) J3 + K2 (K,2) 0,3) (CH2N) 3 (K,2) (L,1) J2 + K3 (K,3) (J,2) Table 2: Revision of the Combination Properties of UNIFAC Groups Combination Properties Groups (M,1) (CH3) ( J ,2) (CH2) (CH) (J,3) (C) ( J ,4) ( J ,2) (K,1) (CH2C1) (CH2NH2) (CH2CN) (J,3) (K,1) (CHC1) (CHNH2) (HCON(CH2)2) (CCl) ( J ,4) (K,1) (K,2) (J,2) (CH2CO) (CH2COO) (CH20) (CONHCH2) (CONCH3CH2) (FCH20) (C2H402) ( J ,3) (K,2) (CH-O) (CHNH) (CON(CH2)2) (K,3) (J,2) (CH2N) (K,1) (CH2=CH) (OH) (CHsCO) (CH3COO) (HCOO) (CH30) (C5H4N) (COOH) (CHC12) (CH2NO2) (I) (Br) CI-(C=C) (Sill3) (CC12F) (CC1F2) (C2H502) (CHsS) (CONH2) (CONHCH3) (CON(CH3)2) (K,2) (CH=CH) (CH2=C) (CH3N) (CC12) (CHNO2) (C=C) (SiH2) (SiH20) (C4H2S) (K,3) (CH=C) (Sill) (SiHO) (K,4) (C=C) (Si) (SiO) (I,1) (ACH) (ACF) (H,1) (ACCH3) (ACOH) (ACNH2) (ACNO2) (K,1) (H,1) (AC) (J,2) (K,1) (H,1) (ACCH2) (J,3) (K,1) (H,1) (ACCH)
(CH2SH) (CH2NH) (CH2S) (CHS) (CHO) (CH3NH) (CC13) (CH-C) (HCCIF) (C4H3S) (C5H3N) (COO)
(ACC1)
378 into the molecule (after the first one, for not cyclic structures. This synthesis concept can be formulated as follows: K _<'NJJ
(cyclic)
K - 1 <_NJJ
(non cyclic)
(2) (3)
Where NJJ = Number of J-J bonds. These conditions are valid for both intermediate and final structures. A total number of J attachments balance, could be obtained as follows" X.i i Ji = 2 N J J + N J F
K
wh e n
<
NJF
(4) or
~ri i Ji = 2 N J J + N J F + 2 (K-NJF)
when K > NJF,
(5)
where the number of J free attachments is given by: N J F = ,]3 + 2 J4 + 2 (non cyclic and J_> 1)
(6) or
N J F = J3 + 2 J4
(cyclic)
(7) Considering the attachment definitions and equations 2 to 7 and taking into account that NJF = 0 when J=0, the general feasibility criteria obtained are presented in Table 3. Table 3" Feasibility criteria for linear and cyclic branched structures K < NJF K > NJF Non cyclic structures K SJ 2 K S J + NJF Cyclic structures K SJ 2 K S J + NJF J=0 K
3.
PROPERTY PREDICTIONS
3.1. Pure component properties The pure compound physical properties prediction methods proposed in the literature, with a few exceptions [16,17] are not based on UNIFAC groups. Pretel et al. [16] revised the Lydersen method, using a data base of 700 compounds. The selected compounds were written in terms of conventional UNIFAC groups and correlations were derived for the prediction of densities, critical properties and boiling points. 3.2. Solution properties The analysis of solvents and products from an environmental point of view requires the knowledge or prediction of solution properties like solubility in water, Henry constants in aqueous solutions, octanol/water partition coefficients. Considering the particular behaviour of the interaction of acids, alcohols and other hydrogen bonding organic compounds in aqueous solutions, the thermodynamic modeling of these compounds
379 should take into account association effects. However it has not been the case in most previous studies of solvent molecular design. The present program incorporates the GCAEOS model [ 13-14], a group contribution with association equation of state, in addition to the UNIFAC method for computing activity coefficients. 4.
COMPUTER AIDED MOLECULAR SYNTHESIS
Using the feasibility criteria of Table 3 an efficient combinatorial synthesis of branched molecules is implemented on the basis of metha groups, i.e. groups with the same combination properties (first column of Table 2). In the synthesis of linear molecules the intermediate structures have two free attachments. But the number of free attachments (NFAs) in branched intermediate structures is larger:
NFA = 2 + N V 3 + 2NV4
(non cyclic)
(1 O)
(cyclic)
(11 )
or
NFA = NV3 + 2NV4,
where NV3 is the number of groups of valence three and NV4 of valence four. The larger number of free attachments of the intermediate structures, greatly increases the size of the synthesis problem in comparison to the synthesis of linear structures. To reduce the problem size, the ECOFAC computer program executes the following steps: 1. Definition of the desired product property constraints and performance index; 2. Selection of the set of intermediate and terminal groups in an interactive way; 3. Synthesis only of feasible metha- Intermediate Molecular Structures (IMS) with NFAs from 2 to 8 metha-groups that corresponds to the selected intermediate groups. A maximum number of 12 groups in Final Molecular Structures (FMS) is allowed. Then, each metha-IMS is replaced by all different possible combinations of the selected groups to form "true" IMSs. 4. Pre - FMSs are obtained by adding (NFA - 2) terminal groups to each IMS. 5. Screening of the pre-FMSs according to the physical property constraints. 6. Termination of SMSs by adding to each accepted IMS different combinations of two terminal groups that preserve their molecular feasibility. 7. Screening of the synthesized SMSs products according to the physical constraints. 8. The remaining products are ranked in accordance with molecular complexity and performance index, indicating the predicted product properties. In the case of solvent design it should be noted that between steps 3-4, 4-5 and 6-7 the synthesis procedure eliminates all intermediate and final structures with unknown binary interaction parameters, reducing in this way the size of the combinatorial problem and the computing time. In any case the synthesis program runs in a very short time in a standard PC. The synthesis procedure results are illustrated with an example of solvent design for the separation of benzene from hexane by liquid extraction, using only molecular weight and solvent loss constraints for the intermediate structures. In this example, 16 groups are selected for the synthesis procedure: intermediate (10) and terminal groups (6). A
380 summary of the results of the synthesis procedure is given in Table 4. The best solvents selected are triesters. In our laboratory we have confirmed the excellent properties of a triester (triacetin) for this application. The computer time required for running this example is less than one minute in a PC (Pentium MMX 166 MHz RAM 32 MB). Table 4: Solvent design for separation of benzene from hexane by liquid extraction 2344 Number of metha- intermediate structures generated 10552 Number of metha- pre final solventes 101934 Number of pre-final solvents 81303 - Pre-final solvents rejected by MW restriction 14475 - Pre-final solvents rejected by lack of binary parameters 4120 - Pre-final solvents rejected by solvent loss constraints 8823 Number of final solvents generated 277 Number of final solvents that satisfy all physical constraints Selectivity Distribution Coefficient Best Solvents 8.8 0.85 (CH3)(CH2)2(CH2COO)2(HCOO) 7.5 0.76 (CH3)3(CH2)2(C)(HCOO)3 REFERENCES
1. R.Gani and E.A.Brignole, Fluid Phase Equilibria 13 (1983) 331 2. E.A.Brignole, S.Bottini, R.Gani, Fluid Phase Equilibria 29 (1986) 125 3. K.G. Joback and G.Stephanopoulos "Designing molecules possessing desired physical property values" Proceedings FOCAPD'89, Snowmass, CO, 1989 4. O.Odele, S.Machietto, Fluid Phase Equilibria 82 (1993) 47 5. V.Venkatasubramanian, K.Chan, J.M.Carutheres, Computer Chem.Eng 18 (1994) 833 6. N.Churi, L.E.K.Achenie, Ind.Eng.Chem.Res. 35 (1996) 3788 7. P.M.Harper, R.Gani, P.Kolar, T.Ishikawa, Fluid Phase Equilibria, 158-160 (1999) 33 7 8. E.J.Pretel, P.Araya L6pez, S.B.Bottini, E.A.Brignole, AIChE Journal 40 (1994) 1349 9. R.Gani, B. Nielsen, Aa. Fredenslund, AIChE J. 37 (1991) 1318 10. Aa.Fredenslund, J.Gmehling and P.Rasmussen, "Vapor liquid equilibria using UNIFAC", Elsevier Scientific, Amsterdan, 1977 11. Aa.Fredenslund, J.Sorensen, Ch.4 , "Group Contribution Methods" in "Models for Thermodynamic and Phase Equilibria Calculations", editor S.I.Sandler, Marcel Dekker, Inc., New York, 1994 12 S. Skjold-Jorgensen, Ind.Eng.Chem.Res. 27 (1988) 110 13. H.P.Gros, S.Bottini, E.A.Brignole, Fluid Phase Equilibria 116 (1996) 537 14. S.Espinosa, G.Foco, A.Berm6dez, T.Fornari, Fluid Phase Equilibria 172 (2000) 129 15. R.C.Reid, J.M.Prausnitz, B.E.P61ing ,"The properties of gases and liquids", 4 th Ed. Graw Hill Inc., New York, 1987 16. E.Pretel, P.Lopez, A.Mengarelli, E.Brignole, Latin American Applied Research. 22 (1992) 187 17. L.Constantinou, R. Gani, AIChE J, 40 (1994)1697
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
381
Evolutionary Synthesis of Almost Closed Conservational Processes B. Csukas and S. Balogh Research Institute of Chemical and Process Engineering, University of Kaposvar Egyetem u. 2., 8200 Veszprem, Hungary. Email: [email protected]; [email protected] The synthesis of a "hypothetical" metabolic model is shown that follows the pattem of the citrate cycle (Szentgy6rgyi - Krebs cycle). This class of problems is characterized by the possible, altemative biochemical pathways, accompanied with incomplete, uncertain knowledge about the important parameters. The model is recognized and identified by a generic simulator collaborating with a genetic algorithm. The investigated problem is isomorphic with many synthesis tasks, where good solutions have to be selected from amongst the possible, altemative production routes, considering the prescribed rates of the input and output flows between the given system and the environment. Consequently, based on the experiences obtained from the behavior of the natural systems, the notion of the Almost Closed Conservational Processes (ACCP) has been introduced. 1. INTRODUCING THE ELABORATED METHODOLOGY
1.1 Direct mapping of structural models onto executable software The existing dynamic simulators are not prepared for the rapidly growing variety of the new processes, e.g. in the field of the forthcoming biological engineering [1,2]. The conventional tools need a detailed physical and chemical database, while these data are often not available, especially for the most important new materials and processes, as well as for the detailed biochemical process networks. On the other hand, it is not effective and straightforward to write new software according to any new need, individually. In the practical realization of Structural Modeling [3,4,5], the conservational processes are described by the dynamic database of the "passive" balance elements and "active" elementary transitions. The "passive" balance elements correspond to the finite amounts of the various conservational measures, while the "active" elementary transitions determine the related increases and decreases of the additive quantities during a time. The informational processes are defined by the dynamic databases of the "passive" signs and "active" rules. All of the structural models can be executed by the same general kernel program that supports the flexible and robust solution of many difficult problems.
1.2 Combining generic simulation with genetic algorithm in evolutionary problem solving Contrary to the basically open loop characteristics of the well known mathematical programming and heuristic methods, the synthesis (design, planning, scheduling, etc.)
382 problems can effectively be solved by means of the evaluation feedback between the detailed simulation model and a genetic algorithm that manipulates over the generic code [6]. In this collaboration, the simulator has to be able to generate and execute the dynamic model of the various systems from the genetic code, containing the description of the generic knowledge. The direct mapping of the conservation based structural models onto executable programs seems to satisfy this demand. On the other hand, the applied genetic algorithm has to be able to evolve according to multiple objectives from a relatively small population, using a limited number of evaluated variants. The method solves this by combining the Pareto Genetic Algorithm with the Deterministic Crowding and with the continuous consideration of every simulated variant. The schematic flow diagram of the algorithm is shown in Fig. 1.
Figure 1. Schematic flow diagram of the combined algorithm 2. E X A M P L E PROBLEM: E V O L U T I O N A R Y SYTHESIS OF A SIMPLIFIED METABOLIC MODEL 2.1 The biochemical process to be modeled
The described methodology has been tried for the solution of a simplified example, derived from a realistic problem. The real system is the citrate (SzentgyOrgyi- Krebs or TCA) cycle [7]. It is to be emphasized that the necessary parameters are not available. Nevertheless, it makes possible to demonstrate how the similar problems can be solved with the available incomplete and uncertain knowledge. It is an example for building a model from almost no quantitative data. The detailed conservational model is synthesized from the smallest possible elementary reactions and transport processes. It means that a usual bimolecular enzyme reaction consists of at least five equilibrium reactions supplied by the optional additional elementary processes describing the effect of the cofactors and effectors. The citrate cycle runs in the mitochondria, while there is component transport between the mitochondria and the cytoplasm. There are altogether 100 components in the mitochondria
383 and 24 components in the cytoplasm taken into account in the simplified model. The main classes of the components (i.e. the passive elements) are specific substrates of the citrate cycle, the general biochemical components, the acceptors, the enzymes, the other controlling components, and the various forms (associated compounds) of the enzymes (see illustrative examples in Table 1). The most important 63 elementary processes (i.e. the active elements) are 1~ 1, 1-->2, 2--->1 or 2-->2 equilibrium reactions, characterized by a kinetic parameter and an equilibrium constant. All of these elementary reactions are assumed to be either first or second order ones in both directions (see Table 2). In addition to the reactions, the component transport between the mitochondria and the cytosol is described for the consumed and produced components. In the simplest model, we assume a self-controlled component transfer that corresponds to the idealized state of a developed organism. TABI~ 1. Emnlles for the passived e t m ~
Name of conla(menls citrate ( ~ ) isocitrate ( ~ ) cis-ac~tate ( ~ )
T ~
Code SOT SICT SCAC
Code SL'IT1 SOq2 SOT3 etc. SOT4 acetyl-CoA( ~ ) SACD SOT5 CoA-SH(general~ ) OEDA SCIT6 NADH~ conpx~) ~ ..~r7 v~er~ ~ ) ~-120 SOq8 etc. SICT1 citrate ~ (~) ECTS SICq2 isocitm~ d h y ~ (~) HCD SICI3 g l ~ aehyaogemse(mzyn~) SICT4 i, etc. SICT5 i, ZCrS_SOXA( ~ ) F_s SOXA S(I~I zcrs SOXA_SACO( ~ ) F_ClS SOXA SAOO S ( / t 2 F__L-~ SCIT (KDA ~cE~._SCIT_(K3OA(ad&~) ||
ii
||
2. ~ e s
for the active domnls
Fxltfilibrimn~ c a l reaction F_LTS+ SOXA=FLTS SOXA FL-TSSOXA+ SAO:)= ECIS SOXA SAO:) ECTS SOXA SAO:)=ECIS SCIT (K3OA ECHS SOT OEOA =FL-qS SCIT+OEDA SCIT= EL'qS+ ~e~-~ ECTSSCIT+ SCIT= FLqS_SCIT_SCIT(-) Fs S~)(-) zcrs + C~TP- EC~ C~Te (9 EACH+~ = EACH_Cr~ (+) EACH (SEES+SOT=EACH CF[~ SCIT EACH ~ SCIT=EACH CFES ~ A C EACH CF~ ~ C =EACH C F ~ SICT EA(]-I (SEES SICT=EACH CFES+SICT EICD+C~TP=n~_GATP(-) EICD+~=n~~(+) EICD C A I ~ + ~ = E I C D C_.exY)P(3qAD
etc.
The amounts of the consumed and produced components are the best known data that can be used for the evaluation of the synthesized alternative models. There is a considerable gap between the necessary and available data. We would need all of the initial quantities and concentrations, as well as the kinetic and equilibrium parameters for every elementary process. In the contrary, even the quantitative data about the reference measures (e.g. the amount of mitochondria) are not known exactly. We know that the global functioning of the citrate cycle corresponds to the overall Equation of AcCoA + 3 NAD + + FAD + GDP + P~ + 2 H20 --> ---> HS-COA + 3 NADH + H § + FADH2 + GTP + 2 CO2
384 It is known that this reaction produces the majority of the metabolized carbon dioxide. Consequently, from the respiratory data and from the estimated amount of the mitochondria we can calculate an overall reaction rate that is approximately 3E-4 mmol/(g mitochondria * s). Surprisingly, or rather thanks to the excellent organization and to the built-in self-control of the natural processes, the first simulations were already quite feasible. Good examples for the self-control are the inhibition of citrate synthase by the produced citrate, or the inhibition of succinyl transferase by the succinyl CoA. Remembering the difficulties, accompanying the first trials of the "artificial" reacting system, the biochemical network looks another world, where the stiffness and the other kinds of problems are less dangerous. Of course, the principle of "garbage in, garbage out" is still valid, however here the "interpretable bad" results help to refine and to tune the model. The solution of the detailed conservational model is rendered by the paradox feature that the inhibitions and promotions are described by the most detailed model, while the available knowledge belongs to the simplified gross reactions and transportation. However, we can overcome these difficulties by the transformation of the conservational model into a combined conservational / informational one. The typical subsets of the elementary reactions, describing an enzymatic process can be replaced algorithmically for the conservational process of the gross reaction and for an associated rule that models the effect of the various cofactors and effectors on the reaction rate. This transformation helps to reduce the size of the problem, and the number of parameters to be identified, as well.
2.2 The possibility space, the fitness evaluation and the run of the genetic algorithm The possible discrete and continuous features of the investigated problem are contained in the so-called possibility space. It describes the classes of the alternative properties and property sets, as well as the optional forbidden combinations of the genetic properties. For the continuous parameters, the user must declare only the upper bound, the lower bound and the suggested initial value. The genetic code is the list of the ordinal number of the properties in the individual classes, supplemented by the actual proposed values of the continuous parameters. The work of the genetic algorithm is controlled by the size of the population, as well as by the applied reproduction, selection and mutation operators. The evaluation of the variants is based on minimizing of the differences between the prescribed and calculated consumption and production of the components, participating in the gross reaction. The prescribed values come from the overall reaction equation (see above). The simulated results are "measured" in the developed pseudo-steady state condition of the simulated model. During the evolutionary run the simulator does not prepare a detailed output, however the genetic algorithm reports about the individual and average fitness values of the subsequent populations. In Fig. 2 the change of the average values obtained for the most important objectives is illustrated, as the function of the population number. Fig. 3 shows the change in the error of the carbon dioxide production of the subsequent variants. 2.3 Simulation of a proposed good enough solution (testing of the results) In the structural model based simulation we can prepare a very detailed output that shows the change of each concentrations, as well as the rate of every elementary process.
385
Figure 2. Change of the average values
Figure 3. Error of CO2 production
A characteristic part of the simulated result can be seen in Figs. 4 and 5. The concentration trends of some major components (Fig. 4) show the evolution of the almost steady state values from the arbitrary initial conditions. The rates of the elementary reactions (Fig. 5) are in accordance with the overall functioning of the citrate cycle.
Figure 4. Concentration trends
Figure 5. Process rate trends
386 3. CONCLUSIONS 3.1 Recognition of models with the generic / genetic method of Structural Modeling The generic dynamic simulation of the structural models combined with a multicriteria genetic algorithm has been successfully applied for the synthesis of a detailed metabolic model from the available and very limited data set. 3.2 Lesson of nature for engineering synthesis: Almost Closed Conservational Processes The biological systems of various levels show many examples for the case where instead of the raw materials and goal products, the appropriate intermediate materials have the keynote role. The Almost Closed Conservational Process (ACCP) means a complex network of recycles that efrom a set of easily usable and easily recoverable intermediate materials, eproduces a set of goal products, ,while these goal materials or their final derivatives after the use can be transformed into the intermediate forms, .with the a given or minimum necessary environmental input and output.
The conscious design and control of the ACCP is fundamental in the future engineering, because the global natural system is an ACCP itself, with the solar energy as the single environmental input. The large scale and long-term character of ACCP synthesis needs new methodologies and the reconsideration of the existing goal functions. It is trivial in life sciences from agriculture to medicine that human being rather controls than designs the natural processes, however this way of thinking must be adopted by all kinds of engineering in the near future. It is worth mentioning that the described methods can be extended to the simulation of the artificial evolution in the closed space of finite sources, where the evolution is limited by the conservational processes themselves. These investigations may contribute to the development of the "engineering genetics" of the artificial systems that should replace for the "diagnosis and therapy" approach of the environmental control today. REFERENCES 1. G.N. Stephanopoulos, A.A. Aristidou and J. Nielsen, Metabolic Engineering, Academic Press, San Diego, 1998. 2. J. Yin, Chem. Eng. Progress, 95 (11), 1999, pp. 65-74 3. B. Csukfis, K. Varga, R. Lakner, Hung. J. Ind. Chem., 24 (2), 1996, pp. 107-130. 4. B. Csukfis, S. Perez Uriza, Hung. J. Ind. Chem., 23(4), 1995, pp. 277-287. 5. B. Csuk~ts, E. P6zna, Hung. J. Ind. Chem., 24(1), 1996, pp. 69-80. 6. B. Csuk~is, S. Balogh, Computers in Industry, 36, 1998, pp. 181-197. 7. G. Michal (ed.): Biochemical Pathways - an Atlas of Biochemistry and Molecular Biology. John Wiley & Sons, 1999, pp. 43-44.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
387
Analysis of azeotropic distillation columns combined with pervaporation m e m b r a n e s A.M. Eliceche a , P.M. Hoch a and I. Ortiz b PLAPIQUI - CONICET, Chem. Eng. Dept., Universidad Nacional del Sur, 8000 Bahia Blanca, Argentina #
a
b Dpto. de Quimica, Universidad de Cantabria, 39005 Santander, Espafia
The main objective of this work is the analysis of azeotropic distillation columns, when a liquid side stream with the distributing non-key component is treated in a pervaporation membrane and the retentate is recycled to the column. The objective is to separate the pure distributing non-key component from the pervaporation membrane, thus helping to improve the purity in the top and/or bottom products. The operating conditions of the column such as reflux ratio, product and side draw flowrate are selected optimally. A systematic methodology for the selection of the optimum operating conditions of the azeotropic distillation column in the hybrid distillation/pervaporation system for Methyl tert-Butyl Ether production is presented. The significant reduction in the operating cost due to the optimisation of the debutanizer column is reported. 1. I N T R O D U C T I O N Debottlenecking and azeotrope breaking are fruitful fields for hybrid membrane systems. Pervaporation is an interesting membrane separation alternative, because it is generally less energy-consuming than distillation. It is not influenced by the equilibrium between components, making azeotrope breaking easier than using a sequence of distillation columns. The separation is based on a selective transport through a dense layer associated with an evaporation of the permeants. This phase changing is usually obtained by lowering the partial pressure of the permeants at the downstream side of the membranes to vacuum pressure. Recent patents propose hybrid distillation/pervaporation technologies for azeotrope breaking processes involving the separation of alcohols and ethers (Chen et al.1). Hbmmerich and Rautenbach 2 have studied the integration of pervaporation and vapour permeation into the Huels process. They analysed the # This w o r k was carried out under research grant PICT 14-04065 from A N P C y T - Argentina.
388 influence of the operating conditions in a hybrid distillation-pervaporationvapour permeation system for the Methyl tert-Butyl Ether (MTBE) production. Gonzfilez and Ortiz 3 carried out experimental work and reported a rigorous model for the pervaporation membrane to separate methanol and MTBE. However, the formal optimisation of the debutanizer column with a pervaporation membrane to treat the side stream has not been a t t e m p t e d previously. The optimum operating conditions such as reflux ratio and product flow rates are calculated solving an optimisation problem to minimise the operating cost. 2. M O T I V A T I O N This work was motivated by the possibility of revamping the Methyl tert-butyl ether (MTBE) sector of a refinery. Methyl tert-butyl ether is used as a high octane fuel additive. The production process of Methyl tert-butyl ether consists of a reaction sector, where i-C4Hlo is combined with methanol to form the ether, and a separation sector where all the MTBE must be separated from unreacted methanol and C4's. Unreacted methanol forms azeotropic mixtures with MTBE and butanes. A sequence of azeotropic distillation columns was used to break the azeotropes, thus recovering MTBE and methanol to be recycled to the reactor. A hybrid distillation-pervaporation process seems an attractive alternative, as it combines the advantages of both methods. The use of hybrid systems can improve the cost of the traditional Huels separation sequence as reported recently (HSmmerich and Rautenbach2). 3. H Y B R I D D I S T I L L A T I O N / P E R V A P O R A T I O N P R O C E S S Different configuration for the hybrid distillation/pervaporation process can be used, locating the pervaporation membrane to treat the feed, products or side stream. In this work the pervaporation membrane is located to t r e a t the side stream and remove the distributing component as a permeate, helping to improve the top and bottom purity. The separation of pure MTBE as a bottom product of an azeotropic distillation column from a mixture of C4's, methanol and MTBE is performed by means of a combined distillation column and pervaporation membrane process. The process being studied is shown in Fig. 1. The azeotrope formation of methanol with both MTBE and C4 limits the purity of the products. A high purity of MTBE is required in the bottom product (B) of the column that separates a multicomponent mixture (F1) including MTBE, C4's and methanol. A side stream (E) of the column, rich in methanol, is processed through the membrane. The membrane selectivity allows the methanol in the column sidestream to be permeated and then condensed and recycled to the reactor (Permeate liquid stream), thus helping to improve the MTBE bottom product purity. The retentate is recycled to the column (F2).
389
Figure 1: Schematic flowsheet for the distillation/pervaporation process. The objective function to be minimised is the operating cost of the process (Co). It is a s s u m e d t h a t cooling water is used to condense the distillate of the column and medium pressure steam is used to heat the bottoms of the column 4. C 0 -- Ccolumn -+- Cmemb = ( C c -F C r + C p )
[$ /
h]
(1)
C~ = Co,~Q~ Cr = Co,rQr
(2)
Cp -- C o,pQp where Qc is the heat [kJ/h] withdrew from the condenser of the column, Qr [kJ/h] the heat added to the reboiler of the column, and Q, [kJ/h] the heat withdrew in the pervaporation unit to condense the methanol. For the operation of this condenser a refrigerant fluid is needed, because the methanol condenses at very low t e m p e r a t u r e s (-5 ~ at the permeate pressure, usually 0.02 bar). The refrigeration cost to condense the permeate is a function of this temperature. 4. CASE S T U D Y The composition, temperature and pressure of the fresh feed are shown in Table 1, where C4's stands for a mixture of i-butane, n-butane, i-butene, 1butene, 2-butene trans and 2-butene cis. The true composition of C4's was used in the simulation.
390 Table 1" Feed to the column
Component
Flow rate 418.973 26.084 92.923 537.98
C4's Methanol MTBE F1 [kgmol/h] Temperature [K] Pressure [bar]
351.15 6.00
The purity and recovery of MTBE in the bottom product required is 0.99. The process was simulated with HYSYS ~. The membrane module is modelled with a splitter unit followed by heat exchangers to simulate the process of expansion and condensation of the permeate. The pervaporation takes place when a vacuum of 15 torr is maintained at the permeate side. For the start-up a vacuum pump is needed, but when steady state is reached, vacuum is maintained by the permeate condensation. The distillation column has 22 theoretical stages plus a reboiler (stage 0) and a condenser (stage 23). The HYSYS 5 selected optimisation option was the Mixed method. It starts the optimisation with the Box method using a very loose convergence. After convergence, an SQP method is then used to locate the final solution using the desired tolerance. The optimisation results are shown in table 2. A significant cost reduction of 36 % is achieved optimising the operating variables of the debutanizer column. The reflux has been reduced by 40 % at the solution point compared to its value at the initial point. Side s t r e a m E is withdrew from the plate were the liquid composition of methanol reaches its m a x i m u m value. The original locations of the fresh feed, side stream extraction and r e t e n t a t e recycle plates were modified to improve the objective function, while m a i n t a i n i n g the total number of stages fixed. There are important decisions to be made regarding the location of the fresh feed, side stream and recycle of the r e t e n t a t e stream. The systematic selection of the number of plates requires the incorporation of integer optimisation variables. The procedure developed by Hoch and Eliceche 6 treat the number of stages as continuous optimisation variables, allowing the simultaneous selection of feed and side stream locations with the flowrates in a nonlinear programming problem formulation. The implementation of this option is currently being studied.
391 Table 2: Optimisation results for the example shown. Initial point Optimum point R 2.14 1.2704 B 107.6 92.92 E 132.5 126.9 NF1 12 20 NF2 10 14 NE 18 16 X MTBE,B 0.8638 0.9907 0.9970 0.9900 Rec MTBE,B C [S/h] 69.4506 44.123 Cost r e d u c t i o n 36 %
The optimisation of the operating conditions for the hybrid distillation/pervaporation system using a rigorous model of the pervaporation membrane, following the work of Gonzalez and Ortiz 3 will be implemented. The operating conditions of the distillation column and the pervaporation membrane would then be chosen simultaneously. 5. CONCLUSIONS A systematic methodology for the selection of the optimum operating conditions of hybrid distillation/pervaporation system has been presented. The numerical robustness to find the solution of the simulation of the separation process has been greatly improved with the incorporation of the pervaporation membrane, with respect to the simulation of the distillation column on its own. This improvement allowed a successful implementation of the selection of the optimum operating conditions. Numerical results are reported for the hybrid configuration analysed for MTBE production. An important reduction of 36 % in the operating cost of the distillation/pervaporation process has been achieved, quantifying the improvement that can be expected if a systematic optimisation is carried out. The main improvement that can be expected by optimising the operating conditions of the debutanizer column are shown. Similar results can be expected in other applications of hybrid distillation/pervaporation systems. LIST OF SYMBOLS B Cc
Cp Cr
CO,C Co,p
Bottom flow rate, [kgmol/h] Condenser operating cost [S/h] Pervaporation membrane operating cost [S/h] Reboiler operating cost [S/h] Condenser operating cost coefficient [$/kJ] Pervaporation membrane operating cost coefficient [$/kJ]
392 Co,r D E F1 F2 NE NF1 NF2
Qc QR QP R Rec MTBE,B X MTBE,B
Reboiler operating cost coefficient [$/kJ] Distillate flow rate [kgmol/h] Sidestream [kgmoYh] Fresh feed flow rate [kgmol/h] Retentate flow rate [kgmol/h] Location of the side draw Location of fresh feed Location of the recycle stream (retentate) Heat withdrew from the condenser of the column [kJ/h] Heat added to the reboiler of the column [kJ/h] Refrigeration needed for the pervaporation module [kJ/h] Reflux ratio MTBE recovery in the bottom product Liquid composition of MTBE in the bottom product
REFERENCES
1 - Chen, M.S., Zionsville R. and J.L. Allentown, U.S. Patent 4774365, 1988. 2.- HSmmerich U. and R. Rautenbach. Design and optimization of combined pervaporation/distillation processes for the production of MTBE, J. Membr. Sci., 146, 53-64, 1998. 3- Gonz~lez B. and I. Ortiz. Mathematical modelling of the pervaporative separation of Methanol - Methyl tert-Butyl Ether mixtures, submitted to Ind. & Eng. Chem. Res, 2000. 4- Seider, Seader and Lewin. Chapter 10: Profitability Analysis, in Process Design Principles, John Wiley and Sons (ed.), 1999. 5 - Hyprotech, HYSYS user manual, 1999. 6- Hoch P.M. and A.M. Eliceche, "Optimal Design of non-conventional distillation columns", Process Technology Proceedings Vol. 10, Computer Oriented Process Engineering, p 369-374, 1991.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
393
Nonlinear Bilevel Programming: A Deterministic Global Optimization Framework Z. H. Gtimti~ and C. A. Floudas* Department of Chemical Engineering, Princeton University, Princeton, NJ 08544-5263, USA A novel technique that addresses the global optimization of the general nonlinear bilevel programming problem is presented. The technique is based on a relaxation of the feasible region and a branch and bound framework, utilizing the basic principles of the deterministic global optimization algorithm, ~BB [1-4]. For problems that involve twice differentiable nonlinear functions and inner problem constraints that satisfy the linear independence condition, epsilon global optimality in a finite number of iterations is theoretically guaranteed. 1. INTRODUCTION The bilevel programming problem, BLPP, is an optimization problem that is constrained by another optimization problem. This mathematical programming model arises when two independent decision makers, ordered within a hierarchical structure, have conflicting objectives. The decision maker at the lower level has to optimize her objective f(x,y) under the given parameters x from the upper level decision maker, who, in return, with complete information on the possible reactions of the lower, selects the parameters so as to optimize her own objective F (x, y). Note that the upper level decision maker is limited to influencing, rather than controlling, the lower levels outcome. Thus, the general BLPP is formulated as follows: min
F(x,y)
(1)
x
s.t.G(x,y) l-l(x,y)
< =
0 0
min f(x, y) y
s.t.g(x,y) h(x,y) xEX
___ 0 : 0
c R nl , y E Y
c R n2
where f ,F " RnlxR n2 --+ R, g - [gl,..,gj] " RnlxR n2 -+ R J, G - [ G 1 , . . , Gj,] " RnlxR n2 -+ R J', h - [hi, ..,hi]" R n l x R n2 --+ R I, H - [H1,..,HI, ] 9R n l x R n2 --+ R I' . The BLPP model has been employed in many and diverse areas that require hierarchical decision making, including centralized economic planning problems [5], civil engineering transportation network design problems [6], and chemical engineering problems such as chemical process design with equilibrium [7-9], plant design under uncertainty [10], flexibility analysis 9Author to whom all correspondence should be addressed.
394 [11] and process design with controllability issues [12]. Given that the BLPP applications are many and diverse, effective solution algorithms are of critical importance. The linear BLPP has the favorable property that the solution is at an extreme point of the feasible set, and can be exploited by enumeration techniques. However, this does not hold for the nonlinear BLPE The conventional solution approach to the nonlinear BLPP is to transform the original two level problem into a single level one by replacing the lower level optimization problem with the set of equations that define its Karush-Kuhn-Tucker, KKT, optimality conditions. However, the KKT optimality conditions are necessary and sufficient for defining the optimum of the inner level problem only when convexity conditions and a first order constraint qualification are satisfied. When the inner problem constraints are nonconvex, the KKT conditions are only necessary, and local or suboptimal solutions may be obtained. A further difficulty arises in locating the global optimum of the resulting single level problem after the KKT transformation. The bilinear nature of complementarity conditions introduces nonconvexities even if the original problem is linear. Furthermore, when the inner problem is nonlinear, the equations that define the stationarity constraints are also nonconvex. Hence, even if the KKT conditions are necessary and sufficient for the inner problem, the global optimality of the transformed single level problem can not be guaranteed unless a global optimization algorithm is introduced. These difficulties related with the KKT-type solution approaches, which are the most efficient and widely used methods for the solution of the BLPP, confine them to the identification of only local solutions when nonlinearities are involved.
2. THEORY 2.1. KKT Optimality Conditions The KKT optimality conditions are equivalent to the inner optimization problem assuming that f, h, and g are smooth, f and g are convex, h is linear in y at fixed x for every x E X, and one of the first-order constraint qualifications, such as linear independence, Slater, KuhnTucker or weak reverse convex condition, holds in terms of x at a feasible point y*. The bilevel programming problem is transformed into a single level problem of the form: min xy
s.t.
(2)
F(x, y) G(x, y) _< 0
H(x, y) _< 0
hi(x, y ) - 0 i C I, af(x'Y) + ~x'JagJ ~=1 ahi a--V- j=, gj(x, y) + sj = O, j E J, = o, j c J,
~j, sj >_O, j E J, xEX,yCY.
(s)
(cs) (cs)
Note that the resulting single problem (2) is nonlinear and nonconvex due to the stationarity (s) and complementarity conditions (cs). If the original bilevel problem is linear, the complementarity conditions are the only nonlinearities in the single level transformed problem.
395 For the convex form of (1), solution methods in the literature generally require the following conditions at fixed x [13,7,14]: (a) f , g, h are continuous and twice differentiable functions in (x, y); (b) the linear independence condition holds at y c Y, such that the gradients of the inner problem equality and active inequality constraints, Vxgj(x,y), Vj C Ja, Vxhi(x,y) Vi C I, are independent; (c) strict complementarity condition holds at y C Y; and (d) the second order sufficiency condition holds at y C Y. Under the assumptions (a)-(d) on the functions in (1), the inducible region, IR defined by the set {(x,y) : (x,y) E ~ , y E RR(x)}. is continuous [15]. Assumptions (b) and (d) assure that the global optimum is also unique. Further, the KKT conditions are necessary and sufficient. However, the resulting single level problem is nonconvex due to the complementarity and the stationarity conditions. Optimization methods for the general nonlinear BLPP include the relaxation and active set strategy techniques [7], which also assume the conditions (a)-(d). Note that the KKT conditions can no longer guarantee global optimality of the inner problem for fixed x. This means that, even if the transformed problem is solved by a global optimization approach, global optimality of the transformed single level problem can not be guaranteed. Hence, methods for the solution of the general nonlinear BLPP that are based on the KKT optimality conditions are bound to be local. The following section presents the main concepts that we have used in order to overcome the limitations of KKT-type methods. 3. C O N C E P T U A L F R A M E W O R K To assure that KKT optimality conditions are both necessary and sufficient for obtaining the global optimum of the inner problem, the functions f and g must be convex and h must be linear at fixed x. Condition 1: If for fixed x, assumptions (a)-(d) hold, f and g are convex and h are linear in y, then the KKT optimality conditions are necessary and sufficient for obtaining the global optimum of the inner problem. If condition I does not hold, then KKT conditions are only necessary. Thus, when the nonconvex inner problem is replaced with its KKT optimality conditions, and the resulting single level problem is solved to local optimality, an upper bound on the global optimum of the BLPP is obtained, provided that the linear independence condition holds. 3.1. Underestimation for the BLPP A lower bound to the global optimum of the BLPP can be found as follows: the feasible region, ~ defined by the set {(x,y): G(x,y) _< 0 , H ( x , y ) = 0,g(x,y) _< 0, h(x, y) -- 0}, can be enlarged in such a way that the infeasible points within the convex hull are included into the feasible set. This can be done by utilizing the basic principles of the deterministic global optimization algorithm ~BB [1-4], to underestimate the nonconvex functions over the (x, y) domain (see [ 16]). Based on the underestimation of every term, a convex underestimator for any given twicedifferentiable function can be obtained through a decomposition approach. For the nonlinear functions, valid underestimators are generated by the decomposition of each nonlinear function into a sum of terms belonging to one of several categories: linear, bilinear, trilinear, fractional, fractional trilinear, convex, univariate concave, product of univariate concave or general nonconvex. After the terms are identified, a different convex underestimator is constructed for each
396 class of term, and a lower bounding function is obtained. See [16] for rigorous calculation methods of underestimators. The equality constraints of the inner problem must be linear for the KKT conditions to be necessary and sufficient. The bilinear, trilinear, fractional and fractional trilinear terms are replaced by new variables that are defined by the introduction of additional convex inequality constraints. Thus, if the equality constraint involves only this kind of variables, the resulting problem is linear. If this is not the case, and convex, univariate concave, or general nonconvex terms exist, the constraint is simply eliminated by a transformation into two inequality constraints: h(x, y) <_ 0 and - h ( x , y) <_ 0, which are added to the set of inequality constraints and convexifed individually as described above. Note that the KKT optimality conditions of the inner convexified problem are necessary and sufficient, and define its global optimum, provided that the linear independence condition holds.
3.1.1. Linear Independence Note that in order to replace the convexified inner problem with its equivalent KKT optimality conditions, a first order constraint qualification such as the linear independence condition of the inner problem constraints at the optima must be satisfied. Otherwise, the transformed single level problem may be infeasible or it can not be guaranteed that it is a lower bound. A simple linear independence check can be made by testing whether the best (x*, y*) values obtained from the solution of the original nonconvex upper bounding problem result in linearly independent active constraints for the convexified problem. After the KKT transformation of the convexified inner problem, the resulting single level problem is still nonlinear and nonconvex due to the complementarity (cs) and stationarity (s) conditions, and nonconvexities in G, H, E The complementarity conditions are transformed into a set of MI(N)LP equations as described below. The resulting MI(N)LP problem is solved to global optimality by using one of the deterministic global optimization algorithms SMIN-ctBB or GMIN-ctBB [ 17,18]. The solution is a lower bound on the original BLPP minimum.
3.1.2. Complementarity Conditions The complementarity condition constraints are one of the major difficulties in solving the transformed single level problem. They involve discrete decisions on the choice of the set of inner problem active constraints. The active set changes when at least one inequality function and its multiplier are equal to zero. With the change in the active set of constraints, the feasible space of the inner problem, at fixed x, also changes. Furthermore, the overall feasible space changes as it is composed of different regions that correspond to different active sets. To overcome this difficulty, the ideas of active set strategy [ 11] can be employed, so as to reformulate the complementarity conditions. In this case a binary variable, Yj, is introduced and associated with each inequality constraint, j E J, depending on whether it is active or inactive. Thus, the inner problem feasible regions belonging to different active sets can be visited simultaneously as the outer problem decision vector x changes.
3.2. Global Optimization of MINLPs The Special structure Mixed Integer Nonlinear ctBB, SMIN-t~BB [17,18] algorithm is a deterministic global optimization method based on a branch-and-bound framework. Epsilon convergence is guaranteed for convergence to the global minimum of problems that involve separable MINLP functions that are twice-differentiable in continuous variables. A valid upper
397 bound on the global solution is obtained by solving the nonconvex MINLP to local optimality. A lower bound is determined by solving a valid convex MINLP underestimation of the original problem. Convergence is obtained by the refinement of the feasible space into smaller regions, in which convex underestimators are generated [17,18]. The General structure Mixed Integer Nonlinear o~BB, GMIN-o~BB algorithm, on the other hand, is also based on a branch and bound framework, but is further applicable to problems without the restriction of the separability of the integer variables.
3.3. Branching and Bounding After upper and lower bounds on the original BLPP problem are obtained, the initial region of (x, y) is partitioned into smaller regions, in the following way: Tighter lower bounds to the problem are obtained by dividing the initial feasible region into two subregions by using one of the branching rules that are developed within the deterministic global optimization algorithm, o~BB [1-4]. After branching, minimization is performed in each subregion. The smallest minimum for all subregions of the original feasible region is the overall lower bound. At the next iteration, only the subrectangle responsible for the overall minimum is further branched on. Hence, a nondecreasing sequence of lower bounds is produced. A non-increasing sequence of upper bounds is created by locally solving the original nonconvex bilevel problem in each subregion, where the best upper bound is the minimum of all upper bounds calculated in previous iterations. The upper and lower bounds bracket the global minimum. The branch and bound framework also includes a fathoming step, in which any subregion with a lower bound higher than the current upper bound is removed from further consideration. The steps of the proposed global optimization framework for the nonlinear bilevel optmization problem are presented in the following section.
4. GLOBAL OPTIMIZATION ALGORITHM Step 1: Set the lower bound, z L~ = - ~ , upper bound, z UP = co, iteration counter, k = 1, and select a convergence tolerance ~. Step 2: Substitute the original inner optimization problem with its KKT optimality conditions, (KKT), employ the active set strategy for the complementarity conditions of problem (2), and solve the resulting single stage MI(N)LP optimization problem [11 ]. Step 3: Solve the resulting problem by using a local MINLP optimizer, such as MINOPT [ 19], which will yield the upper bound, z UP. Step 4: Develop convex underestimators of the nonlinear terms in the functions g(x, y) (these include the transformed equalities) and f(x,y), using the basic principles of the deterministic global optimization algorithm, o~BB [1-4]. Denote the underestimated functions as fC(x, y), gC(x, y). Step 5: Establish tight upper and lower bounds on inner variables that participate in nonconvex terms that are underestimated [20]. Add the simple bounds thus obtained to the set of constraints
[20]. Step 6: Substitute the KKT optimality conditions that are necessary and sufficient for the solution of the convexified problem, and employ the active set strategy. The result is an MI(N)LP
398 problem of the form: z-
min F(x,y) xy
s.t. G(x,y) _< 0 H(x,y) - 0
hl(x,y ) - 0 i ~ I , Of C(x' Y)
~)--------~ +
J
I l v-~ c t)hi (x, y)
c
V )~.ci)gj(x'y) z._~" -j 9~
j=l /)fC(x,y)
~y
~
--
OY
+ t , ldi i=1
~
~ O, a
~)gj (x, y) I /)h~(x, y) < O, Oy - E P c /)--------~ -
j=l
i=1
gjC(x,y) + sjC _< 0 j E J,
- g j ( x , y ) - s~ < 0 j E J,
s
-vrf < o jcJ, -v (1 - ( ) < o j
~,
sj_c>o, j c J ,
j,
x E X,y E Y,Y C {0,1}, where Sjc are the slack variables associated with the inner problem convexified inequalities, Ljc are the associated Lagrange multipliers, and Yf are the binary variables involved in the active set strategy. Step 7: Check if the linear independence condition is satisfied at the best upper bound value obtained. If not, terminate. If satsified, continue to Step 8. Step 8: Solve the resulting MINLP problem to global optimality by the SMIN-t~BB or GMIN~BB algorithms [ 17,18]. If the optimum outer objective function, z*, is higher than the current lower bound, update z ~'~ - z*. Step 9: If z UP - z L~ _< e, stop. The global optimum is obtained. Else, go to next step. Step 10: Branch on a selected variable that participates in one of the nonlinear terms. The branching stragegy has a significant effect on the performance of the algorithm, and the options include seven alternative branching strategies implemented within the ctBB global optimization algorithm, (see [2,3]). Return to Step 2. 5. COMPUTATIONAL STUDY Consider the following simple fractional BLPP with linear constraints [21 ]:
min--8Xl - x
4 X 2 -Jr- 4 y l - -
40y2 -- 4y3
1 + X l + x2 + 2yl - Y2 + Y3 s.t. min y 6 + 2Xl + Yl + Y2 - 3y3 s.t. - Y l + Y 2 + Y a + Y 4 - - 1 2Xl -- Yl + 2y2 -- 1/2y3 + Y5 -- 1 2x2 + 2yl -- Y2 -- 1/2y3 + Y6 -- 1 Xl,X2,Yi >_ 0, Vi = 1,..,6.
399 The problem can be put into a more tractable form by rearranging the inner objective fractional term through the introduction of new variables Wl, w2 and substituting: WlW2 - (1 + Xl + x2 + 2yl - y2 + y3), and w2 -- 6 + 2Xl + yl + y2 - 3y5. The resulting inner problem has a linear objective wl, and two additional constraints that define Wl and w2. Note that the WlW2 term is bilinear, hence the inner problem is a nonconvex BLPP. The bilinear term can be underestimated by introducing a variable co that replaces every occurrence of WlW2 in the problem and satisfies the constraints that define its convex envelope [3]. The resulting inner problem is convex. The bounds on w3 are estimated as described in Step 5.Replacing with its corresponding KKT conditions that are necessary and sufficient, and introducing a binary variable Y j for every inner constraint j, the transformed single level problem which is a lower bounding MILP 9 Y4, , 9 Y6, , W 1 , w2, W 3 ) = problem that is solved to global optimality at (x~, x~, y~, y~, Y3, Y5, (0.0, 0.9, 0.0, 0.6, 0.4, 0.0, 0.0, 0.0, 0.1125, 5.4, 0.9), F/~8 - - - 2 9 . 2 . The upper bounding problem is obtained by replacing the inner level problem with its KKT optimality conditions without underestimation. Solving the MINLP to local optimality, F U8 - - 2 9 . 2 - F/~8 = F*, and the algorithm terminates. 6. CONCLUSIONS A global optimization algorithm for the solution of the general nonlinear bilevel programming problem that involves twice differentible functions is presented. The approach is based on a relaxation and branch and bound framework. The relaxation is accomplished by an enlargement of the feasible solution space of the bilevel problem. The resulting relaxed optimization problem is solved to global optimality by using the deterministic global optimization algorithm, ctBB. Consequently, a lower bound is obtained. An upper bound to the global minimum is obtained by transforming the original problem into a single level one without the relaxation and solving for local optimality. After upper and lower bounds on the global solution are obtained, the initial region of the problem variables is partitioned into smaller regions by using one of the branching rules that are developed within the deterministic global optimization algorithm, o~BB. REFERENCES
1. C. S. Adjiman, C. A. Floudas, Rigorous convex underestimators for general twicedifferentiable problems, Journal of Global Optimization 9 (1996) 23-40. 2. C.S. Adjiman, S. Dallwig, C. A. Floudas, A. Neumaier, A global optimization method, (xBB, for general twice-differentiable constrained nips I. theoretical advances, Comp. Chem. Engng. 22 (9) (1998a) 1137-1158. 3. C.S. Adjiman, I. P. Androulakis, C. A. Floudas, A global optimization method, txBB, for general twice-differentiable constrained nips II. implementation and computational results, Comp. Chem. Engng. 22 (9) (1998b) 1159-1179. 4. I.P. Androulakis, C. D. Maranas, C. A. Floudas, o~BB: A global optimization method for general constrained nonconvex problems, Journal of Global Optimization 7 (1995) 337363. 5. B. E Hobbs, S. K. Nelson, A nonlinear bilevel model for analysis of electric utility demandside planning issues, Annals of Operations Research 34 (1992) 255-274.
400 6. H. Yang, S. Yagar, Traffic assignment and signal control in saturated road networks, Trans. Res. A. 29A (2) (1995) 125-139. 7. P.A. Clark, A. W. Westerberg, Bilevel programming for steady-state chemical process design-I, fundamentals and algorithms, Comp. Chem. Engng. 14 (1) (1990) 87-97. 8. P.A. Clark, Bilevel programming for steady-state chemical process design-II, performance study for nondegenerate designs, Comp. Chem. Engng. 14 (1) (1990) 99-109. 9. Z.H. Gtimtis, A. R. Ciric, Reactive distillation column design with vapor/liquid/liquid equilibria, Comp. Chem. Engng. 21(S) (1997) $983-$988. 10. M. G. Ierapetritou, E. N. Pistikopoulos, Batch plant design and operations under uncertainty, I&EC Res. 35 (1996) 772-787. 11. I. E. Grossmann, C. A. Floudas, Active constraint strategy for flexibility analysis in chemical processes, Comp. Chem. Engng. 11 (6) (1987) 675-693. 12. D. D. Brengel, W. Seider, Coordinated design and control optimization of nonlinear processes, Comp. Chem. Engng. 16 (1992) 861-886. 13. A. V. Fiacco, Sensitivity analysis for nonlinear programming using penalty methods, Maths. Prog. 10 (1976) 287-311. 14. J. E. Falk, J. Liu, On bilevel programming. 1: general nonlinear cases, Mathematical Programming 70 (1995) 47-72. 15. K. Shimizu, Y. Ishizuka, J. E Bard, Nondifferentiable and Two-Level Mathematical Programming, Kluver Academic Publishers, 1997. 16. C. A. Floudas, Deterministic Global Optimization: Theory, Methods and Application, Vol. 37 of Nonconvex Optimization and Its Applications, Kluwer Academic Publishers, The Netherlands, 2000. 17. C. S. Adjiman, I. P. Androulakis, C. A. Floudas, Global optimization of mixed-integer nonlinear problems, AIChE J. 46 (9) (2000) 1769-1797. 18. C. S. Adjiman, I. P. Androulakis, C. A. Floudas, Global optimization of minlp problems in process synthesis, Comp. Chem. Engng. 21 (1997) $445-$450. 19. C. S. Schweiger, C. A. Floudas, MINOPT: A modeling language and algorithmic framework for linear, mixed-integer, nonlinear, dynamic and mixed-integer nonlinear optimization, Vol. Version 3.1, Princeton University, 1998. 20. C. D. Maranas, C. A. Floudas, A global optimization method for Weber's problem with attraction and repulsion, in: W. W. Hager, D. W. Hearn, P. M. Pardalos (Eds.), Large Scale Optimization: State of the Art, Kluwer Academic Publishers, 1994, p. 259. 21. H. J. Calvete, C. Gale, The bilevel linear/linear fractional programming problem, European Journal of Operational Research 114 (1999) 188-197.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
401
Systematic Methodologies for Chemical Reaction Analysis M. Hostrup a and S. Balakrishna b'l aDepartment of Chemical Engineering, Technical University of Denmark, Building DK-2800 Lyngby, Denmark
228,
bMC Research & Innovation Center Inc, 444 Castro Street #505, Mountain View, CA 94041, USA Efforts in new process development are often centered on the chemical reaction system. The development of an efficient experimental plan and an optimum condition recipe is the goal. In order to facilitate this, we have developed a systematic framework for reasoning and analyzing reaction systems and the available experimental data. This has increased R&D productivity by better targeting the research, through concurrent modeling and experimentation. Following the experimentation and development of an adequate reaction model, the system also provides tools for simulation and reactor optimization. The framework contains two components, the Knowledge Asset or the Reaction Database, and the Analysis Toolkit, which facilitates analysis of the reaction system. The Reaction Database includes a rich representation of reaction chemistry and kinetics in a relational database. The Analysis toolkit contains four modules. They include the Reaction Browser, Kinetic Estimator, Virtual experimenter, and Reactor Optimizer. The design of the software closely mimics the R&D workflow, therefore this tool has become part of the research lab infrastructure and has resulted in productivity gains in the experimentation. Furthermore, reactor optimization has led to improvements in the maximum yields for several processes. 1. INTRODUCTION In this paper, we describe software developed to aid the chemist in new process development. Consider the workflow in new process development as shown in figure 1. A chemist has found a candidate reaction route to a target compound. On starting the experimentation, he discovers several undesired events such as catalyst deactivation, parallel or successive byproducts etc. In order to proceed now with the experimental plan, he must reconcile his observations with his hypothesis of the reaction mechanism. The Kinetic Estimation component of this software provides the logical and computational framework to easily represent and solve this problem. Here, given the current experimental data, one can quickly and conveniently reason with several reaction hypotheses. The most likely reaction mechanism and kinetics is then used to develop the next experimental plan. The next experiment can be designed at the point of maximum productivity (conditions at which the model predicts the best yield) or at points for model discrimination. The cycle of experimentation and model-based reasoning continues until the chemist is satisfied that the 1Corresponding author, email: [email protected]
402 observations support the chemical reaction hypothesis. At this point we consider the model to be validated. Which Reaction Set gi~,es R e ~ e t i ~ Bro Best Yield for target ? (Search) Search for Kinetics, Pathways...
Given Expts and Chemistry, Find Best Fit Reaction
<5 Kinetic Estimation
Model
Validate Model and Best Design for Next Experimenl
Find Optimum Conditions fi~r Maximunl Yield.
Reaction Kinetics Database Knowledge Asset
Virtual Expts
Knowledge Asset Reactor Optimization
T a s k s utilizing Asset WorkFlow
<~y
Figure 1" Software Architecture and R&D WorkFlow Following model validation, when the reaction mechanism and kinetics are fixed, the chemist may now want to look at optimal operating policy to maximize yield or some performance index. The logical and computational framework for this is provided in the Virtual Experimentation or Simulation component and the Reactor Optimization Component. Information transfer from the lab data to the final optimization is seamless reducing repeated entry and minimizing inconsistencies. Furthermore, as the models are validated, they can be stored in the knowledge base to be used by future researchers. The software has been designed to mimic the research workflow. In the following sections, we explain each of these components in detail. 2. R E A C T I O N K N O W L E D G E BASE
The reaction knowledge base is an electronic reference for the collection of reactions stored in it. In designing the database, we were careful to allow for general kinetic expressions, which may be either empirical or derived from fundamental molecular or microkinetic considerations. This is important in an industrial setting where the complexity of the model is dictated by the objectives of the project at hand. The knowledge representation is based on fundamental and derived information as shown in figure 2. The fundamental objects are the Compounds, Reactions, Catalyst and Conditions. The Compounds object contains information about the chemical species, the Reaction Object Contains information about the reaction and its properties including thermodynamics etc. The Catalyst object contains catalyst properties and finally the Conditions object stores the conditions under which the system is being studied including the temperature and pressure limits, the phase behavior at reaction conditions and so on. Each of these fundamental objects also has link tables, which are a mapping between these objects. For example, a catalyst is a mixture of compounds located in the compound object, the reaction is a transformation of the
403 compounds in the compound object and so on. A detailed description of the database is out of the scope of this paper. Fundamental Objects 1 1-.....Iq Compounds
Collection Object Reaction System 2
for
.~
Reaction
I"I~Pro pe/tie s ~
.........[Reactions
~ Reaction
to
.........i
c t ly t
Dependent Information Properties of 1 in Collection
r /
Set
Perform
ance
Per Collection, Catalyst, Condition Yield Reaction Kinetics Catalyst D eactivation .......
t-]P'Properties
Conditions
[--ID~roperties Figure 2" Knowledge Base Components A reaction system is a logical collection of these fundamental objects. Therefore, we introduce a Reaction Set object which acts as an aggregator of all the information in the fundamental objects pertaining to just the current reaction system. The intention with the reaction set is to group all information relevant to the reactions for the current system. To better illustrate the framework, we introduce the "Methanol Synthesis" Reaction Set example. Methanol is produced from synthesis gas (a mixture of CO, CO2 and H2), with copper oxide catalysts, the conditions of operation are 5-10 MPa and 200-300 ~ Three main reactions has to be taken into account in this synthesis (Graaf et al. (1988)). RI: R2: Water R3:
CO + 2 H2 <::>CH3OH C02 + 3 H2 <::>CH3OH + H20 gas-shift reaction: CO2 + H2 <=>CO + H20
In figure 3, the detailed structure of the knowledge base is illustrated, with data from the "Methanol-Synthesis" process included. The tree diagram only includes part of the full knowledge base structure. Here, note that the individual reactions R1, R2 and R3 for this system are part of a Reaction Set named Methanol Synthesis. All dependent information for reactions R1, R2 and R3 can now be given in the context of their existence in the Reaction Set "Methanol Synthesis". Now, given a reaction, its kinetics may be represented uniquely only when we know its associated Reaction Set, Catalyst and Condition. For example, Reaction R1 may have different kinetics (if empirically derived) if it belonged to another reaction set, where another compound (for example, Ammonia) was present. Likewise, consider the case where experiments for the same set of reactions were made with two (or more) different catalysts, as is the case for the Methanol-synthesis (often Cu-Zn-A1 oxide or Cu-Zn-Cr oxide catalysts or catalysts from different manufacturers). For this case, the structure of the database allows for simultaneous storage of kinetics data for all catalysts just by the existence of different "SetCatCond" keys, as is indicated in figure 3.
404
m
~.
m
.
~ , Reac~
..+... ~ < l~CO,:,.~.~o,.§ ; ;
-- -- -- 1
-- _ _ ~ / k
/
] T
Metl~rn-lD CO:-ID l~ett~h~:llDD VHO_}-ID D ""
~""
I /
t
/
- labl
+-
.........
9
t
--?--
-
]
/
| /
+
Mapp,ng
"-
" - w - f
~
I
~
/ Name:Methanol synthesis ] / Keyproduct:Methanol J
[ ...."
I IIdentificati~frame
[ I
[
t
~
RI~I'~
Kinetics key
~ : "u',~t:r~
~
I
--,O.n..ca.on,.a~
-
l~/~S~ "~
S Condition . > '--.
__~__ ~e=cat~ion=m~ __ ~ __ __ __~ r ~ . . C L e v _ ~ . .
/
. . . . ps.,o~.,,
I MethS'vn-lD I H'O'ID
--.
~____
Mapping tab,
~ , ~ Catalys >
I -'
I ISetlD:LowPressure I [ Temp:200-300 "C
ID
~
Name
Notes
Uatl
~lO1
CuZnA1 oxide catalyst
L;at2 JiffY22
UuZnC ..........
.
I 1~;;~7o~; "
lye,
'/
~]
~, key for grouping all data relevant to ! unique combination of reaction set, / :atalyst and condition / ~et: Methanol synthesis :at: MK I01 catalyst :ond: 2-6
::'~'~ [
I Kinetic expression "] L~.... =k4?~o'"-k_,c~'to.
MPa, 200-300"C
Yield frame
Yield:95 %
Conversion: ....
Li:t
' ..: . .
J
[ I I J
5%
Figure 3: Simplified tree diagram for the reaction knowledge base. The catalyst, c o n d i t i o n objects together with the r e a c t i o n s e t defines a common "SetCatCond" key which groups data like kinetics, yield and catalyst deactivation. This allows us to store simultaneously the yield and kinetics information under different conditions and catalysts. The tables for the kinetic parameters in the database are general arrays and are designed to provide flexibility for any variety of kinetic expressions. A software application that uses the database simply maps the array into its custom kinetic framework. Our application software for kinetic estimation for example, allows for both simple mass action expression as well as a generalized Langmuir-Hinshelwood-Hougen-Watson (LHHW) expression. For the specific rate only the Arrhenius equation has been used, but again the database structure is not limited to this equation. The reaction database has a very rich representation of reaction chemistry and one obvious use is as a server for a r e a c t i o n b r o w s e r . Here, we have developed a prototype search and sort tool to browse reaction paths for a given target. The user can specify a target compound and then search the tree to develop different pathways to this target. On another note, several comprehensive commercial (MDLI (2000)) and literature databases for reaction systems have been available for quite some time. However, most of these databases do not store reaction kinetics in a systematic way. The focus in these databases has been more towards browsing overall experimental information about these reaction systems. In this study, we have attempted to include the characterization of the kinetics in a systematic way along with the other relevant information. We use the commercial databases for idea generation since they provide structure based searches, while our reaction database is used to store company specific reaction information and also detailed reaction kinetics information.
405 3. TASK MODULES - PROJECT EXECUTION In addition to the knowledge in the database, the software has modules for Kinetic Estimation, Simulation and Reactor Optimization. In using these modules, other information is also required, for example: reactor data, experimental data etc. All this data relevant to a project is stored in a project file. Also data for the current reaction set from the database being used in the project is stored in the project files, thereby making the project files independent of the database. Therefore, project files can easily be shared among coworkers, even though they don't have synchronized reaction databases. This is important in a company where different researchers are located at different sites and the project file for kinetic estimation can be shared among different researchers although their reaction databases may be different.
3.1 Estimation Module The Estimation module takes in experimental data, and guesses from the Chemist on the reaction mechanism to reconcile the observations with the hypotheses. A suite of standard lab reactor models is available to choose from for the experimental setup. Due to the uncertainty at this stage, significant modeling flexibility is provided to the user in terms of providing the guesses on the reaction mechanism, kinetic expressions, quasi-equilibrium reactions and so on. The user also provides bounds on the kinetic parameters and reaction orders along with any special weighting options. A graphical user interface guides the estimation project, maintaining consistency and not requiring the user to write model equations. The resulting nonlinear optimization problem (Differential Systems are converted to Algebraic Equations by Orthogonal Collocation on Finite Elements), is solved by the simultaneous approach (Cuthrell and Biegler (1987)). The optimal estimates and several diagnostic graphs are then presented to the user to test the validity of the mechanism, given the experimental observations. 3.2 Simulation(Virtual Expts) and Optimization Modules The Simulation Component allows the user to perform what-if scenarios and to design the next experiment, given the current reaction mechanism and kinetics. Here, the user provides the reaction conditions and the Simulation Component presents the reactor profiles and conversions to the user. The Optimization Component allows for the graphical formulation of reactor optimization scenarios. Here, the user is provided with rich options for dynamic optimization. For example, in batch systems, the user can request the best fed batch addition and temperature profiles while imposing state constraints. Control Vector Parametrization is provided as an option to enforce smooth control profiles if so desired. The reactor models and solution algorithms are the same for both the estimation and optimization components; the change only lies in the degrees of freedom. This is facilitated by applying the simultaneous infeasible path optimization approach, discussed in Cuthrell and Biegler (1987). 4. APPLICATIONS AND DISCUSSION Applications of this software in our company have ranged from petrochemicals to pharmaceuticals. Several projects involve only reasoning to generate better experiment designs, while others involve the entire chain from Estimation to Optimization. However, with the availability of a systematic tool to quickly run the gamut from mechanism reasoning to reactor optimization, there is often a tendency to develop the most complex
406 reaction models, even at the early stage of uncertain information. This often results in parameter non-uniqueness, since the information content of the experiments is lower than the information required (complexity) by the model. While some features are incorporated in the software to prevent these traps, one aspect of the training is to ensure that the model building is an incremental process. It is first important to reason the behavior from a macroscopic perspective and add detail or additional complexity to the model only when it adds value and there is relevant data to estimate this detail uniquely. Secondly, the judgement of data quality and model formulation is still a critical responsibility of the user. Laboratory data should be analyzed thoroughly to check for atom balances. Sometimes, residuals of these balances may be undetected species if instrument analysis is not an issue. Often, several compounds that are detected may be unknown. By inspecting the profiles of these unknown compounds, one can make good guesses on their location in the reaction network. In situations where two or more compounds can only be measured together (possibly due to overlapping GC/LC peaks), the estimation software allows for this by only weighting the combined sums of errors for these grouped compounds. Model formulation is also important due to the very nonlinear nature of these problems, for example, reaction order bounds should be fairly tight when estimating the reaction order. The user is also provided with access to scaling options, initialization techniques, warm starts, and nonlinear solution tuning options for the more difficult problems. In conclusion, the software has been used effectively at our company. Quick model development and reactor dynamic optimization has resulted in yield improvements for target species. While most of the components in this system are not necessarily new, the framework in which they have been tied together, and the modeling flexibility has resulted in productivity benefits at our research lab. REFERENCES
1. Cuthrell, J.E.; Biegler, L.T. On the Optimization of Differential Algebraic Process Systems. AIChE J. 1987, 33, p1257 2. Graff, G. H.; Stamhuis, E. J.; Beenackers, A. A. C. M. Kinetics of Low-Pressure Methanol Synthesis. Chem. Eng. Sci. 1988, 43, p3185 3. MDL Information Systems, Inc. (2000), San Leandro, California, USA, (www.mdli.com)
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
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An Efficient Approach to Quantify Process Feasibility based on Convex Hull Marianthi G. Ierapetritou a aDepartment of Chemical and Biochemical Engineering, Rutgers University, Piscataway, NJ 088854, USA This paper addresses the problem of quantifying process feasibility (i.e. the capability of feasible operation under changing operating conditions). The basic idea is to evaluate the convex hull that is inscribed within the feasible region and determine its volume based on Delaunay Triangulation. The proposed approach not only provides another feasibility measure but also an accurate description of the feasible space of the process. The approach has been applied to convex and one dimensional convex problems and work is in progress to extend the ideas to non-convex systems. 1. INTRODUCTION Production systems typically involve significant uncertainty in their operation. Variability of process parameters during operation and plant model mismatch (both parametric and structural) could give rise to suboptimality and even infeasibility of the deterministic solutions. Consequently, plant flexibility has been recognized to represent one of the important components in the operability of the production processes. Approaches that exist in the literature to quantify the flexibility for a given design involve the deterministic measures such as the resilience index RI, proposed by [4]Saboo et al. [7], the flexibility index proposed by Swaney and Grossmann [9], [10] and the stochastic measures such as the design reliability proposed by Kubic and Stein [5] and the stochastic flexibility index proposed by Pistikopoulos and Mazzuchi [6] and Straub and Grossmann [8]. A small example is studied in the next section to motivate the need of developing a new approach for quantifying the range of feasibility for a given process. Section 3 introduce the new approach and the detailed procedure that is proposed to determine the new metric that describes the feasible region. The motivating example is used to illustrate the basic steps of the proposed approach and the obtained results in section 4. 2. MOTIVATING EXAMPLE The example presented here is a modification of example 2 used by Pistikopoulos and Ierapetritou (1995). The design is described by two parameters (dl,d2) whereas Zl,Z2 are the control variables and 01,02 are the uncertain parameters. The constraints of the problem are the following: fl = - 1 . 3 + 1.601- 1.6zl + 2.14z2 _< 0 f2 -- 02 - 2.502 + 12zl + 2z2 < 0
408 f3 -- 2 . 6 1 Z l - d l < 0; f4 = 3.14z2- d2 < 0
f5----Zl _~0; f6---~--Z2--~0 0_<01 _<2" 0 ~ 0 2 ~ 8 The design examined first corresponds to dl = 3.5, d2 -- 0.0. For this problem first the active sets are identified based on the fact that three constraints should be active at a time since there are two control variables, (Grossmann and Floudas, [3]). For each of the active sets, the feasibility function is determined and plotted in Figure 1. The flexibility index problem is then solved
Fig. 1. Feasible Region of Design 1 (dl = 3.5, d2 - 0.0)
using the active set strategy approach proposed by Grossmann and Floudas [3], that results in a value of F=0.278. This value describes the rectangular which is shown dark shaded in Figure 1. Note that although the actual feasible region of the design is the light shaded region, the flexibility index identifies only a small percentage of it and thus it largely underestimates design's feasibility. Moreover, the flexibility index does not provide always accurate results for comparing different designs. For example we are considering next the design that corresponds to dl - 2.5, d2 -- 3.5. For this design the feasible region is evaluated based on the determination of the different active sets. The results are illustrated in Figure 3. Note that this design has smaller feasible region since active set 5 constraint moved to the left and thus restrict more the design's feasible region. However, the flexibility index of this design is F=0.278, the same as for the design 1. The presented results for the specific example clearly motivate the need of developing a new approach for determining the feasibility range for a given process and a new metric for evaluating its flexibility that can be used for comparison between different designs.
409
3. PROPOSED APPROACH To illustrate the basic concept of the proposed approach let's assume that the specific design is described by a set of inequality constraints fj(d,z, 0) < 0 assuming that the equality constraints have been eliminated for ease in the presentation. In these constraints d represents the set of design variables, z the set of control variables that can be adjusted to accommodate variations on the uncertain parameter vector 0. Following the ideas of Grossmann and Halemane [4], one can map the feasible region into the uncertainty space by evaluating the feasibility function" ~(d, 0)
=
min u Z,U
f j(d,z, O) <_u, Vj C J Alternatively, the same result can be obtained by identifying all the active constraint sets. The question that arises in this stage in many different engineering problems is how to describe and then quantify the feasible range. The first case we are going to address is the convex case, the extension to the one dimensional convex case will be discussed in section 5. The main idea in the proposed approach is first to determine points at the boundary of the feasible region and then to evaluate the convex hull defined by those points. For the case of convex feasible region the convex hull determined in this way is guaranteed to be inscribed within the feasible region. However, for the nonconvex case there is no guarantee that this would be always the case. To overcome this difficulty an extension of the basic proposed approach is presented in section 5, whereas the general nonconvex case would be a subject of future publication. For the convex case the following steps are considered: (1) Solve the problems of determining the maximum deviation from the nominal point towards the vertices (keK) of the expected range of uncertain parameter (0i) variability (e.g. region defined by the bounds of the uncertain parameters). (2) Solve the problems of determining the maximum deviation from the nominal point by varying one uncertain parameter (0i) at a time. (3) Determine the convex hull based in the points obtained from steps (1),(2) applying the Quickhull algorithm which is described below. (4) Determine the volume of the convex hull obtained from step (3) using Delaunay Triangulation (the basics of the Delanay Triangulation are explained in the sequel). At the same step and using the vertices of the expected range of uncertain parameter variability the volume of the overall expected region is determined. The ratio between the feasible convex hull and the expected region "volumes" is then evaluated to represent the new metric for comparing design's feasibility. Volume of the feasible convex hull Feasible Convex Hull Ratio (FCHR) - Volume of the overall expected range Note that at step (3) the linear functions describing the faces of the convex hull are also determined. As will be illustrated in the next section, this provides a very accurate description of the feasible space for a specific design or process which is of great importance in many different engineering problems.
410
3.1. Quickhull Algorithm The convex hull of a set of points is the smallest convex set that contains the points. There exist a number of different algorithms in the computational geometry literature that are designed to compute the convex hull for a given set of points. A review can be found in Berg et. al. [2]. Recent work on convex hull has been focused on variations of a randomized, incremental algorithm where points are considered one at a time. Quickhull algorithm, (Barber et. al. [1 ]), proceeds using two geometric operations: oriented hyperplane through (na) points and signed distance to hyperplane. The correctness of the algorithm has been proved by Barber et al. [1]. They also provide empirical evidence that the algorithm runs faster when the input set of points contains no extreme points. The detailed description of the algorithm as well as the programming implementation can be found and downloaded from the website of the geometry center in Minneapolis: http://www.geom.umn.edu/software/download/
Delaunay Triangulation The basic definition of the Delaunay Triangulation is the graph that has a node for every Voronoi cell and a straight line connecting two nodes if the corresponding cells share an edge. The computation of the Delaunay Triangulation is based on the same iterative procedure as the convex hull estimation. In particular, a Delaunay Triangulation in R a can be computed from a convex hull in R a+l as described by Barber et.al. [1]. The same sofware described above has the cabability of evaluating Delaunay Triangulation.
4. EXAMPLE REVISITED The example presented in section 2 is used here to illustrate the basic steps of the proposed approach and the results obtained. The problem involves convex constraints so there is no need for function convexification. The active sets of the problem have been identified and the feasible region is described by the shaded region shown in Figure 2. The steps of the proposed analysis are then followed. Step 1: Solve the optimization problems of maximizing the deviations from the nominal point towards the edges. The problems solved have the following form: max ~i
s.t. fj(d,z, Ol,02) <_0, j - - 1,...,6 01 = 0N-4- ~iA01:k
The solutions to the above optimization problems give rise to the square points shown in Figure 2. Note that these problems correspond to the vertex enumeration subproblems (Swaney and Grossmann [9]), proposed to evaluate the flexibility index. Consequently, at this step one can utilize the results to evaluate the flexibility index that corresponds to the minimum of the results (5i), where i denotes different vertices. For this example it is found that F=0.278 limited by the flexibility point as shown in Figure 2.
Step 2: Solve the problems of maximizing the deviations from the nominal point by varying one uncertain parameter at a time. The optimization problems solved at this step
411
have the following form: s.t.
fj(d,z, Ol,02) <__O , j -
max 8 1,...,6
O 1 - oN-+- ~iAO1i o2 = 0 ~ o r 0 1
-
0~
In this way the points illustrated with the circles in Figure 2 are determined. Step 3" Quickhull algorithm is applied to identify the convex hull based on the points obtained from steps 1 and 2. The output of the algorithm is the set of linear constraints describing the convex hull. gl -- 02 -- 8 < 0; g2 -- --01 ~ 0; g3 = 02 -- 1.301 _< 0; g4 -- 02 -- 7.501 + 7.0 <_ 0; g5 - - 0 2 - - 5 . 7 1 0 1 + 4 . 4 <_ 0; g 6 - 02--5.8801 +4.63 _< 0 Step 4: The volume (area in this case, since the problem involves two uncertain parameters) is evaluated at this step using the Quickhull algorithm that utilizes Delaunay Triangulation to determine the volume of the convex hull. The volume of the overall expected range is also calculated. For this example is found that the total volume is 16 units whereas the feasible convex hull has a volume of 10.9 units. This results in a ratio of 0.68 that corresponds to the percentage of design feasibility.
Fig. 2. Proposed approach for Design 1 (dl - 3.5, d2 -- 0.0)
The same procedure is applied for Design 2 (dl -- 2.5, d 2 - 3.5). As pointed out in the previous section the flexibility index for this design has the same value as the flexibility for Design 1 equals to F=0.278. This is due to the fact that the limiting active set is the same in
412 both cases. However, as shown in Figure 3, Design 2 has smaller feasible region than Design 1.This is correctly identified by the proposed approach which determines a ratio of feasibility equal to 0.64 for Design 2 compared to 0.68 for Design 1. It is important to point out that the convex hull offers a very accurate description of the feasible region for both designs.
Fig. 3. Results for Design 2 (dl = 2.5, d2 = 3.5)
Remark: It should be pointed out that the proposed ratio(FCHR) corresponds to the percentage of feasibility based on the overall expected range of uncertainty and n o t the true feasible region. As can be observed from Figures 2, 3 the percentage that corresponds to the feasible convex hull is much higher in terms of the actual feasible region. Acknowledgements The author gratefully acknowledges financial support from the National Science Foundation under the NSF CAREER program CTS-9983406. REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
C.B. Barber, D.E Dobkin, and H. Huhdanpaa. ACM Trans. Math. Soft., page 469, 1996. M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf. Sprieger, 1997. I.E. Grossmann and C.A. Floudas. Comput. Chem. Engng., 11:675-693, 1987. I.E. Grossmann and K.E Halemane. AIChE J1, 28:686-694, 1982. W.L. Kubic and EP. Stein. AIChE J1, 34:583, 1988. E.N. Pistikopoulos and T.A. Mazzuchi. Comput. Chem. Engng., 14:991-1000, 1990. A.K. Saboo, M. Morari, and D.C. Woodcock. Chem. Engng. Sci., 40:1553-1565, 1983. D.A. Straub and I.E. Grossmann. Comput. Chem. Engng., 17:339, 1993. R.E. Swaney and I.E. Grossmann. AIChE J1, 26:139, 1985. R.E. Swaney and I.E. Grossmann. AIChE J1, 31:631, 1985.
European Symposiumon ComputerAidedProcessEngineering- 11 R. Ganiand S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rightsreserved.
413
Design, Sizing and Modeling of a Reactive Extractive Distillation Unit and Solvent Recovery System L. Jim6nez * and J. Costa Chemical Engineering and Metallurgy Department, University of Barcelona, Marti i Franqu6s 1, 08028-Barcelona, Spain A new process to value a methanol + methyl acetate (MeOH + MeAc) azeotropic mixture by reactive extractive distillation with n-butanol (BuOH) using o-xylene as entrainer has been developed. High purity MeOH and butyl acetate (BuAc) were obtained as products. The aim of this work is to synthesize (batch/continuous comparison, residue curve map analysis and solvent selection), to size and rate the equipments (steady state simulation of the azeotropic separation, the reactive extractive distillation unit and the solvent recovery system) and perform the process dynamic modeling (control strategy, parameter tuning and robustness analysis). The preliminary economic study carried out reveals that the process has no profitability. As expected, the process bottleneck is the reactive distillation unit and, in particular, the control strategy to achieve the coupled targets. 1. INTRODUCTION The cost of chemicals, energy and waste treatment continues to rise as the cost of raw materials and fuel increases. Achieve a more efficient use of natural resources leads to highly integrated and dependent plants (recycle streams, energy/water recovery networks, process integration, etc...). Modem plants are no longer designed and operated as a sequence of almost independent unit operations. This integration may suppose difficulties in running and controlling the plant. Therefore, much interest in the implementation of techniques with a significant impact, such as upset propagation and plant-wide control techniques had emerged. In the one hand, a by-product processing plant is more elaborate and costly today than ever, because it must be designed to satisfy the severe raw materials quality requirements. On the other, a recovery system for chemicals is a business opportunity to classical treatment that mixes the economic balance with the stringent environment regulations. Within this philosophy, a reactive extractive distillation p r o c e s s was designed to produce BuAc from MeOH + MeAc azeotropic mixture, thus integrating the process by the MeOH re-use in the poly(vinyl alcohol) process (Jim6nez, 1997) and the BuAc production. Moreover, BuAc, as an aliphatic oxygenated solvent, has not been affected by the restrictions imposed by Volatile Organic Compounds legislation. MeAc + BuOH
( o-~yt >
BuAc + MeOH
t Present address: Chemical Engineering Department, ETSEQ, University Rovira i Virgili, Carretera de Salou s/n, 43006-Tarragona, Spain; e-mail: [email protected];Fax: (34)-977559667
414 I ~
ICMM20
Fig. 1. Process flowsheet diagram. T100: azeotropic distillation; T200: reactive extractive distillation; T300 and T400: solvent recovery system. ASPENPLUS TM, ASPENSPLIT TM, DynaPLUS TM and SPEEDUP TM tools, its striking
synergic effect and capabilities were used to develop this work. Process simulation does not alleviate the need for rather accurate physical property data and models. In particular, this situation becomes important when the design specifications for high purity products are based on terms of maximum impurities at ppm-level (Carlson, 1996). Physical properties were estimated using NRTL model with Hayden-O'Connell equation of state for vapor-phase non-idealities. Vapor-liquid equilibrium, chemical equilibrium and kinetic parameters were regressed from experimental data (Jim6nez 1997). 2. PROCESS SYNTHESIS Process flowsheet diagram (Figure 1) depends mainly on the separation technology. In this case, select the classical unit operations leads to capital and energy penalties due to the nonfavorable chemical equilibrium (equilibrium conversion at operating conditions range from 31 to 36%) and azeotrope constraints, as stated in the residue curve map analyses section. Therefore, high reflux ratios and large recycles flowrates are involved to achieve process specifications. The main advantages of the new process, based on reactive extractive distillation, are that side-reactions can be avoided, the chemical equilibrium constraint can be by-passed, the separation problem is simplified and there are significant investment savings. As the feed is rich in MeOH, a preliminary distillation column (T100) was selected to recover all MeOH (helping the equilibrium reaction forward) and to vaporize the feed to T200 (improving the reactants contact).
2.1. Batch/continuos comparison The BuAc composition achievable for regular column is far less than the obtained by the inverted procedure at the same operation time. Although the production period continues for the regular case, BuAc composition can not achieve high values. From here is clear that regular column presents a poor performance comparing to inverted operation. Mujtaba and
415 Macchieto (1994) also recommend complex or inverted configuration for BuAc production by esterification. 2.2. Residue curve maps analysis
In design problems, technologies such as residue curve maps (RCM), multicomponent azeotrope prediction, automated entrainer selection and shortcut column design had proven very powerful (Doherty et al, 1979). Distillation schemes designed using these technologies often feature significant improvements over process designs arrived at with more conventional approaches. An accurate analysis of the quaternary non-reactive RCM diagram (MeOH + MeAc + BuOH + BuAc) reveals that there is just one distillation region. The distillation boundary present in the five-component reactive RCM, using o-xylene as entrainer, is fortunately located far-off the operating condition, even during start-up and shut down. Hence, there is no need for any boundary-crossing strategy (Wahnschafft et al., 1992). 2.3. Solvent selection
The aim of this section is to determine the re-esterification solvent that best performs this two tasks: break the azeotropes and improve the MeAc/BuOH contact in the reactive area. A preliminary selection, according to heuristics and physical property estimation by UNIFAC and UNIFAC-Dortmund was made. Experimental work, using headspace gas chromatography, was designed and selectivity at infinite dilution (Sij~176 for over 30 systems (alcohol + acetate + solvent) was studied. The Sij~~ criterion (the higher the selectivity, the better the solvent) is useful to cluster the solvents into different groups, but a definitive selection can not be stated. To ponder the industrial application of the promising solvents, the importance of peripheral properties, that is, properties that are of interest when selecting a solvent but often do not directly affect the separation, were discussed. Such properties include safety, cost, density, viscosity, dielectric constant, heat capacity, miscibility with water... Weighting all this consideratios and parameters, alquil-benzenes, and in particular, oxylene was selected as the best alternative. 3. SIZING AND RATING OF EQUIPMENT A critical factor to succeed in reactive distillation modeling is the accuracy in the estimation of the liquid hold-up. For structured packing, this value is about 10-15% of the void fraction, but it is strongly dependent upon the components, the system fluid-dynamics and the operating conditions (Kister, 1989). In addition, the liquid hold-up has a significant effect in systems in which there are competitive (i.e. alcohol dehydration) and/or consecutive (hydrolysis to acetic acid) reactions. High efficiency structured packing were selected for the reactive area due to the excellent mass and heat transfer properties. In a certain range around the operating conditions, it can be assumed that the reaction controlling step, for the column diameters involved, is the surface reaction (De Garmo et al., 1992). The columns diameter were rated using both the capacity factor and the flooding percentage (Kister, 1992), in order to check data and/or correlation inaccuracies. As vapor and liquid flowrates vary widely along the equipment, different packing were used in the same column to avoid any flooding point and to reduce investment. Designs in which columns
416 have variable diameters were rejected because maximum divergences were about 8%. Sensitivity analyses with multiple variables were performed as optimization method (feed stage, solvent feed stage, solvent flowrate, reaction stages...). 4. DYNAMIC MODELING Initial conditions were fixed as the steady state stationary values. A 50% hold-up in all reflux drums and intermediate tanks was fixed. No time-considerations were implemented for the start-up period, and therefore the steady state was reached instantaneously. The presence of multiple steady states, as is well known in MTBE production, may greatly complicate the operation of a distillation column. Current research explores the implications for entrainer selection and column design and seeks an extension of the analysis to reactive distillation systems. In this study, no multiple steady states were discovered. 4.1. Control strategy As pointed out before, the main process constraint are the operation conditions optimization for the extractive reactive distillation unit: heavy impurity recovered overhead will impurify the MeOH, and light impurity leaving as bottoms product will impurify the BuAc. Preliminary tests show that ratio controls slightly bigger than the stoichiometric for reactants and near 4:1 for the entrainer assist to achieve a virtually total BuOH conversion and minimize these problems. The low catalysts' thermal stability (acid ion-exchange resins) acts as an additional constraint (any premature deactivation will lead to shutdowns). To establish the control scheme, common strategies for reactive, extractive and azeotropic distillation had to be mixed-up (Luyben, 1992). For high purity MeOH and BuAc distillation columns (T100 and T400), total impurities analyses (-log XimpuritiesVS T) were carried out to know if direct or indirect action for the controller could be fixed (Figure 2). Moreover, as stated in Figure 3, differences in temperature profiles (AT vs stages) were examined to determine the control strategy for temperature (best sensors location, need of average values, non-sensible variable...). As a result, inferential control strategies were implemented at T100. 150
Stage -.
1
Stage 11
-.
Stage 21
Stage 31
. . . Stage 36
--
Stage 40
30
145
20
--
+10% T(feed)
-.
-10% T(feed)
.~
--
+10% w(feed)
-.
-5% w(feed)
[ /
....
5 % ON
--+5%
x(light)
+5% rr 140
.......................
"-.
(.)
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~'\ ",
125
3.0
32
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36 -log X ,mpunbes
3.8
,,,,i [~i
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""~ "~;
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x(light)
lO
11 130
i
.... SOjoQ N
---5%
21
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4 ~
-10
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Fig. 2. Temperature profile for impurity analysis in T400 distillation column.
W 'v.
-20 Stage
Fig. 3. AT profile for typical process upsets in T200 distillation column.
417
4.2. Parameter tuning and robustness analysis Dynamic behavior facing common process upsets was studied to detect transient behavior (start-up, shut down) or inverse responses that will lead to severe control problems. Standard PID controllers with an additional derivative filter parameter were set to avoid signal noise. On-line analyzers with first order behavior and five minutes time lag were used. Temperature and pressure sensors were supposed to be instantaneous, although typical time constants are around 30 seconds. Once the control strategy was fixed, the controller parameters were tuned with the Auto Tuning Values (ATV) method. In this method, the controller integral and derivative actions are deactivated and the gain is set to infinite. Maximum and minimum control actions are fixed in a narrow range around the set point. Figure 4 shows the typical system behavior to a small disturbance in the set point, in which the target variable (bottom flowrate) changes instantaneously from the maximum to the minimum action, while the operation variables exhibits an harmonic performance. Based on this sinuous diagram and the Ziegler-Nichols method preliminary controller parameters can be estimated. Analysis of performance facing typical upsets in process variables (up to +10%), such as feed flowrate, feed pressure, feed temperature, feed composition, reflux flowrate, reboiler duty, condenser duty and operating pressure were carried out (Figure 5). 5. COST E S T I M A T I O N Modular cost estimation based on index from Chemical Engineering, Oil and Gas Journal, Engineering News-Record and Marshall & Swift was performed. Chemicals cost was based on a six-month average from Chemical Marketing Reporter. Labor, inflation and utilities cost from EU were used. As reactive distillation systems are not common in industry, a 20% contingency factor was fixed. For this particular separation, the cost of the solvent recovery system is significant and, in some aspects, controls the cost of the process. Cost estimation was divided into four steps: unit operations costs, operating costs, fixed costs and project profit.
Fig. 4. ATV method for the reboiler level control in T-400 distillation column.
Fig. 5. T400 performance for increase in feed temperature.
a 5%
418 The economic analysis shows that the designed project has no profitability on its own, but a more extensive and realistic analysis should compare the actual process (conversion into diluted acetic acid, with sulphuric as catalyst) with the proposed solution. 6. CONCLUSIONS Reactive distillation is a promising approach for the MeOH + MeAc separation. Although vapor-liquid equilibrium and kinetics agree with experimental data, pilot plant data are strongly recommended to validate simulation predictions. As global results, the following are the more remarkable: BuOH conversion (99.91%); MeOH purity (99.82%); o-xylene in MeOH (15.2 ppm); BuAc purity (99.81%); o-xylene in BuAc (51.2 ppm); heat utility (44.0 Mcal/Ton MeOH); cold utility (43.1 Mcal/Ton MeOH). As expected, non-symmetric behaviors were detected during the dynamic modeling for all the variables; that is, the effects for a positive or negative disturbance in the process variable do not have reverse and symmetric effects. Detailed analysis of the reactive and extractive distillation unit should be performed due to parameter coupling. Upset propagation analysis of the whole process should be achieved. As a helpful tool, this research involved a unique blend of experiments, makes extensive use of software for computer-aided design and computer-based graphics analysis, which is a rather clear approach. ACKNOWLEDGMENTS
The authors would like to thank DGICYT, CIRIT and Aspen Technology for providing the necessary facilities. L. Jim6nez thanks Fundaci6n Caja de Madrid for the financial support. REFERENCES
1. 2. 3. 4. 5. 6.
Aspen Technology Inc., AspenPlus TM Reference Manual, Cambridge, Ma, USA, 1998. Aspen Technology Inc., AspenSplitTM Reference Manual, Cambridge, Ma, USA, 1998. Aspen Technology Inc., Dynaplus TM Reference Manual, Cambridge, Ma, USA, 1998. Aspen Technology Inc., SPEEDUP TM Reference Manual, Cambridge, Ma, USA, 1998. Carlson, E. C., Don't Gamble with Phys. Prop. for Sim., Chem. Eng. Prog., 35-46, 1996. De Garmo, J. L., Parulekar, V. N., Pinjala, V., Cons. Reac. Dist., Chem. Eng. Prog., 4350, 1992. 7. Doherty, M. F., Perkins, J. D., On the Dynamics of Distillation Processes-III The Topological Structure of Ternary Curve Maps, Chem. Eng. Sci., 34, 1401-1414, 1979. 8. Jim6nez Esteller L., Simulation and Design of a Re-esterification Process of MeAc with BuOH by Reactive Distillation, PhD Thesis, University of Barcelona, Spain, 1997. 9. Kister, H. Z., Distillation Operation, McGraw Hill Inc., 1989. 10. Kister, H. Z., Distillation Design, McGraw Hill Inc., 1992. 11. Luyben, W. L., Practical Distillation Control, Van Nostrand Reinhold, 1992. 12. Mujtaba, I. M., Stuart, G., Macchietto, S., ALAMBIC- A Software Package for Optimazing Design Columns, DYCORD+'95, Denmark, June, 1995. 13. Wahnschafft, O. M., Koehler, J. W., Blass, E., Westerberg, A. W., The product Comp. Regions of Single-Feed Azeot. Dist. Col., Ind. Eng. Chem. Res., 32, 2345-2362, 1992.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
419
M o d e l i n g a Multi-Step Protein Synthesis and Purification Process: A Case Study of a C A P E Application in the Pharmaceutical Industry David Kahn a, Richard Plapp a and Ajay Modi b aEli Lilly and Company, Lilly Corporate Center, Indianapolis, IN 46285, U.S.A. bAspen Technology, Inc., Ten Canal Park, Cambridge, MA 02141, U.S.A. This paper presents a case study in the use of a CAPE tool for the development of a protein synthesis and purification process. The paper has three main objectives. First, it discusses the use of the software in representing the process and in determining the overall material balance and flows. It indicates how this information is used in meeting environmental and safety regulations and to assist with process troubleshooting. It also describes the unit operations typically employed in bulk protein manufacturing, how these are modeled with the software and learning points from the model development process. Second, the paper briefly discusses the context and environment within which the software was deployed at the company. Business drivers, project structure and linkages to other corporate efforts are all described. Third, the paper discusses the evolution of the tool and the direction of its technical development, including expanded CAPE functionality and utilization within the pharmaceutical industry.
1. BUSINESS DRIVERS FOR CAPE IN THE PHARMACEUTICAL INDUSTRY The pharmaceutical industry faces several significant challenges today. Increased pressure to lengthen the protection accorded by patents has resulted in the need for rapid and effective process development. Improved development reduces costs and leads to processes with higher economic efficiency. In the current climate of increasing globalization, economic efficiency will dictate whether a manufacturer will be able to remain competitive, an issue of special relevance to the pharmaceutical industry, which faces a long government approval process for its products. The use of CAPE tools can reduce the duration of the development cycle as well as support the development of processes that are more efficient, environmentally friendlier, safer and more cost effective by allowing more alternatives and options to be evaluated [1 ]. They can support recipe development, material selection, equipment sizing and design, scale-up and process optimization. The use of these tools can lower the barriers between development and manufacturing, thus shortening the facility delivery process. They can also support engineering workflows and technology transfer among laboratory chemists, pilot-plant engineers, process engineers and others who play a role in process development. Batch Plus T M [2] is one such tool currently available.
420
2. PROCESS DESCRIPTION In the protein business many of the products utilize batch manufacturing in multi-step processes. Fermentation is typically used to make the protein while chromatography, tangential flow filtration (TFF) and drying are often used to isolate and purify it.
2.1. Protein Expression A recombinant-derived cell typically expresses the protein. The cells are fermented and then harvested. The unwanted cell parts are then separated from the desired protein. The protein is then purified and chemically altered using chromatography, tangential flow filtration and chemical reactions. A recombinant-altered cell has DNA added to its plasmid to give it instructions to do something it normally does not. In the pharmaceutical industry, the cells are given instructions on how to express the required protein chain. In some cases, the cell cannot make the exact product required; therefore a protein that is close to what is needed is produced. This "similar" protein is then altered and purified to give the final product, using multiple unit operations. When the protein is relatively small, e.g., a peptide or polypeptide, the use of solvents does not irreparably damage the structure of the protein and reverse-phase chromatography can be used to provide an extremely pure product. 2.2. Protein Purification Chemistry and Equipment Chromatography, enzymatic reactions, tangential flow filtration and drying are typical unit operations used in biotech processing. Chromatography is a separation technique based on the protein's affinity to adsorb to and desorb from a solid surface (resin). There are several types of chromatography such as ion exchange, reversed phase and size exclusion. A typical batch process would call for an equilibrating solution to be run across the column to prepare the resin for the protein solution. The protein solution would then be charged onto the column. This might be followed by a wash to remove any material not bound to the resin or weakly bound to the resin. Next, the column is eluted with either a gradient or isocratically, the mixture of proteins then desorb based on their relative affinity for the resin and elution fluid. When the product protein leaves the column it is then segregated from the rest of the material for forward processing. At the end of the elution one or more fluids are run across the column to remove any remaining material; this is referred to as regeneration. Tangential flow filtration is a separation technique used to do size separation. In order to be separated, the difference in size of the molecules needs to be an order of magnitude or greater. A membrane is used to separate molecules of different sizes. Based on the size of the membrane, TFF can be referred to as reverse osmosis, ultrafiltration, microfiltration or viral filtration. The basic principles are the same for all membrane sizes. The fluid is circulated across a membrane with higher pressure on the feed side than the permeate side. The molecules that are small enough to go through the membrane are the permeate and those retained by the membrane are the retentate. Typically, a large quantity of low concentration material is circulated across the membranes until a portion of the carrier fluid and smaller unwanted proteins pass through the membrane leaving the retentate more concentrated. Sometimes it is desirable to change the carrier fluid, and therefore, the retentate would be diafiltered. The new buffer would be added to the retentate as permeate is leaving, thus keeping the concentration of the protein constant.
421 Drying can take several forms including spray, vacuum and lyophilization. The technique used most often is lyophilization since most proteins cannot be crystallized. In lyophilization, the protein solution is frozen, then a vacuum is pulled and the product is slowly warmed. During the warming period, the protein solute is removed by sublimation. At the end of the cycle all volatile material has been removed, leaving a stable powder. 3. CHALLENGES FOR PROCESS ENGINEERS AT LILLY The challenge for the process engineer is to quickly translate the development process to a scaled manufacturing process. At the manufacturing scale, many factors that are transparent during the development can become rate limiting, at best, and detrimental to the process, at worst. The rate of heat transfer for a reaction, the size of a column, the quantities of raw materials required could all be issues when the batch size is changed significantly. A macrolevel look at the equipment and utilities required is usually done with a spreadsheet. The spreadsheet would list the manufacturing steps (TFF, chromatography, reactions, etc.), expected yields, step sizes (mass in), expected capacity, column dimensions, ultrafiltration membrane area, vessel working volumes, raw material requirements and production costs. Once a draft of the manufacturing process is determined, issues and feedback are typically given to development to have new ideas incorporated into the production scheme. Once most of the design criteria are agreed to, it is then possible to start creating a model of the process. The model usually takes several iterations. Each process may be modeled separately, then combined with the other steps and finally linked to raw materials and utilities. One of the larger challenges is to determine the environmental effects of the process. In most cases a mass balance must be determined which details VOC emissions, fugitive and point source, along with solid and liquid wastes generated. 3.1. Description of CAPE Efforts at Lilly At Lilly, spreadsheets have long been the tool of choice for almost any kind of computational engineering problem involved with process analysis. Only in the past four to six years have commercial off-the-shelf tools been considered for wide deployment to support process engineering efforts. Seeing a growing CAPE market, Lilly decided in 1996 to license a suite of engineering software tools with both batch and continuous process modeling capabilities. In 1997, after completing a selection process, Lilly chose AspenTech as its primary vendor of choice for several reasons including the availability of many CAPE applications in its Aspen Engineering Suite.
After the selection and procurement were completed, the software deployment phase began late in 1998. The company's central engineering organization managed the initial deployment as expertise with the software developed and continues to develop today. The current deployment focuses specifically on the Batch Plus tool for performing material and energy balances, emission calculations and graphical and text-based recipe descriptions. Currently, the effort to utilize Batch Plus as a standard tool is being driven by the evolution of the technology (bug-fixes and functional enhancements), developing the infrastructure, and learning how to apply it with a critical mass of users [3]. The Batch Plus infrastructure includes several central support functions such as: on-site training, internal user group
422 meetings, troubleshooting support, bug tracking and model-development support that also encompasses contract modeling. Although the deployment was initially targeted to the company's small molecule, i.e., "synthetic" late-stage development and manufacturing, environmental obligations later drove the deployment to include the large molecule, i.e. "protein" side of late-stage development and manufacturing. Future applications at Lilly may include fill-finish manufacturing and technical knowledge transfer and scale-up in mid and late stage development. The case study presented in this paper provides one example of the benefits and current limitations of utilizing the Batch Plus technology in a protein purification process. 4. PROJECT SCOPE AND DESCRIPTION The scope of this project was to determine a mass balance for solids and liquids entering and leaving the process. The types of generated waste in the process are aqueous, nonaqueous and urea. The aqueous waste can be sent to the local sewage treatment plant for further processing while the non-aqueous waste is sent to a liquid incinerator and the urea waste is sold to a vendor. Due to the nature of the product being manufactured, any amount of product in the urea waste would exclude it from sale to our vendor. The characterization of the waste stream quantities and components was required to ensure we remained within our operating permits. Due to time constraints, this needed to be done quickly and in a manner that allowed updates to be done with minimal effort. 4.1. Mass Balance- Method Review and Selection
Several methods for determining the mass balance were considered: hand calculations, spreadsheet and available software. Using hand calculations was quickly eliminated as a viable method due to the number of calculations required, the need to re-work entire sections if quantities of buffers or concentrations were changed and the difficulty of ensuring accuracy and widespread review. The advantage of using a spreadsheet is that it is easily tailored to the specific process being modeled and can be set-up to the engineer's particular style and needs. Once completed, it is fairly easy to update changes in quantities and concentrations in the spreadsheet. The disadvantages are that the spreadsheet is typically difficult for others to use, it is time consuming to set-up, the calculations need to be checked and it can be difficult to add or change sub-steps. The three software options considered were SuperPro Designer TM, Batches TM and Batch Plus TM. Batch Plus was chosen primarily because it is an Eli Lilly supported tool and is becoming the corporate standard. Process engineers and consulting engineers from Lilly's Engineering Technology Center have been working with AspenTech to improve the product (specifically the biotech capability) and so this was an excellent opportunity to review recent upgrades and give input on future improvements. As a result, it is expected that application of the model by future engineers will be fairly straightforward. Also, if needed at a later date, the air emission calculations could be programmed into the model without much additional effort. One potential benefit not yet capitalized on is the link to our historian, Aspen Process Explorer.
423
4.2. Model Development The model development was segmented into smaller activities: information procurement, recipe detail, backbone assembly, workarounds, model refinement, error checking and updates. The information gathered for use included flow documents, batch records, buffer recipes and equipment diagrams. Once the information was gathered, agreement was reached around the level of detail that would be required and how to construct the model (hierarchy, standards, etc). In this situation, since timelines were a significant constraint, focus was placed on the mass balance and some of the details required to get accurate emission calculations and cycle time estimations were left out of the model. The backbone of the model was then assembled and reviewed by an engineer experienced with the software but not the process. During the assembly it became apparent that some operations could not be directly modeled. For example, the use of a pre-column that goes directly to another column could not be done since a column output could not become another column's input. To remedy this situation a tank was added to the model so the pre-column sent its output to the tank, which was then the input for the next column. Model refinement and error checking was performed by an engineer versed in the process but with little experience (other than a two-day training course) in using Batch Plus. The combination of experts in both modeling and processing made for efficient time use. Error checking was done using the spreadsheet mentioned above, making sure the inputs went somewhere and using the Batch Plus Run History report. This report helped to determine that more was going into some vessels than they would hold or when a vessel feeding a column didn't have enough material in it to meet its charge requirements.
4.3. Model Outputs Since the output reports were available as Excel spreadsheets, this made further data manipulations or use by others more transferable. In this case, the basic material balance needed to be captured, and specifically, any waste quantities. This was done using macros to adjust the stream table. One of the drawbacks of the stream report was that if 100 g of 15% by weight NaOH was used, for example, the stream table showed it as 15 g NAOH and 85 g water. If it is desired to total how much 15% NaOH would be required to process a lot or campaign, then the user would need to take the NaOH total and back-calculate the amount since the report captured quantities by pure components only and not by mixture.
4.4. Learning Points The abilities of Batch Plus to model a process are extensive. Therefore, deliberate decisions as to what is needed and how the model will be used are required. It worked better on this project to limit the initial scope with the potential of building onto the core model later as needs evolved. This allowed for a small success, with the building blocks for larger, more complicated models firmly planted. The use of the having a modeling expert design and program the backbone of the model with a process expert later refining it made the modeling go smoothly and the task reasonable for both parties. The model can now be easily updated for process improvements.
424 The software proved that it could provide a valuable output. The mass balance was reported in a manner that was understandable, supportable and maintainable. A few "workarounds" were required for chromatography and tangential flow filtration but these were fairly straightforward. As feedback from this case and others is integrated with future releases, the utility of the program will increase for biotech process modeling. This model used relatively little of the software's potential functionality but it demonstrated that expanded use in the future would likely be valuable. The next evolution of this model will likely be for cycle-time analysis and emission calculations. 5. EVOLUTION OF SOFTWARE Batch Plus is an advanced simulation and data management tool for the modeling of complex, recipe-based batch processes such as those found in the pharmaceutical, agricultural chemical, biotech and specialty chemical industries. The tool supports a wide spectrum of engineering workflows over the full lifecycle of a process. The evolution and technical development of Batch Plus is expected to continue along several dimensions that expand its ability to support these workflows and address significant business problems. This section describes some of these enhancements. Batch Plus contains a comprehensive library of unit operations and models that supports the incremental development of a recipe at different scales and permits the construction of a single composite model of the entire process. These models range from shortcut models that perform mass and energy balances to rigorous models that allow predictions of process performance. The enhancement and extension of the operations library is one path along which the evolution of Batch Plus is expected to continue. Each successive version of the software has included new operations and models or enhancements to existing ones. The protein purification process model described in this paper made extensive use of the chromatography models in Batch Plus. The updates and enhancements to the unit operation models that rendered them appropriate for use in this project were made in direct response to Lilly's requests. As a result of Lilly' s recent biotech modeling efforts, additional functionality has been requested including non-linear elution gradients, regeneration with material originating from a vessel, factor-based calculation for concentration operations, hierarchical material and energy balance reporting and the ability to utilize the "custom" operation with biotech equipment. A related path along which Batch Plus is expected to develop is the continued expansion of its modeling coverage for secondary, packaging and solids processes. In the pharmaceutical industry, a distinct manufacturing process is usually responsible for producing and packaging a drug in its final finished form. This secondary process, as it is often referred to, has as its starting point the active ingredient or bulk chemical produced by the primary process. Several new features were implemented in a recent version of Batch Plus that allows it to support engineering projects involving secondary processes [4]. These include new unit operations such as mill, granulate, freeze-dry, screen, tablet and fill, the ability to represent particle size distributions for solid materials and the ability to account for the consumption of packaging materials, e.g., bottles, cartons and vials, in the process.
425 In the coming years, the pharmaceutical, biotech and other batch process industries are expected to face several new challenges in complying with the regulatory requirements promulgated by environmental bodies such as the US EPA and state agencies. The new rules that are being formulated will introduce several new calculation methods that have to be implemented at an early stage of process development. The Pharma MACT rules are an example of such a new regulation. The US EPA 1978 CTG equations, the US EPA 1994 ACT equations and the recent US EPA MACT equations are currently implemented in Batch Plus [5]. These emission models allow users to estimate the vapor emissions from the batch operations in a process. These calculations may be carried out for either one batch or multiple batches, thus allowing the user to determine how emissions will build up over an entire production campaign. The incorporation in Batch Plus of additional emission models and methods for the characterization of other waste streams is another path along which the software is expected to develop. Information is a key resource in today's commercial environment and organizations that have the ability to manage their information effectively will have a competitive advantage over those that do not. Effective large-scale information management can yield tremendous business benefits by preventing loss of information, hand-off delays and duplication of effort. The development of an enterprise database architecture for Batch Plus is another dimension along which the software is evolving. This architecture will allow the tool to act as a central repository of process data as well as permit multiple users to work concurrently on the same project by sharing process information and data. 6. S U M M A R Y
This paper described the application of Batch Plus in a biotech process development project. The use of the software in representing the process and in determining the overall material balance was discussed. The unit operations typically employed in bulk protein manufacturing, how these were modeled with the software and learning points from the model development process were described. The paper also discussed the context and environment within which the software was deployed. Business drivers, project structure and linkages to other corporate efforts were all described. Several conclusions were drawn regarding the application of Batch Plus in this project. Key among these was that it was worth using the tool because of the numerous benefits derived. The overall potential of the software was recognized as well as the fact that it was also evolving. The paper concluded by discussing some of the dimensions along which the tool was evolving and the direction of its technical development. REFERENCES 1. G. Pisano, "The Development Factory," Harvard Business School Press, Boston, 1997. 2. A. Modi and R. Musier, "Systematic Batch Process Development and Analysis via an Advanced Modeling and Simulation Tool," In "Proceedings of Batch Process Design and Operations Conference," AIChE Spring Meeting, New Orleans, 1998. 3. R. Plapp and R. Dargatz, "Global Deployment of the AspenTech Process Analysis Toolkit: One Pharmaceutical Company's Approach," Aspen World, Orlando, 2000.
426 4. A. Modi, Y. Wang and S. Sundaram, "Modeling Secondary, Packaging and Solids Processes with Batch Plus," AIChE Spring Meeting, Atlanta, 2000. 5. A. Modi, P. Cherukat and S. Sundaram, "Environmental Calculations During Batch Process Design and Development via an Advanced Software Tool," AIChE Spring Meeting, Atlanta, 2000.
EuropeanSymposiumon ComputerAidedProcessEngineering- 11 R. Ganiand S.B.Jorgensen(Editors) 9 2001 ElsevierScienceB.V.All rightsreserved.
427
New designed TSA bed with cooling jacket for purification and regeneration of benzene and toluene Daeho Koa, Mikyung Kim a, I1 Moon at, and Dae-Ki Choi b aDepartment of Chemical Engineering, Yonsei University, Seoul 120-749, Korea bEnvironment and Process Technology Division, Korea Institute of Science and Technology, Seoul 130-650, Korea This work presents a new design of a thermal swing adsorption (TSA) process cooling jacket installed, under the condition of operating the continuous two-bed TSA process. The new designed cooling jacketed TSA bed enhances the adsorption efficiency, as well as saves the consuming rates of purge gas consumption and regeneration energy compared with the previous TSA bed. The target species are benzene and toluene. As a result of dynamic simulations from start-up to cyclic steady state (CSS), the adsorption ratios are 96% for benzene and 100% for toluene in the cooling jacketed TSA bed, while the adsorption ratios are 82% for benzene and 98% for toluene in the previous two-step operating TSA bed. 1. INTRODUCTION A fixed bed of TSA has been one of the most conventional cyclic processes including adsorption and desorption steps for purifying and regenerating strong adsorbates. Even though there have been many theoretical and experimental studies on TSA, little has been published about the cyclic operation of nonisothermal multicomponent TSA bed and the temperature controlling cooling-jacketed TSA bed. Therefore this paper proposes the novel TSA process in which the adsorption temperature is maintained closely to the desired temperature by the cooling jacket surrounding the bed. This study performed rigorous dynamic simulations by computing the variables representing the highly nonlinear dynamic behaviors involving the exponential terms. The main objectives of this study are 1) to improve the adsorption efficiency of multicomponent cyclic TSA processes, 2) to find a suitable operating condition for the continuous two-bed operation, and 3) to save the consuming rate of the purge gas quantity and that of the regeneration energy. 2. PROCESS DESCRIPTION A general TSA operation consists of two steps (adsorption and regeneration) or three steps (adsorption, regeneration and cooling) as shown in Fig. 1(a).
tCorresponding address : Tel) +82 2 361 2761, Fax) +82 2 312 6401, Email) [email protected]
428
Fig. 1. Operations of TSA processes The feed gas is supplied to the column at ambient temperature during the adsorption step. The hot gas flows in counter-current direction during the thermal regeneration step in this study. The adsorbate-free gas at ambient temperature is used in the same directions as for the heating, if the cooling step is adopted after the regeneration step. Nitrogen is used as a carrier gas of the feeding step, and a purge gas of the regeneration step as well as the cooling step. For the continuous two-bed TSA operation, the adsorption time should be greater than or at least equal to the sum of the regeneration time and the cooling time. The adsorbent is chosen as an activated carbon (Sorbonorit B4, 12-14 mesh). An operation of the jacketed bed is shown in Fig. 1(b). The jacket uses water as a coolant to maintain the gas temperature within the bed at a desired adsorption temperature during the adsorption step. When cooling the bed with the jacket during the feeding step, the coolant at atmospheric temperature, which is the same as the feed gas temperature, flows into the jacket and flows out of the jacket at a little higher temperature than that of the inlet coolant stream. Consequently, even though the regeneration temperature is very high, the temperature within the bed comes to the desired operating temperatures faster and is maintained more closely to those temperatures than any other processes during the adsorption step. The jacket discharges the coolant just before the regeneration step begins, so that there's no cooling water but stagnant air at atmospheric temperature within the jacket during regeneration step. Model simulation is carried out in the gPROMS modeling tool. The method of lines (Schiesser, 1991) adopted in this work. 3. MATHEMATICAL FORMULATION The assumptions are 1) the ideal gas law, 2) the constant gas velocity through the bed, 3) the negligible pressure gradient across the bed, 4) the negligible radial variations of temperature, concentration and velocity, 5) a linear driving force model, 6) the temperature-dependent extended-Langmuir equation, 7) the uniform physical properties of the gas and the feed gas, and 8) the constant physical properties of the adsorbent and the column wall. With these considerations, the following equations are formulated:
429
'
-~=Du
&2
Tg,, Oz 2
8Tga"= KL cqrga" 2 t~t Cpgpp Oz 2 OT,.o,,u = h,. at
a~.
" C,,.,',op
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8t
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at
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(1)
(2)
(3) (4)
(s)
On, = ki(n ~ _ n,) Ot
(6)
n7 = m, (T, ohd)bl (T.,.ol,d)PY, , where m(T) = mo exp(M, / T) and b(T) = B o exp(B1/ T) 1 + ~ b, (T,.o,d ) ryi i
(7)
NR - NR , tR
where N R -_
(8)
ER = ER , tR
where E R = NRCpg (Treg..... ,.... - Tr4 )
mt-----L-R Ph Vhed
(9)
The overall mass transfer (ki) of equation (6) follows the following four-step mechanism" 1) fluid film transfer, 2) pore diffusion, 3) adhesion on surface, and 4) surface diffusion. The values of the parameters of Langmuir isotherms are listed in Table 1. The purge gas consumption (NR) and the regeneration energy required (ER) have been generally used to evaluate the regeneration efficiency (Basmadjian, 1975; Huang and Fair, 1989; Hwang et al., 1997; Kumar and Dissinger, 1986; Schork and Fair, 1988). Thus we adopts the variables which are the consuming rate of the purge gas quantity (~rR) and that of the regeneration energy (/)R) in equation (8) and (9). Tab!e 1:..Isothe.rm constants.for temperatureTdependent Langmuir equatio n Adsorbates No [mol/g] M1 [K], B0 [l/Pa] gl [gl ....... Benzene ...............Z889x 103 .... 79.11 5.429x 101~ 5,292 Toluene 1.828x 10.3 254.10 1.626x 10.8 4,818 4. RESULTS AND DISCUSSION
This study conducted the dynamic simulations of the two and the three-step (if needed) operating TSA processes with non-isothermal condition based on non-equilibrium theory, to
430 check their dynamic phenomena. The target processes are the followings: 2-step Operating conventional TSA process when regeneration temperature is 533 K (2OC TSA) 2-step operating Cooling Jacketed TSA process when regeneration temperature is 533 K -
-
(cJ TSA) 3-step Operating Conventional TSA process when regeneration temperature is 533 K (3OC TSA) The CSS convergence appears at 4th cycle for 2OC TSA and 3ra cycle for CJ TSA and 3OC TSA, resulting from the dynamic simulations. Table 2 describes operating step times and proper bulk fluid velocities of the regeneration step determined by dynamic simulations to satisfy the conditions of running the continuous two-bed cyclic processes. Though the exact cost evaluation is necessary in the future, this study focuses on the analysis of the dynamic phenomena and adsorption efficiency of these processes. Table 3 explains that the cooling jacketed bed (CJ TSA) improves the adsorption efficiency compared with the two-step operating previous bed (2OC TSA), because the lower temperature makes it easy to adsorb the components. If cooling operation is adopted, the adsorption efficiency is good but this case (3OC TSA) needs high consuming rates of purge gas quantity and regeneration energy as shown in Table 4. As can be seen in Fig. 2, the gas temperature within the bed is maintained closely to the atmospheric temperature (293.15 K) in case of CJ TSA and 3OC TSA, while that within the bed shows high temperature in case of 2OC TSA, during adsorption step. -
Table 2. Operating conditions of cyclic TSA processes at CSS (uA = 45.2cm/s, U/acket = 17.2 cm/s)
Operating va[iables '
....
2OC TSA1 57.83 57.83 0.0 115.66 65.59
tA [ m i n i t1r [min] tc [min] tcycte [min] uR(=uc)[cm/s]
CJ TSA1 ....... 41.8 41.8 0.0 83.6 65.18 ................
.......
3OC TSA1 ........ 57.83 23.88 33.95 115.66 141.22
Table 3. Ratios of amount adsorbed at CSS to that at 1st cycle Processes 2OC TSA
Ql_css~1~ / QH~,.,c~ct~ Q2.css ~ct~ / Q2.1~.,~ct~
0.82
0.98
CJ TSA
0.96
1.00
3 0 c TSA
1.00
1.00
Table 4. Regeneration efficiency at CSS of each process Processes NR [mol/g/min] /)R [J/g/min] 9
20C TSA
8'89x 10-3
6.34x 101
CJ TSA
8.84x 10"3
6.30x 101
3OC TSA
1.90x 10"2
1.33x 102
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A x i a l p o s i t i o n = 15. 0 c m Axial position = 20.0 cm
A x i a l p o s i t i o n = 15. 0 c m Axial position = 20.0 cm
(a) 20C TSA
(b) CJ TSA (c) 3OC TSA Fig. 2. Gas temperature distribution within TSA bed
5. CONCLUSION In view of the continuous cyclic two-bed operation, the following facts were drawn: - The devised cooling jacketed TSA bed can be used as an alternative instead of the threestep operation, - The proposed new-jacketed TSA system enhances adsorption efficiency by 17.07 % for benzene and 2.04 % for toluene, compared with the previous two-step operating TSA bed without jacket, even though the purge temperature is very high, and - The jacketed TSA bed saves the consuming rate of the purge gas by 53.47 % and that of the regeneration energy by 52.63 %, compared with the previous three-step operating bed without jacket. REFERENCES
1. Schiesser, W.E. The numerical method of lines. Academic Press: New York, 1991. 2. D. Basmadjian, On the possibility of omitting the cooling step in thermal gas adsorption cycles. The Canadian Journal of Chemical Engineering, Vol. 53, (1975) 234. 3. C.C. Huang, and J.R. Fair, Parametric analysis of thermal swing cycles for multicomponent adsorption. AIChE Journal, Vol. 3 5, No. 10, (1989) 1667. 4. K.S. Hwang, D.-K. Choi, S.Y. Gong, and S.Y. Cho, Adsorption and thermal regeneration of methylene chloride vapor on an activated carbon bed. Chemical Engineering Science, Vol. 52, No. 7, (1997) 1111. 5. R. Kumar, G.R. Dissinger, Nonequilibrium, nonisothermal desorption of single adsorbate by purge. Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, (1986) 456. 6. J.M. Schork, J.R. Fair, Parametric analysis of thermal regeneration of adsorption beds. Ind. Eng. Chem. Res., Vol. 27, (1988) 457. Notation aair : ratio of the log mean surface area of the insulation to the volume of the column wall [1/cm]; ajac~t : ratio of the log mean surface area of the jacket to the volume of the jacket [1/cm]; as : particle external surface area to volume ratio [1/r aw : ratio of the internal surface area to the volume of the column wall [1/cm]; Cpg : heat capacity of gas [J/mol/K]; Cm : heat capacity of solid (0.47 J / g / K ) ; Cpw :heat capacity of wall (0.34 J / g / K ) ; Di.side bed :
432 inside diameter (2.2 cm); Djacket :jacket diameter (4.2 cm); Dz,i : axial dispersion coefficient [cm2/s]; Do,,tsidebed: outside diameter (3.2 cm); Dp : particle diameter (0.154 cm); hs : heat transfer coefficient of adsorbent (1.0x 10-2 J/cm2/s/K); hw : heat transfer coefficient of column (6.0x 10 .3 J/cm2/s/K); i : component identifier(/= 1 for benzene, i = 2 for toluene); KL : axial thermal dispersion coefficient [cm2/s]; L : packing bed length (25.5 cm); Lt: total bed length (40.0 cm); m : molar gas flow [mol/min]; n : moles adsorbed [mol/g]; ni*: moles adsorbed at equilibrium with y* and kI; Qi, lst cycle : adsorption amount of component i at 1st cycle; Q~,css ~yde : adsorption amount of component i at CSS cycle; R : gas constant [J/mol/K]; Rb : bed radius [cm]; Ta~r : ambient temperature (293.15 K); Tfeecl: feed temperature (293.15 K); 1"gas : gas temperature within the bed [K]; Tja~ket : coolant temperature within the jacket [K]; Tjacket, inlet: inlet temperature of coolant into the jacket (293.15 K ); tA : adsorption time [min]; tR : regeneration time [min]; tc : cooling time [min]; t~ycte : cycle time [min]; Trey: reference temperature (293.15 K); T~olgd: solid temperature [K]; Tw~u : wall temperature [K]; u : interstitial bulk fluid velocity [cm/s]; uA : interstitial bulk fluid velocity during adsorption step [cm/s]; uR : interstitial bulk fluid velocity during regeneration step [cm/s]; uc : interstitial bulk fluid velocity during cooling step [cm/s]; Uj~ket : interstitial bulk fluid velocity of coolant [cm/s]'~ [-fair : heat transfer coefficient of air (1.6x 10 -4 J/cme/s/K); Ujacket " overall heat transfer coefficient from the exterior of the column wall to the coolant within the jacket (4.0x10 -3 J/cm2/s/K); Vbea" bed volume [cm3]; Vj'a~ket" volume inside of the jacket [cm3]; yl " mole fraction of benzene (3.0060x 103); y2 " mole fraction of toluene (2.0004x 10-3); v~o~,,, 9volume flow rate of the coolant (100 cm3/s); c : bed void fraction (0.44); Cp : particle void fraction (0.67); pg : gas density [mol/cm~]; AH~: isosteric heat of adsorption [J/mol]; pp : particle density (0.714 g/cm3); p~ : column density (7.8 g/cm3).
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
433
Dynamic optimization for air separation plants A. Kr6ner a, Th. Kronseder a b, G. Engl a, O. V. Stryk c aLinde AG, Process Engineering and Contracting Division, Dr.-Carl-von-Linde-Str. 6-14, D-82049 H611riegelskreuth, Germany bTechnische Universit~it Mtinchen, Lehrstuhl fur H6here Mathematik und Numerische Mathematik D-80290 Mtinchen, Germany CTechnische Universit~it Darmstadt, FG Praktische Informatik/Simulation, D-64283 Darmstadt, Germany
Linde's process simulation and optimization tool OPTISIM | has been extended with optimization features for the formulation and the solution of optimal control and dynamic parameter estimation problems. The optimization problem is solved with state of the art SQP methods. Sensitivity calculations and appropriate discontinuity handling are essential parts of the algorithm. Computation of optimal load change policies for an air separation plant and parameter identification for a gas bottle filling process show the applicability of the chosen approach. 1. I N T R O D U C T I O N While steady state optimization and dynamic simulation have become state of the art tools for process simulation and design at Linde, dynamic optimization is now gradually gaining ground. This development was driven by the need for safer and more economical operation of transient processes. Two major dynamic optimization problems are addressed in this paper: 9 Model parameters such as heat transfer coefficients or pressure drops that are not available from steady-state design calculations must be estimated. Given a time series of measurements the model parameters are adjusted such that simulated and measured values deviate minimally in a least squares sense. 9 The optimal variation of control variables must be determined off-line as a function of time. In this case, unknown control parameters are adjusted such that an objective function is maximized or minimized, while constraints, e.g., on the purities or flowrates are satisfied. In the first part of this work, the properties of the dynamic simulation and optimization model developed in Linde's process simulation tool OPTISIM | [3] are discussed followed by a description of the applied numerical methods [2]. Two applications are presented in the second part.
434 2. M A T H E M A T I C A L M O D E L
2.1. Dynamic process model Dynamic modeling of a chemical engineering process results in a large-scale set of semiexplicit differential-algebraic equations (DAEs) of the form
0
--
f ( y , z , p , u , signq)
=
g(y,z,p,u, signq).
(1) (2)
The state variables x = (y,z) consist of differential variables y E I~ ny and algebraic variables z E I~n~, hence nx - - n y nt- n z. For petrochemical plants the size of Eqns. (1),(2) may reach nx -- 100000 equations. The functions f E R ny, g E 1~nz are piecewise continuously differentiable. For numerical simulation, the model parameters p E ~np a r e given constants, the control variables u E R nu are given functions of time t, and the state variables must satisfy consistent initial conditions x(to) = xo. Discontinuities in the model equations are described as zeros of the switching functions q = q(y,z, p , u ) E I~nq The numerical solution of the DAE strongly depends on its differential index [5]. Currently, there are no numerical integration methods for solving general systems of index greater than 2 directly and reliably. DAEs considered in this work have an index of at most 2. Further problems in the numerical solution of the DAEs result from discontinuities in the model equations. These are due to tabular data, piecewise constant control variables or operation dependent change of model equations. As each discontinuity must be treated as the initial point in time of a new initial value problem, consistent initialization is needed both, at the initial time of integration to and after each discontinuity. .
2.2. Optimization problem The optimization strategy implemented in OPTISIM | uses a direct single shooting approach in which optimization steps and dynamic simulation are performed alternatingly (feasible path method). The controls u in Eqns. (1), (2) subject to optimization are parameterized u c-~ a(p,t) introducing shape parameters/3. The resulting finite dimensional optimization problem is of the form (3)
minimize
J[/3]
-
#p(y(tf),z(tf),~,tf)
subject to
~(t)
-
f(y(t),z(t),p,a(~,t),t),
y(to) - yo E I~ny,
(4)
0
-
~(y(t),z(t),p,a(~,t),t),
z ( t o ) - zo E R nz,
(5)
0
>_ ~ t ( y ( t ) , z ( t ) , p , a ( p , t ) , t ) ,
to < t < t f .
(6)
To simplify notation, constant model parameters p not affected by the optimization and switching functions q are not stated explicitly. Both, however, are considered in the implemented algorithm. Functions f and ~ are introduced accordingly. Without loss of generality the objective is stated in Mayer form. Dynamic parameter identification problems result in a nonlinear least squares objective function (weighted 12 norm of deviation from measurements). This specific form of q~ is exploited in the implementation.
435
3. NUMERICAL METHODS 3.1. Dynamic optimization Using a time grid to - t h < ... < tht -- tf, the infinite dimensional path inequality constraints (6) are transcribed into nt.n h inequality constraints
0 >_ h (y(th),z(th),~,a(p, th),th) ,
i-- 1,...,nt.
(7)
The applied commercially available sequential quadratic programming (SQP) methods [ 1], [4] make use of gradient information ~d0 and dh calculated from sensitivites of the solution w.r.t. /3, i.e.,
~ff E ]~ny• r ( t ) - i)y(t)
,
s ( t ) - -~z(t) ~
c
]~nzxnp.
(8)
3.2. DAE system integration and sensitivity calculation For numerical solution of the model equations a modified BDF method with variable stepsize and order is available for semi-explicit DAEs with index 2 [5]. The sensitivities (8) at time tn+l - tn + rln+l are given by the linear system
0-
( q----n-n~+llny ctk d- fy J~ ) ( )Fn+l + ( --(rn+lqt-q-~+~?n+l) t~k + f p + fa@ ) gy
gz
Sn+ l
(9)
~p --1-~aap
~ r
D
Fn+l
which is obtained by total differentiation of the discretized model equations w.r.t./3. and r are estimates of the sensitivity r and its time derivative r at time tn+l based on k previous time steps. The coefficient t~ depends on the order k of the BDF method. Matrix D is available in decomposed form at no costs from the BDF corrector iteration.
3.3. Consistent initialization Solving DAEs numerically with arbitrary initial values y(t0), z(to) will not lead to a continuous solution. Consistent initial conditions must satisfy the original DAEs (1), (2) and, in case of semi-explicit DAEs with index greater than 1, also higher total differentials w.r.t, time of some of the equations [5]. Under certain restrictions, steady-state conditions can be used as consistent initial conditions. They are defined by the nonlinear equation system
00-
f(y(to),z(to),~,a(~,to)) ~(y(to),z(to),p,a(~,to))
(10) (11)
which originates from (4) and (5). Steady-state conditions are assumed in [to - e, to + e], e > 0. To initiate a continuous transient solution starting from a steady state, controls a(/),t) must satisfy higher order continuity conditions. Differentiation of (10) and (11) yields a linear system of equations for the sensitivities of the initial values
fz)(r(to) fp +fat~p ) ^ s(to))+(~p+~a@ gz
436
Fig. 1. Flow diagram for an air separation plant. where all functions are evaluated at t = to. After discontinuities at time tev during integration, consistent values for y(tev+), Z(tev+) and the sensitivities r(tev+), S(tev+) must also be computed. Robust and efficient methods are employed in OPTISIM | for consistent initialization of index 2 DAEs and the sensitivity equations. 4. A P P L I C A T I O N S The optimization algorithm implemented in OPTISIM | is applied to optimal load-change control of an air separation plant (ASP) and the dynamic identification of heat transfer coefficients in a batch filling process.
4.1. Load change policy for an air separation plant An ASP consists of three major parts: Feed air preparation, cooling and rectification. A standard configuration as shown in Fig. 1, comprises a high pressure column T 1, where the feed air is crudely separated into two fractions, one of which is the liquid nitrogen product (DLIN), a low pressure column T2, where highly pure nitrogen (GAN) and oxygen (GOX, LOX) are produced, and an argon column T3 with crude argon product (crude Ar). All process steps are tightly coupled through material and energy streams. The task is to decrease the load of the plant from 100 % air input to 60 %. The load change takes about one hour, the time of operation is from to = 0 [s] to tf = 6000 [s]. It is of utmost importance for stable operation and product quality that several purity restrictions are not violated during the load change. The air separation plant is modeled by a semi-explicit DAE system with index 2 consisting of about ny = 900 differential and nz = 2600 algebraic equations. The purity restrictions result in lower and upper bounds Xi,min _~ xi(t) ~_ X/,max for six state variables (cf. Tab. 1), i.e., nh = 12 in Eqn. (7). Five constraints refer to product quality. The sixth constraint is a stable operation constraint. The nu = 5 control variables describe the positions of valves a - e. Instead of a full parameterization of the control history, e.g., by piecewise polynomial functions, the optimization of already implemented control schemes which use a global parameterization
437
Table 1 Lower and upper bounds of purity constraints. description name min max oxygen fraction in liquid oxygen product 0 2 LOX 0.997 1.0 oxygen fraction in gaseous oxygen product 0 2 GOX 0.997 1.0 0 2 DLIN 0.0 5.0.10 - 6 oxygen fraction in liquid nitrogen product 0 2 GAN 0.0 5.0.10 -6 oxygen fraction in gaseous nitrogen product argon fraction in argon product Ar crude Ar 0.965 1.0 oxygen fraction in feed to argon column 0 2 ArFeed 0.90 1.0
Fig. 2. Purities and air flow for original parameter setting (values are scaled to lower and upper bounds from Tab. 1).
Fig. 3. Purities and air flow for optimized parameter setting. For legend see Fig. 2.
of u - t~(/~,t) is investigated. The controls are parameterized by np = 9 parameters. The state variable inequality constraints are discretized with a time grid of n t - - 10 equidistant points yielding n t . n h - - 120 nonlinear inequality constraints of the nonlinear programming problem (NLP). The objective is to maximize an integral term describing product gain. The Mayer form (3) is obtained by adding an additional differential equation. However, the operators are in the first place interested in finding a feasible control within the constraints for this highly complex plant. The starting values for the optimization parameters/~, for which the time histories of the relevant purities (Tab. 1) are displayed in Fig. 2, lead to a breakdown of the ASP, caused by several variables violating their bounds. A solution computed with optimized parameters is displayed in Fig. 3. All purities are now feasible within their lower and upper bounds which is most important. 4.2. Parameter identification Industrial gases, e.g., Oxygen, Nitrogen or Argon, are distributed at retail in high pressure gas bottles. At the filling station bottles are pressurized from vacuum to 200 bar with gas from a high pressure gas tank. In order to accelerate the filling process the heat balance of the bottle system must be investigated. To determine the heat flows in the system, coefficients for gas-bottle heat transfer need to be known. As no model for prediction of heat transfer coefficients in a fast pressurized gas volume is available from the literature, heat transfer coefficients must be determined from mea-
438
Fig. 4. Non-insulated bottling process, fitted and measured data for bottie (T_Bottle, T_BM) and piping (T_Pipe, T_PM) temperature.
Fig. 5. Insulated bottling process, simulated and measured data for bottle (T_Bottle, T_BM) and piping (T_Pipe, T_PM) temperature.
surements. The newly implemented algorithms are applied to a model tuning problem where constant heat transfer coefficients are identified based on data from experiments. Dynamic data reconciliation on measurements will be the scope of future work. A single depressurized gas bottle is filled from a bottle battery. The valve is opened at time zero and closed after 120 seconds when pressure equilibrium has been achieved. Temperatures at the filling pipe and at the outside of the single gas bottle are measured. Based on measurements from an experiment with a non-insulated bottle, heat transfer coefficients within the bottle are determined by dynamic parameter identification. The optimized results (solid lines) and the 8 measurements (., A) entering the optimization are shown in Fig. 4. YS_V01 denotes the valve position. The simulation model for the bottle battery, the single gas bottle, piping and valves has nx = 144 equations and an index of 1. Three heat transfer coefficients are fitted. The value of the objective function is reduced by 17%. A comparison of measurements at 9 points in time from an experiment with an insulated bottle with simulation results using the heat transfer coefficients determined above show the applicability of the fitted parameters to a related case (Fig. 5). REFERENCES
1. NAG Fortran Library Mark 18. The Numerical Algorithms Group Ltd., Oxford (1997). 2. Engl, G.; Kr/Sner, A.; Kronseder, T.; von Stryk, O.: Numerical simulation and optimal control of air separation plants. In: Bungartz et al. (eds.): Lecture Notes in Computational Science and Engineering, Springer, Berlin (1999) 221-231. 3. Eich-Soellner, E.; Lory, P.; Burr, P.; Kr/Sner, A.: Surv. Math. Ind. 7, 1 (1997) 1-28. 4. Gill, P.E.; Murray, W.; Saunders, M.A.: SNOPT: A SQP Algorithm for Large-Scale Constrained Optimization. Report NA-97-2. Dept. of Mathematics, University of California, San Diego, California (1997). 5. Brenan, K.E.; Campbell, S.L.; Petzold, L.R.: The Numerical Solution of lnitial- Value Problems in Differential-Algebraic Equations. SIAM, Philadelphia (1996).
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. J~rgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
439
TRIZ-Based Creative Retrofitting of Complex Distillation Processes --An Industrial Case Study Xiao-Ning Li, Ben-Guang Rong and Andrzej Kraslawski* Department of Chemical Technology, Lappeenranta University of Technology, P.O. Box 20, FIN-53851, Lappeenranta, Finland. *E-mail: [email protected] An integrated approach is presented for the retrofitting of complex distillation processes. It is founded on the combination of TRIZ-based (Theory of Solving Inventive Problems) creativity support method and thermodynamic analysis. A three-step hierarchical algorithm is formulated. It has been applied to an industrial case - retrofitting of butadiene extractive distillation plant. Two optimal flowsheets are obtained by using of the proposed method. 1. INTRODUCTION The essence of process retrofit is to improve the performance of the existing plant. Process retrofit can be even more difficult than grassroot design, because of the constraints resulting from the use of the existing equipment and the combinatorial complexity of redesign. The multi-objective character of process retrofit is an additional difficulty. Traditionally, process retrofit problems are solved based on grassroot design methods like heuristics, evolutionary methods, thermodynamic analysis and mathematical programming. In order to better handle the process retrofit problems, recent research was focused on multi-objective analysis (Dantus et al., 1996) and combined approach like the thermodynamic and algorithmic one (Kovac et al., 1995). However, the aspect of innovation in process retrofit problems, manifested by handling the contradictions and the conflicts among several objectives and technical constraints, is not tackled by the existing methods. In this paper, an integrated approach combining TRIZ-based theory and thermodynamic analysis is proposed for creativity support of the process retrofitting problem solving. It is illustrated by an industrial complex distillation process - the butadiene extractive distillation plant. 2. M E T H O D O L O G Y The main idea of TRIZ method (Altshuller, 1998) is to remove the contradictions identified when solving the inventive problems. TRIZ has been based on the extensive studies of patent information. As a result, there have been extracted 39 universal characteristics identifying any technical system and 40 principles for the conflict resolution. The principles used for inventive problem solving are generic suggestions. The characteristics and principles built a contradiction matrix. The simultaneous improvement of some characteristics usually causes the deterioration of the others. It is the source of conflicts. The basic idea of TRIZ-based
440 method is to identify the emerging conflicts. The conflicts generated by any technical system could be overcome by use of the contradiction matrix. Many contradictions and conflicts are generated by trading-off the retrofitting activities and the existing constraints. The TRIZ-based approach is adopted in order to systematically identify retrofit targets and remove process bottlenecks. All the characteristics of traditional distillation systems and the principles that govern the decision-making in process design are extracted to formulate the contradiction matrix. The bottlenecks of process retrofitting can be identified and represented by the contradictions. Next, the contradictions could be removed by examining all the available principles. When the identified conflicts are solved then the nearly optimal alternatives are generated. Finally, thermodynamic analysis is implemented for the further improvement of the generated alternatives. The integrated approach, combining the TRIZ-based creativity support method and thermodynamic analysis, can handle very well the multi-objective aspects of the retrofitting process. Moreover it extends the solution space from traditional distillation schemes to new, non-traditional ones. A three-step hierarchical algorithm is formulated for applying the methodology including analysis, search and implementation phases (Fig. 1).
Fig. 1.The algorithm of problem solving using the integrated approach 3. CASE STUDY An industrial process of butadiene extractive distillation is studied with the main objectives of the improvement of the environmental impact, energy efficiency and total plant cost. 3.1. Flowsheet
The feed - Crude C4, composed of seventeen hydrocarbons, is difficult to be separated by common distillation system. The process is realised in the system (Fig. 2), composed of six traditional distillation columns: 1st stripper (DA- 102); butadiene withdrawing column (DA-104); 2 nd stripper (DA- 105); 1st common distillation column (DA- 106); 2nd common distillation column (DA-107); solvent refining column (DA-108) and two extractive distillation columns (DA101 and DA-103).
Fig.2. The flowsheet of the industrial case
441 3.2. Retrofit targets
The multi-objective analysis requires retrofit activities to be carried out in an integrated way. Not only economic criteria but also environmental sustainability, flexibility, operability and dynamics have to be considered simultaneously. Due to the complexity of the units and the intensive energy consumption, in the case under discussion, the retrofit targets are focused on reducing capital costs, optimising energy consumption and improving sustainability. TRIZ contradiction matrix for distillation processes is composed of 30 characteristics and 29 general and specific principles (Rong et al., 2000). To deal with the retrofit targets, several characteristics are chosen in the matrix: capital cost (No.24), operating cost (No.25), complexity of the system (No.23), environmental impact (No.26), and flexibility (No.28). 3.3. Generation of process retrofit alternatives
The general conflicts can be formulated by analysing and matching the above-mentioned characteristics. Four main general conflicts have been identified. Next, the conflicts are removed by the appropriate principles identified thanks to the contradiction matrix (Table 1). Table 1 The general conflicts and the suggested principles Conflicts Cells Suggested principles 1-change method, 3-change agents environmental aspect/ 26• operating cost 29-balance computer & human role, 5-mass integration 9-mutifuncional units, 4-heat integration operating cost/capital cost 25• 15-side-stream column, 7-change reflux ratio 22-decomposition, 24-simplification operating cost/flexibility 25• 23- complex design, 25-analysing design process 22-decompostion, 23-complex design complexity/flexibility 23 • 24-simplification From the suggested principles, a few most useful ones are chosen, based on the heuristic rules for the retrofitting problems (Grossmann et al., 1987). Principle 3 (change agents) and principle 4 (heat integration) are preferred for the improvement of the operational performance of the existing flowsheet. Then strategies of decomposition and process integration implied by principles 22 (decomposition) and 23 (complex design) are applied in the search for the promising flowsheet structure. A hierarchical decomposition method is formulated by applying those principles, where the heat integration problem is solved in an inner synthesis step while the process flowsheet is optimised in an outer step. When applying those principles to remove the general contradictions, twelve further contradictions are formulated at the subsequent levels. 3.3.1. Solvent evaluation
As principle 3 suggested, DMF (dimethylformamide) has been used as the solvent in the extractive distillation process. It has been compared with other solvents like ACN (acetonitrile) and NMP (N-methylpyrrolidone). The excellent solubility and selectivity among butadiene and other C4 hydrocarbons have suggested that the process operation and control would not be so difficult task. Furthermore, its aqueous solution is not corrosive and toxic. Therefore DMF is the preferred solvent from the environmental point of view.
442
3.3.2. Heat exchanger network Heat integration is aimed at generating a network that will operate at minimum cost thanks to maximum recovery of the available energy and optimal use of the existing equipment. When realising heat integration, six conflicts have emerged involving the following characteristics: flexibility, operating cost, capital cost, heat exchanger number, reboiler number, temperatures and pressures. Then the appropriate principles used to cope with these conflicts are identified in the conflict matrix (Table 2). The recycled solvent, by merging the bottom flows of two strippers, is the main heat source due to the high temperature and flow rate. Thus, the most important issue for heat matching is to maximise the heat recovery from the solvent flow. All heat sources and heat sinks of this process are shown in Table 3. Table 3 The heat sources and sinks in the existing flowsheet No. Heat sources and sinks H1 H2 H3 H4 H5 H6 H7 C1 C2 C3 C4 C5
evaporator of C4 feed 1st reboiler of 1st extractive distill, column 2ndreboiler of 1st extractive distill, column 1st reboiler of 2ndextractive distill, column reboiler of 1st common distill, column 1st reboiler of 2ndcommon distill, column 2"d reboiler of 2nd common distill, column solvent flow from the two strippers bottom flow of 2ndextractive distill, column 1st condenser of 1st stripper condenser of 2nd stripper condenser of solvent refining column
Heat load (106kcal/h) 0.929 1.904 1.385 0.483 1.183 2.4712 0.7156 7.04 0.483 0.7343 0.3511 0.1256
Initial temp. (~ 42
Final temp. (~ 50.5
78
119
70.18 48.93
80.15 49.15
58.08
61.31
162.15 144 119 156.77 152.06
40 85 85 42.35 85
Principle 17 (heat matching among columns) is adopted first for effective use of existing energy sources. Principle 6 (change pressure and temperature) and principle 7 (change reflux ratio) suggest the further considerations for adjusting operational parameters. Temperatureenthalpy (T-H) diagram is the tool for analysing and visualising the energy flows among the concerned heat and cold streams. The T-H diagram of solvent flow and its matching flows suggest that the minimum approach temperature ATmin is too high around 27~ Through decreasing ATmin to 12~ the more efficient heat matching strategy is proposed by additional use of the existing heat sources including the condensers of the 2 nd stripper and the solvent refining column. Table 2 The conflicts and the principles for heat exchanger network Conflicts Cells Principles exchanger no./reboiler no. 16• 15 20,17,19,16 reboiler no./capital cost 15• 8,9,7 reboiler no./temperature 15• 14,6 reboiler no./pressure 15• 10 14,6 operating cost/capital cost 25• 20,17,9 flexibility/exchanger no. 28• 16 4,6,11 .
.
.
.
Table 4 The conflicts and the principles integration Conflicts Cells energy flow/complexity 11 • complexity/capital cost 23• capital cost/operation cost 23• complexity/temperature 23• complexity/pressure 23• 10 flexibility/complexity 28•
for process Principles
22,23,24,28 9,15,13 4,7,16-20 3,1,2,6 1,3,2,6
22,24,23,25
443 3.3.3. Process integration In this case, the work is focused on improving the topological structure and on considering the complex columns. For this purpose, the subsequent conflicts are identified and removed by the appropriate principles via contradiction table (Table 4). As principle 25 (analysing design process) and principle 22 (decomposition) suggested, for the sake of systematic analysis, the whole process is divided into the following four modules: extractive part (the 1st and 2 nd extractive distillation columns); solvent purification part (two strippers and the butadiene withdrawing column); product purification part (two common distillation columns); solvent refining part (the solvent refining column). In the second part, it has been observed that the butadiene withdrawing column and the 2 nd stripper have the similar composition, temperature and pressure distributions based on simulation results. A complex, thermally coupled distillation column is proposed to combine two above-mentioned columns as suggested by principle 23 (complex design) and principle 20 (thermally coupled schemes). The temperature and pressure of the complex column should be changed correspondingly as implied by principle 6. This modification can not only save the number of the heat reboilers but also decrease the energy consumption. The thermodynamic approach is used to explore the potential bottlenecks of the third part by studying the Column Grand Composite Curve (CGCC). 3.4. Thermodynamic analysis and evaluation of the retrofit alternatives Thermodynamic analysis of the CGCC can provide the required information for potential column modifications. Fig. 3 shows the profile of the CGCC of the 2 nd distillation column. It is clearly seen that there is a sharp enthalpy change near the feed location, which suggests that the feed state should be changed from liquid phase to vapour one. The strategy can be implemented through changing the sequence of two common distillation columns, then the top vapour flow of the 2 nd extractive distillation column directly enters into the 2 nd c o m m o n distillation column without condensation. The heavy impurities are removed away first, then the light ones are removed in the 1st common distillation column. The modification produces another thermally coupled column and eliminates the condenser of the 2 nd extractive distillation column. The new CGCC suggests that the sharp enthalpy change is improved and the load ofreboiler is greatly reduced by 38.5% (Fig. 3). water
of condensed steam
O
heat load of reboiler (existing) D
58 L) o%,
h
,
54
C4=,
i
:ii
i
C4= . . . . .
'
i'
i
c--~
50
......~
"'!
D~
4
46
"1 426e6-2e6--
2 Zie6--
2 8e6--
3 2e6
- 3 6e6
H (kcal/h)
Fig. 3.The modification of CGCC of the 2 nd common distill, column
!
"
i O
*
...............................i...................................
DMF i ! ............................................. :.............................
i
......................................................................................... Fig. 4.The proposed optimal flowsheet
444 When getting the optimal retrofit flowsheets, heat integration is carried out for efficient use of energy in the proposed alternative flowsheets. Two optimal flowsheets have been obtained based on the different heat matching strategies. Fig.4 presents one alternative with the new structure and the heat matching strategy. Simulation results show that the saving of the consumption of steam utility for those two optimal flowsheets are of 23.7% and 27.7% respectively compared with the existing plant. Based on the above procedure, a tree structure pattern including three levels of contradictions is formulated to illustrate the TRIZ-based process retrofit as shown in Fig.5.
Extractive Distillation --~ . . . . . . . . . . Plant . . . . . . . . . . . .
~9 . . . . . . . . . . . . . . . . . . . .
environmentI loper, cost flexibility ] ]complexity oper. cost Ilcapitalcost] I ~176 flexibility [ selection
2- oc s inte.~ation
2 (DMF)
intez~ration
116xl lSx ?A
I I
9- 128x161
h~at matchTmg
thermally co led distill, column thermally coupled column instead of the 2"a strip_p_erand butadiene withdrawing column
3
g
CGCC of thermodynamicanalysis
4, changing sequence of two common distill columns
~' feed stat e changmg of 2"~ column
c~conflicts -~principles 1,2,3hierarchical steps Fig. 5.The tree-structure pattern of the TRIZ based process retrofit
4. CONCLUSIONS An integrated approach combining TRIZ idea and thermodynamic analysis is proposed for process retrofitting which is realised by a three-step hierarchical algorithm. TRIZ method is used to generate the promising alternatives by identifying the corresponding principles in order to remove the recognised contradictions and conflicts. The identified principles are used as the tools in search for the optimal flowsheets. Thermodynamic analysis could be applied to the search of potential retrofitting targets and possible solutions for further improvement. The presented approach and algorithm are applied for the retrofitting of the butadiene extractive distillation plant. A hierarchical tree-structure procedure is formulated to illustrate the integrated approach. Two optimal retrofitting flowsheets are obtained that allowed to decrease the consumption of steam utility 23.7% and 27.7% respectively. REFERENCES 1. M.M. Dantus and K. A. High, lnd. Eng. Chem. Res., 35, 4566-4578, 1996. 2. A. Kovac and P. Glavic, Comput. Chem. Engng., 12, 1255-1270, 1995. 3. G. Altshuller, 40 principle: TRIZ keys to technical innovation, Technical Innovation Center, Inc. MA, USA, 1998. 4. B. G. Rong, A. Kraslawski, L. Nystr6m, Computer-Aided Chemical Engineering, 8, 625630, Elsevier Science, 2000. 5. I.E. Grossmann, A. W. Westerberg and L. T. Biegler, Proc o f I st Int. Conf. on Found. o f Comp. Aided Proc. Oper., 403-422, Park City, Utah, July 1987. 6. X. N. Li, Study on Process Integration of Butadiene Extractive Distillation Plant, M.Sc.Thesis, Qingdao Institute of Chemical Technology, China, 1999.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
445
Retrofit of Refinery Hydrogen Systems F. Liu and N. Hallale Department of Process Integration, University of Manchester Institute of Science and Technology, PO Box 88, Manchester, M60 1QD, United Kingdom Several trends in the petroleum industry are leading to increased demand for hydrogen in oil refineries. Previous work developed a methodology for the analysis of hydrogen distribution systems. An automated approach has now been developed to include pressure issues in the design. The method is based on optimisation of a reducible superstructure. Retrofit options are decided automatically through optimisation. 1. INTRODUCTION Hydrogen is vital for oil refiners to face the trends caused by the tighter environmental requirements and heavy-end upgrading. Also a capacity increase or change in product slate of an existing refinery is often constrained by the hydrogen availability. Reducing the sulphur composition in fuel means more hydrogen is necessary for deeper hydrodesulphurisation. More hydrotreating is needed to achieve high-cetane diesel. In the meantime, lower aromatic gasoline specification will decrease the operation severity in catalytic reformers, reducing the by-product hydrogen formed. Another increase in hydrogen demand is caused by bottom-of-the-barrel upgrade. Because of stricter limitations on pollutant emissions, there has been a sharp reduction in the fuel oil market. On the other hand, market trends indicate a very large increase in diesel oil and jet fuels production. Hydrocracking can play an important role in heavy-end conversion because of its considerable flexibility and high quality of the products, but requires large amounts of hydrogen. Alves (1999) proposed a pinch-based approach for targeting the minimum hydrogen utility. A general simplified model of hydrogen consumers can identify the sinks and the sources in hydrogen systems. The flowrate and purity for sinks and sources in the network can be extracted from operating data. These can be plotted as composite curves, which will then give targets for the minimum hydrogen utility. The method is similar to energy targeting in heat exchanger networks (Linnhoff, 1993). To achieve the target from hydrogen pinch, the network can be designed by using Linear Programming. The objective function is to minimise hydrogen utility. However, the hydrogen targets from the surplus diagram may be too optimistic as they do not consider pressure. In reality, a source can only feed a sink if its pressure is sufficient. This can be accounted for by assuming new compressor installation, which however can lead to impractical design and unrealistic compression costs. In this work, we outline an automated approach for retrofit of hydrogen systems, with attention to pressure and compressor constraints.
446 2. NEW A P P R O A C H - AUTOMATED DESIGN 2.1 Network superstructure After the decomposition of hydrogen systems, we can build a superstructure including all the links between sources and sinks. A simple example in Figure 1 demonstrates the decomposition procedure. The inlet pressure of the sinks and the outlet pressure of the source are assumed fixed at their design values.
The basic mass balance constraints have been applied in equation (1) to (6). The existing compressors are decomposed into sinks and sources. Unlike the sinks and sources of the hydrogen consumers, both the flowrate and the purity of compressor sources are variables, which leads to non-linear items in the hydrogen balance equations (4) and (5). There are maximum flowrate constraints imposed on the existing compressors in equation (6). These will be equal to the design flowrate plus any spare capacity that may exist. Banning the matches between sources in lower-pressure levels and sinks in higher-pressure levels reduces the size of the superstructure and ensures that the pressure limitations are included. We can also impose practical constraints e.g. forbid links between processes for reasons of plant geography or contamination.
~., Fs~,.j = Fs,.* ,
VEr
(1)
J
~_F~,sk = Fsk*,
VSk
(2
!
(3) i
Z
-Z
j
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~ Fcompj < FMo~i.... Comp"
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,
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Ysr F. YSkI 200psi~ S i n k A 1600psi 1500psi~ SinkB 2200psi 1700psi~ ~ ' ~ ~ 1 6 0 0 p s i ~
200psi
2200psi~
200psi
1 6 0 0 p s i ~
1500psi
2200ps, [ ~ i (a)
~
1700psi
(b)
Figure 1. (a) A simple existing hydrogen network (b) Superstructure linking sources and sinks
447 2.2 Retrofit options and costs The insights show that it is possible to save hydrogen by piping changes, extra compression to overcome pressure difference and purification. The new compressor model can be built as in equation (4) and (5) by considering no maximum flowrate limit (other than manufacturers limitations) and variable operating pressures. The inlet pressure must be the lowest pressure over all the sources feeding the unit. The power of new compressor is decided by equation (7), which could be simplified to a linear equation once the operating pressures are fixed.
PcoMP= Y'T'R'N(Y--A~ (y--(;~] rr'N-1 I FC~
(7
A purifier can be a pressure swing adsorption (PSA) unit, membrane or cryogenic process. This work assumes that a PSA is used. With specified recovery (R), and product purity (yp), the PSA can be modelled by mass balance equations as Figure 2. The inlet pressure of PSA (Pin) must be the lowest pressure over all the sources feeding the unit. The pressure drop (AP) over a PSA is low and so the product pressure is usually very close to the feed pressure. The residue is released at very low pressure. Other purifiers will have different pressure characteristics.
.
Fi, purifier, P,
I
Feed
~ikF F
feed flowrate__~J
/'yF feed purity ~ ~P0n=min(P,)
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] L~
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I
]
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I
-p
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I Residue Residueflowrate FR=FF_Fp "-
Residue purity y. =
~
FF-yF - F yP'..~ 9 .v ~ ~k
Figure 2 PSA model To find the optimum retrofit design automatically, the objective function has been set as minimum cost (operating and capital). The retrofit options will be selected systematically by considering the cost trade-off. The investment costs of retrofit o p t i o n s - compressors, purifiers and piping changes can be described as linear equations (8) to (10). The coefficients can depend on operating conditions (e.g., pressure and temperature) in different parts of the system. I,,,~ = (a,,,E + bp,,~" A). L (8 I p,,,= a,,,, + b,,,, F,,,,
(9
i~,o =aco~+bc,o .Pco~
(10)
The operating costs of hydrogen systems are dominated by hydrogen utility cost, compression cost and exported fuel value. The automatic retrofit design requires Mixed-Integer Non-linear Programming. The initialisation from the relaxed solution by Mixed-Integer Linear Programming (Quesada and Grossmann, 1995) can improve robustness and help to locate good optima.
448
235~ I co. I
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44.35 92.00% 'k_/~
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~ 8.78 I.cu I
,, _., 10.66 ~
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,41
!
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I
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*
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Flows in MMscfd
,[, ~]' 12.(
kerosene hydrotreater
CNHT cracked naphta hydrotreater NHT naphta hydrotreater
12.80
NHT
6.5~ 60.00%
DHT diesel hydrotreater
HDA hydrodealkylation I Fuel
Fuel fuel gas system
Figure 3. Case study - existing hydrogen system
3. C A S E S T U D Y
An existing hydrogen system in a refinery is shown in Figure 3. Currently, 45 MMscfd of hydrogen are produced from the hydrogen plant. The objective of retrofit design is set as maximum operating cost saving. The utility prices are: hydrogen utility $2.0/MMscfd, electricity $0.03/kWh and fuel cost $2.5/MMBtu. This case study assumes that all the existing compressors have 5% spare capacity, and that a maximum of one new compressor and one new PSA (produces 99% hydrogen, recovery = 90%) will be allowed. A payback of less than two years is specified.
I H Plant l 92.00% 28.61
J 23.5
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Figure 4. Case study - optimum retrofit design
449 Table 1. Cost breakdown Existing network Operating costs Hydrogen Compression Fuel export Total Investment costs Compressor PSA Piping change Total
Retrofit design $20.9 million/yr $1.87 million/yr -$6.24 million/yr $16.5 million/yr
$32.9 million/yr $1.77 million/yr -$12.2 million/yr $22.5 million/yr
$1.00 million $ 7.02 million $1.78 million $ 9.8 million
The optimum retrofit design is shown in Figure 4. To minimise the operating cost, both a new compressor and a PSA are used, as well as introducing some new pipes. The dotted lines show the piping changes and the new units. The new compressor is used to accommodate the increased recycle requirement for the NHT, as well as to compress a feed to the PSA. The cost breakdown is shown in Table 1. The possible saving of operating cost after retrofit is 6 million US$, and the payback period is 1.6 year. Often, refineries have a limited capital budget for projects. The hydrogen system shown in Figure 2 is analysed subject to the budget constraint less than 5 million US$. The possible saving of operating cost after retrofit is 3.5 million US$ which is lower because of the smaller capital investment and the payback period is 1.4 year. The retrofit design of the system (Figure 5) shows that if the capital budget for retrofit project is limited to 5 million US$, the best investment is in a PSA and not a compressor.
35.40 0.19 H2Plant J 92.00% ~2-------]~_~.-
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,~
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99.00% .............................................. ]PSAI 3.70--[" 19.29% { .
...............
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....... ~_ ............
75-00%~8.65 65"00~
_--T-.-;-.--~
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I CCR I75.00% 23.5
'4' ] j NHT
. . . . . . . . . . . . . . . . . .
60.00%Jr6.56
3.81
i ~Fuel Figure 5. Case study - retrofit design with limited capital budget
450 4. CONCLUSIONS Efficient hydrogen integration in refineries is very important. Previous work developed a methodology for the analysis of hydrogen distribution systems. Targets are set for hydrogen recovery and hydrogen plant production. However, the approach has neglected pressure constraints leading to an overly optimistic solution. An automated approach has now been developed to include pressure issues in the design. The method is based on optimisation of a reducible superstructure. The objective function of optimisation is set as minimum costs to consider hydrogen saving and penalty simultaneously. Multiple practical constraints can be implemented to achieve economic, realistic designs. Retrofit options (for example, additional compression, purification and piping changes) are decided automatically through optimisation. The case study demonstrates the potential of the method for cost savings with specified payback times and/or investment limits. It can also be applied to debottlenecking. NOTATION F F* ly I a,b PCOMe FpUR
flowrate flowrate as required or offered purity investment costs cost coefficients compression power feed flowrate of purifier
A L T N r/ r 7
cross-sectional area of pipe pipe length inlet temperature number of stages isentropic efficiency pressure ratio ratio of heat capacities
Sub-script i j Sr Sk Comp COMP P UR PIPE
source sink process source process sink compressor source/sink compressor purifier piping
REFERENCES 1. Alves, J. Analysis and Design of Refinery Hydrogen Distribution Systems, PhD thesis, UMIST, 1999 2. Linnhoff, B. Pinch Analysis - A State-of-the-art Overview, Transactions of IChemE (Part A), 71 (5), 1993 3. Quesada, I. and Grossmann, I. E. Global Optimisation of Bilinear Process Networks with Multicomponent Flows, Computers & Chemical Engineering, 19 (12), 1995
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
451
C o m p u t e r - Aided Synthesis of Molecular Mixtures and Process Streams E. C. Marcoulaki a*, A. C. Kokossis b and F. A. Batzias a aDepartment of Industrial Management, University of Piraeus, Karaoli & Dimitriou 80, Piraeus 185 34, Greece bDepartment of Chemical and Process Engineering, University of Surrey, Guildford GU2 7XH, United Kingdom This work considers the computer-aided design of novel solvents to facilitate difficult separation processes. These materials can be pure or within mixtures of substances that decrease the affinity between the components of the process feed. The task is formulated as an optimization problem to generate superior alternatives according to a set of objectives and constraints. The optimization process takes the form of a stochastic search among different molecular configurations and their contributions in the screened solvent blends. The search is combined with available group contribution methods (e.g. UNIFAC models) to predict the thermodynamic and environmental behavior of the designed materials. The method is employed to suggest novel blends of entrainers to replace extraction solvents commonly used in the industry. 1. INTRODUCTION Computer-aided molecular design applications assume an increasing significance over the last years. The advances on computer systems have enabled the development of powerful tools for property prediction by means of molecular and dynamic simulation. Less accurate, group-contribution (GC) methods are frequently the only available choice, when there is lack of experimental data or time. These technologies predict the thermodynamic properties of prespecified compounds, by establishing a structure-to-property link. Molecular design synthesis (MDS) tools reverse this information flow and formulate a mathematical problem for the development of molecular structures with desired behavior. This work considers the design of solvents to enhance the separation of materials in processes like 11- extraction, extractive distillation, absorption, etc. The first optimization tools for entrainer design were developed by Machietto et al. (1991) who presented a generalpurpose tool, assuming a continuous representation of the functional groups in the solution. The continuous representation enabled the formulation of the problem as an NLP, resulting though in poor interpretation of the results. Recently, Pistikopoulos & Stefanis (1998) formulated an M1NLP problem to consider the synthesis of non-toxic substituents for commonly used solvents. The technology was extended (Buxton et al., 1999) to the selection of environmentally benign blends. Hostrup et al. (1999) presented a hybrid scheme of mathematical programming assisted by heuristics to reduce the problem size. Author to whomcorrespondenceshould be addressed
452 Apparently, there are no safeguards to prevent the M1NLP technology from yielding inferior results, even one or two orders of magnitude away from the optimum. The use of smaller groups, constraints and bounds may enable applications, but, at the expense of the novelty and richness of the problem. Marcoulaki & Kokossis (1998) proposed a stochastic design tool, and reported novel molecular structures and significant improvements over other techniques. The authors (2000a) presented a generic representation of the molecular synthesis problem, along with general and systematic search strategies. Applications included large synthesis problems of solvent design using UNIFAC (2000b). The examples launched a conceptual search based on property predictions, or considered economic objectives using process simulation. The methodology was compared to conventional tools and was found powerful enough to encourage applications on even more complicated systems. Marcoulaki et al. (2000) applied the technology to design pollution-preventing alternatives of traditional chemicals for extractive distillation and liquid-liquid extraction. The approach used here adopts the stochastic technology, and extends the representation and search options to address molecular blends. 2. METHODOLOGY The effort here is to identify promising molecular configurations within a blend that satisfies certain process objectives under a set of constraints on the properties. The search strategy follows the principles of Marcoulaki & Kokossis (1999, 2000a) in order to (i) develop targets for the performance of the solvents, thus justify and assess incentives for replacing materials commonly used in the industry, and (ii) present venues to achieve mixtures of novel chemical compounds, thus explain the development of designs that are close to the targets. The search can be adjustable to preferences and arbitrary synthesis objectives and constraints. There is no need to limit the number of groups or components, nor to introduce artificial constraints just to make the optimization applicable. The variety of synthesis alternatives can only be considered using a large number of discrete and continuous variables that have to be optimized. The MDS problem is solved using stochastic optimization in the form of Simulated Annealing.
State Representation In the synthesis framework, each design instance defines a problem state. The design instance is hereby a solvent blend made up of various molecules in certain concentrations. The molecules are represented in terms of series of molecular and composition vectors of functional groups, as defined by Marcoulaki & Kokossis (2000a) for pure compounds. The optimization variables refer to the existence and the number of occurrences of each group in the molecular vectors. Additional variables account for the concentrations in the resulting component mixture. The optimization variables are subject to physical constraints on the feasibility of the representation vectors, including 9 connectivity features, so that the configurations have zero valence, the aromatic rings consist of six carbons, etc. 9 necessary features, so that the group collections conform with the representation and the property prediction models employed 9 desirable features, e.g. to specify limits on the maximum occurrences of certain groups.
453
State Perturbations A set of m o v e s is defined to generate alternatives and monitor the synthesis search. The moves are such to facilitate efficient perturbations from one feasible state to another. With respect to the state representation, the perturbations are divided into 9 b l e n d concentration moves (to alter the concentrations of the solvent constituents), and 9 m o l e c u l a r composition moves (to alter the group configurations in the constituents) The strategy adopted here enforces feasibility without resorting to trial-and-error, and consists of a p r e p r o c e s s i n g stage, a move selection stage and a p o s t p r o c e s s i n g stage. The last stage consists of minor restoration actions so that the molecules and their blends obey the feasibility constraints discussed. The preprocessing is used to set up the domains for the different possible modifications that can feasibly be applied on the molecular vectors. This stage ensures that the infeasible intermediates created during the moves can easily be restored to yield feasible new states. The move set accounts for the gross actions of 9 substitution, where an existing group is replaced by a different group, 9 contraction, where an existing group is removed from the configuration, and 9 expansion, where a new group is introduced, applied on each molecular vector. These perturbations are based on the moves developed by Marcoulaki & Kokossis (2000a), extended to address multiple molecular and composition vectors.
Implementation The algorithm is implemented as an iterative procedure, which perturbs the states based on a predefined stochastic matrix of probabilities, and follows a hierarchical scheme. The entries of the concentration vector are discretized and the random perturbations follow probability distributions similar to the scheme suggested by Marcoulaki & Kokossis (2001). This assumes a bounded continuous variable, to define a quasi-symmetric probability distribution that is biased in the vicinity of the current instance of the variable, and void outside the bounds. The molecular composition moves are applied on a randomly selected molecule of the mixture, based on a uniform probability distribution. The state is simulated to predict the physical and thermodynamic properties relevant to the process performance and the environmental behavior of the materials involved. The simulation problem involves complex nonlinear thermodynamic models (like UNIFAC), and property mixing rules most suitable for each property. The assessment can also consult process simulation components that are interfaced to the optimization tool and the group contribution models/databases. The state evolution and convergence follows the Simulated Annealing acceptance criteria, using the modified Aarts-Laarhoven cooling schedule proposed by Marcoulaki & Kokossis (1999). 3. DESIGN OF SOLVENT BLENDS TO SUBSTITUTE EXISTING SOLVENTS There are a number of favorable attributes that can be associated with an existing solvent. In replacing the solvent all of these attributes need to be considered simultaneously in the formulation of the optimization problem. These advantages may include 9 high distribution coefficient, to increase the separation driving force 9 high selectivity of the solvent towards the solute 9 high solubility of the solute in the solvent phase 9 high boiling point temperature, to allow sufficient driving force for solvent recovery
454 9 9 9 9 9
high density, to enhance immiscibility low heat capacity, to reduce the energy requirements of the entire process low viscosity, to reduce pumping and recycling costs low solvent losses, to reduce the amount of solvent wasted in the raffinate phase high decomposition temperature, to allow high temperatures in the process, etc.
Synthesis case study Consider the "sulfolane" process, an aromatic-paraffinic separation named after the solvent commonly associated with it. The separation is described as a combination of liquid extraction and extractive distillation (Meyers, 1997). A reformate of paraffins, aromatics and naphthenes is first processed to remove the heavier fractions. The remaining hydrocarbons enter the "sulfolane" unit, where the solvent extracts the paraffins and naphthenes from the aromatic raffinate. Data for the feed composition are taken from Ancheyta-Juarez & AguilarRodriguez (1994) and reformed as a solution of groups. The solute mixture of paraffins and naphthenes is represented as a single pseudo-molecule of an average group composition. The raffinate is a mixture of benzene and toluene, represented by a similar vector. Note that the averaged representation yields the correct results when UNIFAC is used, but does not necessarily agree with other GC-methods. The case study considers the design of a solvent that has all the beneficial properties of sulfolane and more. Marcoulaki & Kokossis (2000b) presented a multi-criteria scheme that simultaneously maximized the solute solubility, the solvent selectivity and the solute distribution while minimizing the solvent wasted in the raffinate. The reported solutions were considerably complex arrangements, even when the synthesis objective was replaced by a molecular complexity index (CI). This work addresses better the practicalities, and demonstrates that the use of solvent blends can yield more conventional alternatives of equally high performance. In this example the constraints are modified to include Ss > 3.3 wt./wt. (0.2902) 9 lower bound on solvent selectivity Sb > 0.15 wt./wt. (0.1291) 9 lower bound on solute solubility in solvent Sl < 10. wt. % (78.%) 9 upper bound on solvent losses to the raffinate M > 0.3 wt./wt. (0.2104) 9 lower bound on solute distribution coefficient Ps > 1000 kg/m 3 (1264.) 9 lower bound on solvent density cpL,s < 0.5 cal/g.K (0.40) 9 upper bound on solvent liquid heat capacity LC50s < 2.0 (minor) 9 upper bound on solvent toxicity 9 upper & lower bounds on solvent boiling point temperature (560.K) TBP,S > 500.K & TBP,S < 550.K The values of the constraints are based upon the relevant properties of sulfolane, given in the parentheses. Sulfolane appears very high on solvent losses, which indeed explains the need for an additional unit to purify the aromatic phase and recover the solvent. Since the constraints alone enforce the design of advanced replacements for sulfolane, the process objective is initially to minimize the molecular complexity of the solvent and its constituents. Once the complexity becomes small enough, a different objective can put into action. So the cases discussed here are considering (a) the complexity objective, with the CI formula refined for this application (b) the minimal complexity as the primal scope, while the distribution coefficient presents a secondary objective.
table 1 Results for the molecular com ~lexity objective for mixtures and pure components
CI
Itl
Ss
[9
3.330 3.786 3.786 3.787 3.787 3.586 3.704 3.964 4.036 4.085 5.568 5.569 5.675 5.720 6.348
1121 1121 1120 1119 1110 1128 1143 1150 1168 1161 1146 1299 1274 1066
S! ~
_
. _
5.957 6.014 5.865 m
5.808 g~935 8.248 8~261 8.174 71153 1.565 7~538 z~440 9305 1.100 , , , m
, .
'TBP ,, COL LCS0 504.6- .3096 506.4 .3091 501.9 .3104 5odi~ .3109 505.3 .3137 515.1 .3065 510.7 ,3036 500.8 .3033 514.7 ,2970 507.1 ,3085 515.3 3083 501.0 ,2977 519.9 2765 509.9 ,3409
Table 2 Results with a combination of minimal CI Ss p , SI % TBp 3.376 1079 9.840 501.7 3.590 1 1 0 3 9.899 500.9
vI
3.241
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9.678
505.1
3.402
1091
9.777
502.3
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M
SOLVENT DESIGN (molecules and }heir compositions)
.3726 .3726 .3727 .3727 .4068 .3911 .3949 .3967 .3763 .2612 .3300 .1973 .1959 .2622
6076 ,6075 6077 6077 6633 6376 6439 6468 6135 4259 5380 3217 3194 4275
.3495 CH3-NO2 + .6505 5ACH-AC-CH(NO2).(CH2)3Br ,2680 CH3-NO2 + .7320 5ACH-AC-CH(NO2)-CH2-Br 2582 CH3-NO2 + .7418 5ACH-AC-CH(NO2)-CH2-Br ,2836 CH3-NO2 + .7164 5ACH-AC-CH(NO2)-CH2-B_r ,2934 CH3-No2 + .7066 5ACH-AC-CH(NOz)-CHz-Br ,2117 CH3-CHz-OH + .7883 5ACH-AC-CH(NO2)-CH2-Br ,1411 CH3OH + .8589 5ACH-AC-CH _(NO2)-CH2-Br 1777 CH3COO-CH2-CH2-CH3+ .8223 5ACH-AC-CH(NO2)Br ,1998 CH3COO-CHz-CH3 § .8002 5ACH-XC-CHeqOz)Br 1275 CH3COO-CH2-CH3 + .8725.5ACH-AC-CH(NO2)Br 1.000 CHs(C=Q),CH2-CH2(C=O)-CHzsBr 1.000 CH3COO-CH2-CH2COO-CH2-CH2-Br [.000 H0-CHz-CHz(C=O)-CH2-Br [.000 Br-CH2COO-CH2-CH2-N02 [.000 C1CHz-CHzCOO-CHz-NO2 1.000 CH3COO-CH2CH(-Br)-CH2-OOCCH3.
.
.
.
.
.
.
molecular complexity and maximal distribution coefficient (M) for solvent mixtures CPL LC50 Sb M SOLVENT DE_SIGN (molecules and their compositions ) .3265 2.687 .4297 .7007 .6861 5ACH-ACCH(Br)NO2 +.3139 CH3(CH2)3-CH2 -CH2-OH .3111 2.813 .4287 .6990 .7569 5ACH-ACCH2CO-Br+.2431CH_3-(CH2)3-NO2 8092 CH3-(CH2)2-CH(Br)-CH2-NO2 3253 3.544 ,4153 6771 +. 1908_.5ACH-AC-CH(CH3)N02 9625 CH3-CH2-CH(Br)-CH2-CH2-NO2 3145 3.482 ,4124 6724 + .0375 4ACH-(ACCH3)AC..CH2-CH(CH3)2 9844 Br-(CH2)4-CH2-NO2 2984 3.048 ,4055 6611 _+ .0156 4ACH-(ACCH3)ACCH2-(CHz)2-CH3 9
3.485
[.978 1.994 [.954 [~939 [.985 [.868 2.000 1.657 1.851 1.840 1.315 1.784 1.764 1.784
Sb
,.
456 Each stochastic experiment included 20 to 30 runs. The length of the Markov chain is 150 for single molecules and 350 for mixtures, and the parameter 5 controlling the annealing speed is 0.06. The solutions are reported on Tables 1 and 2 for cases (a) and (b), respectively. The optimization results involve binary mixtures, unless the problem is restricted to single molecules (last six solutions of Table 1). The search is efficient enough to ensure the designs satisfy the synthesis constraints, and therefore present significant benefits over sulfolane. The designs of Table 1 show that the use of solvent blends yields simpler molecular arrangements. Most of the generated blends involve nitromethane (in composition 25-35% mole/mole) with aromatic nitro-bromides. Similar groups appear in the solutions for pure compounds, though here the groups are forced to reside on the same molecule and increase its complexity 9Since process performance is not included in the objective of case (a), there is no account for high or low distribution coefficients, selectivities, solubility, solvent losses, etc. The setup of case (b) uses the previous results to get a reasonable CI threshold, set here at maximum 50. Below this value a process objective is activated, like the maximization of the solute distribution coefficient (M). The first two designs of Table 2 set the M target at 0.7. The experiment also yielded slightly inferior solutions that approach single molecules with 96-98% purity. Similar to case (a) NO2 and Br groups appear in most of the optimal solutions. 4.
CONCLUSIONS
This work presents optimization technology for the design of novel molecular configurations. The technology appears an extension of the stochastic tools developed by Marcoulaki & Kokossis (2000a), to address molecular blends. The use of mixtures instead of pure molecules increases the complexity of the optimization problem and the technologies involved in the optimal search. Nevertheless, the solution space is extended, thus the final designs are better in their performance and simpler in their molecular structure. Certainly, the introduction of additional components in the solvent mixture presents an issue of concern that one needs to address at the subsequent design stages of validation and experimentation. The method is illustrated with a case study involving the design of entrainer blends for the replacement of commonly used industrial solvents. The synthesis constraints consider a series of separation-enhancing features, as well as desirable properties and environmental behavior. A hierarchical multi-objective approach is illustrated to address the complexity of the molecular designs prior to consulting a suitable process objective formula. REFERENCES
Ancheyta-Juarez and Aguilar-Rodriguez. Oil and Gas J., 91 (1994) 5, 93 A Buxton, A. G. Livingston and E. N. Pistikopoulos. AIChE J., 45 (1999) 817 M. Hostrup, P. M. Harper and R. Gani. Computers Chem. Engng, 23 (1999) 1395 S. Machietto, O. Odele and O. Omatsome. Trans. IChemE, 68 (1990) 429 E. C. Marcoulaki and A. C. Kokossis. AIChE J., 45 (1999) 1977 E. C. Marcoulaki and A. C. Kokossis. Accepted at Trans. IChemE (2001) E. C. Marcoulaki and A. C. Kokossis. Computers Chem. Engng, 22 (1998) S 11 E. C. Marcoulaki and A. C. Kokossis. Chem. Engng Sci., 55 (2000a) 2529 E. C. Marcoulaki and A. C. Kokossis. Chem. Engng Sci., 55 (2000b) 2547 E. C. Marcoulaki, A. C. Kokossis and F. A. Batzias. Computers Chem. Engng, 24 (2000) 705 9 " A. Meyers. Handbook of petroleum refining processes. 2n . Ed., McGraw Hill, NY, 199 7 E. N. Pistikopoulos and S. K. Stefanis. Computers Chem. Engng, 22 (1998) 717
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rights reserved.
457
A tool for optimal synthesis of industrial refrigeration systems F. Marechal I and B. Kalitventzeff 2 LASSC -University of Liege, Sart-Tilman B6a, B-4000 Liege (Belgium) A method for selecting the best refrigerants and the optimal configurations of the refrigeration system starting with the definition of the refrigeration requirements of a process is presented. The method proceeds in three steps: 1) identifying the most important temperature levels using an exergy minimisation method, 2) identifying the most suitable refrigerants to be used to generate a refrigeration system superstructure, 3) using MILP optimisation extract a set of refrigeration system configurations including compressors and heat exchangers characterisation to obtain the complete list of streams to be considered for the Heat Exchanger Network design (fourth step of the method not treated here). 1. I N T R O D U C T I O N The optimal insertion of energy saving technologies in industrial processes is a key issue for the rational use of energy in the process industry. Among the energy saving technologies, the refrigeration system is a technology area that introduces several degrees of freedom. The goal is to identify the best refrigerant(s), the optimal temperature levels and the best cycles configuration as well as the selection of the best compression technology to satisfy the refrigeration requirements of a process at minimum cost. Most of the tools developed to solve this problem use MILP (Shelton and Grossmann, 1986; Vaidyaraman and Maranas, 1999) or MINLP (Colmenares and Seider, 1989) software tools while graphical methods allow to understand the refrigeration cycle integration and deduce heuristic synthesis rules (e.g. Townsend and Linnhoff, 1983). The major difficulty related to the operational research based methods is the poor interaction between the engineer and the software: all the problem constraints should have been defined before starting the solving procedure. The heuristic based methods suffer from the combinatorial nature of the problem. In this context our goal has been to develop a method to guide the engineer from the problem statement to the solution with a list of computer aided steps leaving him the choice as much as possible of interacting with the solving procedure. 2. THE M E T H O D Starting with the definition of the process requirements defined by a Grand composite curve, the first step of the method consists in identifying the most important temperature levels to be considered in the refrigeration system (P1). Knowing these levels, the list of possible refrigerants is determined based on thermodynamic and operation criteria applied to a refrigerants data base. A superstructure of the refrigeration system is then generated systematically and is modelled using the Effect Modelling and Optimisation approach (Marechal and Kalitventzeff, 1997) that applies the heat cascade principles to model the ideal
Now with LENI-DGM, Swiss Federal Institute of Technology, EPFL, 1015 Lausanne, Switzerland, mail to: [email protected] 2 Belsim s.a. All6e des Noisetiers, 1, B-4031 Angleur, mail to:[email protected]
458 heat exchange system and a MILP technique to extract the optimal process configurations (P2). The model of the refrigeration system is defined in such a way that it will include the classical configurations of the cycles allowing the calculation of refrigerants and pressures cascade systems. Obviously the solution to such a problem is not unique, multiple solutions are generated and will be compared to obtain the final solution. After this step, the list of streams to considered in the heat exchanger network is known and the heat exchanger network structure will be determined (P3) using classical methods : heat load distribution, pinch design method and/or optimisation. Compared to fully automatic solvers, the proposed method allows a high level of interaction between engineer and software.
2.1. Step I : Identify the most important temperature levels of the requirement. To identify the optimal temperature levels, we first discretise the overall temperature range with a fixed precision. For each temperature in the list with nk elements, we introduce a heat contribution with a constant temperature (Tk) and an unknown heat load (qk). This contribution will be a hot stream above the pinch point and a cold stream below. The optimal heat load of the new utility streams will be computed by minimising the exergy losses in the heat transfer. In general terms, the exergy losses associated to a stream that changes its thermodynamic state from Tk+l to Tk is computed by : Aex~ =qk(1 - T ~ )
where Ttmk= T~+I-T~
Tl,,k
ln(~--~z~-)
with To the reference temperature and Tlmk the
logarithmic mean temperature of the temperature interval k. This expression simplifies in q~(1-w T--~) for the exergy losses of a heat load qk supplied at a constant temperature Tk. I k,
Introducing as constraints the heat cascade definition including the DTmin contributions of all the hot and cold streams defining the process requirements, we obtain a linear programming problem (P 1) whose objective function is the overall exergy losses of the selected heat levels. (P1)
Minimise~-' q~ 1-Rk ,Yk ,qk
~=1
subject to : heat balanceof the temperatureintervalk q, + ~ Q,k + R,+~- R~ =0
Vk = 1..... nk
i=l
Vk = 1..... nk
qmink Y~< qk < qmaxk Yk y~e{0,1} y~ < Nmax3, k =kt, +1
R~=0, R,,+~=0
y~ < Nrnax2,~y ~ < Nmax~ k=k. +1
(A1) (A2)
k=l
andR k >0
Vk=l ..... n~ +1
To the classical problem formulation, we add contraints (A1) to force any heat load qk tO be significant with respect to the total heat requirement (e.g. 10% at least); and contraints (A2) in order to limit the number of such levels in the different temperature domains. P1 is a MILP problem where n is the number of hot and cold streams; Rk the energy to be cascaded from the temperature interval k to the lower temperatures; Qik the heat load of process stream i in temperature interval k; (Qik > 0 for hot streams and < 0 for cold streams); Nmaxl the maximum number of levels to be found below ambient temperature; Nmax2 between ambient temperature and pinch temperature and Nmax3 above the pinch point; Yk is the integer
459 variable associated with the use of heat requirement qk; qmink (qmaxk) are the minimum (maximum) values accepted for the heat requirement qk. The formulation concerns the whole temperature range. It will also be applied to define the hot utility requirement and identify the most appropriate steam levels to be considered. For cold utility requirements, the values of qmink and qmaxk will be negative. In order to exploit the exergy potential of the self sufficient zones, qmink will be equal to -qmaxk for the temperatures in self sufficient zones. The result is a list of heat consumptions (above the pinch point) and heat productions (below the pinch point) that will be used to define the energy saving technology candidate. Above the pinch point (kp), the results concern the steam network and the heat pump systems, between ambient temperature (ka) and kp, these concern the organic Rankine cycles and the heat pumps, below the ambience these concern the refrigeration system. The results for the candidate levels found for a prototypical refrigeration example is given on figure 1.
Figure 1 : identification of the refrigeration levels and the possible refrigerants 2.2. Step 2 : Identify the best refrigerants to be used Knowing the best temperature levels to be considered, the next step concerns the identification of the list of refrigerants to be used. The algorithm presented on figure 2 performs a systematic examination of all the available refrigerants. The goal is a) to identify the feasible refrigerants, b) to compute the number of possible interstage levels. These values will be used to generate the refrigeration system superstructure. At this stage, the list of possible refrigerants (located on figure 1) will be proposed to the user via a web interface where he will be able to choose the acceptable ones for his process, applying additional rules based on other criteria: e.g. toxicity, safety, stability .... Simulation of the elementary refrigeration cycles Having defined the accepted refrigerants and temperatures, the goal is to determine the configuration of the refrigeration system. It means to define the list of refrigerants, condensers, evaporators and compressors to be considered, the way these are interconnected and to compute the flowrates in the different stages of the refrigeration system. For this purpose, the refrigeration system will be modeled by a set of elementary refrigeration cycles composed of one compressor, one hot (condensation) and one cold (evaporation) stream effects as shown on figure 3 for a 3 stages industrial refrigeration system.
460
O r d e r r by increasing Tboil
r
COMMENTS ON THE ALGORITHM The refrigerants are ordered according to increasing boiling point, this allows to reject all the refrigerants after the first refrigerant is rejected. The temperature list Te results form (P 1). For evaporation at Te, a new refrigerant is added if its saturation pressure Pevar(Te) is between 0.7 bar and 2 bars. For condensation at Tk, we compute the Pcondr(Tk), the saturation pressure of the refrigerant r. The condenser is introduced is accepted if the compression ratio is < 20, and if Tk is lower then Tcrit- 10~ As the major temperature levels have been computed with the minimum exergy losses method, we introduce as an option the possibility of adding temperature levels for each interstage level. Assuming that centrifugal compressors will be used, we compute the number of stages (Nstages)assuming that the speed limit of the fluid at the outlet of a stage must be lower than 90% of sound speed (or): -
I
I I
-
-
-
! I T* t = T k + D Tm i,n/2 r,k l ~ - - " ~ > ~ . . ~
~
k--k-,
/ [ ComputeN
...... ( T . = > T : )
w R zT~x withwthe N.,.,,g~= --uTand u =0.9or withc~=,l~l Pmol
}
yos update Thst ,y Add new sub-cycle r, Pcondr(Tk), Pevar(Te) ii
I
massic specific work, Pmol the molecular weight of the fluid, rhs the isentropic efficiency assumed to be 80%, C~
i
[ .....
Ce -R
Figure 2 : algorithm for refrigerant selection
Figure 3: representation of a 3 levels refrigeration cycle by elementary cycles Problem P2, as formulated here below, is a MILP problem where f,~j is the flowrate in the elementary cycle between evaporator level i and condenser level j using the refrigerant r; Cr,r(T j) the liquid fraction resulting from the flash of refrigerant r at temperature Tj from the previous temperature level of refrigerant r; AHvaPr(Tk,P(Tk))
the heat of vaporisation of
refrigerant r at temperatureTk and the equilibrium pressure at Tk; nr is the n u m b e r of refrigerants; nt the number of temperature levels identified for the refrigeration system; Qek,r the heat load of the evaporator with refrigerant r at the temperature level k; QCk,r the heat load
461 of the condenser with refrigerant r at the temperature level k; W c r the mechanical power consumption of compression for the refrigeration cycles using refrigerant r; Tk is the real temperature corresponding to the corrected temperature in the heat cascade. Therefore, Tk =T*k-DTmin/2k for the evaporator temperature and Tk =T*k+DTmin/2k for the condenser, Tk is used to compute the saturation pressure, the compression power and the head load of vaporisation. nw
Minimise ~ ( YwClw + f wC2 w) + Cel * EL i - Cel o * EL,, Rk ,Yw ,fw
(P2)
w='~"~
nr t
n
+~_. YWk * C l w r +WCr*C2w r +~.,(yck,r*ClCk,r +QCk,r*C2Ck,r)+ r=l
(Yek,r*Clek.r +Qek,r*C2ek.r k=l
k=l
subject to 9 heat balance of the temperature interval k
s163163163 w=l
r=l
Vk = 1..... n k
Rk+,-R ~ = 0 r=!
i=1
Ve = 1,..., nt Qee'r--s
~=Ik!k-O~r(Tj)llA* ,
*AnvaPr(Te'P(Te))l=O
Qe~nyee'r <-Qee'r
Vr = 1,..., nr
k-1
Qck,r - ~ { L~k *AHvaPr (Tk, P(Tk ))} = O
Qc~) yC~,r<-QCk,r<-Qc~,Tyc~,~
e=l
Wc,
s
Vr = 1,...,nr
ocr(Tt) r],., 1 g
e=l k=e+l
Vk = 1..... nt Vr = 1,..., nr
m
Wc)mnywr <_We r ~ wcmaXywr ?tw
Vr - 1,..., nr tlr
I~w
nr
Electricity production" Z f w * Ww -- ~_,WCr + ELi - ELo = 0 &consumption: y ] f~ * w w - ~ Wc r + EL i > 0 w=l
r=l
w=l
f minw Yw < f~ < f maXw Yw, ywe{O,1} Rk >O
V k = l ..... n k +l
r=l
Vw = 1,..., n w R l=O,R,,~+l=O
Problem (P2) is an adaptation of the one developed to optimise the integration of existing refrigeration systems (Closon et al., 1999). The adaptations concern the selection of the refrigerant, the sizing of the units and the cost estimation of the refrigeration system.
Calculation of the refrigeration system units If we assume that condensers and evaporators of the refrigeration system will operate near the DTmin conditions, the size of the heat exchangers will be proportional to the heat load of the exchange Qk,r"
A= Q ~ .
U D T min
The heat transfer coefficient U is computed using
correlations depending on the refrigerant and the temperature and assuming a film coefficient for the opposite stream. The compression power (Wcr) is computed for each refrigerant. It is made of the sum of the mechanical power in the different elementary cycles. The units investment is estimated using a classical exponent formulation (Turton et al., 1998). In the model, the investments are represented by a linear function whose coefficients (Clek,r, C2ek,r, C lCk,r, C2Ck,r) are identified by linearising the cost function for an estimated size obtained from the results of problem (P 1). Here, we assume that the compressors are centrifugal but the method can easily be extended to select among the different compressor types. Using the computed size obtained from the optimisation (P2), the engineer will have to identify the type of compressor using heuristic methods like the one proposed by Jandjel (2000) and re-
462 evaluate the estimated investment. In our system, the heuristic method has been implemented as a searching procedure in a compressor data base.
2.3. Step 3 : Solving procedure and generation of multiple solutions Problem (P2) is a MILP problem that extracts the refrigeration system structure and selects the best refrigerants together with the energy integration of the cooling system (air or water cooling) as well as the cogeneration systems. The interest of such a model, is the possibility of using integer cuts to generate multiple solutions to allow comparison and apply sensitivity analysis. For each solution, the compressor selection will be performed. It leads also to multiple solutions. By this approach, the engineer is in full control of the selection procedure and therefore will make the final decision of the investment. Graphical representation of the optimal refrigeration system using the integrated composite curves allows analysis and better understanding of the results. Comments will be given in a more extended publication. 3. CONCLUSIONS The method presented allows performing the synthesis of the refrigeration system for a given process in three steps that combine the use of MILP techniques and heuristics. Compared to fully automatic procedures, this method allows an interaction with the solving procedure to incorporate in it heuristic rules or preferences. We defined a method based on exergy losses minimisation to identify the most important temperature levels to be considered in a composite curve and presented an algorithm to systematically introduce in the problem the most suitable refrigerants to be considred in a superstructure. The Effect Modelling and Optimisation concept allows modelling a refrigeration system superstructure from which MILP generates refrigeration system configurations. At the end of the procedure, we obtain the complete list of streams to be considered in the heat exchanger network.
REFERENCES H. Closon, F. Mar6chal, B. Kalitventzeff and S. Pierucci, Energy integration: Application to an olefins plant, proceedings of ICHEAP 4 : Fourth Italian Conference on chemical and process engineering, pp. 131-134, (1999). F. Mar6chal and B. Kalitventzeff, Effect modelling and optimisation, a new methodology for combined energy and environment synthesis of industrial processes, Applied Thermal Engineering Vol 17 n~ Vol., pp. 981-992, (1997). G. D. Jandjel, Select the right compressor, Chemical Engineering Progress, Vol. July 2000, pp. 15-29, (2000). M. R. Shelton and I. E. Grossmann, Optimal synthesis of Integrated refrigeration systems-I, Computers and Chemical Engineering, Vol. 10, p 445, (1986). S. Vaidyaraman and C. D. Maranas, Optimal synthesis of Refrigeration cycles and selection of refrigerants, AIChE Journal, Vol. 45, No. 5, pp. 997-1017, (May 1999). T. R. Colmenares and W. D. Seider, Synthesis of cascade Refrigeration systems integrated with chemicalProcesses, Computer and Chemical Engineering, Vol. 13, p 247, (1989). D. W. Townsend and B. Linnhoff, Heat and power networks in process design. Part 1: Criteria for placement of heat engines and heat pumps in process networks, AIChE Journal, Vol. 29, n ~ 5, September, (1983). R. Turton, R. C. Bailie, W. B. Whiting and J. A. Shaeiwitz, Analysis, Synthesis and Design of Chemical Processes, Prentice Hall International Series in the Physical and Chemical Engineering Sciences, (1998).
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. J~rgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
463
Design of Reactive Distillation Process for Fatty Acid Esterification Florin Omota, Alexandre C. Dimian a and Alfred Bliek Department of Chemical Engineering, University of Amsterdam Nieuwe Achtergracht 166, 1018 WV, Amsterdam, The Netherlands Esters of fatty acids are produced nowadays in batch reactors. In this study we present a innovative continuous process based on reactive distillation that can be used as a multipurpose configuration allowing the synthesis of a variety of fatty esters. The approach consists in using a chemical and phase equilibrium analysis to identify the feasible design region, and computer simulation to generate process alternatives. Simulation enables also to define the experimental work. Results are presented for the esterification of lauric acid with methanol and 2-ethylhexanol, the lightest and the heaviest in the CI-C8 alcohol series. Two process alternatives can be accommodated in the same hardware, but with different operation procedures: one with alcohol reflux, other with acid reflux. The first is feasible for heavy alcohols, forming heterogeneous azeotrope with water. The second is suited for both light and heavy alcohols, and may be seen as a generic esterification process of fatty acids.
1. INTRODUCTION Esterification of fatty acids is industrially important because fatty esters serve as feed stock for detergents and surfactants. Nowadays these are produced commercially in rather expensive processes consisting of batch reaction followed by distillation for water removal, recovery of excess alcohol and product purification. In this study we present the conceptual design of a multipurpose continuous process based on reactive distillation. The remarkable feature is that this can produce in the same hardware a variety of esters, of both light and heavy alcohols. Because of selective catalyst and mild reaction conditions the only reaction to consider is: CH3-(CH2)Io-C00H + R-OH ~
CH3-(CH2)lo-C00R + H20
(1)
The alcohol can be either methanol or 2-ethylhexanol. An ester purity higher than 99.8 % is desirable. The selected catalyst, based on sulphated zirconia, has a good activity and selectivity at least up to 200 ~ It may be coated on structured packing or other ceramic material that can be placed on the reactive plates in a kind of 'tee-bag' arrangement. Because the esterification is equilibrium limited the main problem is in-situ water removal. Light alcohols, such methanol, do not form azeotropes, and tend to distil preferentially in top. At the first sight, a second column for alcohol recovery is necessary. On the other hand, heavier alcohols may form azeotropes with water. In this case the heterogeneity could be exploited to remove water and shift the reactive process in a feasible region. In addition, the synthesis of fatty esters must take into account high boiling points of long chain components, which impose vacuum or other means to limit excessive temperature.
aCorresponding author, e-mail [email protected]
464 These severe thermodynamic and physical constraints might give the impression that a generic approach would be impossible. However, in this paper we demonstrate that a generic process can be developed, and adapted to produce various esters. Here we present results regarding the esterification of laurie acid with methanol and 2-ethyl hexanol, the first and the heaviest component in the C1-C 8 alcohol series. Innovative ideas and feasibility of the alternative designs has been investigated by computer simulation with ASPEN Plus TM version 10.1. This served also to guide the experimental research. 2. THERMODYNAMIC ANALYSIS
2.1 Pure component properties Table 1 presents the boiling points of key components at normal pressure, and at 240 mmHg, a value to consider in a vacuum process. Vapour pressure data for 2-ethylhexyl laurate is missing. Estimation of nbp by ASPEN Plus TM leads to controversial values: 441 ~ by Joback, 332 ~ by Gani et al., and 332.6 ~ by Chien et al. Because Pv-T data affects significantly multiphase chemical equilibrium calculations, the accuracy of vapour pressure data for all components was tested and completed with own measurementsb. Table 1 - Boiling points for key components in ~ Laurie acid 2-ethylhexanol 2-ethylhexyl laurate 760 mmHg 298.6 184.6 334.5 240 mmHg 254.4 147.5 289.2
A
300
B
250 200 150 E I- 100
El.
E
01ases0,iI.io0
, 0
0.2
0.4 0.6 Xl, Yl
250 200
Q.
50
300 0 o
Taz=70.1~ Yaz=0"9806
o
0.8
1
Methyl laurate 267.2 221.7
9991'_ -
150 100 50
rwo,,0u,00 0
0204000
1
xl, Yl
Fig. 1. VLLE at 240 mmHg: A-water(1)+2-ethyhexanol(2) and B-water(1)+laurie acid(2)
2.2 Phase equilibrium The esterification must take place exclusively in the organic phase in order to preserve the catalyst activity. The formation of a second water phase in the reactive zone must be prevented. On the other hand, two-phase separation would be of help for water removal as top product. Water gives heterogeneous azeotropes with laurie acid and 2-ethylhexanol. Figure 1 presents T-xy diagram calculated with UNIQUAC, with interaction parameters regressed from own experimental data. It is worthy to note a very low solubility of both organics in water, bDetails over experimental data available at [email protected][.
465 although the reciprocal solubility of water is significant. Accordingly, a good yield of water removal could be achieved when the top distillate would consist only of these two binaries. Conversely, a non-negligible water amount returns in the column with the reflux. Table 1 indicates that the lauric ester is obtained always as a bottom product, but the temperature would be excessive, even under vacuum. Therefore, it is rational to get the bottom product as a mixture with a certain amount of alcohol. Note that the vapour-liquid equilibrium of binaries lauric ester/alcohols is not a thermodynamic constraint. However, if the lauric acid is not entirely converted, its separation from ester would be very difficult.
2.3 Chemical and phase equilibrium Inside the reactive zone, chemical and phase equilibria occur simultaneously. The composition can by found of Gibbs free energy minimisation [1-2], as expressed by the relation: AG(T) = RTlnKa(T)=RTIn(KrK~) (2) 2-ethylhe~l laurate
A
w~, 0 . 8 1 . \ ~ , ~ ~ ~ '
+
0.6
'"
~
o.4 "---.~..
X
"~
B
0.8 co
q~,
lauric acid
methyl laurate 1
~cid
+
"O O
0.6
L v
l
\
rLE ._.L.
0.4
k..
/
iVLL
0 0.2 0.4 0.6 0.8 1 2-ethylhexanol Xl (water+acid) Water
0.2
\
\ |
methanol
0.2
i
0.4 0.6 0.8 1 Xl (water+acid) water
Fig. 2. CPE diagrams for lauric acid esterification with 2-ethylhexanol (A) and methanol (B) Good estimation of liquid activity coefficients is essential. This is an issue in itself, and it will be handled in a separate publication. We mention only that an experimental cell has been built to measure phase and chemical equilibrium up to 10 bars and 200 ~ It was found that the difference between experimental and predicted values by Aspen Plus T M is significant, but a major improvement was obtained by replacing vapour pressure estimations with experimental data. Another significant factor was the regression of binary interaction parameters for heterogeneous azeotropes from experimental data over mutual solubility. As an order of magnitude, at the temperatures between 140 and 160 ~ the equilibrium conversion can exceed 80 % starting with an initial equimolar reactants' ratio. The thermal effect of reaction is small, of 3 kcal/mol. Figure 2 presents a generalised representation of chemical and phase equilibrium in suitable co-ordinates [2-4]. It can be observed that a boundary separates homogeneous and heterogeneous distillation regions. The reaction should take place in the homogeneous (left) domain, where the 'residue curves' converge to the ester node. The lines in the heterogeneous region are 'tie lines' linking the composition of liquid phases at equilibrium. The aspect of the
466 heterogeneous domain depends on the solubility in water of the examined alcohol. The boundary sets also the minimum temperature of the reaction zone. Roughly speaking, a temperature above 100 ~ in the reaction zone is sufficient to place the reaction in the homogeneous domain. Contrary, the condensation in the top must occur at a temperature for which the phase equilibrium falls inside the heterogeneous domain.
3. HEAVY ALCOHOL PROCESS (Flowsheet A) Firstly, we investigated the feasibility of the esterification process with a heavy alcohol, namely 2-ethyl-hexanol. As an assumption we considered that the reaction reaches equilibrium on each stage. Figure 3 depicts the flowsheet and indicates some operation parameters. The reactive distillation (RD) column operates at 240 mmHg, and has five reactive stages. Lauric acid (liquid) preheated at 110 ~ is introduced on the high position of the reaction zone, while the alcohol (vapour) is fed at 146 ~ on the low position of the reaction zone. The bottom product with approximately 10% alcohol goes to an evaporator, from which the ester is recovered and the alcohol recycled. The vapour leaving the top of the reaction zone is condensed and cooled to about 70 ~ where two-phase separation occurs. Water is removed quantitatively from the process with only a small amount of alcohol. The organic phase is refluxed to the RD column. Figure 4 presents temperature and concentrations profiles. The reaction is almost completed in three equilibrium stages. The maximum reaction rate is located on the top tray. The energy balance can control the recycle of alcohol. In this way a high internal alcohol/acid ratio can be achieved, such to increase the local reaction rate, while the reactants' ratio in the initial feed is always 1:1. Because of higher local concentration of alcohol on the low position stages, an undesired etherification reaction might become possible. This aspect deserves a special attention in catalyst development. Note that a stripping section is not necessary. At most, one or two non-reactive stages in the top zone may be provided for a safer separation of ester with respect of alcohol and acid. We simulated also a reactive distillation column with the same configuration, but using a kinetic model. Several rate equations were tested, as for the similar esterification of n-butanol with oleic acid, as well as based on own laboratory measurements. The results differ only in the number of reactive stages, between 4 and 10, depending on catalyst activity.
Fig. 3. Flowsheet for lauric acid esterification with 2-ethylhexanol under vacuum
467
Fig. 4. Temperature (A) and concentration profiles (B) for 2-ethylhexanol esterification
4. LIGHT ALCOHOL PROCESS (Flowsheet B) When methanol is used, a similar process with that described above leads to a two column arrangement: RD column and methanol recovery. The investment and energy consumption only for the second column are higher than for RD column, but always below a batch process. However, another strategy is possible based on the observation that the water removal is easy from an heterogeneous azeotrope with lauric acid. Hence, the design must ensure a complete consumption of the alcohol in top. Only lauric acid and water are allowed in the top vapour stream, from which water can be easy removed after condensation and decantation. This principle leads to a second flowsheet alternative presented in Figure 5. A higher number of stages is necessary, in this case approximately 20. Profles of temperature and concentrations are presented in Figure 6. Note that in this case the maximum reaction rate is located somewhere to the middle of the reaction zone. The position depends on the reflux ratio. A middle location corresponds to an optimum reflux, where both acid and alcohol are completely converted. A lower position corresponds to an incomplete consumption of acid, found as impurity in the bottom product. A higher position corresponds to a an incomplete consumption of alcohol, and lost with the top product. Hence, contrary to the normal distillation, when purity increases with reflux, here purity and yield reach maximum at an optimum reflux rate.
Fig. 5. Flowsheet for lauric acid esterification with methanol at normal pressure
468
Fig. 6. Temperature (A) and concentration profiles (B) for methanol esterification 5. GENERALISED PROCESS
From the above discussion it may be observed that the second alternative can be generalised both for lights and heavy alcohols. The acid recycle via water decanting leads to a total consumption of alcohol on the top reactive stage. As a consequence, the process becomes independent on the type of alcohol used. 6. ON-GOING RESEARCH We proceed researches to improve activity and selectivity of zirconia sulphate catalyst in the operation range determined by simulation. Kinetic measurements are tested by simulation with Aspen Plus TM. A laboratory set-up is currently built-up to prove the feasibility of this new process. 7. CONCLUSIONS The paper presents an innovative process for the synthesis of fatty acids esters by reactive distillation. Accurate simulation of simultaneous chemical and phase equilibria combined with laboratory experiments is used to identify the design space. On this basis two alternative designs are proposed, with alcohol and acid recycle, respectively. The last appears to be generic for any type of alcohol. Purity and yield are maximum at an optimum reflux ratio. REFERENCES
1. Castier, M, Rasmussen P., Fredenslund A., 'Calculation of Simultaneous Chemical and Phase Equilibria in Nonideal Systems', Chem. Eng. Sci., 44 (1989) 237. 2. Perez Cisneros, E. S., Gani, R., Michelsen, M. L., 'Reactive separation systems. Computation of physical and chemical equilibrium', Chem. Eng. Sci., 52 (1997) 527. 3. Barbosa, D., Doherty, M.F., 'Design and minimum-reflux calculations for single-feed multicomponent reactive distillation columns', Chem. Eng. Sci., 43 (1988) 1523. 4. Ung, S., Doherty M.F., 'Synthesis of reactive distillation systems with multiple equilibrium chemical reactions', Ind. Eng. Chem. Res., 34 (1995) 2555.
European Symposiumon ComputerAidedProcessEngineering- 11 R. Ganiand S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V.All rightsreserved.
469
Selection of separation sequences by case-based reasoning Pajula, E., Seuranen, T. and Hurme, M. Helsinki University of Technology, Laboratory of Chemical Engineering and Plant Design, P.O. Box 6100, FIN-02015 HUT, Finland The objective of this paper is to introduce a method for finding feasible separation process sequences and separation process structures utilising case-based reasoning (CBR). This means finding the most similar existing separation processes and applying the knowledge of their concept and separation sequencing for solving new problems in the early phases of process design. 1. INTRODUCTION Typical task in process design is to determine the configuration of a separation sequence. When dealing with multicomponent mixtures, the number of possible separation methods, their combinations and process structures to be screened is huge as well as the work involved. The method used, case-based reasoning (CBR), uses existing design cases stored in the database for solving new separation problems. The synthesis method studies the physical and chemical properties of the species present in the mixture and uses the properties presenting most favourable possibilities for successful separation for retrieving the nearest cases to the current problem. The presented method has the advantage of not losing any information because no generalisations are used. 2. M E T H O D O L O G Y It is a well-known fact that in the majority of cases distillation is the most feasible way to separate components (Barnicki and Fair 1990). Therefore the distillation related properties are studied first in the methodology. The strategy is to find first a feasible distillation sequence for the separations where ordinary distillation is possible, and then to solve the remaining separation problems with further reasoning which apply other separation methods than ordinary distillation (steps 2-4 in Table 1). The main steps of this algorithm are presented in Table 1. Step 1." For all components cz's are calculated and the presence of reactive components is also considered. The most similar cases to the current problem are searched from database based on these parameters. When it is possible (and no known cases for better procedures are found) ordinary distillation is applied using the same separation strategy as defined by the sequence of the nearest case found in database. The separation strategy is described in the cases as a set of heuristic rules or as textual description. The separations are classified in the database based on relative volatility (cz) values as easy (~z>=l.2), possible, where mass separating agent (MSA) could be useful (1.1< cz <1.2) and
470
difficult (~ =<1.1) separations. In a simple situation a search using component names would be the most exact, but in a case where exactly the same components are not found, a more advanced approach is more useful (see Chapter 3). If the cases found are equally similar to the current problem, the query should be made more exact using concentration, capacity, component types etc. also as retrieval parameters. Unfortunately a separation problem is often far more complicated than finding only a distillation sequence and also other separation methods than ordinary distillation have to be considered. Step 2: To compare other separation techniques with mass separation agent aided operations we need to select a suitable MSA for each component pair if possible. This has been done using component names or component types as retrieval parameters. Component types are defined as a taxonomy tree (see Figure 1). The closer the components are in the tree the greater similarity value they have. The principle of similarity is discussed in earlier paper (Pajula 2000). Even if a promising MSA is found, also other separation methods are checked. If there is no proven case for certain MSA, a more detailed study should be done by computer simulations or experimentally (See Example 3.1). Alternatively MSA can be searched using solubility parameter, dipole moment and dielectric constant as retrieval parameters. Table 1. The algorithm Step 1 DISTILLATION FEASIBILITY Search: Make a search with c~'s and reactivities as retrieval parameters Refine: Define a more accurate search (capacity and component types also as retrieval parameters) if several alternatives are found. Action: Apply the separation strategy of the nearest case for the separations where ordinary distillation is possible. If ordinary distillation is not feasible for all separations, continue to step 2. Step 2 FINDING A SUITABLE MASS SEPARATION AGENT (MSA) Search: Make a search with component types as retrieval parameters for each remaining component pair. Refine: Define a more accurate search (concentrations, relative solubility parameter, polarity and dielectric constant also as retrieval parameters) if several alternatives are found. Action: Use the found MSA (if any) for defining solubilities etc. for step 3. Step 3 FINDING PARAMETERS FOR ALTERNATIVE SEPARATION METHODS Action: Calculate relative physical property parameters for each component pair that can't be separated by ordinary distillation and compare them to the feasibility limits of different separation methods. Step 4 SEARCH FOR A SUITABLE SEPARATION METHOD AND STRATEGY Search: Make a search using the relative parameters (min and max values) that are within the feasibility limits. Refine: Define a more accurate search (concentration, capacity and component types also as retrieval parameters) if several alternatives are found. If there are still several alternatives left, make economical comparison. Action: Apply the separation strategy of the nearest case to the components that can't be separated by distillation.
Step 3: It is often important to consider also other separation methods than ordinary distillation. Therefore it is necessary to consider all the possible properties that may be utilised in separation processes and make a search based on these. The principle is to apply
471 separation method that utilises the largest property difference of the components to be separated. To do this relative properties are calculated for component pairs and compared to predefined feasibility limits (Jaksland et al. 1995, Qian and Lien 1994). For example crystallisation is considered very feasible if the relative melting point is greater or equal to 1.2 and feasible if it has a value between 1.1 and 1.2. The approach is used for finding the most important retrieval parameters for CBR, i.e. the parameters that show greatest potential for separation of the species that have too small ot's for ordinary distillation. In this way the amount of retrieval parameters is limited to essential ones. Step 4: The search is made using the relative parameter values of step 3 that are within the feasibility limits as retrieval parameters. The separation method or strategy of nearest found case is applied. [Et--'z~ Component I~t-.~ Organic !~)..~:~ Hydrocarbon ~.'..~ Aliphatic hydrocarbon I~---~ Cyclic hydrocarbon E]..-~;;= Organic component with functional groups ~...~:~ alcohol i
.....qlb butanol .... qlb methanol .....qlb ethanol ....ql= propanol
+ . . ~ ester r~...~ ketone ~--.0 Chlorinated hydrocarbons r~...O Inorganic
Figure 1. Part of the component type taxonomy 3.EXAMPLES FOR SINGLE SEPARATIONS Example 3.1: Selection of mass transfer agent
In this example finding potential mass transfer agents for comparisons is demonstrated. Task: Separate n-propanol (50 wt-%) from water using MSA. Purity requirement for n-
propanol product is 90 wt-%. This cannot be reached with ordinary distillation, because water and n-propanol form an azeotrope at 87 ~ with 71 wt-% n-propanol (Smallwood, 1993). Table 2 . Query and closest cases in Example 3A ............. ..................... Query Found 1 . Component 1 type Water Water Component 2 type Aliphatic a l c o h o l Aliphatic a l c o h o l MSA Yes Yes Component 1 Component 2 Solubility parameter Dipole moment / D Dielectric constant MSA's
~Similarity ..........
11.9 1.7 20.1
F0und 2 Water Aliphatic alcohol Yes
Water isopropanol 11.5
Water sec-butanol 10.8
1.66
1.7
18.3 Cyclohexane, Toluene, Butylacetate, Diisopropylether, Diisobutene, Benzene Isopropyl acetate 0.97 0.93
472 The first search is made using search parameters MSA=yes and component types (water, aliphatic alcohol). This gives quite a few cases with similarity value 1. In this case a more specific search is needed to find the most likely suitable MSA. The following additional retrieval parameters for alcohols are used: solubility parameter, dipole moment and dielectric constant. The two closest alcohols found and the MSA's used with them are summarised in Table 2. This is a realistic result for further studies, because at least benzene, diisobutene, diisopropylether and cyclohexane have been reported for n-propanol/water separation (Smallwood, 1993).
Example 3.2: Finding other solvent to replace current mass transfer agent In this example finding an alternative for a MSA by utilising a case base that includes also component properties is demonstrated. Dimethylformamide (DMF) has been separated from water using chloroform as MSA, but other possible solvents are searched. Task: Find a solvent the properties of which are close to chloroform and which is easy to separate from DMF. The search is made using following retrieval parameters: solubility parameter, dipole moment, dielectric constant (these describe solvent's separation capability), solubility in water and solubility of water (the phases should be practically immiscible). Also boiling point and possible azeotrope with water (azeotrope boiling point) are used as retrieval parameters. This is important because the solvent has been separated from DMF (boiling point 153 ~ by ordinary distillation and the distillation column temperature should be high enough so that the temperature of cooling water is cold enough for condensation. The results are presented in Table 3. This is a realistic result because both methylene dichloride and carbon tetrachloride have been reported as MSA in DMF/water separations. Table 3. Que~:::~d closest cases in Example 3:2 .....
MSA . . . . . . . . . . . . . . .
_e.ue~.
.......
Case!
....................
Case2
................
Name Chloroform Carbon tetrachloride Methylene dichloride 76 40 Boiling point ~ 61 8.6 9.7 Solubility param. 9.3 0 1.8 Dipole moment 1.1 0.077 0.077 Solubility in water 0.82 0.008 0.008 Solubility of water 0.2 2.24 9.1 Dielectric constant 4.806 Azeotropes Water wt-% 97 96 99 B.p. ~ 56 66 38 0.95 0.92 :S:!mi[ar!~............................................................................ 4. EXAMPLE FOR FINDING SEPARATION SEQUENCE
Example 4.1. Separation sequence selection- xylenes In this example the method is applied for finding altemative separation sequences when ordinary distillation is not suitable for all separations. To demonstrate the approach, the search is not done by component names but by relative properties. Task: Separate mixture of ethyl benzene (20 wt-%), m-xylene (40 wt-%), o-xylene (20 wt-%) and p-xylene (20 wt-%) to pure products.
473
Step 1" By searching with ot's and reactivities it is found out that ethyl benzene and o-xylene can be separated by ordinary distillation. The strategies used in the two nearest cases found (similarity 1) are: Heuristics 1" Perform difficult separation last and favour 50/50 split. Heuristics 2: Perform difficult separation last and use CES for finding the distillation order. We choose to use CES to define the separation order for these separations, because the other heuristics found doesn't give answer for the separation order in our problem. According to CES, first o-xylene and then ethyl benzene are separated by distillation. For m-xylene/pxylene separation the available ot is too small for ordinary distillation. Step 2." To find out possible MSA aided separation methods for this separation, we'd like to check for potential MSA as presented in Example 3.1. In search no feasible MSA's were found for a case where both components are aromatic and have low polarity. Step 3. For finding other separation methods for m/p-xylenes reasonable retrieval parameters are needed. This means the relative parameters that are large enough to make a separation possible. The potential parameters found are relative melting point ratio and relative kinetic diameter. The calculated values for these are presented in Table 4. Table 4. Some properties and calculated relative propertie s for Example 4.1 ...... T (boiling)~ T (melting)/K CES (U separation) Binary pair ............ R (melting point) ethyl benzene 409.35 178.2 0.54 p-xylene/m-xylene 1.27 p-xylene 411.51 286.4 0.506667 R (kinetic diameter) m-xylene 412.27 225.3 1.3275 m-xylene/p-xylene 1.16 o-xylene 417.58 249.0
Step 4. The database was searched using these relative parameters (R's) for melting point and kinetic diameter and only including cases in which at least one a-value is classified difficult. The results are presented in Table 5 below. Based on the search results two feasible methods are proposed for further research: First separate o-xylene and then ethyl benzene by distillation. After this separate remaining p- and m-xylenes either by crystallisation or molecular sieve adsorption since these methods were found to be potential in the case-base search. Table 5. Search resu!ts0fExample 4. ! Query_..................Foun d 1 .......... ~'s difficult difficult, easy R melting point. 1.26 1.26 max R kinetic 1.16 1.16 diameter max Components xylenes Separation method SimilaritY . . . . . . . . .
Found 2 ........... Found 3 difficult, easy difficult 1.26
'
1.3
FoUnd 4 difficult 1.66
1.16 xylenes
p- and mdichlorobenzene Crystallisation
THF and water Distillation & Crystallisation Molecular molecular (only p-xylene sieve sieve adsorpt, separated) adsorption 1 .......................................1.................................................0!8 ............................................................_0_:7_..........................
474 5. OTHER ASPECTS OF THE METHOD
In these examples no combined operations were included in the case base. These are discussed in an earlier paper (Pajula et al. 2000). As separation methods, for instance hybrid membrane/distillation processes develop further, the case base needs to be updated. This can be done by adding rules or new cases that have low maturity. For instance, if the components can be separated by distillation, the heuristics presented by Rong et al. (2000) are notified. An other way is to create new cases by simulation. For instance, if the mole fractions (as retrieval parameters) in the feed stream are close to those where distillation the mole fractions in the feed are typical values for a complex distillation flowsheet, the case suggesting complex distillation flowsheet is retrieved among other near cases. When creating this kind of cases including new separation methods also process maturity factors (Pajula et al. 2000) and feasibility limits need careful attention. 6. CONCLUSIONS A method for finding feasible separation process sequences and separation process structures utilising earlier design cases was developed. The demonstrations show how earlier design cases can help in selecting process alternatives to be considered especially at the early stages of a design and in this way fasten the design process. Also importance of updating the presented method was considered. The advantage compared to rule-based methods is that all the existing knowledge is available as cases and can be utilised in a non-reduced form. The method is also very flexible because the user can focus the search by defining more accurate search parameters if several nearly similar solution possibilities are available.
REFERENCES
1. E. Pajula, T. Seuranen, T. Koiranen and M. Hurme in S. Pierucci (ed), European Symposium on Computer Aided Process Engineering, Elsevier Science B.V., Amsterdam, 2000. 2. S.D. Barnicki and J. R. Fair, Ind. Eng. Chem. Res. 29 (1990) 421. 3. I. Smallwood, Solvent Recovery Handbook, McGraw-Hill Inc., New York, 1993. 4. C. Jaksland, R. Gani and K. M. Lien, Chem. Eng. Sci. 50 (1995) 511. 5. O.M. Wahnschafft, T.P. Jarian and A. W. Westerberg, Comp. Chem. Engng. 15 (1991) 561. 6. Y. Qian and K.M. Lien. Can. J. Chem. Eng. 72 (1994) 711. 7. R.W. Thompson and C. J. King, AIChE J. 18 (1972) 941. 8. V.M. Nagdir and Y.A. Liu, AIChE J. 29 (1983) 926. 9. B.-G. Rong, A. Kraslawski and L. Nystr6m, Comp. Chem. Engng. 24 (2000) 247. NOMENCLATURE
R
relative volatility relative parameter
p
degree of feasibility (range 0-1)
Rmax maximum value of relative parameter
European Symposiumon ComputerAidedProcessEngineering- 11 R. Ganiand S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rightsreserved.
475
Optimal Multi-floor Process Plant Layout Dimitrios I. Patsiatzis and Lazaros G. Papageorgiou* Department of Chemical Engineering, University College London, Torrington Place, London WC1E 7JE, United Kingdom. This paper presents a general mathematical programming formulation for the multi-floor process plant layout problem, which considers a number of cost and management/engineering drivers within the same framework thus resolving various trade-offs at an optimal manner. The proposed model determines simultaneously the number of floors, land area, floor allocation of each equipment item and detailed layout of each floor. The overall problem is formulated as a mixed integer linear programming (MILP) model based on a continuous domain representation. The applicability of the model is then demonstrated by an illustrative example. 1. INTRODUCTION The process plant layout problem involves decisions concerning the spatial allocation of equipment items and the required connections among them [1]. Usually, these plant layout decisions are ignored or do not receive appropriate attention during the design or retrofit of chemical plants. However, increased competition leads contractors and chemical companies to look for potential savings at every stage of the design process. In general, the process plant layout problem may be characterised by a number of cost or management/engineering drivers such as: (a) connectivity cost by involving cost of piping and other required connections between equipment items or any other network related operating costs (e.g. pumping); (b) construction cost thus leading to the design of more compact plants (e.g. off-shore plants); (c) retrofit by fitting new equipment items within an existing plant and (d) safety aspects by introducing, for example, constraints with respect to the minimum allowable distance between specific equipment items. Here, we concentrate on the process plant layout problem, which has recently received attention from the research community. A number of different approaches have been presented for the single-floor case. The allocation of units to sections created by aisles or corridors was formulated as a graph partitioning problem in [2]. A mixed integer non-linear programming (MINLP) model was proposed by [3], integrating safety and economic aspects. The use of genetic algorithms [4] has also proved to be effective in obtaining good and practical solutions for the layout problem. Continuous domain MILP mathematical models were presented in [5], determining simultaneously orientation and allocation of equipment items. An alternative continuous MILP formulation was also suggested in [6] for equipment allocation, utilising a piecewise-linear function representation for absolute
*Author to whom correspondence should be addressed: Fax:+44 20 7383 2348; Phone +44 20 7679 2563; emaih 1. p a p a g e o r g i o u @ u c l , ac. uk
476 value functionals (distances between equipment items). The assignment of equipment items to different floors, has been considered in [7] by satisfying a number of equipment arrangement preferences and also taking into account vertical pumping and land costs. The partitioning of units in different floors was studied in [8], combining a graph theory approach and a mathematical programming solution procedure. Grid-based MILP mathematical models have been described in [9, 10], considering equipment of different sizes and geometries, based on rectangular shapes. In this work, a general mathematical programming formulation for the multi-floor process plant layout problem is presented. This work extends the single-floor work of Papageorgiou and Rotstein [5], which is based on a continuous domain representation.
2. PROBLEM STATEMENT In the formulation presented here, rectangular shapes are assumed for equipment items following current industrial practices. Rectilinear distances between the equipment items are used for a more realistic estimate of piping costs [3, 5, 10]. Equipment items, which are allowed to rotate 90 ~ are assumed to be connected through their geometrical centres. Overall, the multi-floor plant layout problem can be stated as follows: Given:
* A set of equipment items (i or j = 1..N) and their dimensions (ai, fli) 9 A set of potential floors; k = 1..K 9 Connectivity network 9 Cost data (connection, pumping, land and construction) 9 Floor height, Fh 9 Space and equipment allocation limitations 9 Minimum safety distances between equipment items Determine:
The number of floors, land area, equipment-floor allocation and detailed layout (orientation, coordinates) of each floor So as to minimise the total plant layout cost. 3. M A T H E M A T I C A L F O R M U L A T I O N 3.1. Floor Constraints Each equipment item should be assigned to one floor: Z v~k - 1
v i
(1)
k
where Vik = 1 if item i is assigned to floor k; 0 otherwise. We introduce a new set of binary variables, zij, which have the value of I if equipment items i and j are allocated to the same floor; 0 otherwise. Their value can be obtained by: zij >__ vik + vjk - 1
V i = 1 . . N - 1, j = i + 1..N, k = 1..K
(2)
zij <_ 1 - - vik + vjk
Vi=I..N-I,j=i+I..N,k=I..K
(3)
zij < 1 + Vik -- Vjk
V i = 1 . . N - 1, j = i + 1..N, k = 1..K
(4)
477
The number of floors, N F , will be determined by: NF
>_ ~
kvik
(5)
V i
k
3.2. Distance Constraints The single floor distance constraints presented in [5] are here extended for the multi-floor case: R i j - Lij - xi - x j
V i - 1..N-
1, j - i + 1 . . N " fij - 1
(6)
A i j - B i j - y~ - yj
V i - 1..N-
1, j - i + 1 . . N " f~j - 1
(7)
U~} - Di~ - Fh ~
k(vik - Vjk)
V i - 1 . . X - 1, j - i + 1 . . N " fij - 1
(8)
k
where the relative distance in x coordinates between items i and j, is R i j if i is to the right of j or Lij if i is to the left of j. The relative distance in y coordinates between items i and j, is A i j if i is above j or B i j if i is below j. The relative distance in z coordinates between items i and j is U/~ if i is higher than j or D~j if i is lower than j. The coordinates of geometrical centre of item i are denoted by xi, yi. Parameter fij is equal to 1 if units i and j are connected ; 0 otherwise. Thus, the total rectilinear distance, Dij, between items i and j is given by: D i j -- R i j nt- Lij + A i j + B i j + U~. + Di).
V i - 1..N-
1 , j - i + 1 . . N " f~j - 1
(9)
3.3. Equipment Orientation Constraints The length, li, and the debth, di, of equipment item i can be determined by: li -- o~ioi + ~i(1 - oi) di - c~i + ~i - li
(10)
V i
(11)
V i
where oi is equal to 1 if li=oq; 0 otherwise (i.e. li=fli).
3.4. Non-overlapping Constraints In order to avoid situations where two equipment items i and j occupy the same physical location, when allocated to the same floor (i.e zij = 1), constraints are included in the model that prohibit overlapping of their equipment footprint projections, either in x or y direction: xi -- x j + M ( 1 - zij + E l i j + E 2 i j ) >
xj - xi + M ( 2 - zij - E l i j + E 2 i j ) >_
yi - yj + M ( 2 -
zij + E l i j - E2ij) >_
li + lj
l~ + lj
di + dj
V i - 1..N-
1 , j = i + 1..N
(12)
Y i - 1..N-
1 , j - i + 1..N
(13)
V i-
1..N-
1,j-
i + 1..N
(14)
478
yj - yi + M(3 - zij - E l i j - E2ij) >_
di + dj
V i = 1 . . N - 1,j = i + 1..N
(15)
where M is an appropriate upper bound and Elij, E2~j are binary variables as used in [5]. Note that the above constraints are inactive for equipment allocated to different floors (i.e z~j = 0).
3.5. Additional Layout Design Constraints Intersection of items with the origin of axes should be avoided: xi > li -2
Yi
(16)
yi > _di -2
V i
(17)
A rectangular shape of land area is assumed to be used and its dimensions (x max, yma~) are determined by: li
xi + -~ 5 x max
g i
(18)
di _ yma~ Yi + ~ <
g i
(19)
These dimensions can then be used for the land area, FA, calculations: F A = x 'nax . ymaz
(20)
3.6. Objective Function The overall objective function used for the plant layout problem is as follows: rain
~
~[C~
. Dij + C~ " Di~ + C h " (Rij + Lij + Aij + Bij)]
i ir
+FC1. NF + FC2. NF.
FA + LC. FA
where the first term represents the total connection cost (C~j is the unit connection cost between i and j), and the second and third terms represent vertical (C~ is the unit vertical pumping cost if i is below j) and horizontal (C h is the unit horizontal pumping cost) pumping costs, respectively. The construction cost incurred is described by the fifth and sixth terms ( F C 1 and F C 2 are the floor construction cost parameters), while the last term is associated with the land cost ( L C is the land cost parameter). The above problem is an MINLP model because of the non-linearities involved in the last two terms of the objective function. However, the x max, ymaz variables required for the F A calculations can be discretised similarly to the work presented in [ 10]. Consequently, the nonlinear terms can easily be linearised. Due to space limitations, these linearisation constraints are not presented here. The linearised problem corresponds to an MILP model which can then be solved using standard branch-and-bound techniques. Next an illustrative example demonstrates the applicability of the MILP model.
479
Figure 1" Flowsheet for Ethylene Oxide Plant ,_. . . . . . . . . . . . . . . . . . . . . . F. .L. .O . .O . . .R. . . . . . .1. . . . . . . . .
FLOOR
2
Z68 m 11.42 m
5.22 m I
Z 68 m _.................................
2
1
8:___n_l_ 4 8 .........................................................
Figure 2" Optimal Layout 4. ILLUSTRATIVE E X A M P L E
Consider the ethylene oxide plant (see Figure 1), derived from the case study presented in [3]. Three potential floors are assumed to be available. Connection and pumping cost data are given in Table 1. The annualised floor construction cost parameters, FC1 and FC2, are 3330rmu and 6.7rmu/m 2, respectively and the annualised land cost parameter, LC, is 26.6rmu/m 2, where rmu stands for relative money units. Table 1" Connection and Pumping Costs Connection C~j [rmu/m] C~ [rmu/m] C~.j[rmu/m] (1,2) 200 400 4000 (2,3) 200 400 4000 (3,4) 200 300 3000 (4,5) 200 300 3000 (5,1 ) 200 100 1000 (5,6) 200 200 2000 (6,7) 200 150 1500 (7,5) 200 150 1500 The above example was modelled in the GAMS system [ 11], using the CPLEX 6.5 optimiser for the solution of the MILP model with a 5% margin of optimality. The resulting mathematical model includes 434 constraints, 113 integer and 119 continuous variables. The equipment dimensions and the optimal layout are shown in Figure 2. The optimal solution (equipment orientation and location, equipment-floor allocation) is given in Table 2. It should be noted that
480 two out of the three initially available floors have been chosen. The total plant layout cost is 50817 r m u with the following breakdown: 23% for connection cost; 32.5% for horizontal and vertical pumping costs and 44.5% for land and construction costs. The optimal land area is 400 m 2 ( x max 20m, ymax 2Ore). =
:
Equipment 1 2 3 4 5 6 7
Table 2: Optimal Solution Orientation Location li [m] di [m] xi [m] Yi [m] 5.22 5.22 14.29 5.97 11.42 11.42 14.29 14.29 7.68 7.68 14.29 14.29 8.48 8.48 14.29 6.21 7.68 7.68 6.21 6.21 2.60 2.60 6.21 11.35 2.40 2.40 3.71 11.25
Allocation Floor 2 2 1 1 1 1 1
5. CONCLUDING REMARKS In this paper, the optimal multi-floor process plant layout problem has been considered. A general mathematical framework has been described, which determines simultaneously the number of floors, land area, optimal equipment-floor allocation and location (i.e. coordinates and orientation) so as to minimise the total plant layout cost. The current model can easily accommodate various considerations related to space restrictions and/or safety. The resulting optimisation problem corresponds to an MILP model. Current work focuses on testing the framework to larger examples and investigating alternative solution procedures (e.g. decomposition schemes). REFERENCES
1. J. C. Mecklenburgh, Process Plant Layout, Institution of Chemical Engineers: London, 1st edition, 1985. 2. S. Jayakumar and G. V. Reklaitis, Comp. Chem. Eng., 18 (1994) 441. 3. E D. Penteado and A. R. Ciric, Ind. Eng. Chem. Res., 35 (1996) 1354 4. C. M. L. Castel, R. Lakshmanan, J. M. Skilling and R. Banares-Alcantara, Comp. Chem. Eng., 22 (1998) $993. 5. L. G. Papageorgiou and G. E. Rotstein, Ind. Eng. Chem. Res., 37 (1998) 3631. 6. D. B. Ozyruth and M. J. Realff, AIChE J., 45 (1999) 2161. 7. A. Suzuki, T. Fuchino, M. Muraki and T. Hayakawa, Kagaku Kogaku Ronbunshu, 17 (1991) 1110. 8. S. Jayakumar and G. V. Reklaitis, Comp. Chem. Eng., 20 (1996) 563. 9. M. C. Georgiadis, G. E. Rotstein and S. Macchietto, Ind. Eng. Chem. Res., 36 (1997), 4852. 10. M. C. Georgiadis, G. Schilling, G. E. Rotstein and S. Macchietto, Comp. Chem. Eng., 23 (1999) 823. 11. A. Brooke and D. Kendrick, A. Meeraus and R. Raman, GAMS: A Users's Guide, The Scientific Press, 1998.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
481
Plate Fin Heat Exchanger Design Using Simulated Annealing J.-M. Reneaume a and N. Niclout b a Laboratoire de G6nie des Proc6d6s de Pau, rue Jules Ferry, 64 000 Pau, France E-mail : j [email protected]
b NORDON CRYOGENIE SAS, 25 bis, rue du Fort, BP 87, 88 190 Golbey, France The objective is to propose a tool for the computer aided design of Plate Fin Heat Exchangers (PFHE). The problem of PFHE design is stated as an optimisation problem. Then it is solved using either Successive Quadratic Programming algorithm (relaxed NLP problem) or Simulated Annealing method (initial MINLP problem). Advantages and drawbacks of both methods are discussed. An industrial example is presented. Compared with the classical design method, mathematical programming techniques allow up to 20% capital cost reduction. 1. INTRODUCTION NORDON CRYOGENIE is one of the world leaders of Plate Fin Heat Exchanger (PFHE) design and manufacturing. When designing a heat exchanger, the engineer knows the duty requirement (inlet and outlet temperatures and fluid flow rates) and the pressure drop specifications. Of course thermodynamic and transport properties of the fluids are supposed to be known. Given such information, the engineer has to design the best PFHE. This is a very complex task : how many cores in the assembly, how many layers for each stream, which fins are to be chosen... The engineer experience and his know-how are very important. Tools have been developed in order to help the engineer. Those tools are generally proprietary heuristic-based programs for the computer-aided design of PFHE. COLETH is the tool developed by NORDON CRYOGENIE. Using such a program it is possible to achieve a good approximation of the optimal design. But in order to improve competitiveness, development of efficient and accurate tools for optimal design of PFHE has become a priority in the different companies. In this paper we describe how mathematical programming techniques are integrated in the COLETH program in order to achieve optimal design of PFHE. In the first section, the general solution strategy is briefly described. In the second section, the MINLP optimization problem is formulated and the different solution methods are presented : SQP for the relaxed NLP problem and Simulated Annealing for the general nonconvex and non-differentiable MINLP problem. The numerical parameters of the Simulated Annealing method (initial temperature and temperature reduction factor) are discussed. In the third section, a challenging industrial example is presented and the program abilities are illustrated. Mathematical programming techniques are proved to be very efficient tools since important capital cost reduction are achieved.
482 2. GENERAL SOLUTION STRATEGY Figure 1 briefly describes the general solution strategy [1]. The COLETH automatic sizing mode allows the design of the whole exchanger : fin geometry, core geometry...But this result can only be considered as a starting point for the engineer (if he decides to perform the design "by hand") or for the optimisation procedure. The first reason is that pressure drop requirements are not achieved : additional work is required in order to have an acceptable commercial proposal. The second reason is that, since heuristic-based procedures are used, optimality of the result can not guarantied. Using the sizing mode, part of the geometrical parameters (see next section) are fixed by the optimisation solver. Then the objective function and the constraints are computed. inlet
Input File [ andoutlettemperatures enthalpies transportproperties flowrotes
..... supplieddata
~r
'e..r[,' leaa
optimization variables (initialvalue)
~r
~176
a
t
~
NLPor MINLPsolver
~
optimisationvariablesI
~ objectivefunction nstraints
Figure 1 :General solution strategy 3. O P T I M I Z A T I O N P R O B L E M 3.1.
Formulation
A section is the space between any stream inlet or stream outlet. In most cases, more than 60% of the total duty is exchanged in one section: the main duty section. In the presented works, the fin geometrical parameters are optimised in the main duty section only. In the other sections, fins remain the same as the initial design performed by the COLETH automatic sizing mode. The optimisation problem is stated as follow : Min f(x, y) x,y
s.t. g(x, y) < 0
,(P)
X min ~ X ~ X max
yeY Different objective functions (f) can be minimised. The main one is the capital cost of the PFHE. For some particular applications such as airborne applications, other objective functions are available : total volume of the exchanger or cross section of the exchanger.
483 Optimisation variables (x and y) are described on figure 2. Continuous variables (x) are : core width (xCW), distributor width (x Dw) and number of layers (xL). Discrete/Integer variables (y) are : core number (yCN), fin height (yn), fin thickness (yT), fin frequency (yV) and fin serration length (yS). There are (5.NS + ND + 2) optimisation variables where NS is the number of streams in the main duty section and ND is the distributor number of the core. (4.NS + 1) variables are discrete ones. In order to reduce the combinatorial aspect of the problem, the number of layer (x L) is considered as a continuous variable. For a given example, the discrete values of the y variables are determined as follow. First, let us consider the geometrical parameters (yn, yT, yV, yS): the standard fins (existing fins) of the data bank are enumerated and the discrete values of the parameters are stored in the Y set. This way, we are sure that the tool to build the fin is available : fins are manufactured by NORDON with proprietary tools. Concerning the values for the number of cores(yCN), two or three values are generally considered below and above the initial value (initial design performed by COLETH automatic sizing mode). For symmetry considerations, only even values (except one) are taken into account.
yS yH
Figure 2. Optimisation variables The main constraints of the optimization problem are: 9 banking limit : the ratio of hot to cold stream layers must be nearly equal to one. The PFHE is modelled assuming common wall temperature hypothesis. Thus layer arrangement is not optimised. This constraint ensures that a feasible arrangement will exist. 9 maximum stacking height, maximum number of layers, maximum width and maximum length of the cores : those constraints arise for mechanical and practical reasons : the PFHE is brazed in a furnace with fixed dimensions. 9 fin manufacture feasibility : as seen before the geometrical parameters of the fins are optimised but optimal fins are not necessarily standard existing fins. Each geometrical parameter (height, thickness...) can take standard values but the resulting fin is not necessarily feasible (because of pressure considerations for example). In order to ensure feasibility of the optimal fin, a proprietary (non continuous!) correlation allows the calculation F max of the maximum fin frequency (y ' ) as a function of the fin thickness and the fin height. 9 pressure drops : as discussed before, for given values of the optimisation variables, a PFEH is evaluated by COLETH sizing mode. This exchanger satisfy the duty requirement (target temperatures are reached) but pressure drop requirements are not. An explicit set of constraints must be introduced at the optimisation level. Other constraints are : velocity of the fluid in the header must be lower than a maximum erosion value; operating pressure must be lower than the maximum pressure of the optimal fins which is a function of thickness, serration length and fin frequency; maximum header size; minimum header size...
484 3.2.
Solution methods The first way to solve this problem is to relax all optimisation variables. Since variables are assumed to be continuous, the problem results in a Non Linear Programming Problem (NLP) which is solved using an SQP algorithm [2]. Such a tool is very interesting in order to perform sensitivity analysis (Lagrange multiplier values are available at the solution point) and to outline major trends (see section 4.2). An other advantage is that the solution is achieved in a very reduce computational time. However, because of non-convexities involved in the model, the algorithm may fail to find the global solution of the relaxed problem : this will be illustrated with the example. Even if the relaxed global optimum is achieved, the user has to consider the nearest acceptable value for each discrete variable in order to build an feasible PFHE. Then, there is no theoretical guarantee that the optimal solution of the MINLP problem is achieved. An other difficulty for the SQP algorithm is that many discontinuities arise in the model : for example, when the Colburn factor is evaluated, correlations used for diphasic flows are non continuous. Such considerations induce us to consider algorithms adapted to non-convex, noncontinuous and non-differentiable MINLP problems. Among different possibilities we have chosen Simulated Annealing. The main drawback of this method is that the computational charge is much more important. Thus an important part of the presented work was to reduce the computational charge optimising the numerical parameters of the Simulated Annealing method. Two parameters have been especially optimised : initial temperature (TO) and temperature reduction coefficient (c~). Concerning the initial temperature, different methods have been tested. For a given example, optimisation variables are first randomly moved from initial point. Thus an initial set of exchangers is built in the vicinity of the initial PFHE. The three most efficient methods are: 1. TO is a fraction of the minimum value of the objective function (fmi~): T0=fmin/2, fmin/4... 2. T0 is evaluated according to the method proposed by Aarts and Korst [3] 3. T0 is evaluated according to the method proposed by Maier [4] In our case, the best results (global optimum with the lower computational charge) are generally achieved using the method proposed by Maier with an acceptation rate equal to 0.9. If the temperature reduction coefficient is to low (the cooling rate is important), the algorithm obviously converges to a local minimum. If o~ is close to one, the computational charge is very important. It should be pointed out that, with a close to one, the algorithm is also trapped in a local minimum. In our case the optimal value for the temperature reduction coefficient is 0.8. 4. TEST PROBLEM 4.1.
Description An industrial test problem is described in figure 3. This heat exchanger is a very challenging design problem: there is an intermediary outlet on the first hot stream (H1). This stream is also redistributed in the third section. One should note that layers occupied by stream H2 in section 1 and 2 are occupied by the stream H1 in section 3. This is an example of a duplex heat exchanger with redistribution. The maximum duty section is the second section. The total number of constraints is 49. There are 11 distributors and 28 optimization variables. 16 of them are discrete variables. Values of the discrete variables are shown on table 1.
485
Section I
Section 2
15.0~
Section 3
-11~.6~
Hi 30.0oc
Apm-~=15 kPa
, ~'
H2 30.0oC
AP. . . . 20kPa
I I
C2 28.2~
,
Apm~-20 kPa
[ ~
-118.5~
-183.1~ Apm~5 kPa
-184.9~ C 1
-127.0~
Figure 3 9Test problem The combinatorial aspect of this example is quite important" 1 347 192 combinations. Table 1 9Discrete variable description Variables ...... units ..... Number o'f variabies Values Core Number (ycN). . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . 1"2; 4 Height (yH) mm 3 3.53" 5.1" ... Thickness (yT) Frequency (yV) Serration Length (yS)
mm
3 3 3
m-1
mm
0.20; 0.25" ... 393.7; 492.13; ... 3.175; 9.525; ...
Number of values 3 6 7 22 6
4.2.
Optimal PFHE The main results are presented on table 2. Considering the initial PFHE (computed using the COLETH automatic sizing mode), one should note that pressure drop requirements (given on figure 3) are violated (especially for stream C2). This exchanger is the starting point of the classical design procedure which results is the Commercial Proposal (3 rd column). It is also the initial point of the different optimisations described in the subsequent columns. Table 2 : Main results of the industrial test example Initial PFHE
AP
H, H~ C, C9 Capital Cost [%] [kPa]]
142 89 19 247 91
Commercial
Capital Cost
Capital Cost
Volume
Proposal
minimisation
minimisation
minimisation
(SQP)
(SA)
(SA)
151 124 39 200 87
151 200 50 200 77
150 198 47 200 97
151 200 50 200 100
Total volume [m3]
4.3
5.4
4.0
4.1
3.6
Core length [m]
6.548
7.654
6.658
6.248
6.924
Core width [m]
0.630
0.915
0.630
0.914.6
0.555
Core height [m]
1.039
0.770
0.951
0.716
0.947
*basis "Commercial proposal capital cost
Solving the relaxed NLP optimisation with an SQP algorithm, one achieves a 13% capital cost reduction (4 t~ column). This is obviously a local minimum since a 23% capital cost reduction is achieved solving the initial MINLP with Simulated Annealing (5 th column). Nevertheless this is an interesting tool : since the result is reached in a very few computational charge (3 minutes with a Digital Alpha Server 8200), it can be used as a program for computer-aided design of PFHE in the same way as COLETH. The main difference is that, at the solution point, pressure drop requirement are satisfied. One should keep in mind that considering the nearest acceptable value for each discrete variable is not enough to build a
486 feasible PFHE : with this example, a 12% capital cost reduction is achieved but unfortunately, pressure drop constraints are violated for stream H~ and C2 (constraints saturated in table 2). The best results are achieved using Simulated Annealing for both Capital Cost (5 th column) and Total Volume (6 th column) minimisation. The important point is that those results can be compared to the Commercial Proposal since pressure drop constraints are satisfied and variables have discrete feasible values. Comparing those two results leads to an interesting result : the cost of a PFHE does not directly correlate with the total volume. The two optimal PFHEs are quite different. A 10 hours computational charge is required with this example. For sake of concision, the values of all optimisation variables are not given here. But comparing the optimal results (for both NLP and MINLP problem) and the Commercial Proposal, major trends can be outlined. Fin height increases and fin thickness decreases. Thus pressure drops are minimised. The thermal effectiveness is augmented increasing fin frequency. Those trends are quite general ones. 5. CONCLUSIONS- PERSPECTIVES The proposed work is an example of industrial application of mathematical programming techniques. An efficient tool for computer aided design of plate fin heat exchangers is presented. Mathematical programming techniques are integrated in the COLETH program. The program allows optimization of the fins (height, thickness...), the core (width) and the distributors (widths). Numerous design or operating constraints are included : pressure drops, maximum stacking height, maximum erosion velocity .... Various objective functions can be used: capital cost, total volume .... Most of the heat exchangers configurations can be optimised: intermediate by-products, redistribution .... The user can choose to relax the discrete variables (SQP algorithm is used) or to use a global MINLP algorithm : Simulated Annealing. The values of the numerical parameters of the method are discussed. An industrial example is presented. The program abilities are illustrated : important capital cost reductions are achieved: up to 20%. Future developments will include: optimization of the other heat exchanger sections (in the presented version, only the maximum duty section is optimised), optimisation of the layer arrangement. REFERENCES 1. J.M. Reneaume, H. Pingaud and N. Niclout, Optimisation of Plate Fin Heat Exchangers- A Continuous Formulation; Chemical Engineering Research and Design, Vol. 78, No. A6 (2000) 849 2. D.J. Ternet and L.T. Biegler, Recent Improvements to a Multiplier Free Reduced Hessian Successive Quadratic Programming Algorithm, Comp. Chem. Engng., Vol. 22, No. 7/8 (1998) 963 3. E. Aarts and J. Korst, Simulated Annealing and Boltzmann Machines- a Stochastic Approach to Combinatorial Optimization and Neural Computers, Wiley- New York (1989) 4. R.W. Maier and W.B. Withing, The variation of parameter settings and their effects on performence for simulated annealing algorithm, Comp. Chem. Engng., Vol. 23 (1998) 47
This work is presented with the support and the permission of NORDON CRYOGENIE SAS.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
487
Optimal synthesis of liquid-liquid multistage extractors Juan A. Reyes-Labarta a and Ignacio E. Grossmann b aDepartment of Chemical Engineering University of Alicante. Ap. Correos 99, Alicante E03080, Spain ([email protected]). bDepartment of Chemical Engineering. Carnegie Mellon University. Pittsburgh, Pennsylvania 15213, USA ([email protected]). The purpose of this paper is to determine the optimal design of liquid-liquid countercurrent extractor systems. The proposed method calculates the optimum number of equilibrium stages and flowrates to obtain a specified product separation and recovery. Based on a superstructure representation, the problem is formulated as a Generalized Disjunctive Programming (GDP) model to minimize the total cost of the process subject to design specifications. The robustness and computational efficiency of the model is illustrated with an example. 1. INTRODUCTION Classical methods for the design of extraction systems involve the use of graphical techniques, which combine material balances with liquid-liquid equilibrium relations (Seader and Henley, 1998). However, these methods are restricted to temary systems. Minotti et al. (1996) developed a design method similar to the one for nonideal multicomponent distillation systems where the minimum solvent flowrate to the extractor is determined by a geometric analysis of the composition profiles and their fixed points. Recently, Marcilla et al. (1999) suggested a tray-to-tray design method of a multistage extractor using an analytical extension of the Ponchon and Savarit method for the design of distillation columns. This method solves a sequence of mass balance and equilibrium equations to obtain the number of stages and the optimum solvent flow. Since the optimal design requires not only the number of stages for the extraction, but also selecting the best configuration of extractors and their distribution of streams, the objective of this paper is to present a new optimization model for the optimal design and synthesis of complex countercurrent extraction cascades. The trays are considered as ideal equilibrium stages in which empirical correlations of the solubility surface and the tie-lines data are used. A nonlinear tray-by-tray model based on Generalized Disjunctive Programming, GDP (Raman and Grossmann, 1994), is presented, as well as a description of the solution algorithm.
2. PROBLEM STATEMENT The design problem can be stated as follows. Given is a set of feed streams with known composition and specified separation of solutes in the product stream. The problem consists in determining the optimal number of stages, the feed stream locations, the solvent flowrate and the possibility of placing intermediate solvent streams, lateral feed/products streams and fractional bypasses of the initial feed stream given a specified set of
488 countercurrent extraction stages. The objective is to minimize the total annualized cost of equipment and utilities. The actual number of stages in the optimal extraction cascade is obtained by selecting from among a maximum number of stages that is specified. 3. GENERAL COUNTERCURRENT EXTRACTOR SUPERSTRUCTURE. Figure 1 shows the superstructure for a complex cascade, that considers the possibilities of a bypass (RObyp)of the initial feed stream, lateral solvent feed streams ELi and lateral raffinate feed/product streams (RLk, PLq, respectively) in each stage. The approach proposed in this paper handles the equations through disjunctions for existing and non-existing stages. For existing stages the following equations are considered: i) Total mass transfer balance; ii) Mass component balances (the enthalpy effects are neglected); iii) Equilibrium equations; iv) Summation of mass fractions in each phase equal to 1; v) Relation between total and individual flowrates for all the components present in every stream through its mass fraction. For non-existing or inactive stages the equations considered are simply input-output relations in which no mass transfer takes place (inlet and outlet flows are the same for each phases). Because the mass balances include the trivial solution, the only difference between existing and non-existing stages is the application of the equilibrium equations. A straight forward approach to solve the design problem could be to allow stages to disappear in the cascade by modeling the optimal design problem as an MINLP using Big-M constrains in which equations and inequalities are relaxed for non-existing trays. However, this approach has poor numerical performance, and strongly depends of the initial point used in the calculations. This is due to the nonconvexities introduced by the bilinear terms in the mass balances and the equilibrium relations. Moreover, the resulting M1NLP model is very large, because all equations must be converged whether or not the corresponding tray is selected. Therefore, in this paper we use a GDP model to circumvent these problems. The advantage of the proposed modeling approach is that the nonconvex equilibrium relations do not have to be converged for non-existing trays, making the convergence of the optimization procedure more reliable. Also, by using Generalized Disjunctive Programming (GDP), the computational expense can be reduced, as well as the likelihood at getting trapped into bad suboptimal solutions (Turkay and Grossmann, 1996).
Figure 1. Superstructure of a multistage countercurrent extractor.
489
3.1. Generalized Disjunctive Programming Model. In this section we present the GDP model for the superstructure in Figure 1. Consider the following set definitions for the model: COMP is the set of components c present in the feed. NT represents the set of permanent stages j in the cascade, where the first stage (j=l) corresponds to the initial feed stage, and the last stage (j=n) is the solvent feed stage. NINT represents the set of intermediate stages. K and Q are the set of lateral feed and products streams, respectively, and me indicates a component mass flow. Let {EI,c represent the recovery fraction of species c in the final stream E l , and "CRdef,c the purity of species c in the final raffinate product Rdef. ~PLq,c and TPLq,care the recovery fraction and purity respectively, of species c in the lateral raffinate product PLq. The constraints are given by the following set of equations (1)-(11): 9Purity and recovery requirements in the final products streams El and Raef:
Eml,c >
~El,c "
/
Rmo,c + ~ RLmk,o,c k=l
X R d e f ,c <-- Z'Rdef ,c
1t
Eml,c < ~El,c " Rmo,c + ~ RLmk,o,c k=l
if c = key component in the extract stream
(1)
if c = key component in the raffinate stream
(2)
X R d e f ,c >- Z'Rdef ,c
9Purity and recovery requirements in the lateral products streams PLq: K
PLmq,c >-- ~ P L q , c
"
Rmo,c + __~RLm k,o,c k=l
t if c = key component in the raffinate stream
(3)
X pLq, c >-- "l'PLq, c
9 Global mass balance: E1 + Rdef + Z PLq = Ro + Eo + Z RLk + q=l
k=l
9Mass balances in each stage: Q /~ Rj + Ej + ~ PLq,j = Rj. 1 + Ej+ 1 + Z RLk, j + ELj q=l
9Bypass mass balance:
(4)
ELi j=l
Vj~ NT
(5)
k=l
Ro
=
Ro ext + eObyp
(6) K
(7)
R def = R n + R o byp + _~ _ RLk, byp k=l
9Nonlinear equilibrium relations (Reyes et al., 1999): U?(yj,c, xj,c)-0
Vj~ NT
(8)
9 Component and total flowrate relations (bilinear terms): Fmj,r Fj.Uj,e = 0
VF e {R, E, PLq, RLk, EL, edef}, U -- {X or y} Vje NT, Vc e COMP
(9)
490 9Lateral stream balances: ~
RL k =
n
RLk,j + RLk,byp Vk ~ K
PLq = Z PLq,j
j=l
Vq ~ Q
(10)
'v'j~ NT
(11)
j=l
COMP ~ X j, c = 1
9Mass fractions:
COMP
Zyj, c - 1
c=l
c=l
The constraints in (1)-(11) involve only continuous variables, and are valid for any cascade configuration. The disjunctions in (12) are the ones associated with the discrete choice of enforcing the equilibrium relations in existing trays. This is accomplished with Boolean variable Zj, which can be true or false depending if the stage j is selected or not. The disjunction is as follows: _
-nZj Xj,c = Xj-l,c ; Yj,c = Yj+l,c [
Zj 1 Ej=Ej+l'c'Emj'c=Emj+l'c Equilibrium." trt~(Yj,c,Xj,c)=O v R j = R j _ l , c ; R m j , c = R m j _ l , c
[
Fmj,C= Fj .ujx
RLk,j = O;RLmk,jx = 0 PLq,j = 0 ; PLmq,j, c = 0 ELj =O ; ELmj, c = 0
Vj~ NT '7'C~ COMP
Vk~KVqe Q '7'FE {R,E, PLq,
RLk, EL, Rdef} zt = {x or y}
(12)
_
To avoid equivalent solutions that are due to the multiplicity of representation for a given number of trays, the following logic constraints are added: Vj~ NINT
Zj ~ Zj_ 1
(13)
The objective function involves the minimization of the total annualized cost of equipment and utilities. The capital cost is considered proportional to the sum of stages, and the operating cost of the process a function of the flowrate of the solvent feed stream, n
min
n
OF=(Eo+~_ELj).C E +~.Zj .C n j=l
(14)
j=l
In order to avoid infeasibilities in the solution of the NLP problems with fixed values of the binary variables, the design specifications are relaxed using slack variables, which are minimized in the objective function. 4. SOLUTION ALGORITHM. The proposed GDP model is solved with a modification of the Logic-Based Outer Approximation (OA) algorithm of Turkay and Grossmann (1996). This decomposition algorithm solves the problem by iterating between reduced NLP subproblems and an MILP Master Problem that consists of linear approximations in all solution points given for all the NLPs solved previously, using the big-M formulation to relax the linear approximations of the disjunctions. If we define PNLP as the set of all the previous solved NLP subproblems and NLF as the set of all nonlinear functions in the model: {bilinear terms and equilibrium relations}, the constrains in Equation (12) are replaced in the MILP master problem by the following equations:
491
ps(Knl)+Vps(~nl).(K_Knl)<_MPs .(l_Zj)+svl(P~ t) Ps(g"t)+VP~(gnt)'(g-gnt)+sv2(Pflt)>_-Me, . ( l - Z j ) where
Ps(~ nt)
] Vs ~NLF, Vnl ~ PNLP VjeNT, VceCOMP
(15) represents the value of the nonlinear functions in all the corresponding
Mp~ is a sufficiently large number. Also in each of the nl variable sv(Pnt) is introduced to make compatible all the different
previous NLP solutions, and
linearizations a slack linearizations of the same nonlinear equation P.
5. NUMERICAL EXAMPLE. The proposed design model was tested with an example that involves the separation of a quaternary mixture, using a countercurrent extractor. We considerer one aqueous feed stream Ro, one solvent feed stream Eo and one lateral raffinate product extraction PL1. The feed for this example is a mixture of water (1), acetone (2) and acetic acid (4) with a flowrate of 275 kg/h. The solvent used is pure chloroform (3). The required purity for the raffinate product are mass fraction of acetone and acetic acid, 0.01 and 0.1, respectively, for a minimum recovery of 90% of acetone and a maximum recovery of 40% of acetic acid in the organic product stream El. The specifications for the lateral raffinate product PL1 are flowrate of 100 kg/h, ~PL1,2-" 5% and XPL1,2- 0.04. A maximum number of 10 potential trays are postulated. The results for this example are shown in Figure 2 in which it can be seen that 6 stages were selected, with two lateral extractions of the product. If these options are excluded the cost increases by 3%. The GDP model involves 350 continuous variables, 547 equations and 10 boolean variables. The problem was solved on a 300 MhZ Pentium II PC, using the GAMS modeling environment (CONOPT for the NLP subproblems and CPLEX for the MILP master problems). The total CPU time required was 570 secs.
J0.3644 Y1'c/0.5282 1.0.0595
00
Yo,c YEL3,c
_r
-- =f,o
El= 206 kg/h
Eo = 110 kg/h
R6=Rdef= 79 kg/h
Ro= 275 kg/h PL1,5= 86.8 x
0.59 J0.29
.0
0.8316]
o,c/o.o
~176 x
[~
0.120oJ
0.0085[
PLI,c
1,6= 13.2 kg/h
I0.8721
/
~0.0100 XRdel',C[o.o075 PL~= 100 kg/h v
[0.1104
Figure 2. Superstructure of a multistage ccountercurrent extractor.
ACKNOWLEDGMENTS The authors would like to acknowledge financial support from the Conselleria d'Educacio-Generalitat Valencia of Spain (POST00-11-89).
492 REFERENCES
Marcilla, A.; G6mez, A.; Reyes, J.A. and Olaya M.M.; New Method for Quaternary Systems Liquid-Liquid Extraction Tray to Tray Design, Ind. Eng. Chem. Res. 1999, 38, 3083-3095. Minotti M.; Doherty M.F. and Malone M.F. A Geometric Method for the Design of Liquid Extractors. Ind. Eng. Chem. Res. 1996, 35, 2672-2681. Raman, R. and Grossmann, I.E. Modeling and Computational Techniques for Logic Based Integer Programming. Comp. Chem. Eng. 1994, 18, 563. Reyes, J.A.; Olaya, M.M.; G6mez A. and A. Marcilla. Calculation of Liquid-Vapour and Liquid-Liquid Equilibrium in Multicomponent Systems Using Correlations of Equilibrium Data. Equifase'99, V Iberoamerican Meeting. Phases Equilibrium for Process Design, Vigo (Spain). 1999. Seader, J.D. and Henley, E.J., Separation Process Principles. John Wiley & sons, Inc. New York, 1998. Turkay, M. and Grossmann, I.E. Logic-Based MINLP Algorithms for the Optimal Synthesis of Process Networks. Comp. Chem. Eng. 1996, 20, 959.
European Symposiumon ComputerAidedProcessEngineering- 11 R. Ganiand S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rightsreserved.
493
Development of Novel Process Designs for Simultaneous Oxidation and Denitrification of Wastewaters 9
a*
S.Rlgopoulos , P. Linke b and A. Kokossis bw a Department of Chemical Engineering, UMIST, PO Box 88, Manchester, M60 1QD, UK b Department of Process Integration, UMIST, PO Box 88, Manchester, M60 1QD, UK
A systematic synthesis approach is adopted to the problem of activated sludge process design. The conventional designs as well as all novel schemes for combined oxidation/denitrification of wastewater are explored. The process is optimised using a novel methodology for optimal reaction/separation network synthesis, supplied with a comprehensive and general-purpose kinetic model (IAWPRC Activated Sludge Model No.l). The optimisation results suggest a counter-intuitive optimal design policy: Simultaneous oxidation and denitrification are carried out simultaneously within the same reactor at very low oxygen concentrations. The new designs feature significantly improved nitrogen removal as compared to conventional processes. 1. INTRODUCTION The activated sludge process is the most commonly encountered form of biological wastewater treatment. Despite having received the attention of many researchers, most of the work so far has addressed the optimisation of individual operating parameters whilst no study to date has attempted the overall process optimisation including the basic structure of the process. The primary aims of the activated sludge process are to oxidise the organic content of wastewater into harmless inorganic compounds, and to convert organic and ammonia nitrogen to gaseous components. These two objectives are contradictory: the former process takes place in the presence of oxygen while the later is anoxic and each process imposes additional processing requirements onto the other. Current systems attempting to accomplish both aims employ separate aerobic and anoxic reactors, such as in the empirically designed Ludzack-Effinger and Wuhrmann systems (Van Haandel et al., 1981). This study combines a detailed activated sludge model with a robust synthesis methodology for multiphase reaction-
Current address: Dept. of chemical engineering, UniversityCollege London, London WC1E 7JE Corresponding author. Current address: Departmentof Chemical & Process Engineering, Universityof Surrey, Guildford, GU2 5XH, UK. Tel. ++44(0)1483 876573, Fax: ++44(0)1483 876581. Email: [email protected]
494 separation networks to gain insights into the complex process trade-offs and to seek pathways towards improved designs. 2. MODELLING OF WASTEWATER TREATMENT Modelling activated sludge systems is still an active research area. To date, the IAWPRC Group Activated Sludge Model No. 1 (Henze et el., 1987) is the most complete and is widely accepted by the wastewater treatment industry. It has been developed by a task group of researchers appointed by the International Association on Water Pollution Research and Control (IAWPRC) and subsequently used by many practitioners. Dold and Marais (1986) verified the model under a variety of process conditions. The model comprises of 13 reactants, participating in 7 biochemical reactions (Table 1). This complexity, as compared to the simple Monod model, accounts for the model's flexibility and applicability to different wastewater compositions. The highly non-linear kinetic equations are shown in Table 2. For a detailed model description it is referred to the IAWPRC report (1987) and Henze et al. (1987). Table 1. Reaction paths in wastewater treatment according to the IAWPRC model
Carbonaceous Organics Reaction Path 02 ,NO,X BH
X S
SS .~_02
Ss +NO3-
Organic Nitrogen Reaction Path
>S S
SNH + 02
XBlf ~ X B H
xn.
02 ,NO,XsH
XND
>XB H ..1_N2
Ss +NO3,
>aND,
SND
Xsg
~ SNH
X.. >XBA + NO 3xn. >X8 n + N2 ,
,,,
Table 2. Kinetics of the IAWPRC model
Process rate (ML3T 1)
Process
Aerobic growth of heterotrophs Anoxic growth of heterotrophs
PH Ks + Ss ~u~ Ks +Ss
Aerobic growth of autotrophs Hydrolysis of entrapped organics, Rh
I
Ko,H+So
'uA K ~ + SNH
kh
XBH Kx + - ~
KO,H+ So KN~ +SN~ .ngXB~
II
Ko,A +So
.itKo,H+So) L "! Ks +Ss)lKNo +Suo
Process
Process rate (Mt'sr ')
Decay of heterotrophs
bh XBH
Decay of autotrophs
bAXBA
Ammonification of nitrogen
khSNDXBH
Hydrolysis of entrapped organic nitrogen
XND Rh'~ Xs
495 3. OPTIMISATION METHODOLOGY The activated sludge process is synthesised using a systematic reaction/separation network synthesis framework (Linke et al., 2000) to determine the optimal biochemical reactor network design along with the sludge separation and recycle policies. The methodology builds upon previous efforts in reaction system synthesis (Kokossis & Floudas, 1990; Marcoulaki & Kokossis, 1999; Mehta, 1998) and employs superstructures of generic shadow compartment and separation task units that capture all possible novel and conventional process design options that exist for the multiphase system. All generic units are interconnected in a network of streams through a superstructure scheme in each of the contacting phases and in each pair of contacting phases where state change is possible. The representation provides for all possible mixing, contacting, and reaction/separation features. The superstructures are optimised using Simulated Annealing. The search produces performance targets in the form of stochastic optima with confidence levels appropriate for this problem and a multitude of designs with close-to-target performances, which offers a major advantage in understanding the process trade-offs. A detailed description of the algorithm is provided by Aarts and Van Laarhoven (1985). Two sets of representative values of municipal wastewater compositions, adopted from the IAWPRC report (1987), are studied. Two sets of values of kinetic parameters were tested, both within ranges suggested by the IAWPRC (1987). Oxygen mass transfer is modelled using the film theory. The weighted objective function is formulated so as to minimise the effluent COD and total nitrogen content: Objective = Min
COD +
N xl O0
(1)
No The stochastic nature of the optimisation means that it will yield a different design every time it is operated with different parameters. Yet these results will share some common features, and an insightful examination of them will provide the guidelines towards an improved process. For statistical confidence 16 runs were performed in each case, and about 2000-5000 different designs were evaluated during each run. 4. RESULTS AND DISCUSSION Before looking at the novel configurations, two conventional designs were evaluated according to the objective function (1) to establish a basis for comparison (see Figure 2 and Table 3). It is apparent that, although conventional designs succeed in removing COD, they do a poor job in reducing the nitrogen content. Accomplishing denitrification whilst maintaining high COD removal ratios appears to be a main challenge for process design. Using the first feed composition and parameter set, an optimisation without volume bounds on the reaction equipment yields a target (lowest value of objective function) of 267, corresponding to a 96.8% reduction in COD and 84.8% in N, but with too high residence time to be compared with the conventional processes. For this reason volume bounds are introduced to limit the mean hydraulic residence time up to 8 days, at which the conventional
496 processes were evaluated. Surprisingly the target did not deteriorate significantly, its new value being 271. COD was reduced by 97.4%, and N by 84.9%, a performance much better than those attained by any conventional process, especially as far as denitrification is concerned. An inspection of the structures revealed a striking feature: many structures did not include an anoxic reactor, and yet yielded excellent denitrification results. Moreover, the system appeared to seek ways to hinder or control the dissolution of oxygen, in striking contrast to current practice, which aims at dissolving the maximum amount possible into the oxic reactors. To find out how such excellent denitrification could be accomplished without the inclusion of an anoxic reactor, the detailed oxygen profiles within the aerated reactors (mostly PFRs) were examined (Figure 2). The concentration of oxygen is controlled within extremely low levels (0.1-0.2 ppm), an order of magnitude less than those currently used in industrial practice (2-9 ppm). This leads to the conclusion that both organic matter stabilisation and denitrification processes occur simultaneously in these designs, due to the very low oxygen concentrations - a policy that seems to achieve maximum efficiency when low volume is required. In order to keep the oxygen concentration in such low levels while retaining the necessary rate of oxygen dissolution to fulfil the requirements of the oxidation process, the new designs employed several ways: recycling the liquid phase in order to dilute the oxygen, recycling the gas phase to reduce the driving force and introducing side streams for better control of dissolution. Both liquid and gas recycling are expected to increase the operating cost of the process, though, and in practice this aim may be easier to achieve by proper selection of the aeration equipment, which provides some control over the overall mass transfer coefficient. The effect of feed composition is investigated by carrying out the optimisation with a more concentrated feed; however, the resulting structures are still based on the same design principles and achieved similar high values of N reduction (88%). Any optimisation relies on the set of parameter values used in the model, and though it is desirable to investigate novel designs, it is the conventional ones that were used to determine the kinetics. An additional study was performed with kinetic parameters that assume values up to the extreme end of their range. The trend of the resulting designs was, however, maintained: optimal solutions still relied on carrying out organic matter oxidation and denitrification simultaneously, the only difference being that dissolved oxygen concentrations were even lower. The main features of the novel designs are shown in Fig.2, while Table 3 summarises the numerical results. 5. CONCLUSIONS A superstructure-based stochastic reaction-separation network optimisation methodology has been applied to investigate and optimise the activated sludge process for combined nitrogen and COD removal. The method indicated a new concept for process design: instead of employing separate aerobic and anoxic reactors it is suggested that both processes are be carried out within the same unit. This can be accomplished by the use of very low dissolved oxygen concentrations that allow both aerobic and anoxic reactions to proceed at reasonable rates. The new design policy yielded a significant improvement in
497 nitrogen removal. The ability of the optimisation methodology to deal with complex problems was demonstrated, as well as its potential to detect novel designs based on concepts radically different from those of conventional processes. The optimisation has, however, employed the reaction model within the whole domain of possible configurations and the optimum did not occur in the region where the kinetic parameters were determined. Therefore the new concept should not be regarded as an immediate design proposal but as a direction for further research, both theoretical and experimental, to explore the behaviour of the process into the new unexplored region.
0.4 E r
03
,-
02
~
01
o
I..................................
T ....................................
0
10
i
..................
20 Reactor length (m)
T ..............................................
30
40
Fig. 1. Dissolved oxygen concentration profile in combined anoxic/aerobic reactor
I
I
a) Modified Ludzack Ettinger
-
-
- ~------,---~--:-:
,
b) Wuhrmann
i
-
-
Cobine:aerobi/ano:c
-
-'-
*
Air Water Sludge
c) Optimised Scheme Fig. 2. Conventional processes (Van Haandel, Ekama and Marais, 1981) and general structure of the optimised process for single sludge denitrification
498 Table 3. Comparison of conventional processes and novel optimised designs
Processes Ludzack-Effinger (Fig.3a) Wuhrmann (Fig.3b) Optimised, no bounds Residence time 8 days Residence time 5 days 8 days, concentrated feed 8 days, different kinetics
Objective function 825 698 267 271 310 415 324
COD % removal 98.3 98.8 96.8 97.5 98.2 98 98.5
N% removal 47 55 84.8 84.9 82.8 88.1 80.8
Comments Anoxic reactor first Aerobic reactor first Huge volume Combinedanoxic/aerobic >> >> Very low oxygen conc.
LIST OF SYMBOLS
XI XAUT XHET
XND Xs
Rho bu,ba,kh,ng,nh, Kx,Ko,A,KNo
Inert insoluble organics, ppm Autotrophs, ppm Heterotrophs, ppm Insoluble org. nitrogen, ppm Insoluble COD, ppm Hydrolysis rate Coefficients of the IAWPRC model (Henze et al., 1987)
SNO
SND SNH SI Ss So COD0, No
Nitrates, ppm Soluble organic nitrogen, ppm Ammonia nitrogen, ppm Inert soluble organics, ppm Soluble COD, ppm Dissolved oxygen, ppm Initial COD and nitrogen molar flows, kgmol/hr
REFERENCES
Aarts, E.H.L., & P.G.M. van Laarhoven (1985). Statistical Cooling: A General Approach to Combinatorial Optimisation Problems. Philips J. Res., 40, 193. Dold, P.L., & G.v.R. Marais (1986). Evaluation of the General Activated Sludge Model Proposed by the IAWPRC Task Group. Wat. Sci. Tech., 18, 63. Henze, M., C.P.L. Grady Jr, W. Gujer, G.v.R. Marais, & T. Matsuo (1987). A General Model for Single-Sludge Wastewater Treatment Systems. Wat. Res., 21,505. IAWPRC (International Association on Water Pollution Research and Control) (1987). IA WPRC Scientific and Technical Reports No. 1, IAWPRC, London. Kokossis, A.C. & C.A. Floudas (1990). Optimisation of Complex Reactor Networks - I. Isothermal Operation. Chem. Engng. Sci. 45, 3,595. Linke, P., V.L. Mehta, & A.C. Kokossis. In S. Pierucci: Computer Aided Chemical Engineering, 8 (pp. 1165-1170). Amsterdam: Elsevier Science (2000). Marcoulaki, E.C., & A.C. Kokossis (1999). Screening and Scoping Complex Reaction Networks Using Stochastic Optimization. AIChE J., 45(9), 1977. Mehta, V.L. (1998). Synthesis and Representation of Multiphase Reactor Networks. Ph.D. Thesis, University of Manchester Institute of Science and Technology. Van Haandel, A.C., G.A. Ekama, & G.v.R. Marais (1981). The Activated Sludge Process3: Single Sludge Denitrification. Wat. Res., 15, 1135.
European Symposiumon ComputerAided Process Engineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rights reserved.
499
Middle Vessel Heterogeneous Batch Distillation of an Azeotropic Mixture Rodriguez-Donis I. 1'2, Gerbaud V. 1, Joulia X. 1 1 Laboratoire de G~nie Chimique (LGC, U M R CNRS 5503), 1NPT - ENSIGCT, 18 Chemin de la loge, F-31078 Toulouse cedex 4, FRANCE corresponding author Email: [email protected] 2 Centro de Quimica Farmaceutica, 200 y 21 Atabey Apdo. 16042, Playa, C. Habana, CUBA In this paper, we discuss the operation of heterogeneous batch distillation (HBD) in a middle vessel column (MVC). A feasibility analysis is attempted and simulations are carried out which demonstrates the advantage of MVC compared to regular and inverted columns. 1.
INTRODUCTION Batch distillation is a key technology well suited for multipurpose plants with high added value products and small product volumes. Synthesis and design of batch distillation is often a complex task when azeotropic mixtures must be treated. Most processes involve homogeneous entrainers and regular or inverted columns 1. Much attention has been given recently to a configuration termed as middle vessel column (see 2 and references in 2) that can allow the recovery of several high purity products. We have recently addressed the feasibility of batch distillation processes using heterogeneous entrainers 3 and we discuss here this process in a MVC. HBD synthesis differs significantly from homogeneous batch distillation synthesis 4. Some advantages of HBD processes compared to homogeneous systems a r e : i ) more design alternatives for the separation of non-ideal mixtures, ii) simplified distillation sequences because of less distillation tasks thanks to the l i q u i d - liquid phase split occurring in some part of the column and in the decanter, iii) the addition of less entrainer, iv) the crossing of batch distillation boundaries due to a flexible reflux policy that enables to reflux to the column a mixture different from the distillate through any combination of the entrainer-rich phase and the distillate-rich phase. The importance of the vapour line and the existence of a minimum and maximum amount of entrainer have also been evidenced 4. We determined recently a complete set of rules for the selection of both homogeneous s and heterogeneous 6 entrainers for the separation of minimum or maximum temperature azeotropic binary mixtures or close boiling components from the analysis of ternary residue curve diagrams under assumptions of high number of trays and total reflux/reboil ratio. Rules combine general conditions related to the system components ability to form together azeotropes and information about their boiling points order. The system fixed points stability is needed to determine the ternary diagram and to assess the process operation. In HBD, a feasible entrainer provides a ternary diagram where both original components are or are not in the same batch distillation region and either the heteroazeotrope or one of the original components is a node. When the heteroazeotrope is an unstable node, a regular column is preferred for the first batch task because it leads to the lowest number of batch tasks needed to obtain a sequence of high purity products. Conversely, an inverted or a regular batch process can be used when the original component is a unstable node or a stable node respectively.
500
Figure 1. a) Middle vessel composition path for heterogeneous batch distillation b) Homogeneous batch distillation regions for a middle vessel configuration. 2.
FEASIBILITY OF H E T E R O G E N E O U S BATCH DISTILLATION PROCESS. We consider a MVC suited for HBD with the vapour flow bypassing the middle vessel like in Hasebe et al. 7. Operation proceeds as follow: the initial mixture is splitted between the reboiler (x r) and the MV (xM). Amounts depend on the total tray hold-up and on the reflux drum and reboiler volumes, x0v should be in a basic distillation region where the heterogeneous azeotrope is an unstable node. When the many stages column reaches steady state under total reflux/reboil operation, the reflux and still drums contain respectively the unstable and stable nodes. A liquidliquid demixing occurs in the decanter at the heteroazeotrope composition. The entrainer-rich phase L1 can be recycled to the column while the L2 phase is taken as distillate. The MV path ~ff to xff depends on the reflux/reboil ratios used for the distillation. Separation feasibility analysis of HBD in a MVC can be made using analytical tools developed for homogeneous MVC distillation 8. Cheong and Barton 8 established the MV path under limiting conditions of high reflux/reboil rate and high number of stages. Considering the overall and component mass balance around the whole column and introducing dimensionless parameter ~ e[0,1] and warped time ~, the change in the ~ vector is: dx m
dr
= x
M
- , g x r' - 0
-,~) xB = x
M
- x
P
O)
Where x P is the net product drawn from the column, that is, a L-weighted average of the bottom (stable node) and the top (unstable node) product removals composition, x a4moves directly away from the net product composition x P. Monitoring the L value allows to control the x a4 path and to obtain the desired product in the vessel at the end of the batch distillation. In the case of a heterogeneous system, the net product x Pa~'~"is still a ~,-weighted average of the bottom and the top product removals, but the top product is now a liquid phase L2 resulting from the condensed overhead heteroazeotrope vapour phase splitting. eh~,~ ~,XL2 (1 2 ) x 8 (2) X
~-
m
m
Flexibility is enhanced as both top and bottom products can be located in basic distillation regions different from the middle vessel initial mixture one. As in the homogeneous case, the possible direction of motion for the MV composition is restricted by a two-dimensional vector cone which is defined by vectors (~4_ ~e) and ( ~ - x B) and s is applied to each vectors: dx M
de
= A ( x M - x L2) +
(1 -
2)(x M - x 8)
(3)
501
The MV path for HBD is shown in Fig. la for a w a t e r - acetic a c i d - acrylonotrile system. There is one basic distillation region where the heteroazeotrope is the unstable node and acetic acid the stable one. The first distillate and bottom products are the heteroazeotrope and acetic acid respectively. The middle vessel vector cone of motion is shown on Fig. la. Fig. lb displays the batch distillation regions for a homogeneous MVC with s The first distillate and bottom products are A E and B respectively. According to the feasibility analysis of Cheong and Barton 8, we set the net product x e on the batch distillation boundary. Then, if x M lies in regions 1 to 4, it will end at the fixed points A, B or E. Likewise, x ~ reaches the azeotrope AE if it lies in regions 6 or 7 and fixed points A or E when it lies in regions 5 or 8 respectively. If we consider the heterogeneous case for Fig. lb with s = 0.5, the distillate product is now the liquid phase L2 instead of the unstable azeotrope BE. Hence, ~ will not reach the side A-B anymore but edges B-E (region 3) or A-AE (regions 5 and 6). Anowing ~ to ~ary, Fig. l a shows that x ~ can reach the entrainer vertex. This is more advantageous than a regular batch column for which the entrainer purity is determined by the liquid - liquid equilibrium in the decanter 3. Cheong and Barton 9 and Warter and Stichlmair 1~ gave entrainer screening rules for homogeneous middle vessel processes. For HBD, entrainers suitable for regular or inverted operation a are also suitable for a middle vessel. In fact, a perfect MV HBD entrainer would lead to a ternary diagram where the heteroazeotrope is located in the same basic distillation region that the miscible component. If those two fixed points have a different stability then separation is performed like the process showed in Fig. 1 and pure entrainer can be obtained in the MV.
3.
SYSTEMS AND TOOLS The quaternary system [ acetonitrile - acrylonitrile - water - acetic acid ] is chosen for illustration. H 2 0 - C2H402 has a relative volatility close to unity near H 2 0 inducing a pinch. Literature reports no azeotrope for C3H3N - C2H402 and no information for C2H3N - C2H402. A homoazeotrope is mentioned for C2H3N - H20 and a heteroazeotrope for C 3 H 3 N - H20. N o ternary or quaternary azeotrope is assumed. Singular points stability and boiling temperature are shown on Fig. 1 and 2. [sn] stands for stable nodes; [sa] for saddle and [un] for unstable nodes. The U N I Q U A C model is used to represent phase equilibrium with binary parameters taken from the D E C H E M A tables. Equilibrium consistency is checked against the experimental data available in those tables with ProPhyPlus. As no experimental data exist for C3H3N - C2H402, we use U N I F A C to generate EVE pseudo-values then regressed with UNIQUAC. Behaviour is predicted nearly ideal. Dimerization of C2H402 valour is considered. A slight curvature of the boundary surface is noticed. LLVE holds on the column plates. 25~ LLE applies for both condenser and decanter. Batch distillation simulation is carried out using ProSimBatch| (batch process simulator) and ProPhyPlus| (properties server) 11. The column model consists of usual plate by plate MESH (Material balance, Equilibrium, Summation of fractions and Heat balance) equations which are solved for the whole column, decanter included and taking into account the liquid-liquid demixion 11. Numerical treatment of the Differential Algebraic Equation (DAE) system and discrete events handling is performed with DISCo, a numerical package for hybrid systems Figure 2. Acetonitrile - acrylonitrile - water - acetic acid system
502 Distillate L2 9
/-
H~9,/
t= O (R total) t- 1 lh
[j .7"
r
"N~~
.... . ...... . ./~ ~
Table 1.Operating Conditions for the MVC Simulations
-'~ Liquid Column Pro ,e
Middle Vessel
"~Vapour
$ C z . . . . . . . . . . . .~.. . . . . ~ : ~ ~ ~ t =
C2H402XB=0.992 Recovery" 98.6%
Line
OperationalParameter Initial reboiler charge (mol) Initial molar composition Initial M.V. charge (tool) Initial refluxdrum charge No. of trays in the rectifyingsection No. of trays in the strippingsection Total tray hold-up (mol) Refluxdrtma hold-up (tool) Reflux ratio Reboil ratio Boilerheat Duty (W) Operating pressure (atm) Efficiency oftrays
Values 19.4 0.1; 0.3; 0.6 1.9 20 12 0.6 0.2 5.2 - 68 Total 205 1 1.0
3.1h
CaH3NXMV=0.995
Recovery 70.5% 9
Figure 3. Simulations results for the separation of a ternary mixture by heterogeneous middle vessel distillation. with D A E solver based on Gear's method 12. A phase split can occur in the column and in the decanter where any combination of the aqueous phase and organic phase can be refluxed to the column. The entrainer-rich phase reflux is varied to keep the entrainer rich phase volume constant. The middle vessel is modelled using a large holdup on a selected tray. MV tray efficiency is set to zero so as to simulate the vapour bypass. Plate volumetric holdup is constant. The initial reboiler, MV or reflux drum holdup are predetermined considering the initial feed charge and the molar quantity of the components we expect to find finally in those vessels. 4.
S E P A R A T I O N OF A T E R N A R Y M I X T U R E IN A M I D D L E V E S S E L C O L U M N . Fig. 3 displays the simulation conditions and results for the separation of the w a t e r - acetic a c i d - acrylonitrile mixture with a MVC and shows the MV and the still paths. The initial point of each trajectory is the composition after the reflux total operation (t=0). The column liquid profile joins the MV heterogeneous azeotrope and the still composition. The reflux policy allows the addition of a little amount of C3H3N to the original binary mixture. The direction of the middle vessel motion is related to s values between 0.5 and 0.6. Once C2H402 is almost exhausted in the middle vessel, the MV trajectory is approximately a straight line and its direction is defined by the position of L2 in the ternary diagram (s At t=-2.5h, the reflux of C3H3N is increased significantly to diminish its amount in the decanter (s Therefore, most of it is recovered with a high purity in the middle vessel. Recovery yields and purities are displayed in Fig. 4. The entrainer C3H3N is found mainly on the trays and in the MV when the operation is over. 5.
M I D D L E V E S S E L S E P A R A T I O N OF A Q U A T E R N A R Y M I X T U R E . In this part we assess the feasibility of the separation of the water - acetonitrile- acrylonitrile - acetic acid mixture by simulating HBD process. Optimal separation is not addressed in this paper. C3H3N is the so-called entrainer partially miscible with water and enabling to separate the homoazeotropic C2H3N - H 2 0 mixture in presence of an impurity, C2H402. Simulation results show below that a single batch task is required when using a MVC whereas two sequential batch tasks are needed with conventional columns (two regular or inverted + regular). Simulation conditions are reported in Table 2. Operation steps simulated are: (1) heating of the initial charge
503 Table 2. Simulation conditions for the all column configurations Operational Parameter values
Acetonitrile - acrylonitrile - water - acetic acid Middle Vessel Column Regular batch column 1st task [ 2nd task 37.0; 4.7; 37.0; 4.7; 43.3; 15.0 43.3; 15.0 14.05 25.54 11.49
Inverted + regular Regular Inverted 41.1; 6.8; 37.0; 4.7; 52.1; 0.0 43.3; 15.0
Initial composition(molar %) C2H3N; C3H3N; H20; C2H402 Initial reboiler charge (tool) Initial M.V. charge (tool) Initial reflux drum charge (mol) 18.26 48 No. Of trays in the rectifying section 64 64 15 No. Of trays in the stripping section 64 2.1 Total stage hold-up (tool) 2.1 2.1 2.1 Reflux drum hold-up 1.8 18.26 1.8 Reflux ratio 5.2 - 32.0 total 4.9 46.2 5.0 4.6- 43.0 I Reboil ratio total total total Operating pressure (atm) 1 1 1 Efficiency of stages 1.0 (0.0.) 1.0 1.0 * null Murphree efficiency for the middle vessel tray so as to allow the vapour to bypass the vessel. _
m the still and fiUmg of the trays including the decanter, (2) operation at total reflux until the heterogeneous azeotropic mixture is obtained into the decanter, (3) batch distillation step considering the reflux of acrylonitrile-rich phase to keep the level of light phase in the decanter and a total reboil ratio. Steps 3 is carried out only for the MVC. T h e initial feed lies in the batch region limited by the water vertex, but results are similar if x0v is placed in the other batch region because the unstable heteroazeotrope is c o m m o n to both. For regular batch operation (Fig. 4a), reflux is varied during the first batch task and the still composition moves towards the C2H3N - C2H402 edge. In the meanwhile, a water-rich distillate is r e m o v e d (XH20,mean = 0.954); tangentially to the still path as in regular h o m o g e n e o u s batch distillation with a high n u m b e r of trays. For the second task, R value is set to 5.0 and a distillate with a high C2H3N content (XC2mN,mean = 0.962). XC2H402,~n~I = 0.999 iS got in the still. O n Fig. 4a, the second distillate path does start somewhere near the C2H3N - C3H3N edge because one removes the C3H3N remaining after the first task, thus slightly polluting the C2H3N-rich distillate. F o r inverted batch operation (Fig. 4b), the column is operated under total reflux and reboil ratios. B o t t o m composition moves towards the acetic acid vertex (XC2H4HO2,~nal= 0.995). C o l u m n trays hold mainly H 2 0 and the reflux drum contains a mixture C 2 H 3 N - C 3 H 3 N - H 2 0 . This
C2H402
Distillate path 1st cut ./ \
~/
~
H20 ~ Still path
~
during total
/"
(~_~r c'gFv--~'~-
~..~._..~ .........
\
Still path CU,
\
....
-' _~-,. 1] "~
. ~
~
'",,
~"~'"-.. x .... - ~
a) TWO REGULAR TASKS
i
~
\
/] Disti,ate
.LI P~hcut ~/--- F. e~d ""- ~"',, ~\-ml~ n .... ]
\ ] ~ C3H3N
Bottom path
~
b) INVERTED + REGULAR
02H402
....
Iiiiiii.
~,-'~h ~
~,
"[[
Still /. ", 2nXd a%.,..~ z ~ e O p ;,,,e,,,~ a t 9h /
Reflux drum path 1sttask
",~,~/
H20
o
. iD stillate 2 ncl
task
path
Figure 4. Separation of acetonitrile - water - acetic acid using acrylonitrile by heterogeneous batch distillation in a regular column (left) and in an inverted column followed by a regular column (right). Top and bottom molar composition paths.
504
Still path . . ~
a)
(x~ after total reflux)~//..ilb~r
Distillate path J
~ ; e
~
~
C2HAO2 ~,,il~"\ - ' ~
Middle /--Vessel path I C-H-N "~
b)
/
~02H402 ~
Liquid profile after/"
.\
Final
r-- profilliquid ~L_. C H N
I (xMafter
!d ...........",,Xaze~t I t~ '
"'"",..,.. ~
(xu a f t e r . . . '",, \ I total reflux) - Xazeo~"
x....
"" N ~,3n3
Figure 5. Middle Vessel Heterogeneous batch distillation of acetonitrile - water - acetic acid using acrylonitrile. Still, Distillate and Middle Vessel molar composition path (a). Liquid overall composition profiles after total reflux and distillate removal steps (b). later is then separated using a regular column with variable reflux. As in the case of the first task of Fig. 3a, the still path reaches CeH3N (XC2H3N,~n~]= 0.995) at the end while the heteroazeotrope is obtained overhead, the aqueous phase removed as distillate ( X H 2 0 , r n e a n = 0.962) and the C3H3N -rich phase refluxed during the operation and discharged at the end for recycle to another batch. For middle vessel H B D (Fig. 5), a single batch task is needed. After the total reflux step, the liquid overall composition profile in the column follows closely the batch distillation boundary surface with a near-heteroazeotropic composition boiling overhead (Fig. 5b). C 2 H 4 0 2 stays on the trays below the middle vessel location. The still is then depleted of C3H3N (Fig. 5a). As reflux of the acrylonitrile-riche phase starts, only the decanter stays in the liquid - liquid - vapour envelope. An aqueous distillate is withdrawn (xH2o..... - - 0.958). x Mmoves progressively towards C2H3N (XC2H3N,final -" 0.995) and x r towards the C 2 H 4 0 2 o n e (XC2H402,final = 0.996). C3H3N is left on the trays (20%) and in the decanter light phase (80%). The aqueous distillate is the sole product and therefore represents the net product x p. x ~ path direction moves away from x P, thus reaching C2H3N. The final liquid overall composition profile is shown in Fig. 5b. This demonstrate that MV HBD is able to split a quaternary mixture in a single batch task where two batch tasks are required when using a regular or an inverted column configuration. REFERENCES 1. C. Bernot, M.F. Doherty and M.F. Malone, Chem. Eng. Sci., 46 (1991) 1311. 2. W. Cheong and P. Barton, Ind. Eng. Chem. Res., 38 (1999) 1504. 3. I. Rodriguez-Donis, E. Pardillo-Fondevila, V. Gerbaud and X. Joulia, ESCAPE-10 Proceedings. Ed. S. Pierucci., Elsevier, (2000) 1123. 4. I. Rodriguez-Donis, V. Gerbaud and X. Joulia, Submitted to Ind. Chem. Eng. Res., 2000a. 5. I. Rodriguez-Donis, V. Gerbaud and X. Joulia, Submitted to Ind. Chem. Eng. Res., 2000b. 6. I. Rodriguez-Donis, V. Gerbaud and X. Joulia, Submitted to Ind. Chem. Eng. Res., 2000c. 7. S. Hasebe, T. Kurooka, B.B.A. Aziz, I. Hashimoto and T. Watanabe, J. Chem. Eng. Jap., 29 (1996) 1000. 8. W. Cheong and P. Barton, Ind. Eng. Chem. Res., 38 (1999) 1549. 9. W. Cheong and P. Barton, Ind. Eng. Chem. Res., 38 (1999) 1531. 10. M. Warter and J. Stichlmair, ESCAPE-10 Proceedings. Ed. S. Pierucci., Elsevier, (2000) 691. 11. ProSim SA (France). http://www.prosim.net. 12. A. Sargousse, J.M. Lelann, X. Joulia and L. Jourda, Proceedings of MOSIM 99, (1999) 61.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
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Optimisation of Heat Integrated Distillation Sequences in the Context of Background Process Anupam Samanta and Megan Jobson Department of Process Integration, UMIST, Manchester M60 1QD, UK This paper presents a mathematical programming approach to the optimal design of distillation sequences that are heat integrated with the background process. A new mathematical model is proposed to rigorously optimise the column parameters (pressure, feed condition, etc.). The model simultaneously takes into account all possible heat integration opportunities between columns, between the background process and columns, steam generation, etc. and key heat integration issues, including continuous variation of condenser and reboiler temperatures, interactions between two successive columns due to the difference in operating pressures, practical constraints, capital-energy trade-offs, etc. The results are compared with conventional sequential approaches. 1. INTRODUCTION Synthesis of heat distillation sequences is a very important part of chemical process design due to the potential energy savings. Published literature on the synthesis of heat integrated distillation sequences is based on two main approaches: evolutionary approaches (e.g. Umeda et al., 1976; Linnhoff et al., 1983), and mathematical programming approaches (e.g. Andrecovich and Westerberg, 1985; Yeomans and Grossmann, 1999). There are advantages and disadvantages to both approaches. Distillation sequence synthesis and the heat integration of distillation sequences can be considered sequentially or simultaneously. Although the simultaneous approach exploits interactions between sequence selection and opportunities for heat recovery, the sequential approach is still generally practised due to the computationally intensive nature of the simultaneous approach. The heat integration of a fixed sequence of distillation columns is still a challenging problem. Firstly, there are many heat integration opportunities for a sequence of columns. These include heat integration with the background process, heat exchange between columns, generation of low and medium pressure steam from condensers, etc. Secondly, there are many degrees of freedom for column design, including operating pressure, feed condition, reflux ratio, use of side exchangers, etc. Thirdly, heat integration introduces many issues, such as the variation of condenser and reboiler temperature, the heating and cooling of feed and product streams, practical constraints, capital-energy trade-offs for columns and heat exchangers, etc. Existing methods cannot consider all these issues simultaneously. This paper considers heat integration of distillation columns for a fixed sequence. The column design parameters are optimised while considering heat integration opportunities between columns, between the background process and columns, constraints and capital-energy trade-offs simultaneously. The 'background process' includes all streams for which temperature and heat duties are independent of the distillation sequence design. * Corresponding author. E-mail: [email protected]
506 2. HEAT INTEGRATION ISSUES a.
Continuous temperature variation:
The condenser and reboiler temperatures and the product temperatures vary continuously with column operating pressure. Similarly, the feed condition affects the feed temperature and reboiler and condenser heat duty. Therefore, we need a model to calculate minimum utility cost or minimum total annualised cost that can account for variable stream temperatures. However, most of the existing heat integration models are only applicable when the stream temperatures are fixed. The transhipment model (Papoulias and Grossmann, 1983) and the transportation model (Cerda et al.,1983) are applicable when the stream temperatures are fixed. The pinch location model (Duran and Grossmann, 1986) can treat stream temperature as a continuous variable but the model has major limitations. Firstly, the model is non-linear and non-smooth due to the presence of the 'maximum' function. Secondly, the model frequently leads to suboptimal solutions when isothermal (e.g. condenser and reboiler, etc.) streams are present. To overcome these limitations, Grossmann et al. (1998) proposed the disjunctive pinch location model. However, this model is computationally intensive. b. Constrained matches:
Safety, start-up and control are the primary reasons for constraining matches in a heat exchanger network. For example, it may be necessary to forbid matches between particular streams for safety reasons or because of large distances between equipment. One may restrict the number of exchangers for a condenser or reboiler or forbid inter-column matches to facilitate process control. Since a heat integrated process introduces numerous interactions between process units, process control is always difficult. It is, therefore, important to include these constraints during the synthesis of heat integrated distillation sequences. Although the pinch location model and the disjunctive pinch location model can treat the temperatures of both isothermal and non-isothermal streams as continuous variables, neither model can include constraints. To date, no heat integration model exists that can simultaneously accommodate continuously varying stream temperatures and constraints. c. Interactions between two successive columns in a sequence:
There is an interaction between two successive columns in a sequence because of the difference in the column operating pressures. The pressure of the upstream column affects the feed condition of the downstream column (Fig. 1). For example, if the pressure of column 1 shown in Fig. 1 is greater than that of column 2, then column 2 receives a two-phase feed. As a result, the condenser duty increases and the reboiler duty decreases for column 2. That is, the heat load of column 2 depends on the pressure of both column 1 and column 2.
Pressure difference (PI- P2) 0 >0 <0
Feed q
~
Fig. 1. Interaction between two successive columns in a sequence
Column2 feed quality Bubble point feed Vaporised feed Subcooled feed
507 Previous research reported in the literature on the synthesis of heat integrated columns did not consider this interaction. Instead the feed to the each column was always assumed to be a saturated liquid. 3. M E T H O D O L O G Y
A new mathematical model is proposed to rigorously optimise the operating conditions for a sequence of columns. The model has two parts. The first part consists of heat integration equations that model all possible matches for heat integration. Because the temperatures of reboilers and condensers depend on operating pressures, so do which matches are feasible. Binary variables are used to model feasible heat transfer between condensers and reboilers and between columns and the background process streams. The temperatures of feeds, condensers and reboilers are treated as continuous variables. The second part of the model considers column and heat exchanger design. The overall model is formulated as a M1NLP problem. However, by approximating the column and heat exchanger design equations and capital costs, the model can be formulated as a MILP. The column parameters are calculated using Fenske-Underwood-Gilliland shortcut design equations. For each column, various parameters, such as heat duties and temperatures of the condenser and reboiler, feed temperature, column capital cost, etc., are calculated for a range of operating pressure and feed conditions. The condenser heat load, the reboiler temperature and the column capital cost are approximated by piecewise linear functions. The reboiler heat duty is calculated rigorously by energy balance. The condenser and reboiler are assumed to be isothermal streams i.e. heat is removed from the condenser and heat is provided to the reboiler at the bubble and dew point respectively. 4. O P T I M I S A T I O N RESULTS The approach is illustrated by application to a four component separation problem. The background process is assumed to contain two hot and two cold streams. The temperatures and flow rates of these streams are assumed to be fixed. Since the direct sequence of columns is obtained as the lowest cost non-integrated sequence, this sequence is selected for further study. Table 1 Separation data and specifications Component Benzene Toluene p-xylene . o-xylene
Feedfraction 0.4 0.3 0.25 0.05
Products Pure benzene Pure toluene Pure p-xylene Pure o-xylene
Recovery of all components is 99% Feed flow rate: 430 kmol/hr Minimum approach temperature:10 ~
Table 2 Background stream and utility data Streams H1 H2 C1 C2
Tin (~ 180 130 120 70
Tout(~ 130 100 140 170
MCP (kW/~ 200 400 600 100
Utilities Water LP steam MP steam
Temp (~ 30 150 185
Cost$/kW 4.92 86.9 123.2
Case A: Pressure optimisation without constraints In order to compare the results using the new mathematical model, we considered two conventional approaches to the optimisation of heat integrated distillation sequences.
508
Conventional approaches: In the first approach, the column operating pressures are selected according to the cost of the available utilities. The columns of the fixed sequence are integrated with the background process for these fixed column operating pressures. In this example, both columns are operated at atmospheric pressure. In the second conventional approach, the operating pressures of the columns in a fixed sequence are first optimised by considering only opportunities for heat exchange between columns. Then the columns are heat integrated with the background process at these fixed operating pressures. Table 3 Pressure optimisation results for conventional approaches Conventional approach 2 Utility cost: $1,710,000/yr Annualised cost: $3,030,000/yr Operating pressures (bar): P1- 1.0 P2 = 1 . 0 P3 = 1.3
Conventional approach 1 Utility cost: $1,810,000/yr Annualised cost: $3,140,000/yr Operating pressures (bar): Pl=l-0 P2=l.0 P3=l.0
In the second conventional approach, the operating pressure of column 3 is increased in order to exchange heat between the condenser of column3 and the reboilers of columns and 2. As a result, costs are slightly less than those for the first approach.
The new method: The new mathematical model is used to optimise the column operating pressures by simultaneously considering heat integration opportunities between columns and with the background process. Table 4 Pressure optimisation results by the new method Objective: minimum utility cost Utility cost: $1,600,000/yr Operating pressures (bar): PI = 1.0; P2 = 1.7 ; P3 = 1.3
Objective: minimum annualised cost Annualised cost: $2,910,000/yr Operating pressures (bar): Pa = 1.0 P2 = 1.7 P3 = 1.3
The new approach results in decreased utility and capital costs, compared to the conventional approaches. This is because column pressures are optimised by considering opportunities for heat integration with the background process at the same time as operating conditions are selected. Therefore, the model finds different optimal pressures than with conventional approaches. In this example, the optimal column operating pressures are the same for both objective functions.
Case B: Pressure optimisation including constraints In order to illustrate the interactions between constraints and heat recovery opportunities, the column operating pressures are optimised in a sequential and in a simultaneous mode. In the sequential mode, the operating pressures are optimised first and then the constraints are imposed at these fixed operating pressures. In the simultaneous mode, the operating pressures are optimised while simultaneously taking into account constraints. In this example, condensers and reboilers are restricted to exchange heat with no more than one stream or utility.
509 Table 5 Results of pressure optimisation with constraints by the new method Sequential approach Utility cost: $1,910,000/yr Operating pressures (bar): PI=I.0 P2=1.4 P3=1.8
Simultaneous approach Utility cost: $1,750,000/yr Operating pressures (bar): PI=I.0 P2=l.0 P3=1.3
Table 5 presents the results, which can be compared to the unconstrained case shown in Table 4. Imposing constraints increases utility costs. As expected, better results are obtained when constraints are considered simultaneously, rather than sequentially. In the sequential approach, column pressures are fixed prior to constraints being considered. Therefore, different optimal pressures are obtained by the simultaneous approach. Hence it is important to consider heat recovery opportunities and constraints simultaneously when optimising a sequence of heat integrated columns.
Case C: Feed condition optimisation Feed preheat is frequently considered in the design of individual columns as a means of reducing reboiler duty. The economic advantage of vaporising the feed is not always obvious because it increases the reflux ratio. Although less of the more expensive utility is required by the reboiler, these savings can be overshadowed by the increase in costs due to the increased total heat load. When columns are to be heat integrated with the overall process, this trade-off is even more complicated because the background process streams offer additional heat integration opportunities. The feed condition of a column can be optimised either sequentially or simultaneously for improved heat recovery. In the sequential approach, pressure is optimised first, and then the feed condition is optimised for these fixed column operating pressures. The pressure and the feed condition are optimised together in the simultaneous approach. Table 6 Results for feed condition optimisation Sequential approach Utility cost: $1,600,000/yr Operating conditions: P~ =1.0; P2=1.7; Pa=l.3bar ql= 1.0 q2=l.0 q3 =1.0
Simultaneous approach Utility cost: $1,520,000/yr Operating conditions: P1 = 1.0 P2 = 1.6 P3 = 1.7 qz = 0.2 q2 = 0.07 q3 = 1.0
Table 6 presents the results for feed condition optimisation when no constraints need to be imposed. In sequential mode, a saturated liquid feed is found to be the optimum feed condition for all columns. As a result, there is no change in the utility cost, compared to Table 4. This is because column pressures are fixed in before optimisation of feed condition. Optimising the feed condition of all columns reduces utility costs when the new model is applied in simultaneous mode. In this case, trade-offs are better exploited and as a result different optimal pressures and feed conditions are obtained compared to the sequential optimisation. Constraints can also be considered simultaneously.
Case D: Multiple effect column optimisation Multiple effect columns can be considered to improve heat recovery when the heat load of a particular column is large. This is because by splitting the column heat load, better heat recovery
510 can be achieved by placing the columns above and below the pinch (Linnhoff et al., 1983). In this case, a double effect column for column 3 is considered since its heat load is large, compared to columns 1 and 2. The pressures, feed conditions and feed flow fractions of each column are optimised simultaneously. Results are presented in Table 7 for the unconstrained and constrained cases. Table 7 Results for multiple effect column optimisation No constraints Utility cost: $1,190,000/yr Operating conditions: P1 = 1.0 P2 = 1.8 P3 = 1.1 P4 = 1.7 ql = 0.2 q2- 0.95 q3= 1.0 q4- 1.0 f3= 0.56 f4= 0.44
Constraints included Utility cost: $1,640,000/yr Operating conditions: e l = 1.0 P2- 1.0 P3 = 1.3 P4 = 1.3 ql = 0.2 q2= 0.8 q3 = 1.0 q4= 1.0 t"3= 0.56 f4= 0.44
In both cases, utility costs are reduced compared to the results presented in Tables 4, 5 and 6. When no constraints need to be imposed, the reduction is significant, but the reduction is more marginal when constraints are incorporated. 5. C O N C L U S I O N S A new mathematical method is presented for the rigorous optimisation of distillation column sequences heat integrated with the overall process. All possible heat integration opportunities between columns and between the background process and columns are considered. The model can simultaneously optimise various column parameters while considering key heat integration issues. These issues include continuous variation of condenser and reboiler temperatures, heating and cooling of the feed and product streams, multiple utilities, constraints, interactions between two successive columns in a sequence due to the pressure difference and capital-energy tradeoffs, etc. It is found that even for a fixed sequence, substantial energy savings can be achieved by optimising the column parameters and by considering multiple effect columns, compared to using conventional approaches. This is because heat integration opportunities, between columns and between columns and the background process, and heat integration issues have been accounted for simultaneously using the new model. It is observed that by including the constraints in the model, rather than imposing them only after selecting process operating conditions, significant energy savings can be achieved. As a result, different optimal operating conditions are also obtained. We have also developed a superstructure-based optimisation model for simultaneous sequence selection and heat integration, by including all possible sequences of simple and complex columns in the superstructure.
6. REFERENCES 1. T. Umeda, T. Harada and K. Shiroko, Comp. Chem. Engg., 3 (1979) 273. 2. B. Linhhoff, H. Dunford and R. Smith, Chem. Eng. Sci., 38 (1983) 1175. 3. M.J. Andrecovich and A.W. Westerberg, AIChE J., 31 (1985) 363. 4. H. Yeomans and I.E. Grossmann, Comp. Chem. Engg. 22 (1999) 1453. 5. S.A. Papoulias and I.E. Grossmann, Comp. Chem. Engg., 7 (1983)695. 6. J. Cerda, A.W. Westerberg and B. Linnhoff, Chem. Eng. Sci., 38 (1983) 373. 7. M.A. Duran and I.E. Grossmann, AIChE J., 32 (1986)123. 8. I.E. Grossmann, H. Yeomans and Z. Kravanja, Comp. Chem. Engg., 22 (1998) $157.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
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A New Heat Integration Model for Streams of Variable Temperature and Constrained Matches Anupam Samanta and Megan Jobson Department of Process Integration, UMIST, Manchester M60 1QD, UK This paper introduces a new heat integration model for the case that one or more stream temperatures are variable. The heat integration model is formulated as a mixed integer programming model, where both temperature and flow rate of streams are treated as continuous variables. Both isothermal and non-isothermal streams have been included in the model. The proposed model uses disjunctive logic to quantify the feasible heat transfer between hot and cold streams. The model considers individual matches between hot and cold streams by using fixed temperature intervals. The model can accommodate constrained matches simultaneously. The model may be applied for the selection and optimisation of heat integrated distillation separation sequences, the synthesis of reactor - separator systems, etc. 1. INTRODUCTION Heat integration of chemical processes has been the subject of research in process system engineering during the past two decades. The heat integration problem has usually been treated as a sub-problem of the process synthesis problem. First the process flowsheet is synthesised and the operating conditions are selected by assuming that all the heating and cooling requirements are satisfied by utilities. In the second step, given flows and temperatures of streams, the minimum utility requirement is calculated and the heat exchanger network is designed. The minimum utility requirement can be calculated either by pinch analysis (Linnhoff et al., 1980) or by optimisation based models, such as the transhipment model (Papoulias and Grossmann, 1983) or the transportation model (Cerda et al., 1983). However, this sequential strategy does not take into account interactions between the choice of process operating conditions and opportunities for heat recovery. Papoulias and Grossmann (1983) introduced linear constraints within the MILP transhipment formulation for structural process optimisation. The limitation of this approach is that in order to use the transhipment model, the temperature of streams is discretized. Duran and Grossmann (1986) proposed the pinch location model for simultaneous optimisation and heat integration where both the flow rate and temperature of streams can vary. However, the model is non-linear and non-smooth due to the maximum function, which is handled with smooth approximations. Secondly, the model frequently gives sub-optimal solutions (Grossmann et aL, 1998) when isothermal streams are present. A disjunctive optimisation model was proposed to overcome this limitation (Grossmann et al., 1998). The model uses disjunctive logic to explicitly model the relative placements of streams for all potential pinch locations. The disjunctive pinch location model is formulated as a MINLP problem for nonisothermal streams and a MILP model for isothermal streams. The characteristics of various heat integration models are summarised in Table 1. * Corresponding author. E-mail: [email protected]
512 Table 1 Characteristics of existing heat integration models Models
Stream temperature variation Isothermal Non-isothermal Problem table Fixed Fixed Transhipment Fixed/discrete Fixed/discrete Transportation Fixed/discrete Fixed/discrete Pinch location Continuous Continuous Disjunctive Pinch location Continuous Continuous
Constraints included No Yes Yes No No
Safety, start-up and control considerations give rise to constraints. For example, it may be necessary to restrict the number of exchangers for the condenser and reboiler of a column. It may be stipulated that either the condenser or the reboiler may be integrated with other columns, but not both. Forbidden matches between hot and cold streams are common for reasons of safety and plant layout. Neither the pinch location model nor the disjunctive pinch location model can incorporate process constraints. This is due to the fact that both models calculate aggregated heat load for each pinch candidate; they do not consider individual matches between hot and cold streams. This work presents a new heat integration model for simultaneous process optimisation and heat integration that can take into account continuous temperature variation of both isothermal and non-isothermal streams and constrained matches simultaneously. 2. N E W HEAT INTEGRATION M O D E L The model can be described based on two different cases: a) both the inlet and outlet temperatures of all non-isothermal streams are fixed; b) the temperatures of non-isothermal streams are variable. In both the cases, the flow rates of the non-isothermal streams can vary. The temperatures and heat duties of isothermal streams are always treated as variables. a. Fixed inlet and outlet temperature o f non-isothermal streams
The fixed temperatures of the non-isothermal streams define the temperature intervals for the process. Since isothermal stream temperatures vary continuously, the temperature intervals corresponding to the isothermal streams are not fixed. Therefore, feasible heat exchange between isothermal stream and non-isothermal streams and also between isothermal streams depend on the temperature of the isothermal streams. Even if heat exchange is feasible, the duty of the feasible match will depend on the temperature of the isothermal stream, relative to that of non-isothermal streams. The new heat integration model uses disjunctive logic to identify and quantify feasible heat exchange. The overall model consists of two sets of equations. The first set of constraints defines the energy balance for isothermal and non-isothermal streams in each temperature interval. The second set of equations describes feasible heat exchange and the heat duty of all possible matches between hot and cold streams. The disjunctive constraints are transformed into big-M constraints using binary variables. Three possible cases of disjunctive logic are described below: First consider heat integration between isothermal streams. If the temperature of the hot stream is greater than the sum of the cold stream temperature and the minimum approach temperature (ATmin), then only heat exchange is feasible. Secondly, we consider heat integration between a cold isothermal stream and a hot non-isothermal. If the isothermal
513 stream temperature plus ATmin is less than the outlet temperature of the hot stream then the match is feasible. Furthermore all the heat from the hot stream can be exchanged with the cold isothermal stream. If the cold stream temperature plus ATmin is less than the inlet temperature of the hot stream and greater than the outlet temperature of the hot stream, then the match is feasible but the cold isothermal stream can receive only part of the heat from the hot stream. However, if the cold stream temperature plus ATmin is greater than the inlet temperature of the hot stream then the match is not feasible. For the third case, where heat is exchanged between a hot isothermal stream and a cold non-isothermal stream, similar logic holds. b. Temperatures o f non-isothermal streams are variables Consider the case that the outlet temperature of a non-isothermal stream is fixed, but the inlet temperature is variable. If the temperatures of all other non-isothermal streams are fixed, a set of fixed temperature intervals can be defined. The stream with the variable inlet temperature can be approximated by an isothermal segment and a non-isothermal segment (Fig .1). Linear T- H profile TIN
lout
.....
Approximate T-H profile T~
Tour
:'~-
~
- -~
?
Fixed temperature intervals
Fig. 1. Approximation for variable temperature non-isothermal streams
The inlet temperature of the non-isothermal segment is set to some fixed interval temperature, depending on the temperature interval in which the stream inlet temperature is located. The temperature of the isothermal segment is assumed to be the inlet stream temperature of the non-isothermal stream. The heat duty of the isothermal segment is calculated accurately. When both the inlet and outlet temperature of a non-isothermal stream vary continuously, similar approximations may be made. The non-isothermal stream is approximated by a non-isothermal segment and two isothermal segments.
3. I L L U S T R A T I V E EXAMPLES To illustrate the new heat integration model three examples are presented: 1) the temperature of all the non-isothermal streams are fixed; 2) the inlet temperature of a hot nonisothermal stream is variable, 3) both the inlet and outlet temperature of a non-isothermal stream are variable. In each case, the temperatures of the isothermal streams are optimised in order to maximise heat recovery. Stream data and utility data are presented in Table 2 and 3. It is assumed that the flow rates of the non-isothermal streams are constant and the heat duties of the isothermal streams vary linearly with temperature. These assumptions allow a MILP formulation of the model. These assumptions can be removed, without affecting the modelling and solution procedure. In this case, the model formulation is MINLP. The results are compared with the disjunctive pinch location model (Grossmann et aL, 1998).
514
Table 2 Non-isothermal streams data Example 1 TI (~ 180 130 120 70
HN1 HN2 CN1 CN2
TO (~ 130 100 140 170
Example 2
Mcp 300 300 600 100
TI (~ ? 130 120 70
TO (~ 130 100 140 170
Example 3 Mcp 300 300 600 100
TI(~ ? 130 120 70
TO (~ ? 100 140 170
Mcp 300 300 600 100
Table 3 Isothermal streams and utility data HIl
HI2 CIl CI2
T (~ ....................................... THn = 1.08" Tcn+23 THI2 = 1.03" TCI2+0.33 Ycn (lower bound 80 ~ TcI2 (lower bound 110 ~
Q (kW) QHn = 20. Tcn+1400 QHI2 = 10" To2+9100 Qcil = QHII QcI2 = QHI2
Utility MP steam
Water
Temp (~ 185 30
Cost $/kW.:y r 123.2 4.56
The temperatures of hot isothermal streams are assumed to be independent variables. The cold stream temperatures and the heat duty of hot and cold isothermal streams are assumed to be dependent on the hot stream temperatures. Case A No constraints considered
Three approaches to solve the problem are presented: 1) sequential heat integration, 2) simultaneous heat integration by disjunctive pinch location model, 3) simultaneous heat integration by the new model. In the first approach, sequential heat integration, first the temperatures of the isothermal streams are determined by considering that all the heating and cooling requirements are satisfied by utilities. Heat integration is considered in the next step at the fixed operating conditions. In the second approach, the temperatures of the isothermal and non-isothermal streams are calculated by considering heat integration of both isothermal and non-isothermal streams simultaneously using the disjunctive pinch location model. The new model is also applied to determine the optimal temperatures for the isothermal streams. Table 4
Results for case A (no constraints considered) Example2 Example 1 Sequential heat integration Utility cost ($/yr): 1,700,000
Optimal temperature (~ Tcll = 80 TCI2 = 110
Utility cost ($/yr): $1,700,000/yr Optimal temperature (~ TcI 1 = 80 To2 = 110
Example3 Utility cost ($/yr): $1,700,000/yr Optimal temperature (~ TcI1 = 80 TcI2 = 110
Simultaneous heat integration by the disjunctivepinch location model Utility cost ($/yr): 851,800 Optimal temperature (~ TcI 1 - 80 TCI2 = 144.95
Utility cost ($/yr): 851,800 Optimal temperature (~ TcI 1 = 80 TIHN1 = 180 Toz = 144.95
Utility cost ($/yr): 834,200 Optimal temperature (~ TCI1 = 80 TIHNI = 180 Tci2 = 144.95 T O H N I= 142
Simultaneous heat integration by the new model Utility cost ($/yr): 851,800 Optimal temperature (~ Tcn = 80 To2 = 144.95
Utility cost ($/yr): 851,800
Optimal temperature (~ TcI 1 = 80 Tcrz = 144.95
TIHNI = 180
Utility cost ($/yr): 844,500 Optimal temperature (~ TcIl = 80 TIHNI = 180 TcI2 = 144.95 T O H N I= 135
The results for this case are presented in Table 4. The utility cost using the disjunctive pinch location model is reduced, compared to the sequential approach. This is because the
>1>
simultaneous model exploits the strong interactions between the selection of operating conditions and heat recovery opportunities. Since the model can treat the temperature of both isothermal and non-isothermal streams as a continuous variable, and the problem is formulated as a MILP model, the results that are obtained are the global optimal solution. The new simultaneous heat integration model also reduces the utility cost compared to the sequential heat integration approach (Table 4). In Example 1, the same results are obtained as with the disjunctive pinch location model because the new model does not require any approximations. In Examples 2, the new model gives same result as obtained by the disjunctive pinch location model. In Example 3, the difference in utility cost compared to the disjunctive model is very small. It is to note that the number of fixed temperature intervals in Example 2 and 3, has been kept same as the Example 1. Although the new model employs slight approximations in Example 2 and 3, the difference in utility cost compared to the disjunctive model is negligible. Case B: Constrained matches Next, consider the case that constraints should be taken into account. Either the process operating conditions (stream temperatures) can be selected first, and then the constraints imposed in a sequential manner, or the selection of process operating conditions and constraints can be considered simultaneously. Three approaches to the problem are considered: 1) the sequential approach, where operating conditions are determined first and the expanded transhipment model (Papoulias and Grossmann, 1983) calculates utility requirements when constraints are imposed for these fixed operating conditions; 2) the disjunctive pinch location method, which cannot address constraints simultaneously, therefore the constraints are imposed at the fixed operating conditions; 3) the new model considers constraints and stream temperature selection simultaneously. These three approaches are used for the three examples presented in Tables 2 and 3, where heat integration is constrained to match no more than one stream or utility with any isothermal stream.
Table 5 Results for case B (constraints accommodated) Example 1 Utility cost ($/yr): 1,700,000 Optimal temperature (~ TcI1= 80 YcI2 = 110
Example2 Sequential heat integration Utility cost ($/yr): 1,700,000 Optimal temperature (~ TcI1= 80 TcI2 = 110
Example3 Utility cost ($/yr): 1,700,000 Optimal temperature (~ TcI1= 80 Tci2 = 110
Simultaneous heat integration by the disjunctive pinch location model Utility cost ($/yr): 1,745,800 Utility cost ($/yr): 1,745,800 Utility cost ($/yr): 2,189,300 Optimal temperature (~ Optimal temperature (~ Optimal temperature (~ Yci I = 80 YciI = 80 TIr~l = 180 T c I 1 = 80 TI~IN1 = 180 YcI2 = 1 4 4 . 9 5 YcI2= 144.95 TcI2 = 144.95 TOrn~l= 142
Utility cost ($/yr): 1,364,900 Optimal temperature (~ TcI1= 80 TcI2 = 148
Simultaneous heat integration by the new model Utility cost ($/yr): 1,364,900 Utility cost ($/yr): 1,451,256 Optimal temperature (~ Optimal temperature (~ Tcil = 80 TIrrNl - 180 TcI1 = 80 TIrvN1 = 180 T c I 2 = 148 Tciz = 147 TOrn~l = 155
516 The results for the constrained case are presented in Table 5. The new model considers process design (selection of operating conditions) and constrained matches simultaneously. It therefore exploits interactions between the choice of operating parameters and heat recovery opportunities. The utility costs of the resulting process are significantly lower than for the two sequential approaches. Clearly constraints can significantly influence the optimal operating conditions and heat recovery. Note that the result obtained for Example 1 using the new model is a global optimal solution because the model does not require approximations for the non-isothermal streams. In Examples 2 and 3, although the new model employs slight approximation due to the variable temperature of the non-isothermal streams, the new model obtains significantly better results compared to the disjunctive pinch location model. As the disjunctive model cannot include constraints simultaneously, it misses the benefit of exploring interactions between heat recovery opportunities and constrained matches. However, the solutions obtained by the new model may not be globally optimal. 4. CONCLUSIONS A new heat integration model has been formulated for continuously varying stream temperatures by considering individual matches between hot and cold streams. A key feature of the model is that constraints and selection of operating conditions can be addressed simultaneously. Constraints commonly arise for reasons of safety and operability. The new model uses disjunctive logic to model feasible matches and heat duties for heat integration. The temperatures of both isothermal and non-isothermal streams can be treated as continuous variables. However, the model employs a slight approximation when the temperature of a non-isothermal stream varies continuously. The model is formulated as a MINLP problem. The MINLP model reduces to a MILP model if the flowrate of nonisothermal streams are fixed and the heat duties of isothermal streams vary linearly with temperature. Several examples, involving both isothermal and non-isothermal streams, are solved to show the advantages and disadvantages of the proposed model. It is found that when constraints are to be considered, the new optimisation model generally reduces utility cost compared to the disjunctive pinch location model, which cannot include constraints simultaneously. This new simultaneous optimisation and heat integration model can be applied to the selection and optimisation of heat integrated distillation sequences, the synthesis of reaction and separation systems, etc. In these cases both flow rates and temperatures of the process streams (non-isothermal streams), and the condenser and reboiler temperature (isothermal streams), etc. are explicit optimisation variables. 5. R E F E R E N C E S
1. B. Linnhoffand J.R. Flower, AIChE J., 24 (1978) 633. 2. S.A. Papoulias and I.E. Grossmann, Comp. Chem. Engg., 7 (1983) 695. 3. J. Cerda, A.W. Westerberg and B. Linnhoff, Chem. Eng. Sci., 38 (1983) 373. 4. M.A. Duran and I.E. Grossmann, AIChE J., 32(1986) 123. 5. I.E. Grossmann, H. Yeomans and Z. Kravanja,, Comp. Chem. Engg., 22 (1998) S157.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
517
Tools for Reactive Distillation Column Design: Graphical and Stage-toStage Computation Methods Sfinchez Daza Oscar a'b, P6rez-Cisneros Eduardo a*, Erik Bek-Pedersen c, and Martin Hostrup c aDepartamento de Ingenieria de Procesos e Hidrfiulica Universidad Autdnoma Metropolitana-Iztapalapa M6xico, D. F., C. P. 09340, Fax: (52 5) 8 04 49 00. E-mail: [email protected] bCentro de Quimica, Instituto de Ciencias de la Universidad Aut6noma de Puebla CCAPEC, Department of Chemical Engineering, Technical University of Denmark. 2800, Lyngby, Denmark Based on the element mass balance concept, a graphical design method and a stage-to-stage multicomponent design method for reactive distillation columns have been developed. For distillation columns comprising reactive and non-reactive stages, a simple design strategy based on reactive and non-reactive bubble point calculations is proposed. This strategy tracks the conversion and temperature between the feed and the end stages of the column. An illustrative example highlights the verification of the design strategy through rigorous simulation. 1. INTRODUCTION In recent years, reactive distillation has attracted attention as a highly promising hybrid process. Application of this combined reaction-separation process is considered useful only for reactive systems limited by chemical equilibrium and it has been applied with great success to the methyl acetate and methyl-tert-butyl ether (MTBE) production [ 1-2]. It has been claimed [3] recently that reactive distillation is also applicable to reaction separation processes involving complex reactive systems such as the diesel and gasoline desulfurization. The increasing interest in reactive distillation has been accompanied by development of various simulation algorithms related to the study of operation and control [4,5] of the process. Design of reactive distillation columns (RDC), however has not received the same attention. Most of the existing work related to design of RDCs has been based on the work of Doherty [6] or it has been treated as an optimization problem (MINLP) [7]. In this paper, a method based on the element composition variables that significantly reduce the dimension and complexity of the intrinsic calculation problem and thereby achieving the design goals with less computational effort is presented. The objective of the present work is to introduce a new set of graphical and multicomponent stage to stage computation methods for design of reactive distillation columns. These design methods are based on the element mass balance approach. The methods developed are similar to those typically employed for non-reactive systems. For binary element systems, which may be ternary or higher in terms of mixture compounds, a
518 simple reactive McCabe-Thiele method has been developed. For design of ternary element systems, which are usually quaternary or higher in terms of mixture compounds, a reactive stage to stage calculation method has been developed. For columns comprising reactive and non-reactive stages, the stage-to-stage procedure is used. Also the driving force approach of Gani and Bek-Pedersen [9] has been extendedto application in reactive separation systems. The methods have been tested in a systematic manner with several reactive systems. In this work, only the MTBE reactive system will be highlighted. All design examples have been verified through rigorous simulations. 2. ELEMENT BALANCES AND EQUILIBRIUM CONDITION In the equilibrium based approach, the computation of chemical and physical equilibrium (CPE) is an important step. Using the element-based approach [8], a multi-component physical-chemical equilibrium problem is transformed into a phase equilibrium problem for a mixture of elements (representing the system). In the element-based approach, the CPE problem is solved by minimizing the Gibbs energy of the reactive system NP
man G(n) = ~ fl:l
NC
~ n/#/z/#
(1)
i=l
Subject to the M constraints: NP
NC
Z Z Aj, 7 fl=l
-
0
i=l
where G(n) is the total Gibbs energy of a system containing NC species and NP phases. Equation (2) represents the M independent element mass balances, with the coefficient Aji being the number of times the reaction invariant element j is present in molecule i. The solution of this constrained optimization problem can be obtained through the Lagrange multiplier formulation where the relation between the Gibbs free energy and the Lagrange multipliers is exploited for a robust method of solution. Thus, a fully consistent thermodynamic description of a chemically equilibrated phase is obtained in terms of b, the (element) composition vector, and ~,, the corresponding element potential vector. 3. DESIGN OF REACTIVE DISTILLATION COLUMNS The design methods are divided in terms of the number of elements involved, where a graphical design method for binary element (reactive) systems while a stage-to-stage calculation method is used for ternary element (reactive) systems.
3.1 Binary Element Reactive Systems Consider a "full" reactive distillation column (all stages are reactive stages) operating under chemical and physical equilibrium conditions (see Figure 1). The feed is a mixture of two elements A and B. In the case of a binary element reactive distillation, the reactive stripping section concentrates the less-volatile element (B) in a liquid stream while the reactive rectifying section concentrates the more-volatile element (A) in the vapor stream.
519 3.2 The Reactive Equilibrium and Driving Force Curves A reactive equilibrium curve is the locus of all chemical and physical equilibrium points. For a given element liquid composition Wta, it gives the corresponding equilibrium vapor composition W~A, and vice versa. A reactive equilibrium stage p is represented as a point on the reactive equilibrium curve where WtAp and W~Ap are the liquid and vapor element compositions leaving the stage. The reactive equilibrium curve can be constructed through sequential computation of reactive bubble points. A typical reactive equilibrium curve is shown in Figure 2. A driving force diagram can be used to visualize how the operation of the reactive distillation column should be determined to achieve the best separation with the least energy. A reactive driving force diagram for the MTBE production example is given in Figure 3. On the basis of this, good preliminary estimates can be obtained for design variables such as the number of raective stages, element reflux ratio, and feed stage location. 3.3 Constant Element Molar Overflow Assumption In order to avoid energy balance calculations, it is at least approximately correct for many problems to apply the assumption of constant total element molar overflow. The change of vapor rate from stage to stage at the rectifying section can be derived by writing an energy balance as v v bp -bp+, -
bp+l(H;V+l-H*pV)-bp'(Hp' - Hp_,) *' (H; v -H;')
(3)
It follows that there are two possible conditions which will cause the total element molar vapor flow b v to be constant from stage to stage: H *v= Constant
Condition 1:
H *t-- Constant
(4)
1
Condition 2"
(H*p" - Hp 9~) 9
= - -bp+ r- 1
(5)
Rt~tifying Qc bvl ~ [
_
hF r
bBT
Fig. 1 Binary Element Reactive Distillation Column
oo
oo
02
04
o6
oa
Fig. 2 Reactive Phase Diagram g~A- grA.
1o
0,00
WA, To
IAqu)dComposilion,WA(l-Butone)
1,00
Fig. 3 Reactive Driving Force Diagram WtA- g~A-
520 If b v is constant, it follows that b t is constant. It should be noted that the enthalpies for the different phases are element-based. From condition l, it can be seen that constant element molar overflow will occur if the element molar heats of vaporization of elements A and B are identical, if sensible-heat contributions due to temperature changes from stage to stage are negligible, if there are no enthalpy-of-mixing effects (ideal liquid and vapor solutions) or if the heat of reaction is negligible.
3.4 The Reactive Operating Lines Consider the rectifying section of a simple reactive distillation column, (see Figure 1), performing an element mass balance around this section for element A, the following equation is obtained: WA - - ~ W I + -ff~-
(6)
where bdr is the total element amount distillated at the top of the reactive column when the ratio ffp/bp+l (see Eq. 5) is maintained constant. Equation (6) represents the reactive operating line for the rectifying section. At the point where the feed is introduced to the column, the element flows in the rectifying and stripping sections must change because of the feed entry. The element flows below the feed introduction (stripping section) are labeled b*l and b*V. Performing an element mass balance for element A in the stripping section gives the reactive operating line for the stripping section. b *l t bBy W ; - b ,--7 W A - ~b, v W,~B
(7)
3.5 Ternary Element Reactive Systems In order to start a stage-to-stage calculation method, it is necessary to specify a selected set of variables that satisfy the degrees of freedom completely. It is nearly always the terminal stage for which this can be done. The stage to stage approach applied in the present work for reactive systems is based on the Lewis-Matheson method. If all the element compositions at the bottom stage (Wtj,B) are known (specified), it is then possible to calculate the element compositions of the vapor phase (W~j.B) leaving the bottom equilibrium stage (re-boiler) by performing a reactive bubble point calculation. The element vapor flows are determined from, bjp - bTWjp (8) The element liquid flows from the stage above the bottom stage is then determined from an element mass balance for the bottom stage, l
v
l
l
v
l
bg,p+ 1 - bj,p+ 1 + biB (9) Alternating use of Eqs. (8) and (9) then gives successive vapor and liquid flows for stages going up the column, until a feed stage or side-stream stage is reached. At this point, Eq. 9 is replaced by Eq. 10. bj,p+ 1 - bj,p+ 1 - bid
(1 O)
521 4. APPLICATION EXAMPLE Binary Element System: MTBE production without inert. Combined Reactive Distillation Column Configuration In this example it will be shown how the two design methods are applied. It should be noted that although the reactive system is represented by two elements, the RDC configuration includes reactive as well as non-reactive stages. Therefore, the stage to stage calculation method is also required. Consider a binary element mixture that is 70 element mole percent of element A (ZA= 0.7) is to be separated into 50 element mole percent bottoms product (14/A,~= 0.5) and 99 element mole percent distillate (g/A,D= 0.99) product. The element feed flow rate is 100 moles per hour at T = 300 K and P= 1 atm. The operating pressure of the reactive distillation column is 1 atm. The reflux element ratio (RR) is 2. The physical and chemical equilibrium curve is constructed with the CPE program package [8] considering the S-R-K equation of state for the gas phase and the Wilson equation for the liquid phase. Theoretical reactive stages, a partial reactive boiler, total condenser and a chemically saturated liquid reflux has been assumed. It is desired to determine the optimum feed stage location and the number of reactive and non-reactive stages. In this case, the reaction in terms elements can be written as: Isobutylene(C4H8=)k) + Methanol(CH3OH=B) r
Methyl-Tert-Butyl-Ether(CsHl20=AB)
4.1 Design Strategy Basically, we start the design of the column considering a distillation column with only reactive stages. In this case, the design can be performed graphically with the developed graphical method. Figure 4 shows the determined number of reactive stages needed to achieve the specified separation and the optimal location of the feed stage. Table 1a shows the column design in terms of conversion to MTBE and temperature at each stage. It can be observed that the conversion at the bottom and the top of the column is negative. This means that the reaction is reversed and a decomposition of MTBE has occurred. Thus, even if the column design satisfies the given separation specifications, the design should be rejected as the conversion is not satisfactory. The next step is to introduce non-reactive stages at the bottom of the column and generate a new conversion and temperature profile. Table 1b shows that by introducing two non-reactive stages the negative conversion at the bottom of the column is eliminated (stage to stage calculation method has been used). In addition, if one non-reactive stage is introduced at the top of the column, a final design without negative conversion is obtained. It should be pointed out that the design strategy here is to track the conversion and temperature and force them to attain values such that the RDC is feasible. However, other variables such as occurrence of reactive or non-reactive azeotrope and composition of one key component could also have been selected to switch/add non-reactive and/or reactive stages. 5. CONCLUSIONS Two methods for the design of reactive distillation columns have been presented. Based on the element mass balance approach, a graphical method for binary element reactive systems is applied for the design of a distillation column with only reactive stages. When the number of elements is greater than two or when a "combined" configuration of a reactive distillation column is considered, the stage-to-stage calculation method is more appropriate. In the example presented, a design strategy for a combined reactive distillation column was
522 developed considering the conversion to MTBE and the temperature at each stage has been presented. This strategy required the use of reactive and non reactive bubble point calculations. 1.0
}
~ ......... ~A
0.6
McC~be Thlele Procedure (ELEMENTS GRAPHIC) WILSON Equat ion S-R-K E q u a ~ m o n of State
A VAP
!
i .......
! ..... ~ ' >
',
:
i
~/'
i
~
HTBE REACTIVE SYSTEM
:
j
,' / ', [ P- 1.0 T~- 30O.0 ZF}I~- 0 . 7 0 0 0 ~;ti :/........................ : I F E E D I S LIQUID-VAPOR ZF(2)-0.3000
....... ,L....... ,L__ . ~ ~ _ ~,/, i
~ ....... I 9 i
0.2
ENTER
0.4 TO
0.6
2.00
STAGE ~JLA 1 0.5000 2 0.5216 3 0.5564 4 0.6704 5 0.8919 0.8 ~JLA 1.0 6 0.9820 -
' 0.2
REFLUX=
XB(1) . . . . . 27 XB{2) = 0 0427 XB(3) = 0 9146
0.4
0
(NO-INERT)
ZF(3) = 0.0000
YD (i)- 0.9981 YD(2) 0.0000 YD(3) = 0.0019
Re~ulr~. XL(1) "YV(1) 0.0427 0.2688 0.0927 0.4741 0.2042 0.7179 0.5084 0.9216 0.8788 0.9871 0.9816 0.9981 FEED STAGE = 3
T 317.14 310.76 298.52 279.68 268.24 265.84
FINISH
Fig. 4. Design Specifications and output result for the MTBE reactive System Table 1. Conversion-Temperature Profiles for a Reactive Distillation Column Design a. Only reactive stages b. Two non-reactive stages c. Final Design Stage Conv. T (K) Stage Conv. T (K) Stage Cony. T (K) 0 320.17 l=Reb. -0.526 317.13 1=Reb. 0 320.17 1=Reb. 2 0 308.45 2 +0.048 310.42 2 0 308.45 3 +0.2908 294.33 3 +0.307 297.41 3 +0.2908 294.33 4 +0.0947 275.01 4 +0.095 275.82 4 +0.0959 275.01 0 267.03 5=Con. -0.0009 267.16 5 +0.0000 266.99 5=Con. 6=Con. -0.0038 265.55 The final design obtained has been verified with rigorous simulation and a maximum difference in temperature of five degrees between the simple and rigorous calculations was observed. Considering the speed and simplicity of the simple calculations, the results from the developed methods serve as useful first estimates. The design methods are being further extended using the driving force algorithm of Gani and Bek-Pedersen [9] to obtain as well, energy efficient designs. REFERENCES 1. Agreda V.H.; Partin L.R.; Heise W.H., (1990), Chem.Eng. Prog., 86, 40. 2. Smith, L.A., and Huddleston, M.N., (1982), HydrocarbonProcessing, 3, 121. 3. Hairston Deborah, (1999), Chemical Engineering, Vol. 106, No. 4, 32 4. Abufares A.A., and Douglas P.L., (1995), Trans Ichem.Eng., 73A, 3. 5. Monroy R, P6rez-Cisneros E., and J. Alvarez, (2000), Chem.Eng.Sci., 55, 4925. 6. Barbosa D., Doherty M., (1988), Chem.Eng.Sci., 43, 2377. 7. Gumus Z.H., and Ciric A.R., (1997), Comp.Chem.Eng., 21, $983. 8. P6rez-Cisneros, Gani R., and Michelsen M.L., (1997), Chem.Eng.Sci., 52, 527. 9. Gani, R. and Bek-Pedersen, E., (2000),AIChE J.,46 (6), p. 1271-1274
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rights reserved.
523
Development of Software Tools for Crystallization System Synthesis Joseph W. Schroer a, Christianto Wibowo a, Ka M. Ng a'+, Lionel O'Young b a Dept. of Chemical Engineering, University of Massachusetts, Amherst, MA 01003, USA b MC Research and Innovation Center, Mountain View, CA 94041, USA A systematic framework has been developed as part of an effort to expedite the development of crystallization systems. The framework consists of three components: flowsheet synthesis, experimental efforts, and modeling activities. To facilitate the efforts, various software tools have been developed. These include generation of phase diagrams based on thermodynamic calculations, representation of experimental data on phase diagrams, and simulation of particle size distribution of the crystallizer product. The software tools are modular in nature so that engineers in process development can use any of the tools that they like in isolation and add their own in-house tools as appropriate. 1. I N T R O D U C T I O N In this era of globalization, there is a relentless pressure to shorten the time-to-market in the chemical processing industries. It used to take ten years or so to build a large-scale grassroots chemical plant starting from basic chemistry, while most companies are now aiming for four to five years. Systematic methods for process design and development are therefore essential for producing a reliable, optimal process while minimizing development time and effort. Indeed, approaches and techniques in systems engineering have been firmly established for the design of complete gas-liquid plants [1-3]. With the gradual shift of the chemical processing industries towards high-value-added chemicals, most of which are sold in solid form, it is highly desirable to examine solids processes, especially crystallization systems, from the perspective of process systems engineering [4]. During the last few years, we have developed a family of systematic procedures for crystallization systems synthesis [5-8]. The synthesis of downstream solid-liquid separation system to properly recover the crystalline product has also been tackled [9]. We also considered the related problems of selecting crystallizer operating policies [10], and kinetics and mass transfer effects on crystallization process paths [ 11 ]. To facilitate the systematic development of crystallization systems, various software tools have been developed. The software tools are modular in nature so that engineers in process development can use any of the tools that they like in isolation and add their own in-house tools as appropriate. They are flexible, easily updated, and are distributed on the world wide web for easy accessibility. + Present address: Department of Chemical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong.
524 2. A SYSTEMATIC FRAMEWORK The task of crystallization system synthesis involves three key activities: separation process synthesis, downstream processing system synthesis, and crystallizer design. While in a typical scenario these three activities are performed sequentially, iteration between steps is often necessary to come up with a superior design. In each step, economic evaluation is implemented to identify the most feasible alternatives. These design activities are closely interconnected with modeling and experimental efforts, as depicted in Figure 1. Solid-fluid equilibrium (SFE) phase diagrams play a central role in our approach, since they are used as a basis for decision-making in separation process synthesis. Therefore, before beginning with this activity, it is important to generate suitable phase diagrams based on solubility data or thermodynamic information. In the absence of experimental data, the multicomponent SFE phase diagrams can be calculated by a thermodynamically consistent model. Necessary thermodynamic data include the heat of fusion and melting temperature, which are tabulated for many substances in standard references, and liquid-phase activity coefficients, which can be predicted using excess Gibbs free-energy models such as UNIQUAC. Further details on the calculation and representation of SFE phase diagram have been published by Dye and Ng [5], and Samant et al. [ 12], among others. After appropriate phase diagrams have been generated, flowsheet alternatives for the separation process are synthesized to meet the separation objectives. The number of
Figure 1. Components of the systematic framework for crystallization system synthesis.
525 crystallizers and the order of separation are determined, additional unit operations are selected. In our approach, unit operations involved in the separation process are represented on the phase diagram as movements from the feed composition points to the product composition points. The aim is to purposefully maneuver to the proper regions on the phase diagram to crystallize the desired products. Limitations such as eutectic points are bypassed using a suitable movement. A more detailed discussion as well as rules to aid decision-making is provided elsewhere [7]. Once the flowsheet alternatives have been generated, preliminary economic evaluation and feasibility checks are performed. The range of design variable values which does not violate practical or material balance constraints is determined. The crystallizer type and operating temperature are also selected. Since a crystallizer does not exist in isolation in a processing plant, the downstream processes around the crystallizer need to be synthesized. These may include unit operations such as preconcentration, filtration, washing, deliquoring, recrystallization, and drying. A systematic procedure on downstream processing systems synthesis has been proposed [9]. Necessary unit operations are selected, and destinations of the mother liquor, wash liquid, and solvents are assigned. A solvent recovery system is then considered to recover solvents and remove impurities. Economic evaluation using shortcut models are then performed to screen process alternatives. Finally, the crystallizer must be designed in such a way that the product specifications such as particle size distribution (PSD) can be met. Since PSD also has a substantial impact on downstream processing, it is crucial that the crystallizer design is done by taking into account the operation of downstream processing units [10, 13]. To some extent, population balance equation-based models can be used to predict the PSD of a crystallizer product [14]. Accurate knowledge of crystallization kinetics is critical in such an effort. Due to complexities such as the presence of secondary nucleation and inhomogeneity of the crystallizer content, laboratory experiments are essential in crystallizer design. 3. S O F T W A R E TOOLS FOR CRYSTALLIZATION SYSTEM SYNTHESIS Our objectives in developing these software tools include the requirements from both the process development community and the educational community. In process development, applications are primarily used by experts; hence flexibility and customization are necessary because processes or products may be new, requiring solutions that are often different from previous applications. Also, code reusability from project to project is important for facilitating rapid process development. Educational and informational uses present the additional requirements of ease of distribution to a large audience and robustness, because programs will be used with people unfamiliar with the codes. Rather than having complete dependence on one format or platform, our codes exist in a mixture of three common forms. Table 1 lists the forms and some of their advantages and disadvantages of the computer languages used: Visual Basic*, FORTRAN, and Java**. All of the forms support features of interoperability. FORTRAN programs can be called by Visual Basic programs or Java programs by dynamic link libraries or input/output file manipulation. MS Excel files (including plots) can be generated by Java programs developed by some software vendors.
526 Table 1. Computer languages used in our computer codes Language Visual Basic
Advantages - Integrated with MS Excel, a common and comfortable interface for engineers.
Disadvantages - Programavailable only on selected platforms. Files are large. -
FORTRAN
- Calculation speed is high. Established in engineering computations. -
Java
Platformdependent: must be recompiled for different platforms / operating systems. - User interface is not appropriate for graphical demonstrations.
-
- Low computing speed. - Flexibility: offers more functionality than Programdevelopment requires more Visual Basic or FORTRAN. expertise. Platform independence: can be run on any computer. Portability: can be distributed as applets and run by web browsers. - Encouragescomponent-based development for code reuse and ease of maintenance. -
-
-
Because these codes are part of a library of software tools for crystallization, they must be dynamically linkable, portable, and have the ability to identify themselves and what they do to other programs. For this reason, Java was selected as the language of choice for many of the programs, thus making them accessible on theoretically any computing platform. In addition, Java allows us to develop programs of much greater complexity and features that are not possible with other languages. Table 2 gives a list of some of the prototype codes and their descriptions. Most of the tools are centered around calculating and displaying SFE phase diagrams. Included are programs for calculating phase diagrams of binary, ternary, and multicomponent mixtures. In addition, some codes aid in representing experimental data in a suitable format and in calculating activity coefficients from data. The Java codes have been assembled in a package named edu.umass.ecs.ngdesign. This package is grouped into three additional packages for organization. The package edu.umass.ecs.ngdesign.demos contains demo programs to illustrate the computer codes. This packages relies on classes that reside in the remaining two packages: edu.umass.ecs.ngdesign.sfe and edu.umass.ecs.ngdesign.graphics. The former contains classes for performing solid-fluid equilibrium calculations for producing phase diagrams. The latter contains classes for rendering graphics and plotting on the computer screen. Documentation for the Java computer codes was done using the Javadoc tool. This tool produces the Application Programmer's Interface (API) in an HTML format that has a standardized look. The Javadoc output lists an object's names, arguments, and return type for each method as well as the object's fields, and allows incorporation of comments added by the programmer. This system allows for code documentation and explanation in such detail that other programmers can easily incorporate the codes into new projects.
527 Table 2. Description of some prototype computer codes. Visual Basic/Excel programs
SFE.xls BatchCC.xls ContC.xls FORTRAN
programs
csdB.exe csdC.exe Java
Contains macros for plotting ternary SFE phase diagrams in equilateral triangular coordinates. Also contains demonstrations of SFE calculations. Provides design guidelines. Plots product PSD and supersaturation profile with data from csdB. Provides design guidelines. Plots product PSD with data from csdC. Calculates the product PSD of batch cooling crystallizers for various operating policies. Calculates the product PSD of continuous crystallizers with various configurations.
programs
Package edu. umass, ecs.ngdesign. demos
BinSFEID 9 TernSFEID
9
9 PolyT3 9 Janecke4c 9 ConjugateSalt 9 Molecular
Calculates and plots binary T-x melting point diagrams for an ideal system. Calculates and plots isothermal and polythermal projection phase diagrams for a 3component ideal mixture. Calculates and plots isothermal and polythermal projection phase diagrams for a 3component nonideal mixture system. Calculates and plots polythermal Janecke projections for a 4-component nonideal system on a ternary phase diagram plot. Calculates and displays a Janecke projection of an isothermal conjugate salt pair phase diagram. Calculates all of the fixed points of an N component mixture of an nonideal molecular simple eutectic system. Returns the list of eutectic compositions, temperatures, and the adjacency and saturation variety matrices. Package edu.umass.ecs.ngdesign.sfe Package edu. umass, ecs. ngdesign. graphics
9 JSplot
Plotting routine for graphing ternary phase diagrams.
4. C O N C L U S I O N S In developing a chemical process, we need to synthesize process altematives, simulate process performance, and generate basic and process data. Our strategy is to forge forward with minimum information; then fine tune the design with increasingly accurate models and data [ 15]. To shorten the cycle time for such an effort, the development team has to possess the right software and experimental tools, generate the right data at the right time, and to share such data and knowledge about the process. It is important that the overall development plan as well as the reasoning of any action is clearly understood among the team members. A design framework that formulates the workflow, facilitates process synthesis and analysis, guides experimental efforts, and provides an environment for data sharing is highly desirable. This article considers such a framework for the development of a crystallization system. It helps to predict SFE phase diagrams, synthesize a crystallization system flowsheet, select the solvent, the crystallizer type and the operating conditions, predict the PSD, organize and interpret experimental SFE data, and so on. However, it is not complete. For example, we considered only particle size distribution, but shape, color and other crystal attributes can be
528 equally important [ 16]. Also, other computer programs can be written. For example, we have not yet included crystallization kinetics and transport limitations in our computer codes. These are not major drawbacks, however. By creating modular codes that can be easily linked and modified, we can readily add new tools or customize existing ones to suit specific needs. This feature is useful during project execution, and allows rapid incorporation of new methods into the framework, thus speeding up the entire research cycle. In addition to process development, we hope that the prototype codes with the demos would help educators teach SFE phase diagrams and their use in process synthesis. Work is under way to expand and refine both the framework and the codes. ACKNOWLEDGMENT
We express our appreciation to the National Science Foundation, Grant No. CTS-9908667, for support of this research. Trademark or registered trademark of Microsoft Corporation. Trademark or registered trademark of Sun Microsystems, Inc. REFERENCES
1. J.M. Douglas, Conceptual Design of Chemical Processes, McGraw-Hill, New York, 1988. 2. L.T. Biegler, I. E. Grossmann and A. W. Westerberg, Systematic Methods of Chemical Process Design, Prentice Hall, Upper Saddle River, 1997. 3. W.D. Seider, J. D. Seader and D. R. Lewin, Process Design Principles: Synthesis, Analysis and Evaluation, Wiley, New York, 1998. 4. S. Rajagopal, K. M. Ng, and J. M. Douglas, Comput. Chem. Eng., 16, 675 (1992). 5. S.R. Dye and K. M. Ng, AIChE J., 41, 1456 (1995). 6. D.A. Berry, S. R. Dye and K. M. Ng, AIChE J., 43, 91 (1997). 7. C. Wibowo and K. M. Ng, AIChE J., 46, 1400 (2000). 8. J.W. Schroer, C. Wibowo and K. M. Ng, AIChE J., accepted for publication (2000). 9. W.-C. Chang and K. M. Ng, AIChE J., 44, 2240 (1998). 10. C. Wibowo and K. M. Ng,, submitted to AIChE J. (2000). 11. V. V. Kelkar and K. M. Ng, AIChE J., 45, 69 (1999). 12. K. D. Samant, D. A. Berry and K. M. Ng, AIChE J., accepted for publication (2000). 13. P. J. Hill and K. M. Ng, AIChEJ., 43, 715 (1997). 14. S. N. Tavare, Industrial Crystallization: Process Simulation Analysis and Design, Plenum Press, New York (1995). 15. L. O'Young, L. Natori, T. G. Pressly and K. M. Ng, , Comp. Chem. Eng., 21, $223 (1997). 16. R. Braatz and S. Hasebe, paper to be presented at Chemical Process Control, Tucson, AZ (2001).
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
529
Optimization of ethylene process design G. Sobo6an and P. Glavi6 University of Maribor, Faculty of Chemistry and Chemical Engineering Smetanova 17, 2000 Maribor, Slovenia Process for producing ethylene from a naphtha pyrolysis gas stream has been studied. The influence of components separation was treated with two different simulators using two different sequences of distillation columns. Considerable differences in reboiler and condenser heat flow rates resulted in economic analysis of particular sequences. The same sequence gave the lowest total annual costs in spite of the use of two different simulators compared to the second best option proposed for distillation columns sequence. Use of different models resulted in different heat integrated structures and, therefore, different profits. In general, heat integration resulted in total annual cost reduction between 9 % and 19 %. Total annual costs of the best heat integrated processes were reduced for about 9 MUSD.
1. INTRODUCTION Systematic synthesis of multicomponent separation sequences is an important process design problem in chemical industry. It is concerned with the selection of a separation method and the selection of the best sequence of separators to split a multicomponent mixture into several products of relatively pure species as desired. For solving separation problems in chemical industry, distillation columns are widely employed separators. Distillation sequences can be specified by different methods: heuristic, evolutionary, algorithmic [1, 2, 3]. Columns sequencing in a multicomponent mixture separation has a decisive influence on the economics of the process. The goal of each designer is to find the sequence with the minimum total costs of separation [4]. If there is a system of distillation columns, it is usually classified as a temperature cascade, which depends on the properties of a feed stream. In a distillation column heavy components are separated from light ones. Light components are evaporated and separated as a condensate on the top of the column. Heavy components are removed from the column at the bottom. Separability of a multicomponent system depends on properties of the feed mixture, operating conditions and other additional restrictions. All these factors influence the equipment costs of a particular sequence. Utilities costs have to be taken into consideration, too. In process design a distillation column sequence usually determines the main part of the process costs with regard to the total annual costs. Utility costs depend on the type of the utility (its price) and the heat flow rate in condensers (cold heat flow rate, ~cu) and reboilers (hot heat flow rate, tPHU see Figure 1). The depreciation and the utilities costs represent the operating cost for a single sequence. The best sequence is the one with the lowest total annual costs.
530
Figure 1. Comparison of utilities influence in the system of a five-column (D l-D5) cascade. In distillation heat is supplied to a reboiler and taken away from the condenser. Heat has to be supplied at the temperature above the boiling point temperature of the vapor coming from the reboiler. The heat flow rate in the condenser is given at the temperature below the dew point temperature. In short cut process design boiling and condensation are supposed to occur at constant temperatures. We usually analyse thermally integrated distillation columns in temperature-enthalpy flow rate difference (T-A/) diagram. In the case of two-component mixtures, constant temperatures in the reboiler and the condenser of a distillation column exist and heat flow rates of reboilers and condensers are represented by horizontal lines. We have simulated ethylene production process using two different process simulators, Aspen Plus and Hysys. Two different sequences of six distillation columns were studied. Our intention was to explore whether the use of different simulators would influence the process economics more or the influence of different columns sequences would predominate. In the next step we tried to integrate each of the three variants thermally. 2. HEAT INTEGRATION OF DISTILLATION COLUMNS Distillation of multicomponent mixtures is one of the most common separation operations in the chemical industry. Because of high energy consumption, heat integration is often vital factor when considering multistage separation in distillation columns. In that case columns are coupled to form sequences. These can be specified by different methods: heuristic, evolutionary and algorithmic [1]. The goal of process design is to find the economic most favourable heat integrated sequence. Optimization of distillation columns includes different reflux ratios, pressures, side reboilers/condensers and preheating/cooling of feed mixture [5]. Heat flow rates and temperature levels of different sequences of distillation columns have to be evaluated and out of them the best combination has to be selected. The goal is to reach the lowest possible temperature difference inside the column and the lowest possible heat flow rates. That will result in a better thermal integration possibilities and lower utilities consumption.
531 In the case of a multicomponent separation the predominant utilities consumption usually takes place in distillation columns. It should be pointed out that in our research classical heat integration between distillation columns was considered. This means that total condensers and total reboilers with condenser-reboiler matches were taken into consideration. Every unit using hot or cold utilities contributes to total costs. So the main target in process integration is to make use of every available hot or cold process stream in order to substitute utilities. The more columns are connected to a heat flow cascade, the higher is the potential profit of heat integration. Heat is transferred from a condenser of a distillation column or from a process stream with a higher temperature to a reboiler of another column or to a process stream with a lower temperature. The greatest benefit is to achieve the highest possible extent of condenserreboiler matches. Maximum heat transfer between the two columns is desired. The goal of the process design is to select the structure and parameters to give the minimum sum of the depreciation and operating cost. The general methodology which leads to the best scheme is called process synthesis. We used the thermodynamic method as a first step in the synthesis of ethylene production process case study. Our research was oriented to produce ethylene and five other products of desired purity. We compared three different versions with regard to the total annual costs of the process. 3. CASE STUDY Our research was tested on a separation problem of a six-product mixture for recovery of ethylene and light products from a naphtha-pyrolysis gas stream. The composition of the mixture is shown in Table 1. The objective of the case study is to synthesise a separation train for the recovery of the nine products from steam-cracked naphtha involving about two dozen components using two different simulators. Total flowrate of the feed is 2.095 kg/s, its temperature is 333 K and the pressure is 136.5 kPa [6]. Table 1 Feed compositions and properties Key components Feed amount fraction, xi A hydrogen+ methane 0.48 B ethylene 0.28 C ethane 0.06 D propylene + propane 0.08 E 1,3 butadiene+ trans-2-butene 0.04 F n-pentane 0.06
TB ~ -161.5 -103.75 -88.6 -47/-42.1 -4.45/4 68.73
CES* 53.33 4.78 9.13 4.18 4.13 -
The ethylene separation process can be divided into two parts. In the first part the feed gas is compressed in five stages prior to cryogenic treatment. In the second part the separation of components is carried out in five distillation columns (D2 to D6). The basic process scheme is shown in Figure 2. Two different sequences of distillation columns D-2 to D-6 were studied rigorously. Figure 2 shows the option 2 from the Figure 3. They were chosen because the previous research showed that these two sequences were the most promising [7]. After each compression, the cracked gas is always cooled to 15 ~ (before another flash, Ft-1 to Ft-5). Interstage cooling is required to prevent polymerization and fouling; water and hydrocarbons, condensed at these points are separated from the pyrolysis gas in interstage separators [6].
532
L~_ ~
Ftl
(/~J Fi2 ~_~~[~Ft3 )~[~ Ft4 L)_~_~i-Ft5l ~
D1
F
Figure 2. Basic process scheme for ethylene production- version 2.
AJBCDEF A/BCDEF
....... B/C BC/DEF --~-~_..... ......... D/EF B/CDEF C/DEF-
ElF D/EF--
E/F
Figure 3. Two sequences studied in the ethylene separation process. For the computer simulation of the process process simulators Aspen Plus and Hysys were used. In our research we concentrated on blocks for simulation of distillation columns. Two different models which allow almost identical starting values to be set were used. The ASPEN module DSTWU performs a Winn-Underwood-Gilliland shortcut design calculation [8]. The Peng-Robinson thermodynamic option has been chosen to simulate ethylene separation process with both simulators. The Column Sub-Flowsheet (CSF) model of Hysys is a rigorous model with a wide range of operational parameters setting. It enables very similar definition of starting parameters as in the case of the Aspen Plus simulator [9]. The combination of the DSTWU and CSF models was taken because of the good performances of each. The process parameters obtained by the DSTWU model were then taken as starting values of reflux ratios and number of stages for the CSF model of Hysys. Particular variants were compared using total annual cost estimation. The research was limited to equipment and utilities costs. The economy of different sequences was estimated by using techniques described in literature [7, 10]. The annual operating time for the production of ethylene was 8500 h.
533 RESULTS AND COMPARISON OF MODELS Our goal was to find how simulators and different distillation columns sequences influence the economic analysis. The two different sequences were studied before and after heat integration and their total annual costs were compared. Altogether, three different simulations and economic analyses were taken into consideration. Sequences 1 and 2 (version Aspen Plus1, -2 in Figure 3) were simulated using the Aspen Plus and Hysys simulators. Finally, three versions were compared. The results of the best heat integrated variants are shown in Table 2. (CTE) represents depreciation-, (CTu) utility- and (CTA) total annual cost estimates for the heat integrated process. Version numbers 1 and 3 (Aspen Plus-1 and Hysys-1) show the same process. The CTA values obtained by Aspen Plus are lower mainly because of the lower CTU. Although different simulators were used, both show lower values for CTA than in the case of sequence-2 simulated by Aspen Plus. The CTA savings in all the three cases are very similar, from 17.5 % to 18.7 % according to the non-integrated process. The best two versions are shown in the T-A[ diagram (Figures 4a) and 4b)). Table 2 Comparison of three best versions of heat integrated processes. No. option CTE CTU CTA CTA r e d u c t i o n (S/a) (S/a) (S/a) % 1. Aspen Plus-1 19 792 000 17 868 000 37 660 000 18,73
(a) 1,4
2.
Aspen Plus-2
18 515 000
25 181 000
43 696 000
17,88
1,3
3.
Hysys-1
15 847 000
27 312 000
43 159 000
17,46
1,2
T/~ 150
/PB
hot utility
-'.//
o-I
TI~
[hot utility D-1
150"
m
/~
D-5")/
100 D-61 ' i
50
'
D.
m
D-3
,00
GCC
"
D-6
,
L
j
GCC
~4.
m
,'~ '.i
,,
,
0-4
'
:'4-
-50
-50
D-2
-100
-100
D-2
f
cold utility
cold utility
20
40
60
80
AI/kW
20
40
60
80
A//kW
a) b) Figure 4. T-A/diagram of the sequence 1" a) Aspen Plus-l, b) Aspen Plus-2. Comparison of cases 4a) and 4b) clearly shows that the main differences appear with columns D-3 and D-4 which have better properties for heat integration in the case 4a). These columns are almost totally integrated with each other but there is a great surplus of cold utility demand (columns D-3 and D-4) from external utility in the case 4b).
534 Differences between the three versions tested mainly appear because of the use of two different models (shortcut/rigorous) for simulation of distillation columns. The sequence 1 from the Figure 3 was supposed to be a better option according to CTA than the sequence 2. It had better capabilities for heat integration [7]. The same sequence (sequence 1) showed lower CTA also when using Hysys simulator (compared to Aspen Plus-2). Although the same sequence was applied in the simulation of both simulators significant differences appeared in the final CvA estimation. Differences were bigger between the Aspen Plus-1 and Hysys-1 than in the case of Aspen Plus-2 and Hysys-1. Thermodynamic models were kept the same in both simulators. Different heat flow rates and different temperatures gave different conditions for heat integration. The base process schemes in both simulators (Aspen Plus-1 and Hysys-1) were different, mostly because of the so called thermal properties. The higher CVA savings influence the longer payback period (Table 3). The Aspen Plus-1 option has the greatest CTA-reduction but also the highest investment. 4. CONCLUSIONS Process for separation of ethylene and five other products was designed with simultaneous consideration of economic and technological point of view. As the main product we wanted to obtain all the six key components at the desired purity. Heat integration between the distillation columns and their integration with other process streams is important. Energy savings achieved by this integration could exceed those obtained by mutual columns integration. The basic process scheme was simulated using two different sequences of distillation columns and two different process simulators, Aspen Plus and Hysys. Altogether, three different cases where studied. Each one was thermally integrated and studied according to the total annual costs. These were reduced by 10-19 % with total annual savings from 8.7 MUSD to 9.5 MUSD compared to the non-integrated process. This was achieved mostly by reduced areas of reboilers/condensers but the main savings showed in utilities reduction. REFERENCES
1. M. K. Kattan and P. L. Douglas, A New Approach to Thermal Integration of Distillation Sequences, Can J Chem Eng, 64/February, (1986) 162-170. 2. D. Trigueros, C. Coronado-Velasco, A. Gomez-Munoz, Synthesize Simple Distillation the Thermodynamic Way, Chem Eng, August, (1989) 129-134. 3. N. Nishida, G. Stephanopoulos and A. W. Westerberg, A Review of Process Synthesis, AIChE J., 27/3, (1981) 321-351. 4. G. Sobo6an and P. Glavi6, Optimization of Ethanol fermentation process design, App Therm Eng, 20, (2000) 529-543. 5. W. Rajah and G. T. Polley, Synthesis of Practical Distillation Schemes, Trans IChemE, 73/A, (1995) 953-966. 6. CACHE Case Study, Separation System for Recovery of Ethylene and Light Products from a Naphta-Pyrolysis Gas Stream, Camegie-Mellon University Pittsburgh (1983). 7. G. Sobo6an and P. Glavi6, A New Method for Studying Thermally Integrated Distillation Sequences, Comput Chem Eng, 20, Suppl. A, (1999) 183-188. 8. Aspen Plus, User Guide, Software Version, (1988). 9. Hysys, Reference Volume 1,2, Version 1.1, Hyprotech Ltd., Calgary (1996). 10. C. W. Hui and S. Ahmad, Total Site Heat Integration Using the Utility System, Comp Chem Eng 18/8, (1994) 729-742.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
535
Optimization of an Acidic Chlorine Scrubber with a Rate-Based Simulation Engine W. Steinbach, A. Friedl., H. Hofbauer Institute of Chemical Engineering, Vienna University of Technology Getreidemarkt 9/159, 1060 Wien, Austria email: wstein(afriedl,hhofba)@mail.zserv.tuwien, ac.at The absorption of chlorine from an exhaust gas stream into an aqueous acidic solution of ferrous chloride is modeled. Chemical reaction kinetics as well as mass and heat transfer is taken into account to determine the rate of absorption. The calculation is performed by the AspenPlus - RateFrac TM simulation engine using the ELECNRTL property method. Chemical property data is checked and several parameters based on the electrolyte NRTL activity coefficient model are regressed. A sensitivity analysis is carried out to optimize the operating conditions and the design of a random packed column with respect to the off-gas concentration of chlorine. 1 INTRODUCTION Chlorine is a poison gas, that must be removed from exhaust gases of many processes down to a legal limit of concentration. In steel industry, where HC1 is used to remove rust and scale from the metal surface, C12 can be formed while regenerating the used acid. Therefore a scrubber fed with an alkaline solution of sodium hydroxide and thiosulfate is provided. This chemicals are not needed if the iron-II containing pickling solution itself is used as washing liquid. Because reactions in electrolytic systems are usually fast and the absorption can not be calculated with an equilibrium model, the design of the column in dimension and pumparound depends on the kinetics of the process. Thus, a rate-based simulation taking into account the three terms of kinetics, mass transfer, heat transfer and chemical reaction kinetics will lead to a successful prediction of the absorption process. The scrubber is a random packed column with counter current flow as shown in Figure 1. The gaseous inlet stream can consist of typical combustion gas containing H20, N2, 02, CO2 and small amounts of HC1 and C12. The liquid inlet stream is the spent pickle liquor containing H20, HC1, FeC12 and small amounts of FeC13. Temperature is considered to be 85~ for the gas and for the liquid, the scrubber works at atmospheric pressure.
536 ILIQIN I
9
GAsouT
9
I~,ooo, J
Fig. 1: Principal flowsheet of a chlorine-scrubber 2 THERMODYNAMIC AND TRANSPORT PROPERTIES
AspenPlus TM recommends a predefined property method called ELECNRTL which was used in the simulation. The physical models contained in this property method are listed in table 1. Table 1 Summary of physical models used Common
Vapor pressure Heat of vaporization Surface tension
Vapor mixture Fugacity coefficient, Density Enthalpy, Entropy, Gibbs energy Vapor viscosity Vapor thermal conductivity Vapor diffusivity
Extended Antoine Watson/DIPPR Hakim-Steinberg-Stiel/DIPPR- Onsager-Samara Redlich-Kwong Ideal gas heat capacity / DIPPR, Barin correlation, Redlich-Kwong Chapman-Enskog-Brokaw Stiel-Thodos / DIPPR Dawson-Khoury-Kobayashi
537
Liquid mixture Activity coefficient, Gibbs energy Liquid molar volume Infinite dilution heat capacity Enthalpy, Entropy
Electrolyte NRTL, Extended Antoine, Henry's constant, Brelvi-O'Connell Rackett, Clarke Criss-Cobble Ideal gas heat capacity / DIPPR, Watson / DIPPR heat of vaporization, Criss-Cobble infinite dilution heat capacity, Electrolyte NRTL Rackett / Clarke Andrade/DIPPR- Jones-Dole Sato-Riedel/DIPPR- Riedel Wilke-Chang - Nemst-Hartley
Density Liquid viscosity Liquid thermal conductivity Liquid diffusivity
Reprinted from AspenPlus T M Handbook, 1999 As AspenPlus extends the calculation of physical properties from binary mixture to multi component mixtures the property data for the simulated system must be checked by literature [2,3]. Parameters of the electrolyte NRTL model were correlated to literature data from [3] for the systems H20-C12 and HzO-HC1-CI2. TM
100
800
E E
+,
I--.--.I
-1-
SystemH20-CI2
9
60-
80 ~ B
SystemH20-HCI-CI2
=i4o
,ooc
..
40-r 20
20-
20
i
I
I
I
40
60
80
100
0 I 0
T [~
, 4
, 8
50~
HCI [ m o l l l ]
Fig.2,3" Henry's Constant; Comparison of literature data [3] and AspenPlus results after regression.
3 ABSORPTION MODEL The overall reaction of chlorine is described as follows.
12
538
C12 + 2 Fe 2+ :=> 2 C I + 2 Fe 3+
The kinetic of this irreversible reaction is known to be in the second order (first order with respect to both C12 and Fe2+-ions) [1 ]. By analyzing Hatta's number for irreversible secondorder reaction with reasonable values, it can be found, that the absorption kinetic of chlorine is determined by mass transfer and not by reaction kinetics. Mass transfer coefficients and the interfacial area available for mass transfer in packed columns is calculated using the correlation developed by [4].
{lexpl 14 .e~ k t = 0,051. (Retw) 2/3 9
1/2
.(ap .dp) ~
(1)
"/g'/~t 1 / 3 p/ t
kg = 5,23 . (Reg )~ . (Scg,C12 )l~3 . (ap . dp ~ 2 .
(2)
a p . D g,cl2
R.rg
(3)
The heat transfer coefficient is calculated, using the Chilton-Colbum analogy [5]. kavSc2/3=
(4)
htc Cp,mix
The dissociation of HC1, FeC12 and FeC13 are defined by the following reaction equations. HC1 + H20 ~ FeC12 ~
CI + H 3 0 +
Fe 2+ + 2 CI
FeC13 ==:, Fe 3+ + 3 CI
4 OPTIMIZATION With the method of sensitivity analysis, where you vary one or more variables over a wide range of values you can get a very clear picture of the optimal design parameters and operation conditions. Because the height of a column is often limited due to construction reasons, the variables remaining for this optimization are the column diameter, the liquid pumparound and the size
539 of the packing. The packing is only available with certain sizes, thus two variables, column diameter and pumparound, are varied. The efficiency of absorption is defined by the relation of incoming to absorbed amount of chlorine. The results are shown in the figures 4 to 7. Table 2 Legend for figure 4 to 7
_Symbol
E ~
A 1 9 9
E
97 % 94 % 91% 88%
5
%" 100
4
~
,ag I1) o')
m r
8o
3
60
2
40
r~
"5 -J
"-i
0 0
Fig. 4:
20 0
,
1
Gas
3 4 Charge [kg/rn=s]
0
dv =
1 inch
Fig 5: dv = 2 inch
2
'~n 20
~ ' 6000
E
E
~15
,--, 4000
~. 10
--
.I= ~9
9~ ,
, m
--
2
3
4
3
4
5
~
~-~
o
1
Gas Charge [kglm=s]
~"
o
5
-~
2000
, m
, ~ ,
0
--
0
1
2
3
4
Gas Charge [kglmZs]
Fig. 6:
dp =
1,5 inch
5
0 0
1
2
Gas Charge [kglm=s]
Fig 7:
dp =
3,5 inch
5 CONCLUSIONS The results of the simulation show that the influence of the packing diameter, which is correlated to the specific interfacial area by equation (1), has an extremely strong influence on the performance of the absorber with a given gas charge. This significant results would not have been obtained with a equilibrium based simulation. An absorber could only be designed with a lot of experience and oversizing. The rate-based
540 simulation as shown in this practical example gives us the opportunity to design a scrubber near to its optimum. REFERENCES 1. 2. 3. 4. 5.
H. Hikita et al., Chem. Eng. Sci., 30 (1975) 607. F. Hine, et al., Bull. Chem. Soc. Jpn., 41 (1968) 71. C.C. Chen and L.B. Evans, AIChE J., 32 (1986) 444. Onda et al., J. Chem. Eng. Japan, 1 (1968) 56. F. King, M. Spiro, J. Solution Chem., 12 (1983) 65.
EuropeanSymposiumon ComputerAidedProcessEngineering- 11 R. Ganiand S.B.Jorgensen(Editors) 9 2001 ElsevierScienceB.V.All rightsreserved.
541
An Accelerated Branch-and-Bound Algorithm for Assignment Problems of Utility Systems Alexandros M. Strouvalisa, Istvan Heckl b, Ferenc Friedlerb and Antonis C. Kokossis c* Department of Process Integration, UMIST, P.O. Box 88, Manchester M60 1QD, UK. b Department of Computer Science, University of Veszpr6m, Egyetem u. 10, Veszpr6m H-8200, Hungary. c Department of Chemical and Process Engineering, School of Engineering in the Environment, University of Surrey, Guildford, GU2 7XH, UK.
a
The paper presents a methodology for integrating logic and engineering knowledge within a Branch-and-Bound algorithm with purpose to accelerate convergence. The development addresses assignment problems of utility networks with emphasis on the optimal allocation of units over periods for maintenance. The solver exploits the special structure of the problem to (i) exclude redundant combination of variables, (ii) prioritise the branching of nodes, (iii) provide bounds of nodes and (v) prune inferior parts of the binary tree. Extraction of knowledge and analysis of operations is supported by the graphical environment of the Hardware Composites. Comparisons with commercial MILP solvers demonstrate the merits of customising the solution search engine to the particular solution space. 1. INTRODUCTION The impact of Mathematical Programming and Optimisation proves significant through a variety of applications in design and operations. The Operations Research community contributed considerable part of the available optimisation tools. At their best, they epitomise general theoretical, computational and numerical knowledge in relevance to the different classes of problems considered. The result is application of general-purpose solvers designed to address formulations ranging from financial problems to chemical process design. Despite numerous efforts the proposed interfaces with solvers exhibit inferior performances as they are not capable of capturing the intricacies of the particular application. In the absence of specific knowledge, the use of general heuristics devotes a large computational effort to redundant searches and formulations that artificially expand the solution space. The importance of including logic in the modelling stage was highlighted by Raman and Grossmann (1992) who employed a combination of heuristics and logic to solve MINLP problems. The same authors later (1993) used inference logic to branch on decision variables and (1994) implemented logical disjunctions as mixed-integer constraints. Solution of MILP's is mainly addressed through application of the Branch-and-Bound algorithm. The algorithmic efficiency relies on the selection criteria for candidate problems, bounding, pruning and branching (Geoffrion et al. (1972)). As Forrest et al. (1974) mentioned, important B&B Corresponding author.
542 functions are promptly determined if the user has adequate knowledge of the physical problem. The Hardware Composites (Mavromatis and Kokossis (1998), Strouvalis et al. (1998)) are employed to reveal information and insights of utility networks that would otherwise be impractical or expensive to acquire by algorithmic approaches. The Hardware Composites not only assist in customising and tuning solvers but also analyse solution space properties and their computational impact. 2. PROBLEM DESCRIPTION AND MODEL ANALYSIS The problem considers the maintenance scheduling of turbines and boilers (assignment of tasks to periods). Objective is identification of the optimal sequence to shut-down units for inspection and maintenance with minimum disruption of the utility operation. Switching-off units imposes penalties to objective function as less efficient units or options (i.e. power purchase) are employed to compensate for the ones maintained. Optimisation has to consider demand variations over time, differences in efficiencies of units and feasibility aspects. The formulations yield MILP problems with significant number of variables. Even for moderate networks the problem can become highly combinatorial and expensive to solve. This class of scheduling problems exhibits the block angular structure with periods coupled to each other due to binary variables assigned for the ON/OFF status of units. Binary variables are present in maintenance and individual period constraints. The special model structure is exploited to set up the B&B algorithm. Especially the customised bounding and pruning criteria capitalise on decoupled combinations of maintenance scenarios while investing computational effort on options associated with linking constraints. 3. SOLVER CUSTOMISATION The customisation spans the main stages of a B&B algorithm. The incorporation of knowledge introduces: 1. Assignment of priorities for the selection of candidate subproblems and branching of variables. 2. Bounding with customised use of the LP solver. 3. Customised tests for minimising the enumerated nodes (enhanced pruning). The B&B solver is implemented in C++ with application of L I N X - a simplex-based routine collection- (Fabian 1992) as LP solver. 3.1. Assignment of Priorities As units are switched-off, the imposed to objective function penalties vary with the unit efficiency, the efficiency of the units available to replace them and the demands of the particular period. Each period is affected to a different extent and preferences/priorities are strong functions of the layout of demands. High preferences relate to minor alterations in the operation of the utility network. The period prioritisation is rigorously defined through calculation of penalties associated with the shut-down of single units in available periods. The basic priority lists are then defined by ordered sets PL(u) = (Pi, Pj..... Pn) for unit u, with period Pi assigned to higher priority than Pj,..., Pn (u switch-off in Pi contributes less increase to objective function compared to Pj,..., Pn).
543 Priorities are established among units as well. Their definition is based on a hierarchical analysis reflecting the relative importance of turbines and boilers. The visualisation of solution space by Hardware Composites offers qualitative understanding of unit efficiencies and capacity limits in view of specific periods and entire sets of demands. The unit prioritisation is included in ordered set PU - (Ua, Ub.... ,Uk), with unit Ua assigned with higher priority than Ub.... , Uk (Ua is more important to operation of the system than Ub,...,Uk). The assignment of priorities (period and unit) is referred to as the preprocessing stage and requires the solution of a number of LP's. The resources spent during preprocessing represent the computational cost of prioritising and revealing the structure of the solution space. Based on sets PL(u) and PU the selection of candidate subproblems and branching of nodes is arranged and customised. 3.2. Calculation of Lower Bounds The B&B solver performs bounding by capitalising on information acquired during preprocessing. Instead of calling the LP solver to estimate bounds at every branched node, a more refined policy of solving LP's is adopted. Enumerated nodes are classified as dependent and independent. A node is dependent if it involves the shut-down of more than one unit in a period or if this node is infeasible. Otherwise the node is termed independent. Independent nodes relate to decoupled maintenance combinations (units switched-off in different periods). These nodes qualify for having their lower bounds defined by already available and computationally inexpensive information. On the contrary, nodes of coupled operations (dependent) necessitate separate bounding. In that manner the LP solver utilisation results to reduced resources spent on calculation of bounds. 3.3. Enhanced Pruning Enhanced pruning uses the properties of dependent and independent nodes. For independent combinations tests are made using the priority lists. The tests select the combinations to enumerate and exclude a further enumeration of nodes (as having a guaranteed lower potential). Units associated with identical maintenance periods or infeasibility at a visited node are the ones justified for relaxation of the priority sequence by examining the next period(s) in preference. When a feasible independent terminal node has been reached or an independent node has been pruned then nodes involving lower periods in priority are pruned without enumeration. It is pointed out that enhanced pruning is rigorous and does not compromise on the optimality of the solution.
4. I L L U S T R A T I O N EXAMPLE The utility system of Fig. l(a) includes 9 units (steam turbines TI-Ts, gas turbine GT and boilers B1-B3) while operating horizon consists of 24 equal-length periods. Each period relates to a pair of constant demands in power and process steam as shown in Fig. l(b). Preventive maintenance imposes the shut-down of all units for one period. The optimal scheduling is expected to meet all demands and maintenance needs in the most efficient way. The problem is modelled as MILP with integer variables assigned for the status of units per period. The complete model properties are represented in Table 1.
544 Table 1: Model size for illustration example. Continuous Variables
Binary Variables
Constraints
Non-zero Elements
457
216
754
2257
Fig. 1: (a) Utility network and (b) sets of demands in power and heat. The solution of the problem is addressed in two steps: I) preprocessing of the solution space and II) application of the customised B&B solver. STEP I:
The solution space is analysed to reveal the feasible and prioritised options for the B&B solver to navigate. Preprocessing is applied in conceptual and computational terms. The location of demands on the solution space (Fig. 3) in relevance to hardware limits identifies infeasible scenarios which are excluded from optimisation. Separate LP's are solved to calculate penalties associated to the shut-down of individual units in feasible periods. Penalties in increasing sequence define priority lists PL(u) for each unit u. The penalties are furthermore used in conjunction to conceptual analysis for defining a qualitative importance of units to operations. In that manner the unit prioritisation list PU is also determined. Ordered sets PL(u) and PU formulate the solver matrix (Fig. 2) representing the prioritised solution space. The first column includes all units subject to maintenance arranged according to PU (upper elements - GT, T2, T1, T3.... - relate to higher priority units). Each element-unit of the first column associates to the corresponding period priority list PL(u) defining the rows of the matrix. STEP II:
The customised B&B searches the solution space capitalising on the structure of the solver matrix. The branching of the binary tree initiates from the first elements of the upper rows and proceeds to deeper options only if specific properties hold. The enhanced pruning effectively disregards inferior parts of the tree from enumeration. The result is acceleration of the B&B algorithm compared to the solution of the same model by OSL implemented in GAMS (Table 2).
545
-GT
5
3
15
14
16
13
6
2
7
T2
15
14
4
3
5
16
13
6
2
7
8
17
12 9
T1
5
6
4
2
3
16
13
1
7
24
12
8
17
T3
4
5
3
15
6
2
16
13
14
7
12
8
1
T5
5
18
6
11
4
1
24
2
12
7
3
23
8
BI
5
4
6
1
11
18
24
2
12
3
23
7
20
1
6
2
20
4
5
11
18
24
12
3
23
7
B3
19 20
10
9
22
21
14 17 23
8
18
15
T4
10 21
22
9
17
23
18
11
24
12
Bz
14
8
1
Fig. 2" The solver matrix reflects the B&B customisation. Table 2: Computational results for illustration example.
Nodes: Iterations: Solved LP's: CPU(sec)- 333 MHz: Objective-($/oper.horizon): (Relaxed Objective) Preprocessing Stage" 217 LP's -
Customised B&B 188 76 13.2 1,021,133.4 (1,004,690) 26.2 CPU(sec)
OSL(GAMS) 50,402 150,000 (interrupted) 50,404 1,339 1,022,899.6 (1,005,439.7)
OSL solver invested significant computational effort in searching the solution space. Even at the iteration limit of 150,000 optimality had not been reached due to the suboptimal flat profile the solver was trapped in. Alternatively, the customised B&B performed better in all aspects and identified the optimal schedule after solving (217 + 76) LP's during preprocessing and Branching-and-Bounding respectively. Optimal maintenance is performed according to vector: (GT, T2, T1, T3, Ts, B1, B2, B3, T4) = (5, 15, 6, 4, 18, 4, 1, 19,10). 5. CONCLUSIONS This work reports on the advantages observed from the customisation in optimisation applications. Experience has shown that general-purpose solvers fail to capture and profit from special properties of problems. Ad hoc inexpensive solvers prove superior to expensive commercial packages. Customised solution search engines with built-in intelligence and search technology perform better orders of magnitude. The capability to apply the basic B&B functions (branching, pruning) tailored to the structure of the solution space accelerates convergence and reduces computational cost.
546
Fig. 3: The Hardware Composites represent the solution space of the utility network. REFERENCES 1. Fabian, C.I. (1992) LINX: An interactive linear programming library. 2. Forrest, J.J.H., Hirst, J.P.H. and Tomlin, J.A. (1974) Practical solution of the large mixed integer programming problems with Umpire, Management Science, 20(5), 736. 3. Geoffrion, A.M. and Marsten, R.E. (1972) Integer programming algorithms: a framework and state-of-the-art survey, Management Science, 18(9), 465. 4. Mavromatis, S.P., and Kokossis, A.C. (1998). Hardware Composites: a new conceptual tool for the analysis and optimisation of steam turbine networks in chemical process industries Parts I & II, Chemical Engineering Science, 53(7), 1405. 5. Raman, R. and Grossmann, I.E. (1992) Integration of logic and heuristic knowledge in MINLP optimisation for process synthesis, Computers and Chemical Engineering, 16(3), 155. 6. Raman, R. and Grossmann, I.E. (1993) Symbolic Integration of logic in Mixed-Integer Linear Programming Techniques for process synthesis, Computers and Chemical Engineering, 17(9), 909. 7. Raman, R. and Grossmann, I.E. (1994) Modelling and computational techniques for logic based integer programming, Computers and Chemical Engineering, 18(7), 563. 8. Strouvalis, A.M., Mavromatis S.P., and Kokossis, A.C. (1998). Conceptual optimisation of utility networks using hardware and comprehensive hardware composites, Computers and Chemical Engineering, 22, S 175.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
Retrofit Design of Chemical Processing Application to Petrochemical Industry
547
Networks
under
Uncertainties:
Min-ho Suh 1, Ferenc Friedler2, Sunwon Park 1, and Tai-yong Lee 1. 1 Department of Chemical Engineering, Korea Advanced Institute of Science and Technology, 373-1 Kusong-dong, Yusong-gu, Taejon, 305-701, Korea 2 Department of Computer Science, University of Veszprem, Veszprem, Egyetem u. 10, H8200, Hungary Multiscenario retrofit design of petrochemical processing networks is addressed in this paper. The combinatorial framework developed for process network synthesis can be used to resolve the computational complexity of the retrofit design. Retrofit design of Korean petrochemical industries under product demand uncertainty illustrates the efficacy of the proposed algorithm. We obtain Pareto optimal solutions of two objectives, namely expected cost and worst-case cost. The robust optimal solution of retrofit design under uncertainty can be determined among the Pareto optimal solutions. 1. INTRODUCTION Retrofit design means addition of new units and expansion of existing units to satisfy the economic needs and product demand requirements. In retrofit design of a chemical processing network, decisions on structural variables such as process network configuration and capacity expansions, have to be made under forecasted uncertain parameters, e.g., product demand and material cost data. Since these parameters usually highly affect the profitability of the system, uncertainties should be taken into account in the design. The most common way of representing the uncertainties is to specify scenarios of the expected values of the parameters. Based on the scenario-based approach, multiscenario mathematical model can be driven by the stochastic programming framework. In comparison with the deterministic model that doesn't consider the parameter uncertainty, the stochastic model forms a large-size problem due to the scenario-dependent variables and constraints. Need for an efficient solution algorithm is emphasized in design models considering uncertainties. Moreover, we need to solve the model repeatedly to obtain the Pareto optimal solutions which is an important procedure in decision making under uncertainties. Together with process network synthesis for new process design, the retrofit design of chemical processing network has common binary decision variables of unit existence. The combinatorial framework for process network synthesis was proposed by Friedler et al [1-4]. P-graph theory and combinatorial algorithms * Corresponding author: [email protected]
548 are rebuilt to adapt to the multiscenario retrofit design problem. Retrofit design of Korean petrochemical industries illustrates the efficacy of the proposed algorithm and robust design approach. 2. DESIGN UNDER UNCERTAINTIES
In representing the uncertain parameters, the scenario-based approach is one of the most commonly applied methods. Uncertain product demands are represented as the realizable scenarios and their probabilities in this paper. Absolute robustness concept [5] is applied to the robust optimization of retrofit design problem. The absolute robustness means the cost we accept when the worst-case scenario is realized. Consequently, the two objectives of the stochastic programming model are the expected cost and the worst-case cost. Computational aspects of multiscenario retrofit design model are addressed. 2.1 Mathematical model
In this multiscenario mathematical model, the expected cost is considered to be the objective function by constraining the required range of the worst-case cost. The objective is minimizing the expected cost. min EXCOST (1) The expected cost is calculated using p.,, the probability of scenario s.
s.t EXCOST=~p.,C,
(2)
s
Costs of all scenarios are constrained by the required worst-case cost, C w . C w >_C,, Vs
(3)
C,. is the cost of scenario s and calculated as the sum of investment cost, operating cost, material cost, and transportation cost. Indices i, j, p represent processing unit, material, plant, respectively.
i
p
i
p
"
P ~J~JRP
( j,p,p')et
T:jpp. is the transportation amount of material j from plant p to plant p ' in scenario s when (j, p, p ') is the member of the transportation allowance set t(j, p, p'). Mass balances, operating level limitations, and capacity limitations lead to the following constraints: ~" rauW~.,p i
ZraoW,.ip i
~" T.~jpp.+
~ T,jp..p > MIN.~p Vs, j ~ Jp , p
p':(j,p,p')et
p":(j,p",p)et
p':(j,p,p')et
p":(j,p",p)et
ZTdpp,+
ZT,jp,, p
~__-MAX~jpVs, j
~ Jg,p
(5) (6)
549
~-'raoW.~.ip i
~f'T~jpp.+ p':(j,p,p')et
~QT~jp..p> 0 Vs, j ~ JI~,P
(7)
p":(j,p",p)et
Equation (5) means that the amount of product j, to be produced at plant p in scenario s, should be at least MINv p . Equation (6) means that the amount of raw material j, to be used at plant p in scenario s, should be at most MAX vp. ra o is the mass balance coefficient which is negative when the material j is input to the unit i and positive when the material j is output from the unit i. Equation (8) limits the operating level W,~pto the capacity of units Qip.
W~.gp< Q,p Vs, i, p
(8)
Q~pis the sum of original capacity QOip and expansion capacity QEip. Q,p = QO~p + QE~p vi, p
(9)
QEip has lower and upper bounds which linked to the binary variable. QEL~
< QEip < QEUPipY~p Vi, p
(10)
Equations (11) and (12) represent variable conditions. Y,p ~ {0,1}
(11)
OE,p , O,p , W , , , T~jpp >- 0
(12)
2.2 Complexities in solving the multiseenario retrofit design model This multiscenario model can be driven in the form of MILP with scenario-independent structural binary variables related to the process synthesis part of the design and scenariodependent continuous variables related to the uncertainties. The large number of scenariodependent variables and constraints makes the already complex process design problem more difficult to solve. A possible way of reducing the complexity of the problem is to exploit the combinatorial properties of feasible processing networks as done in the combinatorial framework developed for process network synthesis. 3. COMBINATORIAL ALGORITHM The combinatorial framework of process network synthesis has been extended to the multiscenario retrofit design problem by adapting the combinatorial axiom system and altering the search direction on the P-graph from backward to forward direction. The basic algorithms, including the ABB (Accelerated Branch-and-Bound) algorithm have been extended to solve the multiscenario model by keeping its original efficacy.
550
NITRILE \
.
|
t
r
~
v =~
' ==
r
CAPROLACTAM
~TEREP~
PVC
,
HTHALIC
Figure 1. Maximal structure of one plant in the petrochemical processing networks.
3.1 Retrofit design feature There is no difference between the retrofit design and the process network synthesis in representing the new units. Also there is no limitation on representing two or more units of which inputs and outputs are the same in P-graph representation of retrofit design. In the retrofit design, we represent extended capacities of existing units as capacities of additional units, which have the same inputs and outputs with their corresponding existing units. 3.2 Multiplant and transportation consideration We represent the transportation routes of transportable materials as units with transportation cost. All the plants are regarded as a single processing network and the transportation of materials are represented by the above-mentioned method. The investment costs for units of transportation are nulls. 3.3 Forward direction search It is assumed that all the products should be included in the solution networks when we carry out a process network synthesis using the P-graph theory. But in retrofit design of petrochemical processing networks, we determine the profitable products among the candidate products and some of them need not be included in the solution networks. The potential product concept is adopted to represent the situation of product selection in retrofit design. As shown in Figure 1, all the petrochemical products are produced from Naphtha, the main raw material. The product-oriented search algorithm of the original P-graph theory is changed to the raw- material-oriented search algorithm.
551 3.4 Computational aspect from the multiscenario point of view The acceleration of ABB algorithm is attributable to: (i) the reduction of the initial relaxed problem to the set of combinatorially feasible structures; and (ii) the reduction in the sizes of the individual partial problems [4]. The second scheme is very effective when the ABB algorithm is applied to the multiscenario model because the reducing effect in the sizes of partial problems also is proportional to the number of scenarios. 4. RETROFIT DESIGN OF KOREAN PETROCHEMICAL INDUSTRIES
The new algorithm has been tested for Korean petrochemical industries under product demand uncertainty. In this retrofit design problem, various types of petrochemicals are produced from naphtha. The optimal solutions have been generated for the expected cost, the worst-case cost and the Pareto set of the two objectives. 4.1 Problem description There are four petrochemical complexes which are located in Inchon, Ulsan, Yosu, and Daesan in Korea. Each plant has the same maximal structure as shown in Figure 1. Some intermediate materials can be transported from one plant to another with transportation costs. All the plants have their product demands and three demand scenarios are assumed by the forecast for petrochemical product demand of domestic, China and southeastern Asian market. Retrofit period is 10 years. Scenario 1 expects 20% annual growth rate of synthetic resin (HDPE, LDPE, LLDPE, PP, PS, etc.) market and 10% annual growth rate for the rest of the products in the market with probability of 0.3. Scenario 2 expects 15% annual growth of aromatic derivatives (PS, ABS, Caprolactam, TPA, and pthalic anhydride) and 8% annual growth rate for the rest of the products with probability of 0.3. Scenario 3 expects 10% annual growth of all the products with probability of 0.4. Basic petrochemical processing networks configuration, existing capacity and demand, and material price data are imported from Bok et al [6]. 950
770
760
u~
.~_ 800
"~
8
.,... ~ 750
750
Ill "/40 - - e - - Cost of scenario 1 - . o - Cost of scenano 2 Cost of scenario 3
Expected Cost
730 850
0
2
4
6
8
10
12
Absolute Robustness Enhanced
Figure 2. Behavior of robust optimal solutions.
,
~
860
870
880
Worst-case cost (million $)
Figure 3. Pareto curve for decision making.
552 4.2 Results The problem is solved using both the general MILP solver OSL implemented in GAMS 2.25 and the proposed algorithm implemented in C++. Computation times are 942 seconds and 133 seconds, respectively, in minimizing the expected cost without the constraint of worst-case cost requirement. The computation test was carried out on a Pentium-III, 550 MHz. Robust optimal solutions are obtained by constraining the required worst-case cost as shown in Figure 2. Instead of increasing like the costs of scenario 1 and 3, the worst-case cost of scenario 2 decreases as the absolute robustness is enhanced. Figure 3 shows the Pareto curve for the expected cost and the worst-case cost. Decision maker can determine the best design solution to be invested with his or her preference to the worst-case risk and expected cost over the uncertainty.
5. CONCLUSION Combinatorial algorithm for multiscenario retrofit design of petrochemical processing networks was proposed. Retrofit design of Korean petrochemical industries was carried out using the proposed algorithm and robust optimal design method. This industrial scale problem illustrated the efficacy of the proposed method. Often the long-term design problem of chemical processing networks can be modeled as a multiperiod design model. Our future research will be focused on the extension of the algorithm to the multiperiod model with keeping the efficacy of the combinatorial framework. ACKNOWLEDGEMENT This work was partially supported by the Brain Korea 21 Projects and SK Engineering & Construction. This research was also supported in part by the Hungarian Science Foundation Grant No. T-029309. REFERENCES 1. F. Friedler, K. Tarjan, Y. W. Huang, and L. T. Fan, Graph-Theoretic Approach to Process Synthesis: Axioms and Theorems, Chem. Engng Sci., 47 (1992) 1973. 2. F. Friedler, K. Tarjan, Y. W. Huang, and L. T. Fan, Graph-Theoretic Approach to Process Synthesis: Polynomial Algorithm for Maximal Structure Generation, Computers chem. Engng, 17 (1993) 929. 3. F. Friedler, J. B. Varga, and L. T. Fan, Decision-Mapping: A Tool for Consistent and Complete Decisions in Process Synthesis, Chem. Engng Sci., 50 (1995) 1755. 4. F. Friedler, J. B. Varga, E. Feher, and L. T. Fan, Combinatorially Accelerated Branch-and-Bound Method for Solving the MIP Model of Process Network Synthesis, Nonconvex Optimization and Its Applications, State of the Art in Global Optimization, Computational Methods and Applications (Eds: C. A. Floudas and P. M. Pardalos), pp. 609-626, Kluwer Academic Publishers, Dordrecht, 1996. 5. G. Yu. On the max-min 0-1 knapsack problem with robust optimization applications, Oper. Res., 44 (1996) 407. 6. J-K. Bok, H. Lee, and S. Park, Robust investment model for long-range capacity expansion of chemical processing networks under uncertain demand forecast scenarios, Computers chem. Engng, 22 (1998) 1037.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
553
Optimisation of an industrial scale ethanol dehydration plant: A case study Z. Szitkai, Z. Lelkes, E. Rev, Z. Fonyo Chemical Engineering Department, Budapest University of Technology and Economics, H- 1521 Budapest, Hungary An industrial scale hybrid ethanol dehydration system is modelled and optimised using MINLP. The system consists of a distillation column for approaching the ethanol/water azeotrope and a pervaporation unit producing pure ethanol. The optimal design and operating parameters including number of trays, feed location, reflux ratio, number of membrane sections in series and the number of membrane modules in each section are determined. Compared to an existing plant, 12 % savings in the total annual cost can be achived by applying 32 % more membrane surface, in consequence of a radical decrease in the reflux ratio (3.3 to 1.4) in the column, and by producing less concentrated alcohol in the distillate. Sensitivity of the total annual cost to the specified ethanol yield, overall membrane surface and membrane replacement cost is studied. Although our superstructure enables partial permeate recycling, total recycling of the permeate flow proved to be optimal in all the realistic cases. 1. INTRODUCTION Distillation processes are the most widespread for ethanol dehydration, in the industrial practice. Either pressure-swing or extractive or azeotrope distillation is applied (Widagdo and Seider, 1996; Lelkes et al, 1998), high operational costs have to be faced. Pervaporation is an emerging membrane separation technology with the merit of low operational costs. A promising technology for ethanol dehydration is the distillationpervaporation hybrid system, which alloys the advantages of both distillation and pervaporation. In this article the hybrid distillation-pervaporation process is dealt with. In our MINLP formulation both the distillation and the membrane modules are rigorously modelled. Optimization of the pervaporation system is already presented by Srinivas and E1-Halwagi (1993). They used a state space model, optimised by MINLP, but investigated only membrane networks without a distillation column. Viswanathan and Grossmann (1993) optimised the distillation column with rigorous MINLP modelling. These authors did not consider capital and operation costs but optimised for the number of theoretical trays at minimum reflux ratio. Sander and Soukup (1988) experimentally determined the permeate concentration and flux in the function of feed concentration at different temperatures in a pilot plant of ethanol dehydration.
554 2.
PROBLEM STATEMENT
The aim of this article was to re-design an existing hybrid ethanol dehydration plant and investigate the possibilities of cost reduction. For this purpose our previous model for the hybrid distillation-pervaporation system, that can be represented and optimised with GAMS/DICOPT++ is used (Szitkai et al. 2000, Lelkes et al. 2000). The optimization is to be performed over the design and operating parameters including number of trays, feed location, reflux ratio, number of membrane sections in series, and the number of parallel membrane modules in each section of the membrane train. 3. S U P E R S T R U C T U R E AND M O D E L L I N G PRINCIPLES The superstructure and modelling principles applied for the hybrid system are presented in (Lelkes et al. 2000). Here we give only a short overview of the main features. The MINLP model and superstructure of Viswanathan and Grossmann (1993) has been adopted for the distillation column. Most of the column cost functions are taken from Z. Novak et al. (1996). Our membrane network superstructure considers membrane sections in series, where a membrane section consists of numerous 1/3 m 2 flat PVA membranes. In each membrane section the retentate is collected and fed to a heat exchanger for re-heating. The permeate is withdrawn as a product stream and/or recycled to the column feed. Depending on the mathematical representation of the superstructure (how binary variables are used for representing the existence of the 1/3 m E flat PVA membranes), structural multiplicity may occur that hampers the optimization. The notion of structural multiplicity and the usage of binary variables for avoiding its occurrence in this case are dealt with in (Szitkai et al. 2000). The membrane model is based on experimental data. It considers also the effect of temperature drop alongside the flat module. Using the experimentally determined characteristic pervaporation functions of Sander and Soukup (1988), the differential equations of (Neel, 1995) can be numerically integrated for the potential values of feed concentration (c0) and feed flow rate (J0). The result surfaces [J(c0, J0) ; c(co, J0)] are represented by regression in the form o f J v = 0,999. Jo - 0,031- C O and Cv =0,55"C0 "z~ . Here J and c are the retentate flow rate and concentration respectively. In our previous article (Lelkes et al. 2000) various regression and interpolation techniques are examined. Our membrane network costing assumes that the costs of the membrane network are linearly proportional to the overall membrane surface. In case of capital investment the proportionality constant is 1616.1 USD/m 2. Equations for the variable cost calculations are taken from Srinavas et al. (1993), except that the PVA membrane replacement cost was taken 775 USD/m 2 based on industrial practice. Considering 8000 annual operating hours the following utility costs were used: Table 1: Utility costs low pressure steam (160~ 5 bars) cooling water (AT= 10~ electricity permeate condenser cooling medium
135/1000 kg, 6.225 $/GJ 0.16$/GJ 0.06S/kWh 2.645/100 m 3
555 4.
INDUSTRIAL CASE STUDY Using our membrane model based on the experimental data of Sanders and Soukup (1988) the outlet streams and concentrations for the fixed industrial membrane structure and inlet stream were calculated. The calculated and measured flow rates and concentrations are shown in Table 2. The inlet stream is 1000 kg/hr, its concentration is 94 mass% EtOH. Table 2: Comparison of measured and calculated stream data of pervaporation Product flow Product conc. Permeate flow Permeate conc. (k~/hr) (Mass% EtOH) (kg/hr) (Mass% EtOH) Measured 940 99.6-99.7 60 15 Calculated 921.5 99.6 78.5 28 The relatively good correspondence gives rise to the use of the membrane data of Sanders and Soukup (1988) for the re-designing and optimising the existing hybrid ethanol dehydration plant.
4.1. Base case The hybrid ethanol dehydration plant with fixed industrial inlet stream and membrane configuration was first optimised. The results of this optimisation, that can be seen in Figure l, serves as base case for the later comparisons. It is inportant to emphasise that in the base case just the distillation column and the recycle streams were optimally designed; the membrane configuration is kept identical to the configuration of the existing plant. In the base case 97.5% ethanol yield is specified; this means that 97.5% of the total amount of inlet ethanol is withdrawn in form of absolut ethanol in the product stream. 4.2. Optimally designed hybrid system at 97.5% ethanol yield Second, optimal design for all the distillation column, pervaporation membranes and recycle streams were carried out. The result of this design are shown in Figure 2. It can be seen that the total annual cost of the plant is decreased by 12.2 %. This saving is due to the increased overall membrane surface (from the industrial 324 m 2 to 428 m2), that allows to decrease the reflux ratio in the column from 3.262 to 1.38. It is worth mentioning, that the inlet ethanol concentration to the pervaporation system dropped from 94.56 to 91.44 %. 4.3. Sensitivity analysis on overall membrane surface It has been shown that in case of 97.5% ethanol yield the total annual cost of the plant can be decreased by 12.2 % by increasing the overall membrane surface from 324 m 2 to 428 m 2. Some additional designs were carried out between the optimal and the actual plant overall membrane surface. The dependence of the TAC and the reflux ratio on the overall membrane surface is shown in Figure 3. 4.4. Influence of the specified ethanol yield on the TAC In the industrial practice (base case) 97.5% ethanol yield is set up. Depending on environmental regulations or plant conditions, however, greater or smaller ethanol yields could also be required. Because of this possibility, optimisations with fixed 95% and 99%
556 ethanol yields were also carried out. The effect of the specified ethanol yield on the TAC of the hybrid plant is illustrated in Figure 4. The increase in TAC with increasing yield is due to both changing the reflux ratio and overall membrane surface.
4.5. Partial permeate recycling The superstructure is formulated in a way to enable partial recycling of the permeate stream. In spite of this opportunity, total recycling is found in all the cases. On the onter hand, partial recycling is found optimal when the specific membrane investment cost is decreased by appr. 50 %, in case of 95 % alcohol yield. However, this option results in just 1.2% saving in the TAC compared to the optimal design with total permeate recycling. This is due to decreased mass load of the distillation column and some less diluted feed to the membrane subsystem. 4.6. Sensitivity analysis on the membrane replacement cost Optimal designs were carried out with different membrane replacement costs varying from 40% to 120% around the original price (775 USD/m2). According to the results the cost of the membranes in the investigated price interval does not considerably affect the design and operational parameters of the optimal hybrid system. This is because all in these cases the membrane inlet concentrations are in the range of 9.7 to 9.9 mass% water, which is a narrow range. It is interesting to see that this range is situated near the constraint we applied regarding the membrane's toleration of concentrated alcohol vs. its lifetime. Hence the pervaporation unit works at the same conditions, irrespectively to the cost of the membranes. refl3.~6a2io:
--t"?'~
80 ].,992.7~ ~ r theor. I 94.56 mass% stages!
feed 80 mass% EtOH
D--0.875 m
retentate (product): 920.7 kg/hr 99.7 mass % EtOH qmin=97.5~1o/
12 x 81 pieces of 1/3 m 2 fiat membranes =324 m 2 total (fixed industrial configuration) total permeate recycling
/
1175 kg/hr
I
membrane capital investment : 52,362 USD membrane replacement : 83,936 USD column capital investment : 18,05 USD column operational cost : 219,472 USD
recycled permeate ~ bottom product i ] 72 kg/hr 254.3 k g / h r TAC=373,82 USD/yr ...... 28.96 mass% EtOH 0.087 mass% EtOH Figure 1: Base case, optimised hybrid ethanol dehydration plant with fixed industrial inlet stream and membrane configuration
557
re0 i)
retentate (product): 920.7 kg/hr 99.7 mass % E t O H rlmin=97. 5%
12 x 107 pieces of 1/3 m2 flat membranes = 428 m2 total
84 1046.3 kg/hr theor. 91.44 mass% stages
feed 80 mass% EtOH
total permeate recycling
D=0.679 m
1175 kg/hr m e m b r a n e capital i n v e s t m e n t : 6 9 , 0 5 8 U S D membrane replacement : 110,758 U S D c o l u m n capital i n v e s t m e n t : 13,931 U S D : 134,377 USD column operationalcost
I ~
~ 1
recycled permeate ]bottom product 125.6 kg/hr _ [ 254.3 kg/hr 30.86 mass% E t O H v0.087 mass% EtOH
[
TAC=328 ,124 USD/yr
Figure 2: Optimally designed hybrid system at 97.5% ethanol yield 400
...................................................................
3,5
350 ,- 300
\
D cat)
D 250 -ro o
-3
\
- 2,5
o
-2
200150-
-
O
<
1,5
100-
x
[]
membrane capital investment
A
membrane replacement column capital investment column operational cost
+TAC --~> - reflux ratio
50-
)(
0 300
)(
)(
i
i
i
350
400
450
0,5 500
overall membrane surface in square meters
Figure 3: Dependence of the TAC and the reflux ratio on the overall membrane surface
558 400
"C"
............................................................
9plant membrane cost
350
9plant TAC
n 300
• optimised
membrane cost
c"-I O
)K optimised TAC
250
O optimised column cost
o 200 /x
150 100 94
4, plant column cost
O
m
m
m
i
J
i
i
i
95
96
97
98
99
100
specified ethanol yield (%)
Figure 4" Influence of the specified ethanol yield on the TAC optimised system vs. plant existing in the industry 5. CONCLUSIONS An industrial scale hybrid ethanol dehydration system is modelled and optimised using MINLP. The optimal design and operating parameters including number of trays, feed location, reflux ratio, number of membrane sections in series and the number of membrane modules in each section are determined. In our case study 12 % savings in the total annual cost can be achived by applying 32 % more membrane surface, by a radical decrease of the reflux ratio (3.3 to 1.4) in the column, and by producing less concentrated alcohol in the distillate. According to sensitivity analysis, the replacement cost of the membranes does not significantly influence the parameters of the system. In all the realistic cases total recycling of the permeate flow proved to be optimal. REFERENCES
A. Brook, et al: "GAMS. A User's Guide. Release 2.25., boyd & fraser, USA, 1992 J. Gmehling, U. Onken: Vapor-Liquid equilibrium data collection, Vol. I., Part 1, Verlag+Druckerei Friedrich Bischoff, Frankfurt, 1977 Z. Lelkes et al., AIChE J., 44, pp. 810-822, (1998) Z. Lelkes, et al.,Computers and Chemical Engineering 24, pp. 1331-1336, 2000 J. Neel, Membrane Separation Technology. Principles and Applications, Ch.5,Elsevier, 1995 Z. Novak et al. Computers Chem. Engng., 20, pp. 1425-1440, 1996 U. Sander and P. Soukup, Journal of Membrane Science, 36, pp. 63-475, 1988 B. K. Srinivas and M.M. E1-Halwagi., Computers Chem. Engng., 17, pp. 957-970, 1993 V.N.Stabnikov et al. Pishch. Prom. (Kiev) 15, 49 (1972) Z. Szitkai, et al., ESCAPE 10 proceeding, 9 Elsevier Science B.V. S. Widagdo, W.D. Seider, AIChE J., 42, pp. 96-130, 1996 J. Viswanathan and I. E. Grossmann, Computers Chem. Engng., 17, pp. 949-955, 1993
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rightsreserved.
559
Computer Aided Design and Analysis of Separation Processes with Electrolyte Systems Kiyoteru Takano a*, Rafiqul Gani a, Petr Kolar b, Takeshi Ishikawa c aCAPEC, Department of Chemical Engineering, Technical University of Denmark DK-2800, Lyngby, Denmark bMitsubishi Chemical Corporation, Ushiodori 3-1 O, Okayama, Japan CMitsubishi Chemical Corporation, Chiyoda-ku, Tokyo 100, Japan A methodology for computer aided modeling, simulation, design and analysis, based on thermodynamic insights, for separation processes with electrolyte systems has been developed. The methodology consists of three main parts: a thermodynamic calculation part, a flowsheet design/analysis part and a flowsheet simulation part. The thermodynamic part "creates" the problem and system specific property model package, which involves pure component and mixture property models and corresponding model parameters. The flowsheet design/analysis part generates process (flowsheet) alternatives, evaluates/analyzes feasibility of separation and provides a visual operation path for the desired separation. The simulation part consists of a simulation/calculation engine that allows the validation and screening of process alternatives. In this paper, the algorithms for flowsheet design, synthesis and analysis are presented together with an illustrative case study.
1. INTRODUCTION Solution chemistry and solid-liquid (phase) equilibrium play a very important role in design, synthesis and analysis of crystallization-based separation processes involving electrolytes. The solid-liquid equilibrium (SLE)-phase diagrams can be used to identify the feasible operating paths for a desired product from a specified feed mixture. They also help to identify the separation boundaries, the temperature of operation, the list of solids that are most likely to precipitate and many more. For a computer aided system, the reliably generation of phase diagrams is an important first step that requires the use of an appropriate set of property models. Information from the generated SLE-phase diagrams such as, phase boundary data and saturation point data may be used to solve graphically, the mass balance equations related to a crystallization operation. Therefore, since the graphical solution of the mass balance equations is related to the operational paths on the phase diagrams, the temperature dependent phase boundaries and the solubility index (indicates which solid is likely to precipitate first), it is possible to simultaneously (graphically) design, analyze and simulate flowsheets with *On leave from Mitsubishi Chemical Corporation, Yokohama Research Center, 1000, Kamoshida-cho Aoba-ku, Yokohama, 227-8502
560 crystallization operations. The simultaneous graphical solution not only provides a visualization of the process operation, but also, provides very good initial estimates for future simulations with a rigorous model. Obviously, phase diagram based methodologies are limited to binary, ternary or quaternary electrolyte systems unless the dimension (component matrix) of the problem can be reduced. Since chemical and pharmaceutical processes handling electrolyte systems may involve many components, one way to achieve a reduction of the problem dimension is through a sensitivity analysis. That is, identify the key components in the component matrix. For example, when a crystallization process is operated, the product to be recovered is, in many cases, only one component. A higher dimensional problem can then be reduced to a ternary or a quaternary system by highlighting only the regions where the most sensitive components are present. If experimental data on the phase diagrams is available, the reliability of the selected property models can be verified, and if necessary, the most sensitive parameters can be adjusted to fit these data. In the developed computer aided system, the design problem is classified into three types according to what is known information and what needs to be calculated. In problem type 1, the feed compositions and temperature are considered as known while the solid that can be crystallized first, is determined. In problem type 2, the solid to be crystallized and the feed compositions are considered known, and the temperature at which the desired solid will precipitate, is determined. In problem type 3, the feed composition and the solid(s) to be crystallized are given, the operation path(s) needed to precipitate the desired solid(s) is determined. In this paper, only problem type 3 is considered in detail, as solution of this problem also requires the solution of problem types 1 and 2. A case study is used to highlight the use of the developed computer aided methodology.
2. METHODOLOGY In this paper, only the main steps of the algorithm for design of flowsheets with/without recycle using ternary SLE-phase diagrams are presented. The complete set of algorithms for the computer aided methodology can be obtained from the authors [1 ]. As shown in figure 1, the developed algorithm consists of seven main steps. In the first step, the order of recovery, which is one of the important design variables in the flowsheet design problem, is identified. In the steps 2-4, the crystallizer operations needed to obtain the desired solid products are determined together with the corresponding operation temperatures. In steps 5-7, the question of recycle is resolved. The problems solved by this algorithm may be formulated as, given, a) composition of the fresh feed stream and b) number and ID(s) of solid(s) to be recovered, determine, a) operation path to recover n products, b) amount of solids to be recovered, c) amount of solvents to be added/removed, d) composition of the mother liquor from each crystallizer and, e) composition of recycle streams if they exist. The solution of the above problem according to the algorithm of figure 1 also requires a number of sub-algorithms. For example, an algorithm-1 to identify the location of the feed point on the solid-liquid (ternary) phase diagram (step 3a), an algorithm-2 to calculate the maximum and minimum dilution/evaporation ratio (step 3b) and an algorithm-3 to calculate the maximum mixing ratio (step 6).
561 Algorithm-1 identifies the feed point location with reference to the saturation curves and the phase boundaries given on the phase diagram. Figure 2 shows a SLE-phase diagram where B1 and B2 represent the saturation curves, B3 and B4 the phase boundaries, F the feed location and P1 and P2, the limiting values of F that gives a solid product A within the 2phase region. Algorithm-2 calculates the range of dilution (evaporation) ratio necessary in order to make a crystallization operation thermodynamically feasible. Dilution (evaporation) ratio is defined as, Amount of solvent to be added(removed) Dilution (evaporation) ratio Amount of solvent in the feed Algorithm-3 calculates the composition of the mixed stream when a recycle stream is mixed with a fresh feed stream. It is an important design variable since with increase of mixing ratio, the composition of the mixed stream moves away from fresh feed point and therefore, the phase region corresponding to fresh feed location. Mixing ratio =
flowrate of recycle stream flowrate of fresh feed stream
Figure 3 shows the operation path as a result of mixing between stream 4 and feed F. The mixed composition is located at point P1. In this figure, the operational paths indicate that solid A is obtained by crystallization at 283.15 K, giving point 2 as the mother liquor. Evaporation from 2 to 3 brings the feed for the second crystallizer to point 3 from where solid B can be crystallized at 373.15 K, giving point 4 as the mother liquor. Stream 4 is then recycled and mixed with fresh feed F. Point 1 indicates the limiting composition for feed F to remain in the two-phase region. Note that no rigorous simulation or calculation is necessary to perform the simultaneous design and analysis
Figure 2: Temary SLE-phase diagram showing feed location, phase boundaries and saturation curves
Figure 3: Temary SLE-phase diagrams showing the operation paths with mixing of recycle and feed streams
562
System information *IDs of components, composition *Temperature, pressure
~
~
Productinformation *Numberof products (n) *IDs of products
I Step 1) Decide the order of recovery through Sl o r ruleof thumb. T I Step 2) Specify the temperature Ti to recover salt I (i=l,n).
I I"
" ~pnase ' Step 3-a) Compute the so1"1" la- lquia diagram at Ti and identification o the feed location together with ID(s) of solid(s) to be precipitated. YES
..
Step 3-b) Select operation. Dilution/evaporation ,.1
.
9
NO
Step 3-b) Select operation. Change Ti/pH/feed composition by stream mixing
] Step4~ Operate the crystallizer at the condition where temperature is 17.
YES
Step 5) Recycleis implemented.
NO
Step 6) Identify the range of mixing ratio and specify mixing ratio within identified range. I Step 7) Repeat step 4 f~ i=l' n"
I ,
Above operation path is converted to the continuous flowsheet/batch operation.
Figure 1: Algorithm for design of flowsheet using ternary solid-liquid phase diagrams 2.1 Validation of Generated Flowsheets
In the developed methodology, the simulation engine is mainly used for screening of processing alternatives, for verification of design and for producing the operational data needed for analysis. A generalized crystallizer model, which consists of MPS model and a decanter model, and the properties model package are implemented in the integrated computer aided system called ICAS [2], which allows the user to solve the design and analysis problems in an integrated manner. That is, once the property model package is created, it is
563 automatically available for generation of phase diagrams and for the simulation engine. Once the flowsheets are generated, the flowhseet information together with the mass balance information (from the phase diagrams) is automatically transferred to the simulation engine. In addition to the validation of the generated flowsheet, it is also possible to verify the reliability of the generated flowsheet specifications (design). Since the flowsheet design algorithm is based on the phase diagrams, the reliability of the generated flowsheet specifications is affected by the accuracy of the phase diagrams. Sensitivity analysis determines quantitatively how much the phase separation boundaries and invariant points move when the most sensitive property model interaction parameters is changed. Consequently, this sensitivity analysis implicitly helps in the design of experiments.
3. CASE STUDY In this case study, two organic salts, Glycine and Sarcosine, need to be recovered as solid products from a ternary aqueous electrolyte system. The problem description is given in Table 1.
System Products Problem
Table 1. Problem description for case study H20(1)-Glycine(2)-Sarcosine(3) Glycine, Sarcosine Flowsheet design
f Feed composition Feed temperature
H20 40g/hr, Glycine 40g/hr, Sarcosine 20g/hr 393.15 K, latm
Application of steps 1 and 2: Specify the system and problem The problem is to design a thermodynamically feasible flowsheet for the recovery of two organic salts. Here, the algorithm [3] for creation of thermodynamic property package is used. Table 2 gives the thermodynamic information needed to create the property package where the shaded parts indicate the feed components. The system includes chemical species involving 6 ionic species. Since this system is classified as an "aqueous" system, the electrolyte NRTL [4] is selected for the calculations. In principle, any other model with the necessary parameters could also have been selected. For the selected NRTL model, the necessary model parameters have been estimated using available binary solubility data. The next step is to generate the phase diagram and identify the operation paths. In this example, two flowsheets, one without any recycle streams and the other with recycle streams, have been generated considering Glycine as the solid product to be recovered first. The algorithm starts with the generation of a feasible flowsheet without any recycle.
564
Application of step 2: Specification of operation temperatures to recover Glycine The temperature of the fresh feed stream is 393.15 K. Therefore, operating temperatures should be below this value. In this example, 283.15 K is selected to recover Glycine. Application of step 3-a, 3-b: Computation of solid-liquid phase diagram at Ti and identification of the feed location to recover Glycine In Figure 1, the SLE-phase diagram is generated at the condition, where temperature is 283.15 K. First, the composition (0.4, 0.4, 0.2) of the feed stream to crystallizer 1 for recovery of Glycine is located in the two-phase region, where one liquid phase and Glycine coexist at 283.15 K. According to Rule 1 in step 3-a, neither dilution nor evaporation is required. However, to get a higher recovery rate, some of the solvent needs to be evaporated (Rule 2 in step 3-b). The algorithm calculates maximum evaporation ratio needed to keep the shifted feed point in the same region as original feed point as 0.241. Minimum evaporation ratio is zero. In this case, 0.241 is selected. After evaporating the solvent, the composition of shifted feed stream is (0.528, 0.208, 0.264). Application of step 4: Operation of crystallizer at Ti to recover Glycine First, the liquid stream, whose composition is (0.528, 0.208, 0.264), is sent to crystallizer 1 to recover Glycine. From mass balance, the composition of the mother liquor is (0.124, 0.386, 0.490), which is the feed to crystallizer 2 for recovery of Sarcosine. Repeating steps 2 and 3a, 3b, the operation temperature for recovery of Sarcosine is selected as 373.15 K. The saturation curves at 373.15 K are calculated and added to the SLE-phase diagrams. In order to crystallize Sarcosine first, however, the amount of solvent (water) needed adjustment by evaporating some of the solvent (Rule 2 in step 3-a). The algorithm calculated minimum and maximum evaporation ratio to keep the shifted feed point in the region, where only Sarcosine is crystallized. Minimum and maximum values of dilution ratio are 0.10 and 0.17 respectively. In this case, 0.17 is selected. After evaporating the solvent, the composition of shifted feed stream is (0.149, 0.261, 0.590). Extending the line joining the pure solid product (Sarcosine) and the shifted feed to the saturation curve identified the operation path and the mother liquor composition, the exit liquid stream from crystallizer 2, as (0.172, 0.310, 0.528). Application of step 5: Implementation of recycle stream From the information generated through steps 2-4, the continuous flowsheet that recovers first Glycine and then Sarcosine is generated. In the next step, recycle is considered. In this case, mother liquor from crystallizer 2 is mixed with the fresh feed stream. Application of step 6: Identification of maximum mixing ratio In this step, the maximum value of mixing ratio (defined above) was calculated to be
565 0.6786 and minimum value to be 0 (no recycle is considered) and a ratio of 0.2 was selected.
Application of step 7: Final design specification When the mixing ratio is set to 0.2, the composition of the feed stream to crystallizer 1 becomes (0.3544, 0.3802, 0.2653). Consequently, repeating steps 3-a and step 3-b gave the maximum evaporation ratio to be 0.17 (compared to 0.24 without recycle). Selecting the evaporation ratio as 0.17 gave the mixed the feed composition to be (0.427, 0.253, 0.320). Step 4 is also repeated for the two products in order to obtain the final design specifications. The composition of mother liquor from both crystallizer were found to be the same as that identified before recycle was considered. This is because invariant points corresponding to the two operation temperatures, used as target mother liquor compositions, do not change if the temperatures are not changed. Figures 4a and 4b show the operation paths and the generated flowsheet for the crystallization process. An important point to note here are that the operation paths on the phase diagrams could also be used to represent the sequence of batch operations needed to obtain the same products. Therefore, the algorithm is valid for generation of continuous flowsheets as well as sequences of batch operations.
Figure 4a: Operation paths for the recovery of glycine and sarcosine on the generated SLEphase diagram
Figure 4b: Continuous flowsheet corresponding to the operation paths shown in Figure 4a
566 4. CONCLUSION A methodology for computer aided design and analysis of separation processes based on the thermodynamic insights of electrolyte systems has been developed and highlighted through an illustrative case study. The novelty of the methodology is that all the necessary steps (from property model selection and validation) to final flowsheet validation through simulation of the process alternatives is addressed in an integrated and systematic manner using visual (graphical) computer aided tools. The methodology includes a parameter estimation technique based on the sensitivity analysis (not discussed in this paper) helps to solve problems when the number of experimental data is limited and/or number of components is larger than four [3]. The methodology includes a rigorous simulation engine option that is able to simulate the steady state behaviour of the separation processes. The sensitivity analysis in the simulation part also calculates quantitatively how the accuracy of the generated flowsheet specifications is affected by the reliability of the phase diagram, and consequently it helps the design of experiment. The identified feasible operational paths can be used to generate flowsheets for continuous operations as well as the sequences of batch operations to obtain the same products. Current and future work involves further extension of the integrated system in terms of increased application range of the process and property models, process optimization and new industrial applications.
REFERENCES
1. Takano, K., "Computer Aided Design and Analysis of Separation Processes with Electrolyte Systems", PhD Thesis, Technical University of Denmark, Lyngby, Denmark,
(2000). 2. Gani, R., Hytoft, G., Jaksland C., and Jensen, A. K., Computers & Chemical Engineering, 21, (1997) 1135. 3. Takano, K. Gani, R., Ishikawa, T., and Kolar, P., Chem Eng Res Des, 78:(A5), (2000) 763. 4. Chen, C. C., and L. B. Evans, AIChE J., 32, (1986) 1655.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
Characterization and Simulation Separating Azeotropic Mixtures
of the Pervaporation
567
Process
for
M. E. Torres Alvarez, R. F. Martini and M. R. Wolf-Maciel Laboratory of Separation Processes Development (LDPS). Chemical Engineering School. State University of Campinas, (UNICAMP), CP 6066, ZIP CODE 13081-970, Campinas-SP, Brazil. E-mail: [email protected] In this work, the characterization of a pervaporation process for separating the azeotropic mixture ethanol-water was performed based on the solution-diffusion model, with a membrane selective to water. The transport parameters were derived from experimental data found in the literature, a general expression for the separation factor as a function of the permeate component fluxes was used and the variables involved in this process were identified and analyzed. 1. INTRODUCTION Pervaporation is a separation process where a liquid mixture is vaporized through a membrane [ 1]. In pervaporation processes, the mixture to be separated is put in contact with a selective polymeric film and the permeate is removed in the vapor phase, on the opposite face of the membrane, which is condensed later. The vaporization of the permeate components takes place as a consequence of the partial pressure reduction, which happens due to the decrease in the total pressure on the permeate side; this decrease can be achieved by using a vacuum pump, for example. Pervaporation differs from other membrane processes by the change of phase on the permeate side. Another important characteristic of this process is that the permeate flow is relatively low. Thus, the pervaporation process becomes attractive when small amounts of materials are to be removed from the liquid phase. In order to accomplish this, the membrane must present high selectivity in relation to the component to be removed [2]. In this work, the characterization of a pervaporation process for the separation of azeotropic mixtures was performed. Indeed, the intention here is to increase the knowledge of the process as a whole and the complexity of the variable calculations to, later, be able to use this important separation process for other potential applications. The variables involved in this process were identified and analyzed and the general permeation equations were based on the solution-diffusion model; two assumptions were made about the pressure gradient inside the membrane (pm): flat gradient (the pressure was kept constant) and linear gradient according to [3]. With these assumptions, the permeate flux equations are expressed as a function of the feed pressure, P1, the permeate pressure, P2, the activity coefficient, y, the diffusion coefficient, D, the membrane thickness, g, the vapor pressure, Pv, the molar volume, v, the mole fraction, x, and the temperature, T. Simulation analysis was carried out using experimental data of aromatic polyetherimide membrane [4]. The effects of the downstream
568 pressure on the selectivity and on the flow were analyzed for the azeotropic system ethanolwater. 2. MODEL APPLICATION According to the solution-diffusion mechanism, which characterizes the pervaporation process, different mathematical models are proposed to describe the behavior of the permeate flux according to modifications in the variables of the process. Several empirical equations have been developed to calculate the diffusion coefficient used in such mechanism, from simple linear equations to exponential models. Those equations try to describe the relationship between the diffusion coefficient and the concentration, for instance the Fujita's theory of the free volume [5]; however, the transport equations for this theory are complex and the model parameters are quite difficult to be obtained. Moreover, the models are very sensitive to approximations in these measures [6]. In the present work, the mathematical model of the permeation equations assumes that the diffusion coefficient remains constant throughout the membrane. A software, named PERVAZ, is being developed in this work; it is based on the model presented in [3] (shown in this section) and makes use of a user-friendly platform to help the user interaction with the process. The software is written in Visual Fortran 6 language. The permeation equations adopted here consider that the pressure in the membrane varies linearly, i.e.,
Z(p, -v,)
Pm=Pl + 7
The permeate flux is a function of the chemical potential gradient, according to the expression [3,7]: Ji =
D e m x m d~ti
RT
(2)
dg
Assuming that the membrane is in contact with a liquid phase on the feed side and with a vapor phase on the permeate side, the chemical potentials in the membrane interface and in the adjacent phases must be equal. By substituting a thermodynamic expression of the chemical potential gradient in equation (2) and solving it, for the case considered here, the permeation equation becomes [3]: Dcm
L
/
s fviv"-Pl't)
vi PI - P2 - P2x2'-------~iexp R--T vi (P2 P~ ~'iX~'i Pv,i Ji = g'jtm i 1-- exp{ RT )}
(3)
RT
and, for a binary system, the total permeate flux is: J = Ji + Jj
(4)
569 where the diffusion coefficient, D, the activity coefficient in the membrane, ~tm, the concentration inside the membrane, C m, and the molar volume of the permeating component, v, are considered constant and the vapor behaves as an ideal gas. The system is assumed to be isothermal. In equation (3), the term D C m / g 7 m , called transport parameter, must be calculated from experimental data of the permeate flux and of the concentration. This methodology was used in [3], who used experimental data for an ethanol-selective membrane (theoretical values) and for a water-selective membrane (Nation and cellophane membrane), whereas in this work a water-selective membrane (polyetherimide membrane) is being used with different experimental data. Moreover, the objectives in Kataoka's work were to study the total flux in function of feed pressure and to compare pervaporation and reverse osmosis processes. For a permeation system consisting of a binary mixture, the separation factor can be defined through the following expression [8,9]:
C/,ij =
J~ .x],,
(5)
s Jj "xi, 1
Equation (5) makes possible the calculation of the separation factor at the steady state. For the calculation of the separation factors, Kataoka et al. [3] considered the permeate pressure was zero and derived expressions which were function of the transport parameters and activity coefficients, whereas, in this work, the separation factors have been calculated for values of P2 ~ 0, what means that they are functions of the permeate component fluxes and, consequently, of the permeate pressure. 3. SIMULATION AND CHARACTERIZATION Experimental results of the separation of the azeotropic mixture ethanol/water were studied using a polyetherimide membrane [4], a water-selective membrane. The experimental data of the permeate flux versus ethanol composition in the feed were presented in the form of graphs by Huang and Feng. In the present work, these experimental data were used to determine the different values of the component fluxes as a function of the ethanol composition in the feed. Such values were then used to calculate the transport parameters DC m / g7 m, for the components i andj. The resulting values were 2.326 and 9.126 (mol/mZh), respectively, for the model developed in [3]. It is important to mention that, in this work, the experimental data used were never tested with the presented model and, also, that originally the model was developed for other kinds of membranes. The influence of the downstream pressure on the rate of permeation and the effect of temperature on the flux under different downstream pressures, considering the transport parameters constant, were analyzed. Both the experimental and the calculated values of the permeated component fluxes versus the ethanol composition in the feed are presented in Figure 1. The square, sphere and triangle points represent Huang and Feng's experimental data whereas the continuous lines represent the data calculated in this work. It can be observed that the model represents quite well the experimental data for most of the ethanol composition. As the membrane is water-selective, the permeate flux of water is greater than the ethanol flux, except for very high values of ethanol in the feed. The permeate flux of ethanol increases slightly as the composition of
570
ethanol in the feed increases. This corresponds to a decrease in the permeate flux of water and, consequently, of the total flux (water + ethanol). 12 ~
Exp.
Data
This
Work
11 -I
o
A
EtOH
-
EtOH
101.]
[]
t3
Water
,
Water
o
Total
~" 9 !
Total
g
2
1 0
9
0,0
,
,
0,1
,
9
0,2
,
0,3
9
,
0,4
9
,
9
0,5
,
.
0,6
i
9
0,7
,
9
o,s
,
.
0,9
1,0
EtOH in feed (tool%)
Figure 1. Variation of the permeate flux with ethanol concentration in the feed (Pl=101.3kPa, P2 =0.1 kPa)
3.1. Pressure Influence The influence of the feed pressure on the separation of ethanol-water by pervaporation was studied in [3]. When presenting the influence of the permeate pressure (P2) on the process, they plotted the flux and selectivity versus a relative pressure (P2/Pvapor), keeping the temperature and feed composition fixed, whereas in this work, the behaviour of the permeate flux and the separation factor for different values of P2 and feed composition are shown. According to the permeation model, the behaviors of the flux and of the separation factor can be described in relation to the pressure on the permeate side and to the feed composition, as it can be observed in Figures 2 and 3, respectively.
9 8
7
~ 9
~
g ~
Pa
= 0.4
P2=0
6
~
10 9
~ . . . . . p2 = O. 1
P2=O
8
7
1
P=04
1.0
4 3 2 1 o
0,0
,
i
,
i
i
i
,
i
.
i
,
i
,
i
,
i
.
i
,
0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0
EtOH in feed (~
Figure 2. Variation of permeate flux versus composition of ethanol at different pressures.
0
0,0
0,1
0,2
"
"
"
~
0,4 0,5 0,6 0,7 0,8 0,9 1,0 EtOH in feed (%mol)
0,3
Figure 3. Separation factor versus composition of ethanol at different pressure
Figure 2 shows that the total flux (J) decreases as the ethanol composition increases for all permeate pressure values (P2) studied. The total flux can also be reduced by increasing the
571 permeate pressure, as can be seen in Figure 2. The same behavior can be observed for the separation factor, which decreases as the permeate pressure increases (Figure 3). This means that, the lower the pressure, the bigger the separation factor. For high concentrations of ethanol in feed (above Xl=0.8) and as P2 increases, it can be observed that the separation factor curves reach a maximum and drop, which shows that there is a complex relationship between the variables.
3.2. Influence of the temperature In pervaporation processes, due to the membrane nature, the temperature is an important factor. Therefore, a study of the temperature effect on the total permeation flux was carried out, as can be seen in Figure 4. lO 9
~====0-----~ > ~ c ~ a_~. 8 ~ < ~ ~ - . ~ ~-Az~,,
310 15 K ------280 15 K
7 ~,
6
3 2 1 0
0,1
1
10
Figure 4: Effect of temperature on the total permeation flux (XEtOH= 41.5 % mol) Figure 4 shows that the pressure on the permeate side, P2, and the temperature have diverse effects on the system. The effect of the temperature on the total flux at low pressures (P2 next to zero) is minimum. For higher values of P2, however, the flux is greatly affected by the temperature. For instance, for P2 values around l kPa, the total permeate flux increases seven times when the temperature increases from 280.15K to 310.15K, showing that an increase in temperature compensates the reducing effect that P2 has on the total flux. However, membranes can be sensitive to high temperatures, which can cause their degradation in some cases. In this work, the simulations were carried out with temperatures close to the experimental data ones. 4. C O N C L U D I N G R E M A R K S According to the results, it can be concluded that the model behaves exceptionally well for a water-selective membrane, mainly considering that originally it was tested for other kinds of membrane. Also, it is important to consider the analysis related to the feed composition. It can be observed that the total flux decreases increasing the ethanol concentration, however the separation factor increases. This for a water selective membrane. This must be taken into account in the process characterization for choosing a best separating process for a given application. The influence of the pressure P2 on the rate of permeation was analyzed and it was observed that an increase in the pressure value causes a decrease in both the permeation
572 rate and the separation factor. The effect of the temperature on the flux under different downstream pressures was also analyzed, considering the transport parameters constant. This is not what happens in reality and future studies will be carried out to analyze the influence of the temperature on these transport parameters, however, the results shown until now enable us to better understand the variables involved and their influences on the process. NOTATION C = total molar concentration (mol/m 3) D = diffusion coefficient (m2/h) J = permeating flux (mol/m2h) g - membrane thickness (m) P = pressure (kPa) Pv = vapor pressure (kPa) R = gas constant (kPa m~/mol K) T = temperature (K) v = molar volume (m3/mol) x = mole fraction et = separation factor
~t = chemical potential (J/mol) T = activity coefficient
Subscripts i j 1 2
= = = =
ethanol water feed side permeate side
Superscript m = membrane s = membrane side
ACKNOWLEDGEMENTS The authors are grateful to the Coordenag~o de Aperfeigoamento do Pessoal de Nivel Superior (CAPES) and to the Conselho Nacional de Desenvolvimento Cientffico e Tecnol6gico (CNPq) for their financial support.
REFERENCES [1] J. N6el, P. Aptel, R. Clement, Desalination, 53 (1985) 297-326. [2] V.S. Cunha, R. Nobrega, A.C. Habert, 12 ~ Congresso Brasileiro de Engenharia Quimica, 14 a 17 de setembro, Porto Alegre - R.S. (1998). [3] T. Kataoka, T. Tsuru, S. Nakao, S. Kimura, Journal of Chemical Engineering of Japan, 24, 3 (1991) 326-333. [4] R.Y.M. Huang and X. Feng, Separation Science and Technology, 27, 12 (1992) 15831597. [5] H. Fujita, Fortschr. Hochpolym. Forsch., 3 (1961) 1. [6] R. Rautenbach, R. Albrecht, Journal of Membrane Science, 25 (1985) 1-23. [7] H.K. Lonsdale, U. Merten, R.L. Riley, Journal of Applied Polymer Science, 9 (1965) 1341-1362. [8] J-P. Brun, C. Larchet, R. Melet, G. Bulvestre, Journal of Membrane Science, 23 (1985) 257-283. [9] C.H. Lee, Journal of Applied Polymer Science, 19 (1975) 83-85.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
573
A screening method for identifying economic improvement potentials in retrofit design E. Uerdingen, U. Fischer* and K. Hungerbtihler Safety and Environmental Technology Group, ETH Zurich, CH-8092 Switzerland Fax: +41-1-632-1053, E-mail: {uerdi, ufischer, hungerb}@tech.chem.ethz.ch If an existing production plant has to be retrofitted for a given r e a s o n - e.g. increase of production efficiency- an considerable number of retrofit altematives should be evaluated in a often time consuming process. This paper presents a screening method to analyze the production process in order to shorten the evaluation process. Based on an analysis of the flow path pattern, four different performance indicators are used to rate the economic impact of each component per flow path. Results from this method are used to identify improvement potential in the process studied and narrow down the search space for retrofit alternatives. 1. INTRODUCTION The competition on the chemical market has increased during the past decades. Therefore, to be still competitive, most existing production processes need constant improvement through retrofitting. The tasks can be manifold: e.g. increase of production capacity and efficiency, increase of product quality, decrease of emissions to the environment. Usually, the number of technically feasible altematives for a given retrofit task can be tremendous and depends on the complexity of the chemical plant to be retrofitted. Studying all alternatives may thus be impossible or at best impractical. Yet only a limited number of alternatives will be economically attractive due to constraints such as return on investment, capacities of individual unit operations. Therefore, prior identification and targeting of improvement potential is essential. In chemical engineering literature two basic targeting methods have proven their successful application to retrofit design: (a) mass exchange pinch analysis ([1], [2]) and (b) heat exchange pinch analysis ([3], [4], [5]). In this paper a new concept is presented that is based on flow pattern analysis of components in the entire flowsheet. The new concept uses indicators characterizing the degree of non-ideality in a process flow pattern and the resulting economic implications. These indicators are combined to an overall score and used to screen the process flowsheet for improvement potential through retrofitting. 2. M E T H O D O L O G Y The proposed screening method is carried out in two steps as shown in the flowchart diagram in Fig. 1: (i) analysis of component flow paths in the process and (ii) computation of screening scores for each component flow path. *Corresponding author
574
Figure 1: Scheme of screening methodology for identifying retrofit potentials. 2.1.
Flow path analysis (see Fig. 1)
The first step of the screening method consists in mapping the path along which a component flows through the process. This procedure has to be applied to components either introduced or generated in the process. We refer to this as flow path mapping. Usually, a component in the process only serves a single function (e.g. reactant, solvent, by-product, inert). In case that the same component serves multiple functions in the process (e.g. water as by-product and water as solvent for extraction) it is necessary to identify the flow path for every function (e.g. find flow path of water with by-product function and water with solvent function). Also, a component flow path might include one or more recycles. For methodological reasons flow paths with or without recycles have to be treated differently.
2.2.
Screening scores (see Fig. 1)
2.2.1. Flow paths with recycling While recycling is done on purpose to prevent the loss of raw materials (e.g. unconverted reactants, solvents) it will increase utility cost in the process. We introduce four different indicators to assess process improvement potential related to recycles in flow paths. Two of the indicators are intensive parameters while the other two are extensive ones. The first indicator, the recycle accumulation factor, is a measure for the accumulation behavior of a component in a recycle loop. We define it as the ratio of the recycle mass flow rate to the total recycle loop output.
575
--S1--~
Mixer
--S2--~
TT
Reactor
- - $ 3 - - ~ Separation 1 - - S 4 - - ~ Separation 2 m S 5 - - ~
SR1 SR2
Figure 2: Example flowsheet.
It is computed for each set of component and function involved in the recycle. This indicator corresponds to the recycle ratio which relates the total flow rates of the two streams [6]. A high accumulation factor caused by sharp separation indicates high utility costs. The flowsheet example shown in Fig. 2 is used to illustrate the concept. The accumulation factors A F are calculated with Eq. (1) for recycle loop S2-S3-SR1 and Eq. (2) for recycle loop $2-$3$4-SR2, where rhi,k denotes the mass flow rate of component i in stream k:
AFi,sR 1 = rhi,sR1 / rhi,s4
(1)
AF~,sR2 = rh,,sRz/rh,,s5
(2)
The second indicator is simply the mass flow recycled per component and function. Theoretically, an ideal process does not require any recycling. A high mass flow rate is therefore a sign for a non-optimal process. The third indicator allocates utility costs of mass recycled. Again, this applies to each set of component and function in a recycle loop. Hence, the additional utility costs caused by a recycle stream estimates the maximum possible cost savings - not including heat integrationthat can be achieved by retrofitting. The allocation is performed in a simple manner. The calculation is based on the mole flow rate of a component and function in the recycle stream. The mole flow rate is then multiplied with the utility costs per mole in the feed of a unit operation in the recycle loop. Calculations are carried out only for energy-intensive unit operations (e.g. distillation) in the loop and the results are summed up. If a mole flow of a component changes due to reaction, the change in mole flow has to be taken into account. The last indicator is the ratio of utility costs of mass recycled to the economic value of the component and function in a recycle. The specific value is given by multiplication of its market price with the mass flow rate in the recycle stream. If the utility costs for a component in a recycle stream are equal or greater than its value, the utility cost over value ratio is set to a maximum value of 1.
2.2.2.
F l o w p a t h s without recycling
Component flow paths with a single function and without recycles are handled in a similar way. Instead of looking at the recycle stream in a recycle loop, calculations are based on the mass or mole flow rate of the open flow path. As there is no recycling in this kind of flow path an accumulation factor cannot be computed. Otherwise, the same indicators are used as for flow paths with recycling.
2.2.3.
S c r e e n i n g scores
In a first step all indicators, except the utility cost over value ratio, have to be normalized. The mass flow and utility cost indicators are both divided by the corresponding maximum
576 value obtained in the entire analysis. However, accumulation factors in flow paths with recycling are normalized as follows: A p(norm) = "
i,k
(log(AF~,k)_ log(AF/(kmin)))/(log(AF~kmax)),
(3)
log(AF/(kmin)))
,
,
In Eq. (3) AFi, k refers to the accumulation factor of a component and function i in recycle stream k. The superscript "norm" denotes the normalized value, while "min" and "max" refer to the minimum and maximum value obtained in the analysis. The normalization procedure ensures that all four indicators range in an interval from 0 to 1. The final screening score is the unweighted sum of these indicators divided by the number of indicators used. A high screening value indicates a high improvement potential for retrofitting. We propose to search for retrofit projects in the order of priority established by the screening results. 3. CASE STUDY The presented methodology was applied to a process for the production of an important intermediate for the fine chemical industry. The process consists of two consecutive reaction steps. Three recycle loops provide a high degree of reactant recovery. The process flowsheet is shown in Fig. 3.
3.1. Description Reactants R1 and R2 are fed via stream S1 to the mixing tank. The mixed reactant solution is then fed to the first reactor for partial conversion to an intermediate product IP and byproduct water. Additional water is added with stream S 13 to the reactor effluent $3 in order to dilute the catalyst used in the first reactor. Stream $4 is then split in the first distillation column into stream $5 with all unconverted R1, part of unconverted R2, and the intermediate product IP, while water, catalyst and the rest of unconverted R2 leave the column in stream Sll.
I
S13 [~
I[ Tank
II
Reactor
SR2
,l
.......... .... L Reactor II
S8
, r I
tifier I
SR3 ectifierII
Extractor
SRI I Sll Rectifier
III
Figure 3: Process flowsheet of the case study (acronyms are explained in the
S12
,ct~
text).
577 Table 1: Results obtained in the case study for components in flow paths with recycling. Score Mass Acc. Utility Value of Normalized Indicators Flow Factor Cost Recycled Mass Acc. Ut. Ut.Cost/ Recycled Mass Flow Factor Cost Value [kg/h] [-] [kUS$/a] [kUS$/al [-] [-1 [-] [-] [-I 6'700.0 1.5e+0 1'000.0 6'600.0 0.70 0.37 0.44 0.15 0.42 Reactant 1 R1 4'300.0 9.0e-1 600.0 4'200.0 0.45 0.35 0.26 0.14 0.30 2 290.0 2.2e-2 36.0 290.0 0.03 0.15 0.02 0.12 0.08 3 2'200.0 9.0e-1 300.0 2'100.0 0.23 0.35 0.13 0.14 0.21 By-Product 2 3'600.0 8.8e+1 310.0 10'000.0 0.38 0.58 0.14 0.03 0.28 112 Reactant 2 17.0 4.8e-3 2.4 50.0 0.00 0.08 0.00 0.05 0.03 3 IP Intermediate 2 270.0 3.2e+5 14.0 960.0 0.03 1.00 0.01 0.01 0.26 P Product 2 0.20 3.1e-3 1.1 1.0 0.00 0.05 0.00 1.00 0.26 3 0.60 9.2e-3 2.8 2.9 0.00 0.11 0.00 0.98 0.27 HzO By-Product 1 0.67 1.1e-3 150.0 0 0.00 0.00 0.07 1.00 0.27 I1 Inert 2 790.0 1.1e+2 41.0 0 0.08 0.59 0.02 1.00 0.42 12 Inert 2 500.0 1.7e+4 32.0 0 0.05 0.85 0.01 1.00 0.48 13 Inert 1 300.0 2.8e+2 24.0 0 0.03 0.64 0.01 1.00 0.42 Recycle 1" $2-$3-$4-S11-SRI" Recycle 2: $2-$3-$4-$5-$6-$7-SR2; Recycle 3" $4-$5-$6-$7-$8-SR3 Component Function
Recyle
The remaining R2 in stream S 11 is finally separated from water and catalyst in the third distillation column and recycled in stream SR1. More IP from a different plant (stream S14) is added to IP from stream $5 and converted in the second reactor to final product P and R2. Reactants R1 and R2 are then recovered in the second distillation column (stream SR2). Finally, P is purified by extraction with water from stream $9 and exits the process in stream S10. Remaining R1 and R2 are recovered in the aqueous phase and recycled to the first distillation column (stream SR3). Inert components I1, I2 and I3 enter the process in stream S 14 and mainly accumulate in recycle streams SR1 and SR2. 3.2.
Results
Following the proposed screening method, nine component flow paths with single function and recycling as well as two component flow paths with single function and without recycling can be identified. The screening results from the case study are displayed in Tab. 1 (with recycling) and Tab. 2 (without recycling). The results in Tab. 1 show high scores for inert components I 1, I2 and I3 in either recycle 1 or 2. Although all three recycle mass flow rates of inert components are relatively small compared to R1, high accumulation factors indicate that these components are trapped in the process. As a consequence they only generate utility costs without being valuable. The screening method also yields high scores for the flow path of reactant R1 in recycle 1 and 2. This is due to large mass flow rates caused by low conversion and excess concentration of R1 in the first reactor. Utility costs and recycle value are high for R1. The accumulation factor for R1 in recycle 1 is low, which implies a nonsharp separation in the first distillation column. The remaining cases score lower and therefore are only of second priority. There are two flow paths without recycling in the process as shown in Tab. 2. In both paths water is introduced to the process for different purposes and yield high scores. Water stream S 13 is needed to dilute catalyst salts in stream $3 in order to reduce the risk of congestion and corrosion in the first and third distillation column. The amount of water introduced strongly impacts on utility costs without providing any value. A similar explanation can be given for water in the extraction process although costs are inferior as compared to the previous case.
578 Table 2: Results obtained in the case study for components in flow paths without recycling. Component Function
H20
FlowPath
Solvent/dilution S13-$4-SI1-S12 Solvent/extraction $9-SR3-$4-Sll-S12
Mass Flow
Utility Valueof NormalizedIndicators Score Cost Recycled Mass Ut. Ut. Cost/ Mass Flow Cost Value [kg/h] [kUS$/a] [kUS$/a] [-] [-1 [-1 H 9'500.0 2'300.0 0 1.00 1.00 1.00 1.00 3'000.0 710.0 0 0.32 0.31 1.00 0.54
Since the screening method is slightly different for component flow paths with or without recycling, the scores should not be compared on an absolute level. However, the scores can still indicate overall priorities. Summarizing the findings from the screening method, priorities should be assigned to retrofit project ideas in the following order for flow paths with recycling: (1) recycling of inert components in recycles 1 and 2; (2) reactant R1 in recycles 1 and 2. As both component flow path without recycling score high, they should also be given a high priority. 4. DISCUSSION We presented a new method for identifying retrofit potentials in chemical processes. The method presents several advantages: (i)
If process mass and energy balance data is available the screening can be performed in a short period of time.
(ii)
A simulation is not required, but in some cases necessary to create a process mass and energy balance where plant data is unavailable.
We applied the new method to a case study from the fine chemical industry. The method highlighted several possibilities for improvement options. One of these options, a project to reduce the amount of water used for the dilution of salts, was undertaken in the real plant and proved to be successful. We believe that the proposed method can be complementary to mass and heat exchange pinch analysis as a preliminary study to locate non-optimal aspects of a flowsheet. 5. R E F E R E N C E S
[1] [2] [3] [4] [5] [6]
E1-Halwagi M.M., Manousiouthakis V., Synthesis of Mass Exchange Networks, AIChE J., 1989, 35 (8): pp. 1233-1244. Hallale N., Fraser D.M., Retrofit of Mass Exchange Networks Using Pinch Technology, AIChE J., 2000, 46 (10): pp. 2112-2117. Linnhoff B., Hindmarsh E., The Pinch Design Method for Heat-Exchanger Networks, Chem. Eng. Sci., 1983, 38 (5): pp. 745-763. Tjoe T.N., Linnhoff B., Using Pinch Technology for Process Retrofit, Chem. Eng., 1986, 93 (8): pp. 47-60. Ciric A.R., Floudas C.A., A Mixed Integer Nonlinear-Programming Model for Retrofitting Heat-Exchanger Networks, Ind. Eng. Chem. Res., 1990, 29 (2): pp. 239-251. Levenspiel O., Chemical Reaction Engineering, 1999, 3 rd ed., John Wiley&Sons, New York.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
579
MILP Models for the Synthesis of Protein Purification Processes E. Vasquez-Alvarez and J. M. Pinto1 Department of Chemical Engineering, University of Sao Paulo, S~.o Paulo, Brazil 05508-900 The objective of this work is to develop mixed integer linear programming models for the synthesis of protein purification processes. Formulations that are based on convex hull representations are proposed to calculate the minimum number of steps from a set of chromatographic techniques that must achieve a specified purity level and alternatively to maximize purity for a given number of steps. Models are tested in several examples and present time reductions of up to three orders of magnitude when compared to b i g - M models. 1. INTRODUCTION Depending on the degree of complexity of the mixtures that result from bioreactions, several recovery and purification operations may be necessary to isolate the desired product. The most important operations are chromatographic techniques that are critical for therapeutical products such as vaccines and antibiotics that require a very high purity level (98 - 99.9%). One of the main challenges in the synthesis of downstream purification stages is the appropriate selection and sequencing of chromatographic steps. While there is a significant amount of research on the development of expert systems for the synthesis of purification processes, much less has been reported in terms of optimization approaches. Bryant and Rowe [1] present a review on computational techniques based on artificial intelligence methods that extract information from chromatographic analysis data. Examples of expert systems in biochemical engineering are Lienqueo et al. ([2]-[3]). Mathematical programming approaches for process synthesis rely on the representation of algebraic equations with discrete variables. Moreover, these problems may be modeled with disjunctive constraints. Balas [4] shows that every MILP model may be written as a disjunctive program and vice-versa. Turkay and Grossmann [5] propose the convex hull representation for systems of linear disjunctions as opposed to Big M constraints. Grossmann and Lee [6] describe the convex hull of a disjunction that involves nonlinear constraints. The objective of this work is to develop mixed integer programming models for the synthesis of protein purification processes. Models based on the convex hull representation are proposed to calculate the minimum number of steps from a set of candidate chromatographic steps that must achieve a specified purity level. Moreover, models are generated to select the operations and their sequence that maximizes the purity of product. These are tested in several examples with experimental data.
t Author to whom correspondence should be addressed. E-mail: [email protected]. Financial support received from PADCT/CNPq (62.0239/97 QEQ), from VITAE (B- 11487/GB004) and ANTORCHAS (A- 13668/1-9) (Cooperation Programs Argentina - Brasil - Chile).
580
2. PROBLEM DEFINITION Given a mixture of proteins at different concentration levels and a desired product specification in terms of a minimum purity, the problem is to synthesize a high-resolution purification process, which is usually carried out by chromatography. Selection of these purification operations is based on the efficiency of different chromatographic techniques that may be employed to separate the target protein from the contaminant ones. 2.1. Models for the chromatographic operations The proposed model for each chromatographic step is based on approximations of the chromatograms by isosceles triangles and on physicochemical property data for the target protein and also for the major contaminants. Each chromatographic step is able to perform the separation of the protein mixture based on their physicochemical properties, which define the dimensionless retention times (Kdi,l,). These can be defined in general terms as follows: Kdi, p = f (P,,4,)
V i, p
(1)
The mathematical correlations applied for each of the chromatographic techniques used in the synthesis of the protein purification process can be seen in Lienqueo et al. [2] and Alvarez et al. [7]. The deviation factor for each protein p in technique i is defined as follows: DFi,I, = IKdi, dl, - Kdi,pl
V i, p
(2)
The concentration factor is represented by relationships determined from chromatographic peaks that correspond to the desired protein and to a contaminant.
two
oi if 0 < DFip < m 10
(3a)
CFi,p = (1 +0.02)( 0/2 - 2DFiap ' ) o/2
if -~- <- DFi.l,< cYi 10 2
(3b)
CF,,t, = 2(1 +0.02)
if o_~_/ < DF,.I, < (~i
(3c)
if DFi.p >
(3d)
CFip = 1 '
'
( O i --
DF/,p)2
o/2
CFi.p =0.02
2
0 i
The concentration factors (CFi, p) obtained from (3a) to (3d) are introduced in the synthesis problems that will be described in the next sections.
3. MODELS FOR OPTIMAL SELECTION Consider disjunction (4) that represents the contaminant constraints in consecutive chromatographic steps: Wi v mOi,p = CFi,p * moi_l, p moi, p -- moi_l, p
V i, p
(4)
The first term of disjunction (4) indicates selection of technique i and therefore enforces a reduction in the mass of contaminants. On the other hand, if i is not selected the mass of contaminant remains the same as indicated in the second term. 3.1. Convex hull model for the minimization of the total number of steps (MDla)
581 Model (MDla) has the following structure: (5)
MinS= ~, w i i s. t.:
moi,p=CFi,p *mp,l *wi +mp,l ( 1 - w i )
V p, i= 1
(6a)
moi, p = C F i ,p *mo~q, p +mo2_l,p
'v' p, i=2...I
(6b)
2 p - m o l l , p + moi_l, p
'7' p, i=2...I
(6c)
O<_mo]_l, p <_U * w i
'v' p, i=2...I
(6d)
O<_mo2_l,p <_U * ( 1 - w i )
'v' p, i=2...I
(6e)
m~
(7)
mOl,dp >_SP@ ~_~mol, p, p,
wi ~ {0,1}
1 , mo2_l,p > 0 moi,p, moi-l,p
Vi
Vi, p
(8)
Constraints (6) result from the convex hull formulation of disjunction (4). The mass of protein that remains after the first step is indicated by equation (6a). In the following steps, (6b) to (6e) holds. Each term of disjunction (4) is represented with subscripts 1 and 2. Finally, constraint (7) imposes that the specified purity of the protein of interest must be achieved. 3.2. Convex hull model for purity level maximization (MDlb) Model (MDlb) that maximizes the purity level for a fixed number of steps k* and for which the sequence of operations is irrelevant was obtained by analogy with model (MDla). min S = ~
(9)
mol, p
p*@
s. t.:
(10)
ZWi =k* i
contaminant constraints (6) and domain constraints (8) Objective function (9) minimizes the overall mass of contaminants along the chromatographic steps. Constraint (10) indicates that exactly k* are selected among the candidate techniques. Constraints (6) and (8) are the same as the ones from model (MDla). 4. C O N V E X H U L L M O D E L S F O R S E L E C T I O N AND S E Q U E N C I N G Convex hull models were also developed for the selection and sequencing of purification steps. These models rely on the following general disjunction: v
i=I mp,k+ 1 =CFi,pmp, k
v
mp,k+ 1 = 0
V p,
k=2..K-1
(11)
Disjunction (11) contains I+1 elements for each protein p and order k. The first I terms model the selection of step i in order k, while the last term models no step selection. 4.1. Convex hull model for minimizing the number of steps (MD2a) The resulting model that minimizes the number of steps and determines the optimal selection and sequencing is as follows:
582
Min
S=Z
,~, Yi,k = ~ , k * Z k
i k s.t.:
(12)
k
~ , Yi,t +O~k =1 i
k<_K-1
(13a)
~,Yi,t <1
Vi
(13b)
~ , Yi,k+l < ~ Yi,t i i
k<_K-1
(14a)
Z k 32_~ , Yi,k - ~ , Yi,k+l i i
k<_K-1
(14b)
~., Yi,k" >- Z k i
V k, k' < k
(14c)
'v' k, k' > k
(14d)
k
ZYi,k,+Zk i
<1
~ Z k =1
(14e)
k rap,2 = ~ , CFi,p * Yi,1 * mp,1 i
'V'p
(15a)
mp,k+l =~_~CFi, p .m i,1p ,k -k- rap,2 k
V p, k=2...K-1
(15b)
mp,k _ ~_~m i,p,k 1 2 + mp,k i
V p , k=2...K-1
(15c)
mi,p,k -
V i, p, k=2...K- 1
(15d)
2
'7' p, k =2...K-1
(15e)
i
1
mdp, k+l >-- SPdp. ~,mp,,k+l - U (1-Zk)
k
(16)
V i,p,k
(17)
P Yi, k E
{0,1}
V i, k
mt,,k, Zk, m i,p,k, 1 m p2, k , ~ k >- 0
Objective function (12) selects a minimum sequence for a given purity. Constraint (13a) indicates that at most one step i may be chosen in order k. The slack variable a k is activated if no steps are selected in order k. Constraint (13b) imposes that step i is selected at most once in the sequence and (14a) states that steps are assigned in increasing order. Constraints (14b)(14e) define the last step of the sequence, denoted by Zk. Constraint set (15) relates subsequent steps and is derived from disjunction (11). Constraint (16) enforces the purity specification.
4.2. Convex hull model for maximizing product purity (MDzb) Model (MD2b) is proposed for selecting a sequence with the objective of maximizing the purity level. In this case, disjunction (11) may be reformulated by restricting the number of steps to k* and removing the last term. Model (MD2b) is written as: (18)
min S = ~ mt,,k.+l pcdp
s.t.:
~.,Yi,k = 1
k < k*
(19)
V i
(20)
i k*
~.. Yi,k <1 k=l rap,2 = ~., CFi, p * Yi,1 * mp,1 i
V p
(15a)
mp,k+l =~-~CFi,p *mli,p,k
V p, k = 2,... k*
(21a)
i
583
mp,k " - Z m ~ , p , k i
V p, k = 2.... k*
(2 lb)
m~,p, k <_U * Yi,k
V i, p, k = 2.... k*
(21c)
Yi.k ~ {0,1}
V i, k
ml,,k,
m i,p,k 1
, >m 0
V i, k, p
(22)
Objective function (18) minimizes the amount of contaminants in the mixture. Equation (19) indicates that exactly one step i is selected in order k (Yi,k = 1). Moreover, (20) imposes the selection of step i at most once. The mass of contaminants after the first step is obtained with (15a) and (21 a-c), which result from the reformulation of disjunction (11). 5. C O M P U T A T I O N A L RESULTS The software GAMS 2.25 [8] was used to implement the model and its solution method. The proposed models were solved with OSL [9]. The models are solved for three different examples of increasing size. In Example1, taken from Lienqueo et aI. [2], we consider the purification of a mixture containing four proteins: S e r u m f r o m bovine albumin (pl), Ovalbumin (p2), S o y b e a n trypsin inhibitor (p3) and Thaumatin (p4). Their physicochemical properties as well as the initial protein concentration of the mixture are shown in Alvarez et al. [7]. The purity level required for p l is 98%. Overall, there are twelve candidate techniques that are ion exchange at 4.0, 5.0, 6.0, 7.0, and 8.0 pH levels, gel filtration and hydrophobic interaction. The solutions obtained with the four models contain three steps. Moreover, with models (MDIb) and (MD2b) a maximum purity level of 99.8% is reached. By comparing the optimal results of model (MD2a) to the solution obtained by an Expert System (ES) [2] for 94.5% purity we obtain the same sequence and final purity. However, since the required purity for the MILP models is 98%, cation exchange at pH 8.0 is added. In E x a m p l e 2, we consider the purification of p-l,3 glucanase (8.3% concentration) that must be separated from eight contaminants. Physicochemical data for was provided from Lienqueo et al. [3]. The purification requires 94% p-l,3 glucanase and the steps must be synthesized from 22 candidate techniques (the ones from Example 1 + anion and cation exchange at 4.5, 5.5, 6.5, 7.5 and 8.5 pH levels). Results for model (MD2b) are shown in Fig. 1. Note that this mixture is very difficult to purify. Overall, six steps are required and a final maximum purity of 94.8% is attained. The solution provided by ES [3] is limited to a final purity of 70%. The suggested sequence is hydrophobic interaction (32.7% purity) followed by anion exchange at 6.5 pH (70.3%); these steps were obtained by (MD2b).
Fig. 1. Optimal results for (MD2b) in Ex. 2.
Fig. 2. CPU times for (M1) / (MDI) in Ex. 2.
In E x a m p l e 3, it is required to purify Somatotropin to 98.0% in a mixture that contains 13 contaminants. The candidate techniques are the same ones from example 2. Despite being a larger problem, only two steps are necessary to achieve the desired purity level.
~84
Table 1 presents statistical data for the three examples and the four M I L P models presented in this paper. Note that models (MD1) present a smaller n u m b e r of binary variables, since these are defined simply for the selection of steps. Results are also c o m p a r e d to the ones obtained from Alvarez et al. [7] that are based on B i g - M representations (models M1 and M2). Note that there is an increase in the n u m b e r of continuous variables, due to the disaggregating variables. Nevertheless, there is a significant reduction in the n u m b e r of nodes and in C P U time. Figure 2 (in log scale) presents C P U time as a function of the sumber of steps. T h e r e is a reduction of up to three orders of magnitude for (MD~) with respect to (MI). Table 1: S u m m a r y of statistical data for models (MD1) and (MD2) Model Example Binary Cont. Rows N o d e s CPU time Cont.vars vars variables (s)* (%) 1 12 137 182 49 2.1 +179.6 2 22 577 767 187 13.9 +189.9 MDIa 3 22 897 1192 44 18.8 +190.3 1 12 137 182 90 2.1 +179.6 2 22 577 767 87 9.2 +189.9 MDlb 3 22 897 1192 9 6.6 +190.3 1 168 565 808 48 164.6 +1156 2 288 2170 2473 1096 2603 +2070 MD2a 3 288 3375 3728 26 2467.9 +2077 1 36 109 137 106 3.1 +738 2 132 1045 1171 62457 4434.6 +1800 MD2. 3 44 337 380 0 2.6 +1062
Notation a protein property CFi, p c o n c . factor of contaminant p after step i DFiq, deviation factor for protein p in chrom, step i @ desired protein chromatographic technique (i = 1.... /) I order in the sequence (k = 1.... K or k*) k Kdi,p retention time of protein p in chrom, step i mass of protein p after chrom, step i mOi,p rnp,k mass of contaminant p after chrom, step in order k (if k= 1 denotes initial mass) protein (product + contaminants) (p = 1.... P)
ea,p
s SPdp U W, wi
Y,.k Yi, k
Zk ak CYi
Rows (%) -6.2 -3.4 -3.4 -6.2 -3.4 -3.4 -30.7 -40.8 -41.6 -36.0 -42.1 -42.1
CPU time (%) -56.5 -99.8 -73.1 -50.0 -99.8 -91.4 +909 -96.2 +1016.0 -46.3 -42.7 -49.0
value of property a for protein p objective function variable specified purity level of the desired protein dp upper bound on protein mass boolean var. for selecting chrom, technique i binary var. for selecting chrom, technique i boolean var. for chrom, technique i in order k binary var. for chrom, technique i in order k binary var. that indicates if order k is the last slack variable relative to order k width of chromatographic peak in step i
REFERENCES 1. C.H. Bryant and R.C. Rowe, T r a c - Trends in Analytical Chemistry, 17 (1998) 18. 2. M.E. Lienqueo, J.C. Salgado and J.A. Asenjo, Comp. Appl. Biot., Osaka, Japan, (1998) 321. 3. M.E. Lienqueo, J.C. Salgado and J.A. Asenjo, J. Chem. Technol. Biot., 74 (1999) 293. 4. E. Balas, Discrete Applied Mathematics, 89 (1998) 3 - 44. 5. M. Turkay and I.E. Grossmann, Comput. Chem. Eng., 22 (1998) 1229. 6. I. E. G r o s s m a n n and S. Lee, Comput. Chem. Eng., 24 (2000) 2125. 7. E.V. Alvarez, M.E. Lienqueo and J.M. Pinto, E N P R O M E R ' 9 9 , Florianopolis, Brazil, (1999) 607. 8. A. Brooke, D. Kendrick and A. Meeraus, G A M S a user's guide (release 2.25), T h e Scientific Press, San Francisco, 1992. 9. IBM, O S L (Optimization Subroutine Library) guide and reference (release 2), Kingston, NY, 1991.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
585
A CAPD approach for reaction solvent design Y. P. Wang and L. E. K. Achenie* Department of Chemical Engineering, University of Connecticut, Storrs, CT 06269, U.S.A. [email protected] and [email protected], Phone: (860) 486 2756 Fax: (860) 486 2959 This paper develops a systematic framework for designing optimal reaction solvents while simultaneously considering process constraints, environmental constraints, safety, and material compatibility issues. Solvent attributes for extractive fermentation processes (such as biocompatibility, inertness, and phase splitting) are investigated in this paper. The mathematical programming based solvent design framework is successfully solved for ethanol extractive fermentation using an Outer Approximation approach. For the case studies, the proposed design approach not only identifies a solvent for the new application but also finds a new candidate solvent for ethanol extractive fermentation. 1. INTRODUCTION Many bioreactions such as fermentation or enzyme-catalyzed reactions can be inhibited by high product concentration (Bagherpur, 1998, Kollerup and Daugulis, 1991). This phenomenon that can lead to lower conversion and productivity is called "End Product Inhibition (EPI)". However by removing the product in situ with the aid of a Mass Separation Agent (MSA), one can decrease the effect of EPI. Currently, solvents as MSA for fermentation have been successfully employed for producing ethanol (Minier and Goma, 1982), acetone and butanol (Roffler et al. 1988), propionic acid (Gu et al. 1998), and butyric acid (Evans and Wang, 1990). A good solvent for the above applications not only increases the productivity of the reaction, but also reduces the cost. Therefore selecting a suitable solvent for a fermentation reaction is a very important research problem. Daugulis and his colleagues (1991) developed a database approach for identifying candidate MSA solvents for the extractive biocatalysis reaction. Nevertheless their selections are confined to the available solvents in the database and cannot enforce additional criteria for the choice of solvents. In addition, the database approach does not have a systematic way to account for the possibly conflicting design objectives and the interaction of the solvent with the system. The shortcomings in the database approach motivate the need for a systematic approach that augments the database approach but avoids its pitfalls. In this study, we propose a mathematical programming approach combined with ComputerAided Product Design (CAPD) technique to find the optimal solvent for ethanol extractive fermentation process. We first identify the desirable performance attributes of solvents for extractive fermentation and the corresponding mathematical models. Then we present an efficient solution strategy for the resulting MINLP formulations followed by case studies.
586 2. DESIRABLE PROPERTIES OF SOLVENTS IN BIOREACTION An MSA solvent in extractive fermentation or bioreaction should satisfy the following requirements (Cen and Tsao, 1993): it should (a) be highly biocompatible (i.e. not inhibit the bioreaction or cause any deactivation of the biocatalyst); (b) be chemically inert under the reaction conditions; (c) have high separation factor and selectivity for the product; (d) cause phase splitting; (e) have a suitable boiling point range; and (f) be liquid under reaction conditions. Next, we will discuss the quantitative characterization of these solvent attributes. 2.1. Biocompatibility Biocompatibility is a very important property of solvents for microbial or enzyme-catalyzed reactions. If the solvent is not biocompatible, it can be toxic to the cell or microorganisms. Laane et al. (1985) employed the logarithm of the partition coefficient (logP) of the solvent to measure bioactivity. However, there is no accurate correlation between logP and the structure of the molecule. For the current study, we investigate biocompatibility based on LCso data that has been modeled in a group contribution relationship (Gao et al., 1992): L - log(LCso ) = Z Njcej (1) j=l where LC5o represents the concentration causing 50% mortality in fathead minnow (mol/L), Nj the number of group j in the compound, as contribution of group j, and L the total number of groups in the compound. It is not difficult to see that the higher the - l o g (LC5o), the more toxic the compound is. Therefore we would like to choose a compound with: L -I~176 = Z N j a j
2.2. Inertness to reaction Solvents for chemical reactions must be chemically inert under the reaction conditions (Stoye and Freitag, 1998), i.e. the solvent cannot react with the reactant or the product. The solvent's inertness to the reaction can be verified through the chemical equilibrium constant (calculated from the pure component Gibbs energies). We take a general reversible reaction in (3) as an example to illustrate this issue: A+B r
(3)
If the reaction does not occur, the chemical equilibrium constant for the reaction should be very small. The relationship between the Gibbs free energy of the reaction A G and the chemical equilibrium constant K is AG = - R T l n K (4) where AG = AGc - A G A - A G B . We enforce the Gibbs free energy of the reaction to be greater than zero so that K is small enough. The Gibbs free energy for the solvent can be calculated through Joback's group contribution correlation (Joback and Reid, 1987).
58'/
2.3. P h a s e s p l i t t i n g
The main function of solvents in extractive fermentation is to extract the product, therefore the most important property that the solvent should have is to cause phase splitting so that reactant and product can be separated. We simplify the problem by assuming that two phases are formed after the solvent is added to the system. For predicting liquid phase splitting, Swank and Mullins (1986) evaluated various methods such as Gibbs free energy minimization, Henley and Rosen's direct iteration method, and Michelsen's method. Since Henley and Rosen's method can directly calculate the equilibrium compositions, we apply this approach to enforce phase splitting. When phase splitting occurs, the moles of each component in each phase, nl and hi' must satisfy the material balance equations: ni.+ni'=zi
i = l ..... N
(5)
where zi is the overall mole fraction of component i. Also, the phase equilibrium equation must hold, i.e. r;xj =r;'xi' (6) Defining the distribution coefficients Ri as: ,,
R; : r~ = x__~_t,
Y,
(7)
xi
We can solve equation (8) for [3 (here [3 is the fraction of the liquid in phase l) by the NewtonRaphson method. Since the distribution coefficient depends on the composition, we employ a black-box model for phase equilibrium calculations. For extractive fermentation, the reaction equation is needed. If 13 is between 0 and 1, then phase splitting occurs. Otherwise the original phase is stable.
~i
z,(R, -I)
9 fl + ( I -
fl)R~
=0
(8)
In addition to the attributes mentioned above, the structure of the solvent molecule must be feasible. In this paper, we employ the octet rule modeled by Odele and Macchietto (1993) to enforce the feasibility of molecule structure, i.e. Z(2-vj)nj = 2 m (9) /
where nj, vj are the number and valence of group j, and m=l,O, -1 for acyclic, monocyclic and bicyclic groups, respectively. 3. M A T H E M A T I C A L FORMULATION AND SOLUTION STRATEGY Since the CAPD problem deals with finding the optimal compound with desired properties, the structure of the molecule needs to be determined. Using the approach by Sinha and Achenie (1999), Churi and Achenie (1996), and Duvedi and Achenie (1996), we define a binary variable u o. as:
uo _ f l
ifotherwisethe/-thposition in a molecule has structural group j
(1 O)
where i - 1 .... N m ~ (maximum number of positions in a molecule),j-1 .... M (number of available groups in the basis set). To assure that only one group occupies one position in a molecule, we introduce the following constraint: <1, i : 1 ..... Nmax (11)
Zlli] i
588 We employ the group contribution method (Joback and Reid, 1987) for estimating boiling (Tb), melting (Tf) point, LCso, Gibbs free energy (AG), critical volume (Vc), activity coefficient, and the octet rule for pure component solvents. We can cast the group contribution models and the octet rule in terms of the above notation. The corresponding mathematical expressions can be found in Wang and Achenie (2000). Therefore, the general form of the solvent design problem for bioreaction can be formulated as a mixed-integer nonlinear programming (MINLP) (Viswanathan and Grossmann, 1990). Specifically for our solvent design problem, the binary variable corresponds to uu ' and the continuous variable corresponds to composition, flow rate, etc. In this study, we apply the Outer Approximation (OA) algorithm (Viswanathan and Grossmann, 1990) to find the locally optimal solution of the problem. We solve a sequence of mixed integer linear programs (MILP) and nonlinear programs (NLP) until the objective function starts to increase. We employ a SQP (Sequential Quadratic Programming) algorithm to solve the NLP subproblem and LP_SOLVE (ftp.es.ele.tue.nl/pub/lp_solve) to solve the MILP master problem. 4. CASE STUDIES
Example 1" For the current study, we will consider the extractive fermentation of ethanol in a continuous-stirred-tank reactor (CSTR) at T=298K, P= 1atm, as shown in Figure 1. Pure solvent Ds (/h), Fs (L/h)
v
Extractive Phase Fs'(L/h), Ds' (/h), Ps, S~
Solvent+ethanol+water Ps, Ss Feed (substrate) F0(L/h),Do(/h), P0, So, Xo
Solvent+ethanol+water+cell P,S
Raffinate Phase F' (L/h), D'(/h), X,P,S
Figure 1. Extractive Fermentation of Ethanol The objective function for solvent design in this example is to maximize the extraction efficiency (the ratio of the volumetric productivity of the ethanol recovered in the solvent phase and the total volumetric ethanol productivity), while achieving a conversion greater than 0.75. In addition, the chosen solvent should be liquid under the reaction conditions. This is achieved by putting limits on the boiling and melting points. Choosing the basis set GI=[CH3, CH2, CH, OH, CH3COO, CH2COO, CH3CO, CH2CO, CH30, CH20, CHO], we obtain the optimal results as shown in Table 1. The optimal structure contains two-CH3 and six-CH2, which corresponds to n-octane. The experimental data for n-octane, which is a popular solvent (Flick, 1985), is also shown in Table 1. So far, very few solvents have been employed for the ethanol extractive fermentation process because of limitations on solvent performance. For example, a previous screening study has found that dodecanol is fairly biocompatible, but it is not liquid under the reaction conditions (melting point of dodecanol is 26~ Our solvent design framework can avoid this problem and find the optimal one. It is worth mentioning that the choice of the group basis set depends on the intended application and availability of accurate group contribution models for predicting the properties of interest. In this study, we would like to find a solvent, which is a member of one or more of the following classes: hydrocarbon, ester, ketone, alcohol, and ether. If biocompatibility is not an
589 issue, then the groups-C1, -Br, and -Ar can be included in the basis set. Moreover, to avoid solvents that react with ethanol, the acid group - C O O H is excluded. Table,,,!" Solvent design,,,results for Exampl e ,,,1........................... Experimental Data Optimal Result Structure (CH3)2, (CH2)6 CHs-CH2- CH2- CH2- CH2- CH2- CH2-CH3 n-Octane Name 111-65-9 CAS number 114 Molecular Weight 114 399 Boiling Point (K) 382.4 216.2 Melting Point (K) 179.9 -log(LCs0) 3.11 84.0 Conversion (%) 84.0 0.30 Split factor (13) 0.30 86.0 Extraction Efficiency (%) 86.0 Slightly soluble Water immiscibility Hydrocarbon Solvent type Example 2: Let us consider again the process shown in Figure 1. In addition to the solvent performance discussed in section 2, we would like to find a solvent which can make the substrate conversion greater than 0.75 and extraction efficiency greater than 0.80, but has a minimum mass flow rate. Choosing the group basis set GI=[CH3, CH2, CH, OH, CH3COO, CHzCOO, CH3CO, CH2CO, CH30, CH20, CHO], we get the locally optimal results as shown in Table 2. .......Table 2! Solvent d e s i ~ results for Example 2 ....................... Experimental Data.., Optimal Result structure ...... (CH3)3, (CH), (cISi:co0) CH3-CH2COO- CH (CH3)- CH3 Isopropyl propionate Name 637-78-5 CAS number Molecular Weight 116 116 394.5 Boiling Point (K) Melting Point (K) 184.7 1.97 -log(LCs0) Density (g/L) 780.6 Conversion (%) 76.1 76.1 0.91 0.91 Split factor (13) Extraction Efficiency (%) 83.5 83.5 Objective function (g/h) 143.88 Solvent type Ester The optimized structure shown in Table 2 corresponds to the compound isopropyl propionate. Since there is hardly any experimental data available for isopropyl propionate, it appears that this solvent has not been widely used. At any rate our approach predicts that this is a potentially useful solvent. Therefore, we would suggest that an experimental study be carried out to verify the performance of isopropyl propionate and the accuracy of the CAPD model. Overall, the CAPD technique has the potential of saving time and reducing cost for identifying a solvent. ACKNOWLEDGEMENTS The authors sincerely appreciate the useful discussions with Dr. Gani from the Technical University of Denmark.
590 REFERENCES
1. Bagherpour, K. "Simulation, Design and Analysis of Biochemical Processes", Ph.D thesis, DTU, 1998. 2. Kollerup, F.; Daugulis, A. J. Solvent selection strategies for extractive biocatalysis. Biotechnol. Prog. 1991, 7, 116-124. 3. Minner, M.; Goma, G. Ethanol production by extractive fermentation. Biotech. Bioeng. 1982, XXIV, 1565. 4. Roffier, S. R.; Blanch, H. W.; Wilke, C. R. In situ extractive fermentation of acetone and butanol. Biotechnol. Bioengng. 1988, 31, 135-143. 5. Gu, Z.; Glatz, B. A; Glatz, C. E. Propionic acid production by extractive fermentation. I. Solvent considerations. Biotechnol. Bioeng. 1998, 57, 454-4612. 6. Cen, P.; Tsao, G. T. Recent advances in the simultaneous bioreaction and product separation processes. Sep. Technol. 1993, 3, 58. 7. Laane, C.; Boeren, S.; Vos, K. On optimizing organic solvents in multi-liquid phase biocatalysis. Trends Biotechnol. 1985, 2, 251-252. 8. Stoye, D.; Freitag, W. Paints, coatings and solvents. John Wiley & Sons Ltd. West Sussex, England, 1998 9. Joback, K. G.; Reid, R. C. Estimation of pure-component properties from group contributions. Chem. Eng. Comm. 1987, 57, 233. 10. Swank, D. J.; Mullins, J. C. Evaluation of methods for calculating liquid-liquid phase splitting. Fluid Phase Equil. 1986, 30, 101-110. 11. Odele, O.; Machietto, S. Computer Aided Molecular Design: A Novel Method for Optimal Solvent Selection. Fluid Phase Equil. 1993, 82, 47-54. 12. Sinha, M.; Achenie, L.E.K. Design of Environmentally Benign Solvents via Global Optimization, Computers & Chemical Engineering, 1999, 23, 1381-1394. 13. Churi, N.; Achenie, L. E. K. Novel mathematical programming model for computer aided molecular design. Ind. Engng Chem. Res. 1996, 35(10), 3788-3794. 14. Duvedi, A. P.; Achenie, L. E. K. Designing environmentally safe refrigerants using mathematical programming. Chem. Eng. Sci. 1996, 51, 3727-3739. 15. Flick, E. W. Industrial Solvents Handbook (3rd edition). Noyes Data Corporation, NJ. 1985 16. Wang, Y.; Achenie, L.E.K. Computer Aided Solvent DesignlEthanol Extractive Fermentation Case Study. Biotech Prog. (Submitted) 17. Viswanathan, J.; Grossmann, I. E. A combined penalty function and outer approximation method for MINLP optimization. Computers Chem. Eng. 1990, 14(7), 769-782.
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rightsreserved.
591
Automatic synthesis of complex separation sequences with recycles Stanislaw K. Wasylkiewicz and Francisco J. L. Castillo AEA Technology- Hyprotech 800, 707 - 8th Avenue SW, Calgary, Alberta, Canada T2P 1H5 A method for rapid generation of feasible distillation sequences for a continuous separation of azeotropic mixtures has been developed. It allows crossing distillation boundaries by pressure change, by exploiting their curvature or by liquid-liquid separation in decanters. The method automatically identifies suitable recycling options and calculates recycle mass balances. In the paper a few examples of synthesis of separation schemes for homogeneous as well as for heterogeneous azeotropic mixtures are presented, where various methods of crossing distillation boundaries are explored and recycles are automatically calculated. 1. INTRODUCTION A fast and reliable determination of feasible products, which can be attained in individual separators for a given feed composition, is the basic problem in synthesis of schemes for the separation of azeotropic mixtures. There has recently been a considerable activity to solve this problem for temary azeotropic distillation systems, see e.g. [1-4]. For any number of components and constant pressure, the problem can be solved efficiently by a rigorous feasibility test and split generation method developed by Rooks et al. [5]. We have modified the method to include non-isobaric distillation column sequences [6] and decanters [7] and then implemented it in the computer program DISTIL [8]. In our implementation, we start with a thermodynamic model of the mixture, parameter estimation and regression from experimental data [9]. Then all azeotropes are calculated by homotopy continuation method [10, 11 ]. Using residue curves and information about all azeotropes in the mixture, a structure of distillation regions is identified [5]. This information enables us a rapid generation of feasible distillation separation sequences. Our modification of the split generation method allows crossing distillation boundaries by moving them with pressure change, by exploiting their curvature or by liquid-liquid separation in decanters. 2. RECYCLES In the systematic synthesis of the azeotropic separation schemes, first we determine the feasible products, which can be attained in individual separators for given feed compositions. Then we identify suitable recycling options. From the synthesis point of view, there are two types of recycles: 1. Primary recycles detected automatically during the synthesis. 2. Secondary recycles, where streams are introduced in order to shift the total feed composition and by doing this change functionality of a separation unit.
~92
2.1. Primary recycles During synthesis procedure, a primary recycle is detected when the composition of a product stream from the last added separator: 1. is equal to the composition of a feed stream already present in the partially synthesized sequence ff~rst kind of primary recycle); 2. is not equal to the composition of the existing unit feed although if added to the unit as a second feed will only change flow rates but not compositions of the products (second kind
of primary recycle); 3. is not equal to the composition of the existing unit feed but both streams have the same components and addition of the stream to the unit will not move the unit from the original distillation region (third kind of primary recycle). Main goal of the primary recycles is to reduce the number of separation steps. Units that perform identical tasks should be merged by recycling. For primary recycles of the first and second kind the composition of the unit product streams do not change and only flow rates must be recalculated (linear problem). For other recycle types the downstream unit's product compositions usually changes and have to be recalculated. The problem is usually non-linear especially when products on distillation boundaries are involved in the recycle loops. Our algorithm automatically detects primary recycle streams and their destinations, and calculates recycle mass balances.
2.1. Secondary recycles A secondary recycle stream is introduced in order to shift the total feed composition and by doing this change the functionality of a separation unit. 1. In an external secondary recycle an external species is introduced as the separating agent. 2. In an internal secondary recycle a species present in the sequence feed is introduced as the separating agent. It can be a pure component produced somewhere downstream in the sequence or any other intermediate stream. 3. EXAMPLES A few examples of synthesis of separation schemes for homogeneous as well as for heterogeneous azeotropic mixtures where prepared, where crossing distillation boundaries by moving them with pressure change or by liquid-liquid separation in decanters are explored and proper recycles are automatically calculated.
3.1. Generalized pressure swing distillation The extended algorithm for automatic generation of feasible distillation column sequences [12] has been applied to the equimolar feed of the four component mixture of acetone, methanol, methyl formate and methyl acetate at pressures 1 and 10 atm. More than 100 feasible sequences have been generated each of them separating the mixture into pure components. One of them is shown in Fig. 1. There are three distillation regions for each pressure. Some azeotropes are quite pressure sensitive what can make pressure swing distillation economically attractive. Three primary recycles have been detected and automatically calculated.
593
Fig. 1. Sequence of distillation columns for separation of the equimolar mixture of acetone methanol - methyl f o r m a t e - methyl acetate at pressures 1 and 10 atm.
3.2. Heterogeneous azeotropic distillation The heterogeneous azeotropic distillation process is widely used in industry to break binary azeotropes. One of possible separation system configurations is shown in Fig. 2.
Fig. 2. The heterogeneous azeotropic distillation process for I P A - Water separation using DIPE as an entrainer.
594 Isopropyl Alcohol (IPA) has to be recovered from its mixture with Water (8 % mole IPA, 92 % mole Water, stream F on Fig. 2). Diisopropyl Ether (DIPE) has been chosen as an entrainer to brake IPA - Water azeotrope (67.5 % mole IPA, 32.5 % mole Water at 1 atm). Addition DIPE to the binary mixture creates three additional azeotropes (two binary and one ternary, see Fig. 2), although there is a big miscibility gap, which extends through all three distillation regions. This makes heterogeneous azeotropic distillation feasible by crossing distillation boundaries in a decanter. The algorithm for automatic synthesis of the heterogeneous azeotropic distillation sequence with recycles is as follows. 1. Calculate all azeotropes [10, 11] and distillation regions [5] for the whole mixture (original mixture + entrainer). Find in which distillation region is the original feed (RF). 2. Generate feasible splits [12] for the original feed F (Fig. 2). Select one, which produces one of the desired products in the bottoms B 1 and a mixture close to the binary azeotrope, we want to break, in the distillate. 3. Calculate composition of the second desired product from the sequence (B2 in Fig. 2). Find in which distillation region is stream B2 (RB2). 4. Find heterogeneous azeotrope, for which one liquid phase is in distillation region RF and the other in a different distillation region (crossing distillation boundaries). 5. Select stream D2 (distillate from the second column) close to this heterogeneous azeotrope and in the distillation region RB2. Calculate compositions of the streams leaving the decanter (L1 and L2 in Fig. 2). 6. Guess the recycle ratio in the first distillation column. 7. Calculate overall feed composition to the first column (F1). 8. Calculate distillate composition of the first column D1 (on the distillation boundary between distillation regions RF and RB2). 9. Calculate overall feed composition to the second column (F2) and then its bottom composition (B2). 10. If composition B2 calculated in step 9 is not close enough to the composition B2 calculated in step 3, guess a new value for the recycle ratio for the first distillation column and go to step 7. Since the distillate from the first column D1 is on a distillation boundary (usually curved) there is an iteration loop (steps 7 to 10). To converge it quickly we minimize an error function according to the recycle ratio in the first column.
3.3. Generalized heterogeneous azeotropic distillation The heterogeneous azeotropic distillation process used to break binary azeotropes can be generalized for more component mixtures to cross distillation boundaries. Let us consider the ternary mixture of ethanol (30 % mole), propanol (10 % mole) and water (60 % mole), which has to be separated to pure components. In the mixture at 1 atm, there are two binary azeotropes, two distillation regions and one distillation boundary (Fig. 3). In order to achieve the separation objectives we have to cross the distillation boundary. By adding toluene to the original mixture we increased number of azeotropes by five (Fig. 3). There are three distillation regions. In each region the water - ethanol - toluene azeotrope is the unstable node. In region 1, the stable node is propanol, in region 2 - water, and in region 3 toluene. The overall feed is in region 2. Now, in order to achieve the separation objectives we have to cross the distillation boundary between regions 1 and 2 (shaded in Fig. 3). Fortunately, the unstable node is the heterogeneous azeotrope and a decanter can be used to cross the
595 boundary. The algorithm presented in the previous section can be used with minor modifications. The second sentence in step 2 has to be changed to: Select one, which produces one of the desired products in the bottoms (water in Fig. 4) and a mixture close to the distillation boundary, we want to cross, in the distillate.
Fig. 3. Ternary and quaternary systems with marked azeotropes and distillation boundaries. The column and recycle mass balances and the flow-sheet diagram of the sequence is showed in Fig. 4. Notice that the product from the heterogeneous azeotropic distillation sequence (two distillation columns and decanter) is a binary mixture (ethanol 75 % mole, propanol 25 % mole), which can be easily separated in one simple column (Column 4 in Fig. 4). In the separation sequence, two secondary recycles are exploited, one to shift the total feed composition into the mass balance line between the two desired products (Column 2 in Fig. 4), second to change the functionality of a separation unit (Column 1).
Fig. 4. The heterogeneous azeotropic distillation process for separation of ethanol-propanolwater mixture using toluene as an entrainer.
596 4. CONCLUSIONS The heterogeneous azeotropic distillation process used to break binary azeotropes has been generalized to cross distillation boundaries for any number of components. Two secondary recycles are exploited, one to shift the total feed composition into the mass balance line between the two desired products, second to change the functionality of a separation unit. The algorithm is general, for any number of components, and can be applied at any step in the automatic generation of distillation column sequences. An alternative approach to determine which streams should or could be mixed is using a programming problem formulation. This is work in progress. REFERENCES 1. L. Laroche, N. Bekiaris, H.W. Anderson and M. Morari, Ind. Eng. Chem. Res., 31 (1992) 2190. 2. Z.T. Fidkowski, M.F. Doherty and M.F. Malone, AIChE J., 39 (1993) 1303. 3. O.M. Wahnschafft, J.P. Le Rudulier and A. Westerberg, Ind. Eng. Chem. Res., 32 (1993) 1121. 4. S.K. Wasylkiewicz, L.C. Kobylka and F.J.L. Castillo, CHISA'98, paper No. 135, Praha, Czech Republic, August 1998. 5. R.E. Rooks, V. Julka, M.F. Doherty and M.F. Malone, AIChE J., 44 (1998) 1382. 6. S.K. Wasylkiewicz, H.K. Shethna and F.J.L. Castillo, AIChE Spring National Meeting, paper No. 23g, Houston, TX, March 1999. 7. S.K. Wasylkiewicz and F.J.L. Castillo, AIChE Spring National Meeting, paper No. 136b, Atlanta, GA, March 2000. 8. DISTIL v 4.1 software, AEA Technology- Hyprotech, http://www.hyprotech.com/. 9. S.K. Wasylkiewicz, L.C. Kobylka and M.A. Satyro, Chemical Engineering, August (1999) 80. 10. Z.T. Fidkowski, M.F. Malone and M.F. Doherty, Comput. Chem. Eng., 17 (1993) 1141. 11. S.K. Wasylkiewicz, M.F. Doherty and M.F. Malone, Ind. Eng. Chem. Res., 38 (1999) 4901. 12. S.K. Wasylkiewicz, L.C. Kobylka and F.J.L. Castillo, Hungarian J. Ind. Chem., 28 (2000) 41.
European Symposium on Computer Aided Process Engineering - I l R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
597
M o d e l based t e m p e r a t u r e control of an exothermic s e m i - b a t c h reactor M. Aartovaara Department of automation, Kemira Engineering Oy, P.O. Box 330, 00101 Helsinki, Finland A model based temperature controller of an exothermic semi-batch reactor developed for Kemira Fine Chemicals Kokkola plant is presented. The controller is based on generic model control (GMC) introduced by Lee and Sullivan [1] and further developed for reactor temperature control by Cott and Macchietto [2]. Generic model control uses a dynamic model describing reactor-jacket cooling and heating system with reaction heat estimation based on temperature measurements. A simulation model of the reactor cooling and heating system was constructed using Simulink program to study the controller performance. Both simulation and practical results of GMC-based controller are presented.
1. TEMPERATURE CONTROL PROBLEM This paper presents a control strategy for exothermic reactor for Kemira Fine Chemicals Kokkola plant in Finland. Figure 1 illustrates the reactor system including jacket glycol water circulation for heating and cooling.
Fig. 1. Reactor and jacket cooling and heating system.
598 The reactor is used for highly exothermic oxidation that can be characterized by the following set of reaction equations. These three equations are only a simplification, in reality the reactions are much more complex. Reaction 1 Reaction 2 Reaction 3
A + B ~ C C + D ---) E C + E ~ F
(1) (2) (3)
Components A and D are the initial reactants, B is the oxidizer, E is an intermediate product and F is the desired final product. The reactor is operated semi batch-wise; in the beginning an initial mass of reactants is placed in the reactor and oxidizer is fed to reactor continuously during the whole batch time. All the reactions are exothermic and the third one is the most exothermic of all. When the third reaction begins to dominate in the reactor the production rate of heat changes very rapidly and consequently easily causes a temperature runaway. This temperature runaway considerably deteriorated the operability of reactor and running the reactor required constant attention of operator. Temperature runaway has several unwanted consequences in addition to safety risks. The oxidizer feed must be discontinued for safety reasons when the temperature exceeds a certain limit. Feed can be continued again when the temperature has returned to safe operating range. Due to break in oxidizer feed the batch time is prolonged. Also the reaction yield decreases as a result of unwanted side reactions when the temperature is not at optimum value.
2. S I M U L A T I O N M O D E L OF R E A C T O R AND J A C K E T HEAT T R A N S F E R A simulation model of reactor and jacket heat transfer and jacket glycol water circulation was created using Simulink for studying heat transfer conditions and performance of control strategy. Reactor and jacket temperatures and heat flows were derived from energy balances. Equations for temperatures and heat flows give
dTr ~
D
B
Qj,out-t-Qre .-!-FoxCp,ox(Zox-Tr)
dt
m r
dTj,o.___.___~t= Qj,i. - Q,i,o.t dt
(4)
Cp, r (5)
Vi(,O iC p. j
Qj,in - Fjcp,j(Zj,in
- Tj,out )
Qj,out - UA(Tj,o., - Tr )
(6) (7)
As the figure 1 shows the jacket is heated using steam in the heat exchanger and cooled by passing a portion of the glycol water through the cooling machine. The cooling machine and heat exchanger were not modeled, instead it was assumed that the temperature of the flow
599 leaving the cooling machine is constant and not affected by the flow changes and that only the latent heat of steam is transferred into the glycol water in the heat exchanger. The jacket temperature controller is a regular PID-controller that controls the steam and cooling circuit valves. The jacket temperature controller was constructed to the simulation model according to actual controller in use in the plant automation system.
3. G M C C O N T R O L The framework of generic model control (GMC) was introduced by Lee and Sullivan [ 1]. Cott and Macchietto [2] have developed GMC for reactor temperature control. Liu and Macchietto [3] have reported experimental comparisons between GMC and conventional temperature control strategies. Examples of GMC for batch reactor temperature control were encouraging enough to test GMC for our case. GMC controller requires a dynamic process model. The derivative of the controlled variable is solved from the dynamic model and is used in GMC controller algorithm: t
dx = K1 (Xsp _ X) + K 2 I(xv, dt
x)dt
(8)
o
Variable x presents the controlled variable, reactor temperature in this case, and Xsp the desired value of x, K1 and K2 are tuning parameters. The control performance is described by the combination of two objectives. The controlled variable should have such a rate of change (dx/dt) that the process is returning to its desired value (1 st term in eq. (8)) and also that the process has zero offset (2 nd term in eq. (8)). The relation between reactor and jacket temperature is given in equations (4) and (5). Replacing Tr for x and Trsp for Xsp in these equations together with controller equation (8) gives the desired value of jacket temperature.
[
t
Zjou ' _ Z r _[_mrCp, r K1 ,(Tr,s p _Zr)_[_g 2 I(rr,s p __Tr)dt '
UA
o
t
Qre
(9)
UA
This may be used for the jacket temperature set point if no dynamic compensation for jacket dynamics is required. Heat from oxidizer flow is regarded negligible and thus not included into equation (9). The reactor energy balance is used for estimation of reaction heat (eq. (4)). This approach was presented by Jutan and Uppal [4] and also used by Cott and Macchietto [2]. To minimize the problems of unknown parameters, Qre/UA is estimated instead of Qre: Qre mrCp, r , d T _ _ r +Tr_T UA
UA
dt
1,out
(10)
The derivative of the reactor temperature must be estimated from measurements. Since numerical differentiation is very sensitive to measurement errors a low-pass filter and a highorder difference equation was used to remove high-frequency noise. A low-pass filter and
600 filtered measurements of temperature and reactor mass were also used in estimating Qre/UA with equation (10). The equations for the filtered temperature and the derivative of temperature are At T[(k'-'zf(k-l'"}---(z(k)
dTcr ck, dt
-
_j (k-l) )
rE.
(11)
_18,T(k-1)+9,T,~k-2)_2,T,~l,-s)
ll,T~k, =
(12)
6 * At
where subscript f refers to filtered values and "c is the time constant of filter. In discrete form with filtered signals the controller equation (9) becomes
Zv(,~ = Z (k, Jr-
Ec
UA
pr gl (Zr(,~~ _ Zr(k,)..1_K2
o
(Z~(,~~ - z(k, )At
t
Ore
(k) (13)
UA .r
A more rigorous presentation of generic model control and advice how to select the tuning parameters can be found in the articles of Lee and Sullivan [1 ] and Cott and Machietto [2].
4. S I M U L A T I O N RESULTS The actual heat released by the reaction (Qre) was not known accurately since the reaction kinetics and rates were too complex to be modeled and calorimetric experiments were not carried out. But it was known that there is a sudden change in the reaction heat at certain point of the batch time. Therefore, to test the controller performance an additional heat flow describing the heat production from the reaction was added to the reactor energy balance. The simulation was made for a situation that resembles a normal run as closely as possible. The initial mass of the reactor contents was 10 000 kg, the initial reactor and jacket temperatures were 60 ~ and the oxidizer fed at feed rate of 200 kg/h. The heat production rate from the reaction changes sharply when about 3,5 hours have passed (fig. 2.c). The reaction heat estimation tracks this change rather well. The jacket temperature and set point are plotted in figure 2.b., the deviation from the set point is due to slow tuning of jacket temperature controller and physical constraints. The tuning of the jacket temperature controller has to take into account the restriction set by cooling machine capacity. Regardless of the constraints and slowness of jacket controller only a difference of 3 ~ is observed between the reactor temperature and set point (fig. 2.a).
5. P R A C T I C A L RESULTS Since simulation results proved to be promising the controller was implemented to the plant automation system. Practical results are plotted in figures 3.a and 3.b. There appears to be moderate oscillation in the jacket temperature and set point owing to the tuning parameters of GMC. These results are from one of the first runs made with GMC for oxidation reactor
601 110 90 70 5030
70 68 66
9
x,,,,
64
~
j .......
62
60
i i , 0
1
2
3
4
5
6
7
0
8
1 2
3 4
tin~ (h)
6
7
8
tkne (h)
Fig. 2.a Reactor temperature (grey) and set point (black), ~
300
5
Fig. 2.b Jacket temperature (grey) and set point (black), ~
p;",~
200
100
X
0 0
1 2
3 4
5
6 7
8
time (h) Fig. 2.c Estimated heat flow (grey) and added heat flow (black), kW.
68
65_
5545_
66 A
64
m a r x
fv-v-
,.,v--...__.,~
I
62 60 58
\ i
I
i
i
i
i
i
i
0 1 2 3 4 5 6 7 8 9
25 _ 15_ -
~ ~ i
0
i
i
i
i
i
i
i
1 2 3 4 5 6 7 8 9
time (h)
time (h)
Fig. 3.a Reactor temperature (black) and set point (grey), ~
Fig. 3.b Jacket temperature (black) and set point (grey), ~
602
and therefore the tuning is not final. Yet there is only a small and acceptable deviation of the reactor temperature from the set point just when the heat production rate changes (fig. 3.a).
6. C O N C L U T I O N S A temperature controller for highly exothermic reactor based on dynamic model of heat transfer conditions was studied with simulations and finally applied to the real process. Since the implementation, the generic model controller has been continuously in use for oxidation reactors in Kemira Chemicals Kokkola plant. Later on this control strategy was successfully applied to other reactors with similar features. Both in the original and new cases the implementation of GMC to the automation system was rather easy and straightforward. The actual programming of the controller structure required only a reasonable amount of work from the plant automation personnel. In both cases the new temperature control has remarkably facilitated the daily work of operators and improved the economic result in production.
NOMENCLATURE
variables
subscripts
A = heat transfer area, m 2 Cp mass heat capacity, J/(kg~ F = mass flow, kg/s 1 = latent heat of steam, j/kg m = mass, kg Q = heat flow, W T = temperature, ~ U = heat transfer coefficient, W/(m2~ V = volume, m 3 p = density, kg/m 3 = time constant of signal filter, s
c = glycol water through cooling machine circ = glycol water not passing the cooling machine f - filtered signal h - glycol water through heat exchanger in = entering j = jacket or jacket glycol water out = leaving ox = oxidizer r = reactor re - reaction s =steam sp = set point
=
REFERENCES 1. Lee, P.L., Sullivan, G.R., Generic model control, Comput. Chem. Eng. 12 (1988), 573580. 2. Cott, B.J., Macchietto, S., Temperature control of exothermic batch reactors using generic model control, Ind. Eng. Chem. Res. 28 (1989), 1177-1184. 3. Liu, Z.H., Macchietto, S., Model based control of multipurpose batch reactor - an experimental study, Comput. Chem. Eng. 19 (1995), 477-482. 4. Jutan, A., Uppal, A., Combined feedforward-feedback servo control scheme for an exothermic batch reactor, Ind. Eng. Chem. Process Des. Dev. 23 (1984), 597-602.
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rightsreserved.
603
Adaptive General Predictive Controller for a Nonlinear Bioreactor B. Akay, S. Ertung, N. Bursah, H. Hapo~lu and M~ Alpbaz Ankara University, Faculty of Science, Chemical Engineering Department, Tando(gan-Ankara, Turkey, e-mail: [email protected] In this work an adaptive nonlinear control algorithm based on the use of Hammerstein model was applied to keep the bioreactor temperature at the desired value. The sort of control algorithm is known as a non-linear form of generalised predictive control (NLGPC). It is well known that this control system is a flexible linear form of generalised predictive control (GPC) and it deals effectively with a range of non-linear system. Third order Hammerstein model was used to describe the dynamic behaviour of the system. This model gives the polynomial relation between the reactor heat input and inlet temperature. In nonlinear GPC algorithm, the one-step Newton-Raphson root solution method was used and the parameters of Hammerstein model were calculated on-line in every sampling time step. Successful application of NLGPC based on Hammerstein model was realised to control the bioreactor temperature. In addition the performance of NLGPC was compared with GPC algorithm. The feasibility of NLGPC was tested by means of several experimental studies. 1. INTRODUCTION S.cerevisiae (baker's yeast) is widely used in food and bioproducts. It is possible to produce ethanol and commercial quantities of the yeast itself from S.cerevisiae grown under aerobic conditions with the use of molasses or glucose as carbon and energy sources, respectively [1]. Because glucose was used as an energy and carbon source in the growth medium, an exothermic reaction took place as a result of the glucose consumption of the microorganism under aerobic condition and then the specific growth rate of the microorganism was inhibited by the increasing growth medium temperature. To maintain the growth medium temperature at its set point which ensures maximum yeast productivity and quality in the bioreactor, an adaptive non-linear generalised predictive control was employed experimentally. For a large number of processes such a linearising approach is sometimes acceptable, all of relatively small nonlinearities are effectively linearised by the controller. The resulting performance seems to be satisfactory. However, a large number of different types of nonlinearity can occur in practice [2-7] and nonlinear type of control systems are required to control the system variables. In this study, linear and nonlinear predictive control algorithms were used in order to control the bioreactor temperature. Hammerstein model was utilised. Parameters of linear and non-linear models were determined by using Recursive Least Square (RLS) method. The most suitable values of the NLGPC controller parameters were determined. Satisfactory control of growth medium temperature was obtained by utilising NLGPC control. This researchhas been supported by Ankara University,Researchfund Grant No. 982500008.
604 2. N O N L I N E A R MODEL BASED C O N T R O L A L G O R I T H M
The single input- output (SISO) discrete time Hammerstein model was utilised and given below Ay(t) = BAx(t - 1) + Ce(t) Ax(t) = ~b{Au(t)}
(1) (2)
where y(t) , u(t) and e(t)are output, input of system and disturbance at time t, respectively. It is assumed that C equals 1 for simplicity [8]. The plant to be controlled consists of a linear part cascaded with a nonlinear part. In this work an auto regressive integrated moving average with exogenous (ARIMAX) model was used to describe the linear part, whereas the high order polynomial approximates the nonlinear part. This paper deals with the control of nonlinear system using available plant data. In particular, the use of nonlinear polynomial NARIMAX models for nonlinear process control purposes was emphasised. The most general NARIMAX model structure is defined by Billings and Leontaritis [9] and given in Eq (3).
AAy(t) = BlU(t - 1) + Bz (Au(t - 1))a + ....... + B, (Au(t - 1))" + Ce(t)
(3)
Nonlinear model based control minimizes a cost function as follows;
J(N1,N2,Nu) - E{ ~ [y(t + j ) - w(t + j)~ + NZu2[Ax(t + j -1)] 2 }
(4) j=N 1 j=l where w is the set point, A, is the time - varying weights on the change in the input. The quadratic minimization of Eq. (4) now corresponds to a direct problem of linear algebra [8], with Ax(t) = Ax(t- 1)+ (G TG + AI)-' G T(w - f )
u(t) was found from Equation (2) following equation [ 10]
(5)
by using Recursive Newton Raphson methods with
AUn+l(t) = A u n ( t ) _ [~b{Aun(t)}- Ax(/)] ~b'{Au, (t)}
(6)
Where, the subscript n denotes the order of iteration, such that the (n + 1) th iteration is obtained from the n th iteration. In the present work, the steps used in the application of the nonlinear form generalised predictive control algorithm was summarized as follows : Initial data: A and B polynomial degree, N 1 = 1, N 2 - 10, N u = 1, Z, - 0.005 a) b) c) d) e) f)
Measure plant output (time instant t) Update plant model coefficient estimates using Recursive Least Square Method (RLS) Calculate intermediate variable from eq. (5). Using Newton-Raphson method, calculate the manipulated variable u(t) Apply u(t)to the bioreactor as input Wait for new sampling time and return to (a)
605 3. M A T E R I A L A N D M E T H O D S The yeast S.cerevisiae NRRL Y-567 obtained from ARS Culture Collection (Peoria, IL, USA) was used in the present study. The cells were growth in medium containing (in g/L): Glucose (20), yeast extract (6), K2HPO4 (3), (NH4)2SO4 (3.35), NaH2PO4 (3.76), MgSOn.7H20 (0.52), CaCI2.4H20 (0.01) and 0.2 ml of Antifoam A (pH 5). All the equipment and medium were sterilised in autoclave. Air was supplied continuously to the bioreactor by a sparger after passing it through an air rotameter and microbiological filter. In Figure 1, A schematic diagram of the well mixed bioreactor is shown. The cooling jacketed bioreactor has 1.6 L volume. There are an electrical immersed heater, an agitator, thermocouple, pH probe and dissolved oxygen probe. The temperature of the growth medium is controlled by changing the heater power and this power is adjusted by a triac module which is connected to a computer. The analog signals from the measuring elements are amplified to 4-20 mA and converted to 0-10 V pulses by using an A/D converter. A similar way, 0-10 V control input values from the computer are converted to 4-20 mA by using a D/A converter and are sent to a triac module. Therefore, the average power to the heater is proportional to input 4-20 mA input analog signal.
!
................... I~
i-.i
........
................. I
2
I........I A~I .....i
3
o
i
14
No 1 2 3 4 5
-4
I
i
I sl.,
I
5 ~
i........l0 ........t D'AI
i"ql
Equipment
No
Equipment
No
Equipment
DO meter pH meter Preamplifier Thermocouple Computer
6 7 8 9 10
Agitator Rotameter Microbiological filter Immersed heater Triac
11 12 13 14
Pump Air Condenser Cooling water Reservoir
Fig. 1. Experimental System
12
606 4. RESULTS AND DISCUSSION Temperature control systems are an integral part of biochemical processes that regulate both the quality and the rate at which products can be produced. In this work, linear and non-linear generalised predictive controls were utilised in order to control the temperature of a bioreactor which produced S.cerevisiae microorganism. These control strategies were tested under various conditions. Firstly, the GPC was employed in the system when the reaction took place without any additional load effect (Figure 2). Time variation of the model parameters of ARMAX and changes in the controlled and controlling variables with time are depicted in Figure 2 a,b,c,d, respectively. It was shown that GPC control provided a good temperature tracking with small fluctuationsparticularly its ability to vary the parameters of the controller as the dynamics of the process change. To attain offset - free closed loop performance in the face of such load effect, the controller must possess inherent integral action. It is well-known that the linear and nonlinear generalised predictive controllers maintain an integrator as a natural result of its structure involving the system model.
36E 34
~,
30 0
i
y,~
I 100
I
y
I 200
I 300
t 400
i 500
I 600
I 500
I 600
I 500
I 600
.
. 700
(a) ~ .~ ~ ~
40 30 20 10 0
100
200
I 400
300
.
700
(b) .~< ~.
0.5
a ~
oo
~
-o.s
al\
-1.0 -1.5
-0
100
" I 200
I 300
I 400
(c)
"~ ~ o.os to.oo ~ Lo.o5 m
-0.10
0
bo...,
_
b~,, 100
I 200
700 . , .
-I 300
i 400
1 i 500
I 600
700
Sampling number
(d) Fig. 2. GPC control results (Sampling time= 30 sec, t =sampling timex sampling number) a) Controlled variable (y) and set point (Yset) b) Manipulated variable (heat=Byte x 2.17 cal/sec) c) A polynomial parameters (a 1, a 2 ) d) B polynomial parameters (b o , b I )
607 Secondly, nonlinear generalised predictive control was applied to the bioreactor system without any additional load effect and the results were given in Figure 3. The controller should be tested over a considerable period of time as it is only these conditions where changing bioreactor operating conditions occur that the nonlinear generalised predictive controller over linear controller will become apparent. By comparing linear and nonlinear control results, it is concluded that nonlinear control results were more successful in the present work.
~"
34 36!
"~ 32 30
t--
- -0
Y,et ,,,,~ -----v I . . . .
Y'~
~-'--v I 200
100
.
. I 300
.
.
.
I 400
-----_ = ~ I 500
~.,....'-"'= .,.,,.,-"~"-.~.,. I 600 700
(a) ~.,
~'~
12
I 200
4 0
100
I 300
I 400
I 500
I 600
700
(b) ~
20 16
4 0
"~
0
100
200
400
500
600
700
/ 400
1 500
1 600
700
1 400
i 500
t 600
I 700
(c)
~9 [ o.oo ,~ ~
300
a''x_-
-0.40
al "x_.-__
~. ,~ -0.80 "< ca" -1.20 0
1 200
I 100
t 300 (d)
i
m
o.10 0.05
_
~ o.o0 g-o.os -0.10
_
_bo " ~ b~ " x -
0
I 100
I 200
I 300
(e) Sampling
number
Fig. 3. NLGPC control results (Sampling time = 30 sec, t =sampling time x sampling number) a) Measured output y and set point (Yset) b) Manipulated variable (heat=Byte x 2.17 cal/sec) c) Intermediate variable x(t) d) A polynomial parameters (al, a 2 ) e) B polynomial parameters (b0, b 1)
608 OPERATORS AND NOTATIONAL CONVENTIONS A(z -1 )" Monic polynomial in the z-domain representing the poles of the discrete-time system B(z -1 ) : Polynomial in the z-domain representing the zeros of the discrete-time system
Bi
: Unknown parameter evaluated by RLS method
C(z -1 ) : Monic polynomial in z-domain representing the zeros of the process noise : White noise e(t) : Unit matrix of o unspecified dimension 1 : Loss function for the NLGPC algorithm J : Minimum costing horizon in Eg. (4) N1
N2 N.
: Maximum costing horizon in Eq. (4)
t
: Time, sec : Set-point at time t : Manipulated (input) variable at time t : Intermediate variable in Eq. (2) : Output variable at time t : Control weighting : Mathematical expectation : first difference operator
w
u(t) x(t)
y(t) 2 E A
: Control costing horizon in Eq. (4)
REFERENCES
1. Bailey J.E. and Ollis, D. F. (1986) Biochemical engineering fundamentals 2 nd ed. Mc Graw Hill, New York. 2. J.D. Momingred, B.E. Paden, D.E. Seborg and D.A. Mellichamp, Chem. Eng. Science, 47 (1992) 755-762. 3. D.P. Atherton, Nonlinear Control Engineering, Van Nostrand Reinhold, 1975. 4. K. Anbumani, L.M. Patnoik and I.G. Sarma, IEEE Trans., AC-26 (4), (1981), 959-961. 5. M. Agarwall and D.E. Seborg, Automatica, 23(2), (1987), 204-214. 6. D. Nesic, System and Control Letters, 29, (1997), 223-231. 7. G. 0zkan, H. Hapo(glu and M. Alpbaz, Chem. Eng. Comm., 170, (1998), 199-215. 8. Clarke, D.W., Mohtadi, C. and Tufts, P.S. Automatica, 23 (3), (1987), 137-160. 9. Billings S.A. and Leontaritis I.J. Int. J. Systems Sci., 15, (6), (1984), 601-615 10. Q.M. Zhu, K.Warwick, J.L.Douce, IEE Proceedings, 138, (1991), 33-40
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
609
Optimal control of semi-batch reactors N. Aziz and I.M. Mujtaba* Computational Process Engineering Group, Department of Chemical Engineering, University of Bradford, West Yorkshire BD7 1DP, UK email: [email protected], [email protected] Optimal control of an exothermic semi-batch reactor is considered here. The reactor temperature and flow rate of the feed reactant are used as the control variables of the system which are optimised under three modes of operations i.e. maximum conversion, minimum time and maximum profit operations. Optimal control problems for three modes of operations are formulated and Control Vector Parameterisation (CVP) technique is used to pose these problems as Non-linear Programming Problems (NLP) which are solved using an efficient Successive Quadratic Programming (SQP) based optimisation technique. The optimal results of all three modes of operations are compared and the suitability of each type of operation under specified circumstances is discussed. 1. INTRODUCTION Batch and semi-batch reactors are essential unit operations in almost all batch-processing industries. As far as safety and quality are concerned, semi-batch reactors have advantages over the batch reactors. The former has a good temperature control and has the capability of minimising unwanted side reactions by maintaining low concentration of one of the reactants. The dynamic optmisation of batch and semi-batch reactors have received major attention in the past [1-2]. Depending on the downstream process requirement (plant schedule, quality, quantity, etc.), batch and semi-batch reactors can be operated in any of the three modes: fixed batch time operation, fixed conversion operation, maximum profit operation. Since batch and semi-batch reactors are inherently dynamic, the optimal operation using any of these modes will result to dynamic optimisation problems such as maximum conversion problem, minimum time problem and maximum profit problem [3]. For a fixed batch time the maximum conversion problem will maximise the product. In some industries, the quality (conversion) of the product becomes the main constraint or aim to achieve. Therefore, the minimum time problem in this case will achieve the desired product in minimum time and will minimise the operating cost of the operation. If there are no constraints in terms of batch time and conversion, the.maximum profit problem will determine the optimum batch time and conversion while maximising the overall profit. In the past most authors considered either maximum conversion problem or minimum time problem only. Luus and Hennessy [4] studied the maximum conversion problem for three different types of fed-batch reactors. They determined the feed rate profile which will
* Correspondence should be addressed to I.M.Mujtaba. Email: [email protected].
610 maximise the amount of the product. Abel et al. [5] has considered minimum time problem for an industrial semi-batch polymerisation reactor. In this work, all three types of optimal control problems are considered for an exothermic semi-batch reaction system with the reactor temperature and the flow rate of feed reactant as the control variables. A simple reactor model based on the material balances and reaction kinetics is used. It is assumed that the optimum temperature profile is obtainable by controlling external heating/cooling (not shown in the reactor model). Unlike conventional batch reactors, the reactor volume in semi-batch reactors gives an additional constraint (path constraint) in the optimisation problem. The optimal control policy in terms of feed rate and reactor temperature obtained in this work can be used to design the heating/cooling system of the reactor [6]. There are many techniques available in the literature to solve optimal control problems such as Iterative Dynamic Programming (IDP) [7], CVP technique [3,5] and Luus-Jaakola Direct Search Optimisation Procedure [4]. In this work, CVP technique has been used since it is able to solve all the above mentioned types of problem. CVP technique is used to pose the optimal control problems as Non-linear Programming Problems (NLP) which will be solved using an efficient Successive Quadratic Programming (SQP) based optimisation technique [8,9]. 2. OPTIMAL C O N T R O L OF SEMI-BATCH REACTORS Mathematically the optimal control problems mentioned above can be written as: Min or Max T(t), OB (t) f(t, x'(t), x(t), u(t), v_) = 0 (model) s.t. TL < T _
611
(1)
d(CAV)/dt = - 2 k l C A 2 V d(CBV)/dt = k l C A 2 V - k 2 C B V - k3CBV d(CiV)/dt = kjCBV,
k4CB g -
2ksCB2V + OBCBv
(2)
(3)
i = C, D, E and j = 2, 3, 4
d(CvV)/dt = ksCB2V
(4)
dV/dt = t~B
(5)
kj = kj0exp[(-Ej/R)/T], j = 1, 2 ..... 5
(6)
where CA, Ca, Cc, CD, CE and CF are concentration of component A, B, C, D, E and F respectively; kl, k2, k3, k4, and k5 are the rate constant for reaction 1 to 5 respectively. V is the total volume of the reactant and uB is the feed rate of reactant B. All the parameter and constant values used in the model are; ki0=44.4 l.moll.min l , k20=4500.0 min l , k30=2500.0 min 1, k40=200.0minl , k50=400 /.mol-J.minZ, El/R=2500 K, E2/R=5000 K, E3/R=4000 K, E4/R=3800 K, Es/R=4200 K, CAF=2.0 mol/l, CBF=0.6 mol/l 4. RESULTS The reactor temperature is optimised within the bounds: 20 < T < 95~ and the feed rate of reactant B is optimised within the bounds: 0 < T < 100//min. The reactor volume (Vr) is 1.5 m 3 (1500 /) and the initial values of [CcV, CDV, CEV, CvV] are [0.0, 0.0, 0.0, 0.0] respectively. The initial values of (CAV) and (CBV) vary from case to case. It is assumed that the volume of the reactor content, V(t) is not changed during reaction. Therefore at any time V(t) will be linear and will be given by: V(t)-V(0) + oBt
(7)
where V(0) is the initial amount of the reactor contents. Vent condenser
Stir DB
Coolant/ Steam ~ inlet
Coolant/ ~ Steam outlet
Product Fig. 1. Schematic diagram of a jacketed semi-batch.
612 To avoid overflow of the reactor contents, following endpoint constraint V(tf) < Vr will ensure no overflow at any time t. In all cases, 4 time intervals are used within the total batch operation time. In each interval, the temperature, feed rate of reactant B and the length of the interval are optimised. 4.1. Maximum Conversion Problem The objective of this work is to maximise the desired product D while optimising the reactor temperature and feed rate of reactant B profiles within bounds on the temperature and the feed rate respectively. The fixed batch time tf is 60 minutes. A number of cases were run with varying amount of initial charge of B. Table 1 shows the optimal temperature and feed rate of reactant B profiles and the maximum amount of product D achieved.
Table 1 Summary of the results for maximum conversion problem Initial Case charge(mol) t,m3~ Vr Optimum Temperature Profile \ l A B uB (l/min) 79.14 5.99 3.90 4.13 1 1500 0 1.5 Temp'(~ 90.77 39.87 ! 39.80 [ 75.13
2
3
1500
1500
240
360
1.5
1.5
Prod D (mol)
I 581"~
Time (min) 0.0 6.39 11.39 20.35 60.0 uB (l/min) 8.21 10.62 4.62E-3 4.54 Temp'(~ 25.45 82.38 74.32 76.33 ] I ] I
616"37
Time (min) 0.0 5.09 17.30 22.30 60.0 UB (//min) 8.78 8.13 2.87E-2 7.19E-3 Temp'(~ 25.73 81.71 40.36 74.66 ] I ] I
589.83
Time (min) 0.0
6.42
14.25
19.25
60.0
Case 1 is the base case. Cases 2 and 3 show the effect of initial charge of reactant B to the system. Product D increases when some amount of B is charged initially but decreases with excessive amount of initial B. The production of D depends on both the value of k3 and the amount of B. In Case 1 where there is no B available at the beginning stage, the system operates at high temperature at the beginning of the operation to produce B. The product D will only be produced with the existence of B. Due to this reason, the amount of D achieved in Case 1 is lower compared to the amount of D in Cases 2 and 3. However with excessive amount of B at the beginning of the process (Case 3), the system operates at lower temperature for a longer period to favour reaction B --) D but this reduces the conversion of B from A. The overall effect therefore is a low production of D. 4.2. Minimum Time Problem The objective here is to produce the same amount of D as in Table 1 in minimum time (Cases 1 and 2 only). The optimal temperature and feed rate of B profiles and the minimum batch time required to achieve the product are presented in Table 2.
613 Table 2 Summary of the results for minimum time problem Initial Case charge(mol) (m Vr3) Optimum Temperature Profile A B t~B (l/min) 33.53 8.98E-4 1.33E-3 1.34 Temp.(~ 95.00 56.27 20.00 68.39 1 1500 0 1.5
]
I
I
Final time (min) 50.83
I
Time (min) 0.0 21.46 26.46 32.39 50.83 (l/min) 0.415 0.0 0.0 0.0 Temp.(~ 95.00 95.00 20.00 20.00
DB
2
1500
240
1.5
I
I
Time (min) 0.0
I
39.13
48.69
[
58.69
53.69 58.69
It is found that minimum time needed to achieve the specified conversion in Cases 1 and 2 is less than the fixed batch time of Table 1. For Case 1, the time needed to achieve the same product is 15% lower. This was achieved by operating the reactor at higher temperature thus producing more B and requiring lower feed rate of B (compared to Case 1 of Table 1). The minimum time for Case 2 is very similar to the fixed batch time of Case 2 of Table 1 but initial high temperature operation reduces the total amount of external feed of reactant B (by 95%). The study of these two types of problems shows that if there is opportunity, productivity (amount of product/time) can be improved by choosing an appropriate type of operation. Table 3 Summary of the results for maximum profit problem Case
1
2
Optimum Temperature Profile OB (l/min) Temp.(~
51.59 95.00
8.31 78.77
[
0.0 36.01
I
0.0 78.77
~
Prod D (mol)
Max Profit
563.2
13.54
10.74
12.80
12.14
9.46
11.44
Time (min) 0.0 11.39 19.39 27.39 47.96 OB (l/min) 14.48 9.82 7.79 0.0 Temp'(~ 92.75 78.21 78.18 77.44 ] [ ] 1607.1 Time (min) 0.0
11.48
21.48
Profit $/min Max Min Conv Time
31.48 47.09
4.3. Maximum Profit Problem The objective here is to determine the optimum amount of product, batch time, the reactor temperature and feed rate of reactant B profiles which will maximise a given profit function. The profit function used in this study is given below:
P = {DxCo - (A0-A)xCA- (t~BCBF+B0)xCB + CxCc + EXCE + FxCr}/t - (0.95/60)xVr 0"6 (8) where t is the batch time in minute. A0 and B0 are the amount of A and B at the beginning stage. CA, Cm Cc, CD, CE and CF are the prices of A, B, C, D, E and F with numerical values
614 [0.425, 0.675, 0.5, 2.5, 0.5, 0.625] respectively. All values are in $/mol. The last part of the Eqn. 7 denotes the capital cost of the reactor ($/min) based on the reactor volume in litre. Two cases are run with the same initial conditions as in Cases 1 and 2 of Tables 1 and 2 and the results are compared. The optimal temperature and feed rate of B profiles, and the final product D are presented in Table 3. From Table 3, it is found that the maximum profit problem gives the highest profit compared to the profits based on the results of the maximum conversion (Table 1) and minimum time problems (Table 2) although amount of product D, temperature and feed rate profile of I3 are all different. It shows that optimisation of both time and conversion is important to get the maximum profit of any operation, rather than specify one of them. 5. CONCLUSIONS Optimal control of three modes of operations i.e. maximum conversion, minimum time and maximum profit have been studied for an exothermic semi-batch reaction system. The reactor temperature and feeding rate of one of the reactants are used as control variables which are optimised. A simple model based on the material balances and reaction kinetics was used in this study and it is assumed that the obtained optimal control profiles are implementable (therefore detailed modelling of cooling/heating was not considered here). In all cases presented, the maximum profit operation gives the highest overall profit compared to the profits obtained using the results of maximum conversion (Table 1) and minimum time (Table 2) operations. The minimum time operation was found advantages over the maximum conversion operation, in one case it was in terms of productivity while in another case it was in terms of the total amount of reactant (B) fed in semi-continuous mode. However, the choice of a particular type of optimisation problem depends on the overall schedule of the batch reactor within the whole batch and semi-batch plant and on the requirements of the downstream process. ACKNOWLEDGEMENTS The Fellowship support from the Universiti Sains Malaysia is gratefully acknowledged. REFERENCES 1. D. Bonvin, J. Proc. Cont., 8 (1998), 355. 2. D.W. Rippin, Comp.Chem.Engng., 7 (1983), 137. 3. N. Aziz and I.M. Mujtaba, Advances in Process Control 5 Symposium (2-3 September 1998), IChemE. 4. R. Luus and D. Hennessy, IEC. Res, 38 (1999), 1948. 5. O. Abel, A. Helbig, W. Marquardt, H. Zwick and T. Daszkowski, J. Proc. Cont, 10 (2000), 351. 6. N. Aziz M.A.Hussain and I.M. Mujtaba, Comp.Chem.Engng., 24 (2000), 1069. 7. R. Luus, J.Proc. Cont, 4 (1994), 218. 8. K.R. Morison, PhD thesis (1984), University of London 9. C.L. Chen, PhD thesis (1988), University of London. 10. I.M. Mujtaba and M.A. Hussain, Comput. Chem. Engng., 22 (1998), $621.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
615
F e e d b a c k Control Design by L y a p u n o v ' s Direct M e t h o d Anibal M. Blanco, Jos6 L. Figueroa and J. Alberto Bandoni Planta Piloto de Ingenieria Quimica, UNS - CONICET, Camino La Carrindanga, Km. 7 8000 Bahia Blanca, ARGENTINA, fax: +54 291 486 1600, E-mail:[email protected] The purpose of this work is to introduce a systematic technique for feedback control design. Based on Lyapunov's stability theory a Non-linear Programming Problem is formulated in order to obtain an optimal closed loop design in some sense. The proposed technique is applied to the feedback control design of a classic stirred tank reactor. 1.1NTRODUCTION In this work we deal with the so-called first and second fundamental problems in the theory of automatic control, as introduced by Letov (1961). The first fundamental problem, or stability problem, consists of determining the values of the parameters of the controller, which are required to guarantee stability of a steady state point. The second fundamental problem, or control quality problem, deals with the character of the convergence of the motion in terms of response speed. Both aspects will be considered in the control design formulation, by inclusion of Lyapunov's stability conditions. Lyapunov's direct (or second) method, is the most general available tool for assessing stability of non-linear dynamic systems, described by a set of differential equations. It is based on an energetic approach and neither explicit nor numerical solutions of the equations are required. Besides the stability issue, it also admits the consideration of transient response speed in an indirect way. 2. LYAPUNOV'S STABILITY THEORY In the present section, most relevant issues of Lyapunov's Stability Theory are outlined. See, for example, Vidyasagar (1993) for a complete analysis.
2.1. Lyapunov's linearization method Consider the free, autonomous system: aX dt
- f(~),
f(0) = 0
(1) w
where i , represents the deviation state vector. We can write f ( i ) - A i +f~(i), where A=
, and i l(x) is the residual. Then, it can be proved (Vidyasagar, 1993) that 0 is i=0
516 an exponentially stable local equilibrium of (1) if all eigenvalues of A have negative real parts ( if A is a Hurwitz matrix). Moreover, the quadratic form V ( ~ ) - ~ r p 2 is a suitable Lyapunov function of the system and f?(2) - - g r Q ~ + 2 g r p ~ (~), where" A r p + PA =
-Q
(2)
being Q an arbitrarily chosen positive definite symmetric matrix. Lyapunov equation (2) is solved for P, which is also symmetric and positive definite. Moreover, denoting eigenvalues by Xi and choosen r such that 9
II ,(ql < - -~min , (Q)
VxEBr
R" I NI<,'}
(3)
then I;r(~)<0, whenever ~ e B, and ~ ~ 0. Br provides an estimate of the domain of attraction of 0, that is, the region of the state space where asymptotically stable trajectories are generated. 2.2. E s t i m a t i o n o f t r a n s i e n t s
Consider parameter q, defined as r/= min, -
dV(i)/dt V-~ j,
which may be loosely regarded
as the reciprocal of the largest time constant, which is descriptive of the motion over the region of asymptotic stability and therefore it is a figure of merit for the control system. A large value of q indicates that the system returns rapidly to the origin. In particular, for a Lyapunov system (2), it is found (Koppel, 1968) that 77 : ffLmin(p-1Q)
(4)
These results will be applied in the proposed control system design formulation of the next section. 3. CLOSED LOOP DESIGN PROBLEM FORMULATION The basic general problem of (steady state) design, may be posed as a constrained, non linear, optimization problem (NLP)" min ~(y) Y
s.t.
h(y) - 0 g(y)__ 0
(5)
Y e Y = {YlY'-
617 corresponds to minimize Xmax(P),which is also a desirable objective in order to enlarge the estimate of the domain of attraction of the origin, as can be concluded from (3). The (nonlinear) steady state model of the closed loop system and the Lyapunov equation (2), conform the set of equality constraints of (5). Positive definite condition on matrix P is also required. The resultant problem turns to become an eigenvalue optimization, non-linear semi-definite programming problem. Large-scale eigenvalue optimization and linear semi-definite programming problems can be efficiently tackled via interior point methods. Boyd et. al., (1997) describe how control system analysis and synthesis can be performed via linear matrix inequalities. Some of such theory may be extended to nonlinear non-convex semi-definite programming problems (see Lewis and Overton (1996) for a comprehensive review of these subjects). In this work, a different approach is proposed in order to transform the nonlinear semi-definite programming problem into a NLP.
3.1. Objective function The problem of eigenvalue optimization of a symmetric matrix P=[Pij], min 2max( P ( y ) ) , y
may be reformulated in terms of a slack variable z as follows (Ringertz, 1997):
s.t.
min z Y'~ z > 2~(P(y)) i= 1...n
(6)
Since analytic expressions for the eigenvalues of a problem (6) may be approximated as below:
s.t.
matrix are not in general available,
min z Y'~ z > 2~ax(P(y))
(7)
where 2max is an upper bound of the maximum eigenvalue of P. For a real symmetric matrix /
\
(Jennings, 1977): 2ma~(P(y))<_[pi,(y)+~-']pi~(y)ll J ' " t~:!
~k.
.However, this upper bound may be max
very conservative. A further property of eigensystems (Jennings, 1977) establishes that the eigenvalues of a matrix remain unaltered if a row is scaled by f a n d the corresponding column /
I/
9
~/
.
x.
I"-J,J
I"X
I;
c,e
etter,ou ,
max
(closer from above) if we let f to vary within certain ranges. In view of such property, problem (7) may be rewritten as: min z y,z,f ,
(8)
Problem (8) is a non-smooth NLP because of the absolute value function.
518 3.2. Positive definiteness on matrix P
A symmetric matrix P (n,n) is positive definite if and only if each principal submatrix Pk (k,k) (1 < k < n) has a strictly positive determinant (Noble and Daniel, 1989). As a result, a set of n additional inequalities of the form: [Pkl_< e, e > 0
(9)
k=l...n
should be added to (8) in order to ensure positive definiteness of matrix P. The above ideas are applied to the controller design of a continuous stirred tank reactor in the following section. 4. APPLICATION EXAMPLE In a couple of papers (Berger and Perlmutter, 1964 (a, b)), the authors analyze both, open loop and feedback controlled stability of a chemical reactor, based on Lyapunov's direct method. They apply a limited version of Krasovskii's theorem to study the regions of asymptotic stability, in an analytic way. No closed loop performance issues are considered in their contribution. We applied the methodology of section 3 to a somewhat reacher example provided by Devia and Luyben (1978). The system under analysis is a typical continuous stirred tank reactor in which an homogeneous, exothermic, first order, reaction is taking place. The following three states model describes the dynamics of the system:
dt
-
-
CA. o -
TO-
dTj
l-V-fj) (r~,0 dt -
(10)
CA -
T - ~PC-----~e
p V R C , ( 7 ' - Tj )
UAH (T-Tj) -r~)+p~C~v~
(11) (12)
A classic Proportional - Integral feedback control law is applied, pairing T and Fj as measured and manipulated variables respectively: ar dt - T~p - r
(13) K~
Fj = Kc ( T,v - T ) + --~i- ~ + Fj.,v
(14)
The analyzed example is open loop unstable since the corresponding jacobian matrix is not Hurwitz. Closed loop stability can be achieved within certain ranges of the control parameters Kc and zi. As commented above, slack variable z becomes the objective function, as posed in (8). Closed loop reactor model (10)-(14) in its steady state version, together with Lyapunov equation (2) (which provides ten single equations) conform the set of equality constraints of (5). Four additional inequalities, (9), and bounds on the control parameter values complete the formulation. The above formulation was implemented in GAMS modeling language (Brooke
619 et. al., 1996) and standard DNLP option, to cope with the non-smoothness of the model, was used in its solution. Numerical data and optimization results can be found in tables 1-4. The resultant system is closed loop stable since matrix A is Hurwitz for such controller parameters. Moreover, it can be seen that the bound strategy for eigenvalue optimization works satisfactorily since the maximum eigenvalue of matrix P is 44497.56. F CA0 To Tj0
556 0.50 530 530
cu.ft./h mol A/cu. ft. R R
Table 1: Streams data
Cp Cj p pj c~ E )~
0.75 1 50 62.3 7.08e10 30000 -30000
Btu/lb. R Btu/lb. R lb/cu, ft. lb/cu, ft. 1/hr Btu/mol Btu/mol
Table 2: Physical data
AH V Vj T Tsp Tj CA Fjsp
399.7 668 167 600 600 553 0.245 1920.5
sq. ft. cu. Ft. cu. Ft. R R R mol A/cu. ft. cu. ft./h
0
Table 3: Reactor data z Kc "ci
44497.57 -500 1.363
Table 4: Optimization Results
5. DISCUSSION
Most, widely applied control system design techniques resort to linearization. The proposed approach for feedback control system design, admits a non-linear analysis frame, since it is based on the general non-linear Lyapunov's stability theory. Moreover, no dynamic simulation (or explicit solution) of the system is required for transient quality assessment, since dynamic response speed and the size of the estimate of the domain of attraction are optimized in an indirect way through the eigenvalue optimization. Classical, realistic Proportional-Integral feedback control was applied, although any other control scheme with a formulation expressed in terms of state variables could be considered. Process / Control System design could be carried out simultaneously within the same methodology. In this case, an economic type objective function should be considered, and a multi-objective optimization problem between cost and closed loop performance would probably arise. Future work will consider these topics.
REFERENCES
1. 2.
Berger J. S. and D. D. Perlmutter (1964 a); Chemical Reactor Stability by Liapunov's Direct Method; AIChE J.; 10 (2) pp. 233-238. Berger J. S. and D. D. Perlmutter (1964 b); The Effect of Feedback Control on Chemical
620 Reactor Stability; AIChE J.; 10 (2) pp. 238-245. Boyd S., Crusius and A. Hansson (1998); Control Applications of Non-linear Convex Programming; Journal of Process Control, 8 (5-6), pp. 313-324. 4. Brooke A., D. Kendrick and A. Meeraus (1996); GAMS Release 2.25 5. Devia N. and W. L. Luyben (1978); Hydrocarbon Processing; June 1978; pp. 119-122. 6. Jennings A. (1978); Matrix Computation for Engineers and Scientists; John Wiley & Sons. 7. Koppell L. B. (1968); Introduction to Control Theory with Applications to Process Control; Prentice Hall. 8. Letov A. M. (1961); Stability in Non-linear Control Systems; Princeton University Press. 9. Lewis A. S. and M. L. Overton (1996); Eigenvalue Optimization; Acta Numerica (1996); pp. 149-190; Cambridge University Press. 10. Noble B. and J. W. Daniel (1989); Applied Linear Algebra; Prentice-Hall. 11. Ringertz U. T. (1997); Eigenvalues in Optimum Structural Design, in Proceedings of an IMA Workshop on Large-Scale Optimization (A. R. Conn, L. T. Biegler, T. F. Coleman and F. Santosa, eds.) Part I; pp. 135-149. 12. Vidyasagar M. (1993); Non-linear Systems Analysis; Prentice-Hall. 3.
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rightsreserved.
621
Batch to Batch Improving Control of Yeast Fermentation D. Bonn6 a* and S. Bay JCrgensen a~ aDepartment of Chemical Engineering, Technical University of Denmark, DK-2800 Lyngby, Denmark An extended framework for modeling and control of nonlinear batch processes by means of simple linear models is proposed. The framework is extended to include a batch-to-batch dynamic description of initial conditions and their effect on batch evolution. In addition, the framework is extended with measurement noise. It is shown that based on desired model properties, regularization can be applied to obtain models applicable for control from sparse and noisy data. The potential of the proposed framework is demonstrated on simulated fed-batch yeast fermentations. 1. I N T R O D U C T I O N With an increasing industrial demand for flexible and specialized production methods, batch and semi-batch processes are becoming ever more popular. The typically experienced deviations from normal operation force operation scheduling and plant design to be conservative in order to ensure reliable operation. Consequently, feedback control may be applied to reduce deviations from desired operation and thereby to reduce the conservatism of present operation scheduling and plant design. In addition, implementation of control will not only increase reliability, but also enable process optimization (JCrgensen et al., 2000). Batch operation, nonetheless, has received little attention in academia. The absent academic developments within batch operation are presumingly due to the often highly nonlinear, time-varying dynamic behavior of batch processes, which cause traditional linear time-invariant control tools to be inadequate. The purpose of this paper is to propose an extended data driven modeling framework for iterative learning control of batch processes. Furthermore, the paper presents an identification scheme based on desired model properties. In section 2 the model development and the identification scheme are given and in section 3 an iterative learning model predictive control algorithm is presented. The tool is demonstrated with simulation results and finally conclusions are given. 2. M O D E L D E V E L O P M E N T
The implemented batch modeling approach is based on a framework presented by Chin et al. (1998). A set of simple, linear, and time-invariant models are set up describing complex and time-varying dynamic behavior. Chin et al. (1998) suggested using a set of Finite Impulse Response (FIR) models, here the model may be set up using either FIR models or AutoRe*Corresponding author; fax : +45 4588 2258 and e-mail : [email protected] tfax : +45 4588 2258 and e-mail : [email protected]
622 gressive models with eXogenous inputs (ARX). Operation modes of batch processes will vary with time. Therefore, the number of measurements, which require trajectory tracking may vary with time. Often it will be necessary to include additional measurements for monitoring and estimation purposes, and to measure the quality of the end-product. Consequently, it seems reasonable to define the following variables: Control variable, u(t) E Nnu(t). Controlled output variable, y(t) E ~ny(t), which requires tracking control. Secondary output variable, s(t) C R ''(t), used only for monitoring. Quality variable, q c [~nq, only measured after a batch. Let N be the batch length and define the controlled output sequence as y = [yT(1) y T ( 2 ) . . . y T ( N ) ] T and the control sequence u (from 0 to N - 1 though) and disturbance sequence d in a similar manner. Let the initial condition be defined as Yini =-- y(O). Note, not all initial conditions are measurable and/or physically meaningful. The output sequence is in general related to the input sequence, initial condition, and disturbance sequence by a nonlinear algebraic model:
y = 91/[(tt, Yini,d) - N ( u , d ) + 9 ~ i ( Y i n i ) .
(1)
Let 37be the specified and preferably optimal output reference trajectory and let k be the batch index. Then the output error trajectory, ey, is defined as: ~k+ 1 -- ~kk -- GYAuk+ 1 - Gini y Ayini,k + 1 + W~k,
~-Y-y:~+~,
(2)
w h e r e Auk+ 1 = Uk+l --Uk a n d Ayini,k+ 1 = Yini,k+l --Yini,k. T h e m a t r i c e s G y a n d GiYi are l i n e a r
system approximations and ~ is the part of ~ that reappears in ~+1- Both w~k and ~ are assumed zero-mean i.i.d. (independent and identically distributed) sequences with respect to k. By applying the same modeling approach to the secondary and quality outputs the resulting combined model is" ~
0
g'k+l = ek -- G A u k + I -- Giniek+ 1 + Wk,
(3)
ek -- e'k + vk, where e -- {e yT e sT eqT] T a n d e~ --[AYiT i AsTni]T. To enable usage of a Kalman filter for estimation of the initial condition e~ the initial condition is modelled with a simple batchwise random walk model: o = e0 + v0 ek+l e0 = e ~ ~,
(4)
where v0 is assumed a batchwise white disturbance term, e~ is the measurement of e~ and ~o is assumed batchwise white measurement noise. 2.1. Identification One major drawback of the proposed modeling approach is the immense dimensionality of the resulting set of models. E.g., assuming causality a process with one output and one input sampled 100 times during a batch will be described by 5050 parameters using FIR models versus 198 model orders and assuming all model orders set equal to 3,594 parameters using ARX models. In practice, this immense dimensionality will render any identification problem illconditioned. Fortunately, there is some structural information in the resulting set of models G.
623 Firstly, by causality the upper triangular is 0, secondly, the time variation can reasonably be assumed smooth (L1), i.e., the diagonal elements in G will vary in a smooth manner, and thirdly, the impulse responses can reasonably be assumed smooth (L2), i.e., the column elements in G will vary smoothly. Regularization utilizes such a priory knowledge, reducing the amount of data needed, by introducing constraints or penalties in order to attract excessive degrees of freedom towards values, which satisfy the above specified requirements. Unfortunately, regularization will result in biased estimates, i.e., there will be a tradeoff between variance and bias. The optimal tradeoff is found by selecting the regularization weights ~,i that minimize an average prediction error over the validation batches. Given Nk data sets the unknown parameters x are determined by solving the regularized system in a Least Squares (LS) sense:
0
0
--
[AI
~IL1 x, ~2L2
A--
A11 I11 "
,
b--
"
AUk
(5)
,
buk
where b and A are the system outputs and inputs, respectively. In literature, several approaches to selection of scalar optimal regularization weights have been presented (Hansen, 1996). However, seemingly no work has been reported with vectorial regularization weights based on expected model properties. Furthermore, usage of high quality data for the identification will decrease estimation variance. The quality of batch identification input data can be determined by statistical comparison to a normal reference model obtained through multi-way analysis. 3. MODEL-BASED ITERATIVE LEARNING CONTROL
Based on the above obtained batch transition model, Model Predictive Control (MPC) can be applied for trajectory tracking. Chin et al. (1998) and Lee et al. (2000) presented an Iterative Learning batch Control (ILC) algorithm in a model predictive control framework combining trajectory tracking and end-product quality control. The block diagram of the control algorithm with the extensions introduced in the present paper, i.e., initial conditions and measurement noise, is shown in figure 1. From equation (3) the following periodical, Linear Time-Varying (LTV) state-space model is obtained for control design:
ek,t
0 I
ek(t)= [ 0
ek,t_ 1
Gt-1
nt ] [ ek,tek't] -+-~(t)'
(6)
t--12,...,N,
where ek(t) is the measurement of the output error ek(t), and ~(t) is assumed zero-mean i.i.d. measurement noise. At the start of a batch the states in (6) satisfy:
I,e o o] IN][Gini] ['] I 0
e~_ 1,N
Gini e~ +
I
wk- 1 +
[0] I
vk.
Note that the structure of Ht and e~(t) depend on which measurements are available at sample t. Let ek,t+m[t = ek,t[t -- Gr~AUkmtbe the optimal estimate of the error trajectory of the k th batch given measurements up to time t and m future control moves. The optimal input movements are then
624 -~(N)
~e~ /~
" -x(N)
y(t)
filter
ek (t
-G~"t ~'~
MPC
AUk(t--l)
--qk (N)
~ _ ~ -- qk-l ( N) q(N)
lAuk(t)
t ~'Tidmelear'y
[ Kalman ~ f t e~(O) i ~l _ ~ t- Y k e( OYO () r
Process
; ~ Y (t) - yt (t)
t~--O
-yk(0)
y(0)
Fig. 1. Block diagram of an Iterative Learning batch Control (ILC) algorithm in a Model Predictive Control (MPC) framework with combined output trajectory tracking and end-product quality control. determined by employing the standard quadratic objective function known from conventional MPC control. min
{ k,t+mltQtek,t+ml t +
Aut,,
S.t.
mT
m
(8)
~k,t+ml t -- ~.k,tlt -- GtAUkmt,
where Qt and Rt are weighting matrices. The analytical solution to (8) is given by: AUk, t [G?T Q t a ? _.1_Rt]-I G mT .. m__ t Qtek,tlt,
(9)
where only Auk(t) from AUkmt is implemented. In practice, (8) should be extended to: min AUkmt~gk,t S.t.
{AukmT QtAUkm,t -Jr-2~PLtTAu~,,t -k- s
AtAlgkm ' ~ Bt(Ek,t) '
} ,
(10)
g~,t >__0, where O~ -- G~ TQtGt~ + Rt and RtT -- -- ~T k,t[tQtGr~" By proper configuration of .~ and B both the outputs and the inputs may be constrained. The constraint softening slack variable e insures feasible solutions and St is the corresponding weighting matrix. The presented control technique transfers the refined control error sequence as initial condition for the next batch. Furthermore, it computes the input sequence as a change from the previous batch based on the
625 ~ , 0 . 1 j/
/
,
,
,
,
,
,
,
,
1
2
3
4
5
6
7
8
30
.
.
.
.
.
.
.
.
/
~'o.o5F 0 ITi
O0
9 1 0 ~.
~ ooF .
0 ~-~oo[
. . 2. . .3 .
""
1
.
.
4
.
5
6
"I
13
9
10
g ,~~
~
5
6
7
8
9
10
~ 7
~ 8
9
10
;
~
;
,0
. . . . . .
0
.~~176 I ~
0 / ....
1.4
0
~
90 ~ 1
~
1 ~
2 "r
3 ~
; "r
5 -1-
6 "-r
"/ "1-
8 "i-
9 1-
10
1
2
3
4
5 Time (h)
6
7
8
9
10
..,.-.
~ 2
~ 3
~ 4
~
;
~
~ 5
~ 6
100 ~,,,,.._
i
~
1.2
10
-.
Fig. 2. Validation of noise free FIR model identification. True output, solid line; predicted output, dotted line.
997;
Batch
No.
Fig. 3. Summed squared error sequence, quality, and yield evolution in noise free trajectory tracking of EtOH, OUR, CER, and RQ. No control corresponds to 100%.
control error prediction. This use of information from previous batches provides the control algorithm with an offset elimination action along the batch index (Chin et al., 1998). Thus, through this action the control algorithm is expected to eliminate offset errors both from disturbances that repeat themselves from batch to batch, and from model bias (Chin et al., 1998), i.e., the disturbances represented by w. Lee et al. (1997) have proven convergence for a batch MPC controller based on state-space model (6). This proof can be extended to hold for this algorithm also. 4. S I M U L A T I O N RESULTS To validate the presented modeling and control framework a FIR model was identified on data from yeast (Saccharomyces ceravisiae) producing fed-batch fermentations simulated with a biochemically structured model (Lei et al., 1999). As controlled output variables the ethanol concentration (EtOH), Carbon dioxide Evolution Rate (CER), Oxygen Uptake Rate (OUR), and Respiratory Quotient (RQ) were chosen and the feed rate was chosen as control variable. In fact, RQ _= CER/OUR, but here RQ was modelled as a measurement. The model was identified on data from 10 normal batch runs and cross-validated on 3 normal data sets. The identification was regularized to ensure smooth model parameter evolutions and smooth impulse responses. The regularization weights were optimized by trial and error, minimizing the mean prediction error from pure simulation model cross-validation. As demonstrated in figure 2, the identified model possesses reasonable predictive capabilities. Based on the identified FIR model trajectory tracking of the four controlled outputs was simulated for further model validation. In the control simulation scenario the initial conditions were randomly perturbed -t-10% in every batch runs and the feed concentration was perturbed by a persistent 10% reduction in all batch runs. In figure 3 it can be observed that the controller rejected approximately 90% of the Summed
626 Squared Error Sequence (SSES) in the first batch run and that after having been trained on the first three batch runs more than 97% of SSES with exception of batch run no. 6. The sluggish performance in batch run no. 6 was due to a particularly intractable disturbance direction, i.e., the yeast had been starved before seeding. However, the controller still rejected approximately 80% of SSES and the performance in subsequent batch runs was not affected. In addition to SSES, also the end-product quality and the yield are shown in figure 3, and it can be observed that the quality can be improved by output tracking and that the yield may endure a minute decrease as an expense of high tracking performance. 5. CONCLUSION In the present paper, the framework presented by Lee et al. (2000) has been extended with a dynamic description of the batch-to-batch behavior of initial conditions and with their effect on batch evolution and with the inclusion of measurement noise. The modeling framework results in models with immense numbers of parameters and hence an identification scheme has been developed utilizing information on desired model properties through regularization. Applying this identification scheme, high dimensional models can be obtained from sparse, noisy data. The model state estimation algorithm included in the original framework has been stabilized and extended with the ability to include initial conditions and process noisy measurements. The extended modeling and control framework as well as the identification scheme were validated through simulated trajectory tracking of fed-batch fermentations with highly satisfactory results. Hence, the presented extended framework for modeling and control of batch processes exhibits significant potential for industrial implementation. REFERENCES
Chin, In-Shik, Kwang S. Lee and Jay H. Lee (1998). A unified framework for control of batch processes. '98 AIChE Annual Meeting in Miami. Hansen, Per Christian (1996). Rank-Deficient and Discrete Ill-Posed Problems. Polyteknisk Forlag. Lyngby. JCrgensen, Sten Bay, Lars Gregersen and Dennis Bonn6 (2000). Data driven bioreactor modelling for monitoring and control. In: Book of Abstracts m 3 rd European Symposium on Biochemical Engineering Science. Center for Process Biotechnology. Department of Biotechnology, DK-2800 Lyngby, Denmark. pp. 49-50. Lee, Jay H., Kwang S. Lee and Won C. Kim (2000). Model-based iterative learning control with a quadratic criterion for time-varying linear systems. Automatica 36(5), 641-657. Lee, Kwang S., Jay H. Lee, In-Shik Chin and Hyuk J. Lee (1997). A model predictive control technique for batch processes and its application to temperature tracking control of an experimental batch reactor. '97 AIChE Annual Meeting in Los Angeles. Lei, E, M. Rotbr and S. Bay JCrgensen (1999). A Biochemically Structured Model for Saccharomyces cerrevisiae. Submitted to Journal of Biotechnology.
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rights reserved.
627
Distillation Control using Passivity Duncan P. Coffey, a B. Erik Ydstie, a* Torben R. Andersen, b and Sten Bay JCrgensen b aDepartment of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213 bCAPEC, Department of Chemical Engineering, Technical University of Denmark, DK-2800, Lyngby, Denmark This work will present the design of an asymptotically stable control system for homogeneous multicomponent distillation. The control system will be designed using passivity and thermodynamic based analysis. One design solution will be presented to illustrate the use of passivity and thermodynamics to better understand the distillation control problem. 1. I N T R O D U C T I O N In a previous publication Coffey, Farschman and Ydstie (2000) were able to show that multicomponent homogeneous distillation is open loop asymptotically stable using natural boundary conditions. The goal of this work is to take this analysis one step further and design a distillation control structure that asymptotically tracks given setpoints for temperature, pressure and concentration. One asymptotically stable solution will be illustrated to show how the use of the passivity and thermodynamic theory can be used to better understand the distillation control problem. The configuration presented will be based on the fundamentals of thermodynamics where pressures will be used for mass balance control, temperatures will be used for energy balance control and chemical potential will be related to composition control. The last statement about chemical potential and its relation to composition will be explored to show complications that can occur in azeotropic distillation. There has been an enormous amount of work done on the control of distillation columns. Skogestad (1997) gives an extensive review of previous work done on both dynamic modeling and control. We believe the result in this paper gives the first general stability result for control of multicomponent systems. 2. D I S T I L L A T I O N M O D E L AND O P E N L O O P STABILITY A simple distillation column has two product streams (distillate and bottoms) and one input (feed). The goal of this section is to generate a model for the open loop distillation column shown in figure 1. Due to paper length restrictions only a single tray model will be shown here but a full description of the model used can be found in Coffey et al. (2000). The model will first be written from a typical modeling view point and then from Eulerian and Lagrangian view points. The *Corresponding author
628 two different view points are needed for the stability/control analysis and a detailed description of the methods used can be found in a previous publication (Coffey and Ydstie, 2000). The two different view points are used similar to how they are used in transport phenomena. The Eulerian view point is used to describe the overall mass flows driven by pressure differences and the Lagrangian view point is used to describe how the concentration and temperature change as they move at the speed of the Eulerian view point. Generalized Onsager relations will also be used in the Eulerian and Lagrangian view points and are a method for describing the flows of a system as a function of a phenomenological coefficient (A) multiplied by a driving force (X)(Coffey and Ydstie, 2000). Examples of Onsager like relations are Fourier's law for heat conduction (Q = U A A T ) and Fick's law for diffusion (J -- DAConc). Each tray of the column has a combined vapor and liquid holdup along with two input streams and two output streams. The two input streams are the vapor output from the tray below (Vi+l) and the liquid output from the tray above (Li-1). The two output streams are in turn the vapor (Vi) and liquid (Li) input streams to the tray above and below respectively.
d~ dt
Conventional Model
(1)
=
V i + l h i V 1 - k - L i _ l h i L 1 - ~ h V - L i hL
(2)
=
Vi+l +Li-1-~-Li
(3)
--
rC~+lY(i+l)k+ Li-lX(i-1)k -- giYik -- Lixik,
d~ dt dMik
dt
dvl dt dv e dt
k= 1...nc-1
Lagrangian / and Eulerian e Models
(5)
=
-Al/~+lXVi+ 1 - & i _ l X L i _ l + &iXLi-~- A~Xvi
(6)
:
-AV/+lXV/+l
- mLi_ 1(Oi-1 -if-mti~-Oi +
AviXvi
--
XL i
---
(7)
(8)
Driving Forces Xvi
(4)
(Oi--(_Oi_l
(9)
(_Oi-- (Oi+ 1
(10)
The driving forces used for the trays are fairly straight forward, except for the liquid streams. The liquid streams are not driven by a conventional pressure difference between two points, but by liquid level and gravity over a weir. The way this is modeled is by calculating the height of the liquid on each tray which can be inferred from the pressures changes caused by the liquid height. Passivity (Lyapunov) type analysis is used to determine stability. A system is passive if there exists a non-negative storage function (A) that captures the dynamics of the system and satisfies the following inequality.
A(t) < A(O) +
/o'yTudz
(11)
The storage function that will be used in this work is given here A
-
o3*(v-v*)+S*-S,
vT
--
[U
Vol
M1
...
A>_O Mnc ]
(12) (13)
629
r cocoqlingl
1
"]Condenser ] Tray 1
~,f--1 [Tray n f
~Vn,f
coF
~coD
[
~nf-2 --
1[
~Lnf-1
a
~l Feed Tray nf ] I
~n.f + l ~n f [Tray n f + 11
~nf;2
~nf+l
I
[ Zrayn I coheqtingl "1 Reboiler [ t.
,coB
Fig. 1. Distillation column: n number of trays, n f feed tray ~)S =
OV
9
T
-.1
T
-.nc]
"'"
Fig. 2. Distillation control structure
(14)
T .~
(DT
where v is the inventories, S is the entropy and co is given in equation 14 as the derivative of entropy with respect to the inventories. The co introduced here is the same one introduced earlier as part of the driving force of the Onsager relations. To perform stability analysis the dynamic equations for the system must be substituted into the supply function (or derivative of the storage function).
dA dt
= CO,Tdv ~-=
( d s ) Tdv dvv
d---t
(15)
-(m- m*)r~
(16)
If the supply function is negative then A(t) is decreasing with time and the inequality given in equation 11 is satisfied and therefore the system is passive and stable 9 In this system there are two different view points each with a different model therefore there are two separate supply functions.
dA e dt
N
Bc (17)
i=1
k=l
j=l
630
dA l
N
Bc
-- E yfTAiffi - E - ( o y A j ~ j
dt
i=1
(18)
j=l
Equations 17 & 18 summarized the stability result from Coffey et al. (2000) and show a negative definite supply function when AA is positive semi-definite. This then proves that the simple distillation column described is open loop asymptotically stable. This result will now be extended to the design of an asymptotically stable control system. 3. DISTILLATION CONTROL A control structure will be designed using the ideas introduced by the thermodynamics and then controllers designed for asymptotic stability. From equation 14 it is shown that temperature directly effects the rate of entropy change with respect to the energy and similarly for the other intensive variables and inventories. It is with these ideas that the following control structure is designed. Temperature is used for energy control, pressure for overall mass inventory control, and concentration for component inventory control. The basic control structure will be as follows: Condenser pressure will be controlled by distillate outlet valve/pressure, condenser temperature by cooling fluid flow/temperature, condenser level by reflux flow valve. Reboiler temperature will be controlled by heating fluid flow/temperature, reboiler level by bottoms outlet pressure. Composition control of the distillate and bottoms products will be controlled in a cascade loop with the inner loop being the condenser and reboiler temperatures. The control structure is shown in figure 2. The control structure is not unique but represents a base structure designed from the fundamentals of thermodynamics. The stability of the chosen structure will now be shown. In equations 17 & 18 there are three different types of terms: boundary terms, flow terms, and difference terms from the weir equations. The terms that will be effected by control will be the boundary terms and the flow terms. Control can be implemented in two ways, either by manipulating the flow coefficient (A) or the thermodynamic conditions (X or m). Manipulating the flow coefficient is equivalent to controlling a valve. Manipulating the thermodynamic conditions is equivalent to changing the suction/output pressure of a pump or the thermal conditions of a heating or cooling fluid. In the stability proof that will follow simple proportional controllers will used. Stability of the control implemented at the boundary conditions will be looked at first. The following equation shows the form of the supply function boundary terms that will be changed by control. d A __
dt
"
.. _ ~ T ( A i o ) i - A* (.0~)
(19)
This is in a general form for either manipulation of Ai or mi and which is actually manipulated can be determined later. The proportional control law shown in equation 20 is then substituted into the supply function and results in equation 21.
-Ait.oi -4-A~ m~ dA dt
-....
(20)
-K(oi -~TK~
i
(21)
The supply function remains negative definite and therefore the system remains asymptotically stable with the introduction of the control law. The following equation shows how to manipulate
631 the flow coefficient or valve position as a function of the measured intensive variable.
Ai-
1
(A~0~;§
mi
(22)
The following equations shows how to manipulate the boundary condition and the resulting supply function (here it is assumed Ai -- A*). (0 i dA dt
-....
(23)
- K(oi_ n - (OT_nKT AiK(Oi_n
(24)
Here it should be noted that the sensor and actuator are not located at the same point. The measurement is at i - n and the manipulated variable at i. For the system to still be stable the only requirement is that the system is passive between i and i - n. Since the controller is simply adjusting the amount of dissipation, and can never become non-dissipative, then the whole system will remain dissipative and stable since we know from previous work that the system is stable. The above type of boundary condition control will be used for the condenser pressure control, reboiler level control and both the condenser and reboiler temperature control. The condenser pressure will be regulated by the distillate flow valve, reboiler level will be regulated by the bottoms flow valve, and the condenser and reboiler temperatures will be regulated by the temperature of the heating and cooling fluids. The last controller, level of the condenser, uses an internal flow with open loop supply function term of the form shown below dA dt
....
- f f ? (AiXi - A * X ; )
(25)
The control law shown in equation 26 is again stable when substituted into the supply function in equation 27. - A i X i § A~X[' dA dt
--
-KXi
....
- ffi~Kffi
(26) (27)
The internal valve position for the condenser level control is given here.
A i - (A'X* § K X i ) - ~ i i
(28)
Here the measured variable is a difference in pressures which can be used to determine the liquid level. The same type of control could have been used above for the reboiler level control. Composition control is performed by cascade control with the condenser and reboiler temperatures. The thermodynamic separation is defined by the chemical potentials differences between the condenser and reboiler. The chemical potential can be directly correlated with the temperature. This is easy to see in distillation, the nodes in composition space are the high and low boiling points within a given distillation region. The nodes of an azeotropic system can be the limits of the separation due to a minimum or maximum in the chemical potential space which will correlate with the high and low temperatures. This means when the nodes of the composition space are not the pure components, a component separation in terms of composition can
632 go through a maximum. This results in complications in designing composition control. The chemical potential is the driving force for separation and with its direct relation to temperature it is the proposal of this paper to use temperature as a means to control separation. The goal of composition control is to first control the distillate or bottom products to a reference point and then control the separation by the temperature difference. A simple example of how this would work is in binary distillation where the desired product is a pure distillate. The condenser temperature would then be set to the light key pure component boiling point and the temperature difference would be set to the desired split. If the reboiler temperature is set to the saturation temperature of the feed or below then there will be no distillate product. As the temperature of the reboiler increases (temperature difference increases) internal flows will be generated and distillate product formed. The limit would either be equipment related or of course when the reboiler temperature equals the heavy key pure component boiling point. Due to saturation conditions in the condenser and reboiler the temperature difference is related to the pressure difference. As the temperature difference increases so will the pressure difference. Separation will increase as the pressure difference increases as it is the driving force for the internal vapor flow. The temperature and temperature difference can be updates in the controllers from either concentration measurements, where possible, or from a model of VLE data. 4. CONCLUSIONS The work presented in this paper uses passivity and thermodynamics to design an asymptotically stable control structure for a general multicomponent distillation column. The structure shown in this paper is not unique but is used to illustrate the how the use of passivity and thermodynamics and can be used to better understand the distillation control problem. REFERENCES
Coffey, D. E, Farschman, C. A. and Ydstie, B. E. (2000). Distillation stability using passivity and thermodynamics, Computers Chem. Engng 24:317-322. Coffey, D. E and Ydstie, B. E. (2000). Stability of process networks, ADCHEM 2000 Pisa,
Italy. Skogestad, S. (1997). Dynamics and control of distillation columns-a critical survey, Modeling, Identification and Control 18(3): 177-217.
European Symposiumon ComputerAidedProcessEngineering- 11 R. Ganiand S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rightsreserved.
A Novel Adaptive Multivariable Industrial Reactor
DMC
633
Controller. Application
to an
E. C. Dechechi a, L. A. C. Meleiro b and R. Maciel Filhob'* a p u c P R - Pontific Catholic University of Paranfi (www.pucpr.br), Curitiba- PR, Brazil. CP 16210 CEP 81611-970. Fax. +55-41-316 3033. E-mail: [email protected] bLaboratory of Optimization, Design and Advanced Control (LOPCA). Department of Chemical Processes. School of Chemical Engineering. State University of Campinas (UNICAMP). Campinas - SP, Brazil. CP 6066 - CEP 13081-970, Fax +55-19-7887840. E-mail: [email protected] and [email protected] A novel adaptive control algorithm based on Dynamic Matrix Control (DMC) philosophy with adaptive features was developed and applied in an industrial hydrogenation multiphase reactor. The case study is a phenol hydrogenation reactor used to obtain cyclohexanol (an intermediate for nylon production). This is a three-phase reactor with solid catalyst in the reaction medium. This process has complex heat transfer mechanisms and fast dynamic behavior. The hydrogenation reaction is a process with distributed parameter and presents some important effects as catalyst deactivation, for instance, that are difficulty to prevent and to monitor. The motivation to use adaptive DMC controllers arose from its high potential and robustness when applied to chemical processes, since it couples prediction and adaptation behaviors in a convenient fashion. Besides, there is a lack of tests of this control strategy in the literature. The on-line internal model adaptation is carried out successfully using recursive least square method with an ARMA based model. This algorithm showed a very good performance leading the reactor to be operated safely in large range operational conditions. The results also showed the high efficiency of the developed controller when applied to this industrial process under normal operation. 1. INTRODUCTION The main focus of this paper is to present an advanced software for process control developed and tested on-line in an industrial multiphase reactor. This software, denominated D-AMPC ( D e c h e c h i - Adaptive Multivariable Predictive Control), is based mainly on predictive Dynamic Matrix Control (DMC) [10] philosophy and also on the Self-Tuning Control (STC) [14, 15] to provide adaptive capabilities with constant identification to an input-output multivariable model. Literature presents some papers dealing with the matter of coupling the predictive and adaptive advanced control strategies [7, 8, 12, 13]. However, in those papers the performance of such strategies was tested in simulations of ideal process. The aim of this work is to verify the performance of the novel proposed algorithm in an industrial multiphase reactor.
*Corresponding author ([email protected])
634 The controller was tested in the refrigeration loop of an industrial three-phase reactor that produces cyclohexanol by phenol hydrogenation in the presence of a nickel-based catalyst. This process has complex heat and mass transfer mechanisms, fast dynamic behavior and constant load changes in all input variables (temperature, concentration of catalyst and liquid flow rate). The hydrogenation reactor is a system with distributed parameters and has some effects, as catalyst deactivation, that are difficulty to prevent and to monitor by the usual commercial software for control purposes. The development and application of this novel adaptive multivariable DMC controller (D-AMPC) involves mathematical modeling, advanced control strategies, real time optimization and connection with a real time scheduling. These strategies have been developed to be feasible in real industrial chemical plant. In this work the main objective is to present this novel multivariable DMC controller who was tested under real industrial operating conditions. 2. REACTOR DESCRIPTION Three-phase reactors can be found in several important applications and is verified in the literature a lack of works that cover these complex reactors when compared to other types of catalytic reactors [1-5]. In this case, a catalytic and exothermic reaction takes place inside several modules with concentric tubes. This industrial multiphase (slurry) catalytic hydrogenation reactor is showed in Figure 1. In this reactor, the highly exothermic reaction of phenol hydronization is developed and the heat removal happens due to both the immersion of the reactor inside a steam generator, and the coolant circulation inside the first six reacting tubes. Besides the heat removal generated by reaction, that is accomplished in the steam generator, the first six tubes have an internal refrigeration with coolant flowing in co-current with the reaction medium to maintain the temperature profile controlled inside the tubes. The coolant is composed of the condensed steam from the steam generator to which a make-up stream may be added. Coolant is circulated to each of the six first modules, and each flow rate may be manipulated. The main function of the coolant in this case is to avoid the so-called "runaway" that occurs when the inside temperature of the reactors increases exponentially. This problem can lead to unbalance thermal reactions. Runaway effect is more frequent in fixed-bed reactors, with the possibility of appearance of hot spots, however such condition cannot be discarded in slurry reactor. Besides the possibility of the runaway effect, such a temperature increase provokes an increase in the formation of by- Figure 1 - Schematic diagram of the industrial reactor products, that is, decrease in the selectivity of the reaction. The structural and operational complexity underlying to this reactor is increased by other features associated to its complexity and operational performance, such as catalyst deactivation, incrustation along the reactor (three-phase system) among others. The first six modules are identical in structure and are composed of four concentric tubes. Reactant flows from one module to another through the inner tube passage and through the annular passage
635 formed between the two external tubes. Through the other annular passages, coolant flows in order to control the temperature of the reaction of each reaction tube. The last two modules are composed of two concentric tubes and only reactant is present. These last two reaction tubes are used to prevent the total phenol conversion. In this industrial reactor is observed that in its usual operational conditions high thermal oscillations can occur. These changes are due various factors as catalyst composition in the feed, catalyst deactivation and others typical difficulties of a very complex industrial reactor. These disturbances have high impact in the selectivity, productivity as well as in the oscillations of the temperature of each tube. This complex behavior is the major challenge of the D-AMPC control developed. As such an oscillations may occur frequently in this reactor and its causes are not well understood, a robust control strategy based on predictive and adaptive features has real potential of success in a large range of operational conditions. 3. M O D E L A N D C O N T R O L L E R D E V E L O P M E N T
In this work was developed and tested in the real plant an novel advanced control strategy based on Model Predictive control, more specifically DMC [6]. This novel DMC control was specifically designed for this industrial application and has adaptive capabilities to take in account the dynamic variations that occurs in the real operational conditions. These dynamic variations occur in the input variables (load variables). This controller has a DMC philosophy and its internal model is constantly modified to take into account the dynamic variations. Figure 2. The proposed advanced control algorithm is different from the other approaches dyna~emal identilicalion described in the literature [7-12] in the way that - - the internal model is built up (with no need of step tests) as well as in controller parameter I adaptation procedure. The variations in the dynamic process are observed by a multivariable ARMAX model that is Figure 2 - Structure of the adaptiveDMC controller D-AMPC successfully identified by a Recursive Least Square "RLS" algorithm (estimator block).
i
Defining the parameters vector as O(t - 1 ) = [a,, a2,.... , a,,, b0, b],. ..... , b, b]T, and data vector as
X T( t ) - [- y(t -1), - y(t - 2), .... , - y(t - na), u(t -1), u(t - 2), ...... , u(t - n b -1)],
the
RLS
^
algorithm basically searches estimations, 0, of the unknown parameters in such a way that the cost function, J
=
q~O)'-j y ( j ) -
j
, with 0
j--1
at the time t, b(t)= X r 0 ) 0 0 - 1 ) ,
where q~(t) is a known parameter called "forgetting factor".
This parameter allows that the more recent data have higher influence than the older ones. The parameters of the proposed ARMAX model are 10, 6 and 2 for the order of the involved output, input and noise disturbance variables, respectively. Based on this adjusted parametric model at each estimation time a new model of the DMC controller is generated and an updated DMC gain is generated. The multivariable controller developed was tested in closed loop system with five controlled variables (the most important is the temperature along the reactor length), six manipulated variables (cooling flow rate of
636 each jacketed tubes of the reactor), and three measured load changes (feed measured variables). The parameters utilized for DMC control, Model Horizon (MH), Prediction Horizon (PH), Control Horizon (CN) and Suppression Factor (D, where 10, 10, 6 and 1.0, respectively. 4. RESULTS The main results of this work is focussed on three different parts: i) Development of a real time software to multivariable process identification. This identification algorithm is based on parametric models and was extensively tested with industrial reactor data; ii) Development of a multivariable predictive adaptive controller (DAMPC); ill)Industrial tests with the D-AMPC under usual operational conditions. The results obtained where divided into two specific on-line real tests: Part A: Dynamic model identification and Part B" D-AMPC industrial test 4.1. Part A: Dynamic model identification The dynamic model identification developed to the D-AMPC software was extensively tested under on-line industrial conditions. These tests were carried out in order to test both, the parametric model adaptation, and the internal DMC model generated. Figures 3-9 show the results of a specific preliminary industrial test. This test was made immediately before the inclusion of D-AMPC controller. Figures 3-4 show the dynamic behavior of the input variables. These variables were used to show the performance of the dynamic model identification of the output variables and the generation of DMC internal model. -0,49 0.09_ ~ 0,08-1 ~ 0.07-1 0.06_1
M e a s u r e d ~adables
-0,50
~ Feed flow rate of H2 - - o - - Feed flow rate of Phenol ~ F e e d pressure
! .....
i ~176 0.04-1 0.03-1 0,0
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.
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.
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Behavior of the perturbations variables during the preliminary test of model identification.
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~
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I
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V
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Figure
4 -
Behavior of the input variables (Fj5 and Fj6) during the preliminary test of model identification.
-0,025Temperature at the bottom of the Tube 2 Process data Parametric model Internal DMC convolutionmodel
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i
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5
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25
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Figure 5 -Behavior of the output variable during the preliminary test of model identification.
o
~
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Figure
6 -
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.'
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Behavior of the output variables during the preliminary test of model identification.
637
In the Figures 5-9 can be observed the efficiency of the identification block and the adaptation of the internal DMC model. These results assure that was not needed the implementation of the typically step tests necessary to model based control. o,13 -
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.....
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i
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erna
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5
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Figure 7 - Behavior of the output variable during the preliminary test of model identification,
"
15
'
20
'
25
'
30
"
3~5
Sampling time
time
Figure 8 - Behavior of the output variables during the preliminary test of model identification.
With the results obtained in the Part A, it was possible to implement the industrial test and consequently closed-loop of the D-AMPC controller, Part B. These preliminary results show that the D-AMPC identification block is able to represent adequately the dynamic of the reactor. 4.2. Part B: D-AMPC industrial tests Figures 10-12 show the dynamic behavior of the controlled variables during the all industrial test. D-AMPC in closed-loop action is indicated in the figures by the arrows. 0,012 _
0,1 _
controlled vanable
T~om, 4
Temperature at the bottom of tube 6 --.-i11-- Process data
O,OLO -
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----O--- Sel-panl
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i
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i
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~
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Figure 9 - Behavior of the output variable during the preliminary test of model identification,
-o,1-
0
10
20
30
40 Sampling
50
60
70
80
90
100
-time
Figure 10 -Behavior of the controlled variable and the Set Point during the industrial test of the D-AMPC
The duration of the industrial test was 2.25 hours and all security requirements were completely satisfied. In the Figures 10-12 can be seen that the difference between controlled variable and set point was considerably high before the D-AMPC test. The operators consider this operational behavior normal, which clearly is not acceptable if high performance operation is required. Results show that under D-AMPC controller action, a higher performance of this reactor can achieved. Better performances of the D-AMPC controller can be obtained when a more detailed study on the feasible set-points are accomplished. In fact, great attention is given nowadays to find out suitable values for the set-points, since some of the values normally used cannot be achieved due physical limitations of the system.
638
O,lO
-~
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0001 0,04
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1
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.
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,
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.
,
20
.
,
30
.
,
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.
,
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.
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,
60
.
,
70
.
,
80
.
,
90
.
,
.
100
Figure 11 - Behavior of the controlled variable and the Set Point during the industrial test of the D-AMPC.
-0,020
I 0
9 , 10
9 , 20
9 , 30
9 , 40
.
, 50
.
, 60
.
, 70
.
, 80
.
, 90
.
, 100
.
Sampling time
Figure 12 - Behavior of the controlled variable and the Set Point during the industrial test of the D-AMPC.
5. CONCLUSIONS Based on the performance of the multivariable predictive-adaptive DMC controller (DAMPC) developed, the main conclusions are: i) The identification block with parametric model and the generation of the internal DMC model was very efficient in extensive on-line industrial data acquisition; ii) The adaptive capabilities of the D-AMPC controller developed dispense the step tests in the industrial unit; iii) The D-AMPC controller can be used safely for a long range of operational conditions; iv) The development and industrial test of this DAMPC controller represent an important result because its a complex advanced model based controller applied to a real three phase reactor.
ACKNOWLEDGEMENTS The authors are grateful to FAPESP-Brazil to sponsor the Research Projects at processes n ~ 96/12584-0 and to Rhodia Brazil Ltd.
REFERENCES 1. Santana, P. L., M.; Tvrzsk~ de Gouv~a, M.; Maciel Filho, R.; Dechechi, E. C.; Domingues, A. Submitted to Chemical Engineering Science (2000). 2. Coussemant, F. and Jungers, J.C. Bull Soc. Chim. Belg. 59, 295-326 (1950). 3. Patai, S. Interscience Publishers 1,578-581 (1971a). 4. Patai, S. Interscience Publishers 2, 641-718 (1971 b). 5. Ramachandran, P. A and Cahaudhari, R.V. McGraw-Hill, New York, U.S.A (1983). 6. Dechechi, E. C. Ph.D. Thesis - UNICAMP, Brazil (1998). 7. Maiti, S. N. and Saraf, D. N. J. Process Control, vol. 5, no 5, pp. 315-327 (1995). 8. 0zkan, L. & t~amurdan, M. C., Computers Chem. Engng., vol. 22, pp. $883-$886 (1998). 9. Seborg, D. E.; Edgar, T. F.; Shah, S. L., AIChE Journal, vol. 32, no 6, pp.881-913 (1986). 10. Cutler, C. R. and Ramaker, B. L. AIChE Annual Meeting, Houston (1979). 11. Dechechi, E. C.; Luz Jr., L. F. L.; Assis, A. J.; Wolf Maciel, M. R., and Maciel Filho, R. Computers and Chem. Engng, vol. 22, Supl. pp. $867-$870 (1998). 12. Feng, W.; Genceli, H. and Nikoloau, M. Computers Chem. Engng, 20,S 1011-16 (1996). 13. Lundstr6n, P. et alii, Computers Chem. Engng., Vol 19, no 4, pp-409-421 (1995). 14. Clarke, D. W.; Phil, D. and Gawthrop, P. J. Proceedings of the lEE, vol. 122, No. 9, pp. 929-934 (1975). 15. Astr6m, K. and Wittenmark, B. Addison-Wesley Publishing Company Inc. (1995).
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
639
Computer design of a new predictive adaptive controller coupling neural networks and kalman filter L. Ender a, R. Scheffer b and R. Maciel Filho b aChemical Engineering Department, Regional University of Blumenau, Rua Ant6nio da Veiga 140; CP 1507, Blumenau - SC, Brazil, CEP 89010-971 bLOPCA/DPQ, Faculty of Chemical Engineering, State University of Campinas (UNICAMP), Cidade Universitfiria Zeferino Vaz, CP 6066, Campinas - SP, Brazil, CEP 13081-970 This work proposes a new predictive control algorithm based on constraint neural networks. The algorithm utilises a sequential quadratic programming algorithm to compute the next action of the manipulated process variables. The predictive control parameter, the suppression factor, is optimised on-line by a standard Kalman filter. The new control algorithm was tested on a fermentation process and it shows that the suppression factor can be identified and shows promising results to be applied in complex multivariable systems. 1. INTRODUCTION Most process control applications consist of not only keeping controlled variables at their set-points but also keeping the process from violating its operating constraints. This is particularly important when there are changes in set-points and when the process is multivariable and non-linear. While good advances have been obtained using conventional predictive controller algorithms, such as DMC, when high performance operation are required, more sophisticated controller are required. In many industrial systems, there exists strong interactions among the process variables as well as internal changes, such as deactivation of a catalyst. To cope with these changes a model predictive control algorithm with self-tuning capabilities is needed. 2. CONTROL A L G O R I T H M A predictive algorithm is proposed where the constraints in the manipulated and controlled variables are considered in the on-line minimisation of a quadratic cost criterion. The commonly used minimisation criterion, which compares a reference trajectory, Yref, with the predicted output, y , taking into account the controller movement, is:
J = min
P
arg u
Zj=lZkNl(Yref ,j.k -- Yj,k
)2
m
Nc
+ ~--'~j=lAj~--'~k=l(Uj.k Uj.k-Z~
(1)
where p is the number of controlled outputs, m the number of manipulated input variables,
Npthe prediction horizon, Nc the controller horizon and )~j the suppression factor of the corresponding manipulated input variable uj. The manipulated inputs and controlled outputs are subjected to the following constraints:
640
y,,,,,, < .~(k + i)< y,,,~ Umi,, < u ( k + i - 1 ) < u , , , ~ [u(k+i-1)-u(k+i-l~<_A
~lmax
i=l,...,Np
(2)
i=l,...,N u
(3)
i= 1,...,N u
(4)
The substitution of the classic models used in the prediction along a horizon for a neural network with on-line learning is suggested, due to ability of the nets in representing complex dynamic behaviours, taking into account non-linearity features. The neural network is trained in real time by a closed-loop training scheme as developed by Ender and Maciel (2000). Another real-time training scheme would be the application of recurrent neural networks with an extended Kalman filter training algorithm, which shows superior dynamic identification of complex chemical processes (Scheffer and Maciel, 2000). Adopting the receding horizon technique, only the first control action is implemented and all the calculations are repeated at each sampling time. The minimisation is accomplished by Sequential Quadratic Programming. The suppression factor )~ assures that no exaggerated control action is calculated and influences the systems dynamics. A too small ~, results in large control actions, which can result in a instable response, while a too large ~, results in a sluggish response. It is common to regard the suppression factor as a design parameter, but this can result in a sub-optimal response if the process is subjected to changes. Normally, the parameter is tuned manually until a desired process behaviour is obtained, which can be a very time-consuming procedure. Additionally, different values of)~ might be needed for other operating conditions. Therefore it is proposed to create an automatic estimation procedure for )~ to ensure an optimal value of)~, which will result in a satisfactory control for all process situations.
2.1. The neural network For the development of this work the feedforward architecture with back-propagation learning and sigmoidal function were used. In all the used neural network configurations, one hidden layer was considered, because Hiecht-Nielsen (1989) proved that any continuous function can be approached for any degree of precision using a backpropagation neural network with three layers, if there is sufficient number of active neurons in the hidden layer. The pattems presented to the neural network are composed of information from the last inputs/outputs and the current disturbance. This neural network when used in the prediction becomes recurrent. In this case, the weights are adjusted on-line, at the same time in such a way that it is used in the predictions. 2.2. Estimation of the suppression factor The adjustment or tuning algorithm of the parameter ~, is based on the standard Kalman filter. To be able to adjust ~ a dynamical system has to be created which can observe the state of the parameter ~, as in:
z~ = c , ~ (+ v~)
(5)
641 where Wk and Vk are random variables with a normal distribution of N(0,Q) and N(0,R) respectively. Zk is the measurement related to the state ~,k. Normally the noise of a parameter state is zero, but a small process noise results in a more stable filter. The observation equation of the L can be derived from the minimisation criterion J in equation 1 assuming that the SQP minimisation was successful. The first derivative of J will be zero, when J is in its minimum. It is assumed that only Uk, the implemented control action, is of importance. The last input, Uk-1, is regarded as a constant This can be validated because the receding horizon technique implements the first control action only. This results in the following dynamical system: '~k+l = 2k + wk
Y~N.i
}
(6) :(Uk -- Uk-,)Ak + Vk
The measurement, Zk, in equation 6 is represented by a summation of derivatives multiplied by the error of the process output with the output reference. The estimation of the measurement is done by the right side with E{Vk}=0. Vk is chosen as random variable with a normal distribution of (0,1). As the measurement is actually a mathematical expression, the variance of the measurement noise is chosen 1. This is mainly done to prevent a division by zero in the Kalman filter, which occurs when Uk equals Uk-1. It was already said that for a parameter the process noise Wk would be discarded, but it can be shown that the Kalman filter converges to a constant gain matrix for linear systems (Goodwin and Sin, 1984). )~ is wanted to change in real-time to cope with changing process conditions and an elegant way to do so is to make the process noise a function of an error. It is proposed to make the variance of the process noise a function of error between the reference and prediction of the process output by the neural network:
Wk oc (O,(y,.~y,I _ YANN,' )e )
(7)
In this way the uncertainty of the parameter will be high if the process is far away from its set-point and thus the Kalman filter will be triggered, resulting in a larger adjustment of the parameter )~ by the measurement.
3. R E S U L T S A large industrial fermentation process for the production of penicillin is considered as a case-study. The simulation is based on a model of Rodrigues (1996), which is validated with industrial data. The process is a fed-batch process and falls down in two parts, the growing phase and the production phase. In the growing phase a high sugar level is maintained, while in the production phase it has to be kept low as it inhibits the penicillin production. The emphasis is put on the production phase where the feeding strategy of the sugar substrate has to be chosen carefully to maximise the penicillin production.
642 One input, the substrate feed flow, and four state variables, the bio-mass concentration, the substrate concentration, the penicillin concentration and the dissolved oxygen concentration, were taken to identify the process. The substrate feed influences directly the cellular concentration, the substrate concentration and the penicillin concentration. The dissolved oxygen concentration is also influenced, because of the cellular growth which requires oxygen for respiration. The dissolved oxygen concentration is a vital variable for a good process operation. Rodrigues (1996) mentioned that a dissolved oxygen concentration lower than 30% causes deterioration of the fungi. The dissolved oxygen concentration can be controlled by aeration and stirring, but the latter will destroy the fungi at high rotation speeds. This shows the need for an optimisation algorithm which takes into account such constraints. A single input, single output (SISO) system was set-up to verify the proposed process control scheme. The controlled variable was the dissolved oxygen concentration which was controlled by the rotation speed. Various constraints are applicable here as, maintaining the dissolved oxygen concentration above 30% and avoiding high rotation speeds which destroy the fungi. A white noise was imposed on the dissolved oxygen concentration to simulate measurement noise. It was started with a manual tuning period to determine the stability of the process. In this case, the penicillin production process has a very slow dynamics, for which 9~resulted to be near zero. Note that this is valid for most SISO control problems where the manipulated variable can change at its maximum rate. In the SISO case the proposed control strategy becomes therefore interesting for very sensible and unstable systems only. Still it can be interesting to see if the proposed control scheme is stable and to see if a value different of zero is found or actually that a value near zero is encountered. k has to be greater or equal to zero, and it can be seen from equation 6 that this will not be always the case. Therefore the dynamic system 6 was modified by taking the absolute values of both sides, which will result in positive values for )~. Another possibility is taking the positive value of the resulting )~. Both were implemented but only the first case is shown here. In Fig. 1 it is shown the convergence of parameter )~ for various initial values. It can be seen that the parameters converge sooner or later to the same value and from then on show the same dynamic behaviour. Fig. 2 shows that there is only a minor difference in the beginning for the dissolved oxygen concentration and it looks like that the control is better when the initial L is different from zero but not to far away from its optimal value otherwise it constrains the input action to much. It can be seen that with measurement noise present that the parameter ~ does not stay constant, which is due to the insertion of process noise into the dynamic estimation equation of the parameter )~. But it can be seen that this is also due to the developed identification system. After a set-point change the controller movement will become rapidly close to zero while the measurement calculation will always be different from zero due to the measurement noise, resulting in small changes of)~. This will be problem when the absolute values are implemented, because this results in a parameter drift until a new set-point change or perturbation occurs.
643 0.010
Xo= 0
0.008
" " " X0=l
0.006
.~
+ k~
3
2
0.8 .s ~0.7 o
0.004 ---
, ........
" 1
~
0
'
~o.5
0.002. "~-~,--0.000
~0.6
j
..;,~.....~.~;.~..~. ..............,........................ 0 60 80 100 120 Time (h)
.........:............. .............:............
20
40
Fig. 1" Estimation of suppression factor for different initial values for the servo problem
~0.4 ~ 0
~
2'0
40
~o
Time (h)
X0=0 --X0--1 o
1
go
10o 120
Fig. 2: Controlled dissolved oxygen concentration for the servo problem
In Fig. 3 and Fig. 4 the control system is shown for a complex regulation problem, where a series of step changes is implemented which maximise the penicillin production. The sugar concentration affects the dissolved oxygen concentration due to the cellular growth and due to the energy required in the metabolism for the penicillin manufacturing. Even with a series of perturbations of every 6 and 6 hours the penicillin process is maintained under satisfactory control with an maximum error of 5% after the tuning of the suppression factor. It can be seen that the parameter X is never constant, but as in the servo case never diverges and never shows instability. 0.00100-
0.70
0.00075.
E 0.65 O t-
0.00050.
O
0.00025. / 0.00000
0
o 0.60 t(D
"~-~..._
o~' 0.552'0
4~0
6'0 80 Time (h)
__F 100 120
Fig. 3" Evolution of the suppression factor for the regulator problem
ID
> 0.50 . "5 ~ 0 a
. :50
. . ~,0 ' 6 0 ' 8 0 Time (h)
' 100' 1:20
Fig. 4: Controlled dissolved oxygen concentration for the regulator problem, perturbation in the substrate feed stream
A test was done with no measurement noise present to test the behaviour of X and to see if it would settle on a constant value or settle to zero. In Fig. 5 a comparison is made between the proposed estimation system with the self-tuning method of X and a fixed X. First it can be seen from the dissolved oxygen response (Fig. 6) with a fixed X that there is actually a need for variable X. At some set-point changes the dissolved oxygen concentration is controlled vary well, while at other set-point changes the same value of X results in a small but significant overshoot.
644 0.04 .9
m -,0
0.03 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.02-
;Lo = 0 0 4
.... X, = constant = 0.0
e"
/
0.01 0.00
0
#0
40
60
Time (h)
80
Fig. 5: Suppression factors without measurement noise
100
9
tD t.) t"
1:~0
0
w,v,,w,;v,-.~;--,~,;;
9
g
~I
..... SetpoJnt I o X -004 ->o~0.41 ~ X = constant = 0.03 ,n 0 20 40 .~_ ca Time (h)
60
80
Fig. 6: Control of dissolved oxygen concentration without measurement noise
The results with the proposed tuning system reveal that it is important to tune the suppression factor in such a way that no overshoot or sluggish control in all set-point changes. The first point to considered is if the minimisation criterion is a good starting point for deriving an identification system of the parameter X. X has a huge influence on the calculated movement in the minimisation criterion, but cannot only be determined of it. Therefore care has to be taken with the identification method of X which has to be suitable from the available process information. 4. CONCLUSION The proposed control algorithm has shown to be robust for the analysed disturbances, promising to have a great potential to be used in control strategies of large scale systems. Particularly, the application of the proposed algorithm in the SISO case reveals its potential for multivariable systems where the interactions and thus the suppression factors are more important. It is noted the importance of the adjustment of the suppression factor to be able to cope with change in process operations. The estimation algorithm of the suppression factor, will determine successfully the rate of change of the system due to the control action. This will result in a fast control algorithm with little or no overshoot. REFERENCES 1. Ender, L. & Maciel Filho, R. (2000), Computers & Chemical Engeneering, 24, 937-943 2. Hecht-Nielsen, R. IEEE Int. Conf. On Neural Networks San Diego (1989) v.I, p.593-605. 3. Goodwin G.C. and K.S. Sin, (1984) Adaptive filtering, Prediction and control, PrenticeHall, Englewood Cliffs, NJ 4. Rodrigues, J.A.D. and R. Maciel Filho (1996), Chemical Engineering Science June, v51, il 1 pt B, p2859-2864 5. Scheffer, R. and R. Maciel Filho (2000), in "Application of Neural Network and Other Learning Technologies in Process Engineering", ed. Mujtaba and Hussain, "Process identification of a fed-batch penicillin production process - training with the extended Kalman filter"
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
Function-Behavior Modeling and Diagnosis of Chemical Processes
Multi-Agent
645
Approach
for
Fault
Soo Young Eo a, Tae Suk Chang b, Bytmgwoo Lee c, Dongil Shina and En Sup Yoon a aSchool of Chemical Engineering, Seoul National University, San 56-1, Shinlim Dong, Kwanak Gu, Seoul 151-742, Korea bSamsung SDS Co., Ltd. 707-19, Yoksam-2Dong, Kangnam-Gu, Seoul 135-918, Korea CAutomation Research Center, Pohang Univ. of Science and Technology, San 31, Hyoja Dong, Nam Gu, Pohang, Kyungbuk 790-784, Korea In this paper, we suggest a function-behavior modeling and distributed multi-agent system approach for the diagnosis of chemical processes. In the proposed system, diagnostic agents of individual units communicate by exchanging messages and try to solve the global fault diagnosis problem using collaborative diagnosis method. The benefits of the suggested collaborative problem solving approach are demonstrated by solving the diagnostic problem of a CSTR system with recycle and a first order irreversible reaction. 1. INTRODUCTION During the last two decades many diagnostic methods have been suggested, developed further and resulted into some success. However, still diagnostic systems are not widely adopted by the industry due to some difficulties remaining unsolved in spite of the progress in this area [ 1, 2]. To solve those difficulties, this research proposes and develops a fast, reliable, easy-to-understandable fault diagnostic system, which can be quickly implemented with consistency, for chemical processes. As many seen in the area of artificial intelligence, Multi-Agent System (MAS) technology is emerging to solve problems in complicated, distributed systems, such as Intemet and human bodies. Multi-agent systems have been known to provide distributed and collaborative problem solving environment [3, 4]. In the process systems engineering, there have been some efforts to apply agents in modeling and design of processes in the concept of concurrent engineering [5]. However, few efforts have been made to the application of agents as software components and agenthood in the process fault diagnosis domain. In this paper, the development of a multi-agent system for chemical process fault diagnosis and function-behavior modeling for the knowledge base of the system is presented. This system uses the diagnostic agents for each process unit and makes them communicate with each other to solve the fault diagnosis problem in a collaborative way.
646 2. MULTI-AGENT FAULT DIAGNOSIS SYSTEM
2.1 Overview of the Multi-Agent Fault Diagnosis System Comparison of the conventional, shared memory single-agent approach---centralized and integratedmand the message passing multi-agent approach--decentralized and distributed is shown in Fig. 1. Diagnostic agents are distributed and implemented for each process unit, and each autonomous agent tries to solve problems through the collaboration with others. This collaborative approach is the most prominent difference from conventional centralized approaches. The concept of distributed fault diagnosis using multi-agents in chemical processes is shown in Fig. 2. Diagnostic agents (DAs) corresponding to existing process units are placed in accordance with the topology of the process units. They have a fault detection module, knowledge bases and an inference engine which performs reasoning as well as local diagnosis. The fault detection module has the information on possible fault types, measured and unmeasured variables of individual process unit and can determine the existence of any faults. Knowledge base has the relationship between variables based on function-behavior model [6, 7]. DAs of process units are said to be connected when the corresponding units are physically connected through material/energy flow or when the units are linked by information streams, such as controller signals. In addition, DAs communicate only with other neighboring DAs. DAs of the corresponding units have their knowledge about the relations among state variables in the units represented by function-behavior models of the units. With the knowledge, DAs perform diagnosis by processing the messages from other DAs in the eventdriven way. The messages are invoked from the event such as the detection of symptoms from measured variables, reception of queries from other agents, etc. The communications between DAs are limited to directly linked DAs and this makes the MAS structure very distributed. 2.2 Function-Behavior Modeling Function-Behavior modeling of Oh [6] aimed at fast system diagnosis by simplifying the large chemical processes and resulted in process interpretation similar to the behavior modeling due to the emphasis on grouping process units by their functions. To perform the collaborative fault diagnosis based on process topology, by interpreting process units and their
~ DA: DiagnosticAgent ................~A~~'''" SA:SchedulingAgent
,roce,:
Single-AgentSystem
...
Multi-AgentSystem
Fig. 1 Comparison of single-agent and multi-agent systems.
Fig. 2 Multi-agent diagnostic syste structure.
647 functions as DAs, and their functions and status variables of the units as the contents of the DAs, a different approach from Oh's method is required. Basically, every equipment in a chemical process carries out tasks related to mass, energy and component. Mass, energy, component, etc. become objects of functions of process units; these objects are defined as keyword, and the status of the keyword, keyword status, describes the unit's behavior depending on the function of equipment. For example, the abnormal high mass flow in a pipe may be described as MASS_STATUS HIGH. All process units have such keywords, and each keyword-related behavior of units can be interpreted together with behaviors of other units via keyword. Keyword status such as MAYBE_HIGH, MAYBE_NORMAL, MAYBE_LOW describes the status estimated from that of nearby equipment when the corresponding variable of the equipment is not measured. UNKNOWN is the status that is not measured and cannot be estimated from nearby equipment. In case of keyword COMPONENT which represents component composition, it can be extended as COMPONENT1, COMPONENT2, etc. when many components are processed in the process.
2.3 Diagnostic Agent As DA is not a software component with specified requirement or format, it doesn't have regular structure. However, it has formats containing knowledge base and communication protocol. Knowledge base includes function-behavior model, which enables different local diagnostics from other units. In addition, it contains the inference engine and operation rules with of various message and functions according to the characteristics of units. All DAs share a common communication protocol while they are based on different function-behavior models. Thus, MAS shows consistent performance as communicated information is translated into a common language. Diagnosis by DAs is essentially carried out by exchanging status and local reasoning of neighboring DAs as needed. Therefore, a DA sends queries for what it wants to know, performs reasoning on its status based on the answers to queries, and notifies the user of detected faults of itself and neighboring units. Function-behavior model and activities and role of DA will be explained using a simple process, shown in Fig. 3. This system consists of three process units where input is fed into a
Fig. 3 A simple tank process for demonstration of diagnostic agenthood.
Fig. 4 MADISON for a CSTR process.
648 tank unit through a pipe unit and output comes out through another pipe unit. DA infers causal relationship through communication with neighboring DAs and fundamental selfreasoning. The detecting module of DA is always active, and DAs perform diagnosis by sending and receiving messages when events leading to faults occur. DAs of Equipment_l, 2, and 3 are named DA1, DA2, and DA3, respectively. Let us assume a leak occurs at Equipment_2, a tank. This event causes a symptom of low flow of Equipment_3 (F3) and low level of Equipment_2 (L). The symptom is expressed in the language of function-behavior model as follows: keyword status low of keyword MASS (MASS[low]) is detected in DA2, and MASS[low] is detected in DA3. Triggered by events leading to faults, DA3 asks a query to DA2 in order to verify if this symptom is propagated or intermittent. DA2 replies the answer to the query of DA3, triggered by the event of receiving a query message, and DA3 performs a local diagnosis from receiving reply in the input device. In this case, it can be concluded that the fault is propagated to unit itself based on the fact that MASS[low] is received and the status of itself is MASS[low]. In fact, a DA has many types of messages for processing problems depending on an unmeasured variable exists, a controller states a diagnosis opinion, keyword changes occur as in the heat exchanger, etc. In addition, the knowledge system may be different due to the uniqueness of process necessitated by process units. As mentioned before, multi-agent diagnosis system uses messages where an event of receiving a message triggers diagnosis. Messages will be explained more in the following section. The important functions of DA can be summed up as communication protocol of message passing, reasoning, and report of inference results. DA reports the opinion of each DA to operators by relative importance based on the results of local diagnosis. Inference characteristic of processes or process units is required in this step, and relationship between keywords and related faults should be identified in advance. How to handle physically infeasible keywords and faults in a process unit should be considered as well.
2.4 Certainty Factor (CF) Distributed fault diagnosis shows good performance in portability, expandability, and speed. However, conflicting diagnostic results may exist and should be resolved because local fault diagnosis and communication among agents are inherent in this system [8]. In this paper, possibility of fault occurrence is quantified by introducing certainty factor (CF) to resolve the conficting diagnostic results. As for the weight factor used in calculating CF, Gaussian distribution is assumed in the time domain from the start of the diagnosis to the current time. This approach is suitable for the characteristics of fault diagnosis as the latest diagnostic results take priority over the past results, and even old results are not ignored completely. 3. CASE STUDY The process used in the case study is a classic example for fault diagnosis, and has been used in many researches, including the work of Kramer and Palowitch [9]. The mathematical modeling of Sorsa and Koivo [ 10] was used for this study. There are three feedback control loops in the process, and PI controllers are used. They control the level of reactor, recycle flowrate of product, and reactor temperature, respectively. Level control of reactor is direct and no keyword change occurs, recycle flowrate control is reverse and no keyword change occurs, and temperature control of reactor is direct and
649 keyword change occurs. Therefore, different function-behavior model is required for each controller. 3.1 Multi-Agent Diagnostic System for a CSTR process We implemented a fault diagnosis system development tool for chemical processes, MADISON (Multi-Agent Diagnostic System ON-line), on G2| Expert System Shell. Fig. 4 shows a multi-agent fault diagnosis system constructed by applying MADISON for CSTR process. As process units are DAs, process topology of the real process is preserved in the diagnosis system. Fault diagnosis is carried out through local diagnosis unit-wise, and as a result, the interface is very friendly and easy to understand. A simple model to verify status is formed for PI controller used in the case study. The function of controller is to maintain the set point of controlled variable. This is not limited to PI controller, and is for almost every controller. Here lies the advantage of function-behavior model. Measured values and manipulated variables are used for this purpose. That is, manipulated variables are used to maintain the set points of controlled variables as measured values change. Hence, the relationship among measured values, and manipulated and controlled variables is important, and controller is thought to be problematic when these values and variables change out of accordance with expectation. MADISON does not consider the case of controller failure, and warns possibilities of undetected faults of nearby process units instead. There are three PI controllers in this CSTR process, and the related variables are as follows. (1) Reactor temperature controller: T (ENERGY) - CT (ENERGY-MASS)- FW (MASS) (2) Reactor level controller: L (MASS)- CL (MASS)- FP (MASS) (3) Recycle flowrate controller: FR (MASS)- CR (MASS) Proper control is said to be functioning when all the three variables show the same tendency in reactor temperature and level controllers. Recycle flowrate controller is determined as functioning when the two variables (FR and CR) show the opposite tendency. Generally, when a controlled variable shows deviation, the controller tries to compensate for the deviation error. Therefore, the controlled variable itself is not detected as a symptom, but manipulated variable shows symptom. For example, when the level L of the reactor--the controlled variable of the reactor level controller--starts to decrease, the output signal CL decreases so that the output flow (manipulated variable) of the reactor increases to maintain the level L at setpoint. In this case, LC-DA--the level controller DA--suggests that there is possible fault in the input stream to CSTR and notify it to DA-CSTR as both CL and FP decreased. DA-CSTRmthe CSTR DA--is designed to receive the notification of possible fault in the input stream and to report the another possible fault in the level of CSTR itself (which means that there might exist leak) to the operator.
3.2 Test Case: a CSTR Leak This case is when the leak of 0.5kg per second occurs in the stirred reactor after 200 second during the simulation for 500 seconds. The level could be controlled by the level controller very well and showed no symptom of deviation. After the reactor started to leak, the output signal of the level controller CL decreased and the flowrate of product FP also decreased. As the level was maintained, the temperature of the reactor did not deviate and only the flowrate of product kept decreasing. The diagnosis was performed quickly with exchanges of messages of DAs as soon as the detection of symptoms occurred.
650 Table 1 Diagnoses of CSTR leak. ID of DAs Keyword_status
Diagnosis
CF
CSTR-DA-2
MASS LOW
CSTR leak
0.570
PIPE-DA-6
MASS LOW
Pipe_blocked Pipe_leak
0.444
Control valve stuck low Pipe_blocked Pipe_leak
0.045
-
PIPE-DA-9
MASS_LOW
As a result, CSTR-DA-2 suggested the leak of CSTR as the most probable candidate of fault with 0.57 CF, which is the largest among other candidates. Note that the reactor leak was suggested as a fault regardless of no symptom of reactor level decrease as a result of collaboration and negotiation of neighboring DAs. The final results of diagnosis based on the local diagnoses and CF of each DA are summarized in Table 1. Because the leak was not so serious, the controller could maintain the normal operation conditions to some extent. Hence, the process was not so dynamic and the symptoms occurred somewhat late and the value of CF of each DA had small values around 0.5. 4. C O N C L U S I O N The multi-agent technique offers a powerful alternative for diagnostic problem solving in a lot of engineering applications. In this paper, we suggested function-behavior modeling and multi-agent diagnostic system for chemical processes where diagnostic agents of each unit communicate by exchanging messages and try to solve the global fault diagnosis problem by collaborative diagnosis method. The suggested approach was implemented on the expert system development tool, and a prototype application built showed its effectiveness. Acknowledgements
This work was partially supported by the BK 21 Program supported by the Ministry of Education and the National Research Lab Grant of the Ministry of Science & Technology. REFERENCES
1. W. R. Becraft, D. Z. Guo, P. L. Lee and R. B. Newell, PSE91, Montebello, Quebec, 1991. 2. D. Mylaraswamy and V. Venkatasubramanian, Comp. Chem. Eng., 21S (1997) 940. 3. J. M. Bradshaw (ed.), Software Agents, AAAI Press/The MIT Press., San Francisco, 1997. 4. M. N. Huhns and M. P. Singh, Readings in agents, Morgan Kaufmann, Cambridge, 1998. 5. R. Batres, S. P. Asprey and Y. Naka, Comp. Chem. Eng., 23S (1999) 653. 6. Y. S. Oh, Ph.D. Thesis, Seoul National University, Seoul, 1998. 7. S. Y. Eo, T. S. Chang, D. Shin and E. S. Yoon, Comp. Chem. Eng., 24 (2000) 729. 8. T. S. Chang, Ph.D. Thesis, Seoul National University, Seoul, 2000. 9. M. A. Kramer and B. L. Palowitch, AIChE J., 33 (1987) 1067. 10. T. H. Sorsa and N. Koivo, IEEE Trans. on Sys. Man and Cyber., 21 (1991) 815.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
651
Robust Predictive Control Combined with Nonlinear Observation for Monitoring (Bio)chemical Processes O. Gehan, M. Farza, M. M'Saad and G. Binet, Laboratoire d'Automatique de Procrdrs, L.A.P., E.A. 2611, I.S.M.R.A., Universit6 de Caen, 6 Bd Mal Juin, F-14050 CAEN cedex, France An integrated approach for the on-line monitoring of chemical and biochemical reactors is presented. While controlling the process, the proposed approach also provides on-line estimates of the reaction rates and the non measured component concentrations. The main features of this approach are detailed and illustrated in simulation through a chemical process. 1. INTRODUCTION A systematic and rigorous approach for the on-line monitoring of chemical and biochemical reactors is presented. Two main features of the resulting monitoring approach are worth to be mentioned: 9The control design is carried out using the long range predictive control culture that has been comprehensively compiled by [2]. Of fundamental interest, the design parameter specification is carried out using an iterative procedure that provides an appropriate shaping of the usual sensitivity functions, ensuring thereby the underlying robustness. 9The estimation procedure makes use of recent results on nonlinear observers' design (see e.g. [3]). The corresponding observers are synthesized on the basis of the process mathematical balance models. They do not assume or require any model for the reaction rates and are very successful in accurately estimating these variables. An outline of this paper is as follows: in the next section, we describe the main components of the robust control scheme. In section 3, we introduce (bio)chemical reactor state-space models which are the basis of the nonlinear observers synthesis. Finally, in section 4, the features of the proposed approach are illustrated through a simulation experiment. 2. THE ROBUST PREDICTIVE CONTROL APPROACH
The input-output behaviour of the plant to be controlled is assumed to be approximated using a polynomial approach by the following difference equations :
A ( q - 1 ) y ( t ) -- B ( q - 1 ) u ( t - d
- 1)+ v(t)
and
D(q-1)v(t) = C(q-1)7(t)
652 where q-1 denote the shift operator, A(q-1), B(q-1), C(q -1) and D(q -1) are polynomials in q-l; {u(t)} and {y(t)} respectively denote the control variable and the measured plant output, {v(t) } represents the external disturbances, {3t(t)} is assumed to be a sequence of widely spread pulses of unknown magnitude or random variables with zero mean values and finite variances, d+ 1 denotes the plant model delay in sampling periods.
{eu(t)} and {ey(t)} defined by" eu(t)- u(t)-~A(q-1)y*(t+d+ 1)
It has been shown in [4] that the sequences
ey(t)-- y(t)-~B(q-1)y*(t) and with A* (q-1)y* (t)=B*(q-1)u* (t) represent the input-output errors of the partial state reference model control which is motivated by the ability to specify the tracking and regulation dynamics B* (z-1) denotes the desired partial state referindependently. The pulse transfer function A*(z-1) ence model which provides the reference sequence {y*(t)} from a bounded setpoint sequence 1
{u* (t) } and 13is a scalar which is introduced to ensure a unit static tracking gain, i.e. 13-- B(0)" The underlying control design is carried out using the generalized predictive control approach proposed in [2] and leads to a linear controller that may be written as follows:
S(q-1)D(q-1)eu(t) +R(q-1)ey(t) = 0 with S(q -1) -
(1)
Sa(q-1)Wyd(q-1)Wun(q -1) R(q -1) --Ra(q-1)Wyn(q-1)Wud(q -1) and P,~a(z-1) - Ra(Z-1) '
So(Z-l)
denotes the regulator corresponding to the cascade composed by the plant together with the user defined input and output frequency weighting filters, namely
Wu(z -1) - D(z-1)Wun(Z-1) Wud(Z_ l ) and
Wyn(Z -1 ) ~y(Z -1) --- Wyd(Z_l) " The system input-output behavior is then described by:
y(t) -- ~B(q-1)y*(t) + S(q-1)Vy(t) - T(q-1)Vy(t) d- PS(q-1)(Vy(t) q-Vy(t)) u(t) -- [~A(q-1)y*(t +d+ 1)+S(q-1)Vu(t)- T(q-1)Vu(t) + RS(q-1)(Vy(t) +Vy(t)) where Vu(t) and Vy(t) respectively denote the input and the output disturbances; the terms Vu(t) and Vy(t) are the input and the output noise measurements, respectively; the pulse transfer functions
5(Z -1)
-- A(z-1)S(z-1)D(z -1) Pc(z -1 )
RS(z -1) - A(z-1)R(z -1) -pc(z-l)
T ( z - 1 ) _ z-d-ln(z-1)R(z -1) Pc(z -1 )
pS(z-1) - z-d-lB(z-1)S(z-1)D(z -1) Pc(Z-1)
will be referred to as the usual sensitivity functions. These are genuine performance and stability robustness quantifiers whenever the involved system identification is successfully performed. Their shapes may be refined by properly choosing the predictive control law parameters. The term Pc(q -1) is the characteristic polynomial of the closed loop transfer. Of particular importance, the control system is asymptotically stable if and only if the characteristic polynomial is Hurwitz.
653
3. MODELLING AND ESTIMATION IN 5BIO)CHEMICAL PROCESSES The mass balance equations associated to N components C1,..., CN, which interact through M reactions r l , . . . , rM inside a (chemical) biochemical reactor can be written as follows : (2)
C= Yr(C,t)-DC+DCin-G
where the vectors C, C/n, G E ]RN correspond to the component concentrations, the influent component concentrations and the mass outfow rates in gaseous form, respectively. The N x M matrix Y contains the yield (stoichiometric) coefficients and the vector r E IRM corresponds to the reaction rates. Finally, the scalar D denotes the dilution rate. For biochemical processes, the dynamical balance model is restricted to system (2). In the case of chemical ones, the energy balance is considered and system (2) is augmented by the following equation: TR --
--1-----~AHTr--Q-t-D(TRin-- TR) pRCpR
(3)
where TR and rRin denote the temperature inside and inlet the reactor, respectively. The row vector AH r C IRM contains the reaction enthalpies. The scalars PR and CpR denote the density and the specific heat of the reaction bulk, respectively. Finally, Q is a normalized heat flux. Now, the following system can be adopted as a unified modelling framework for chemical and biochemical reactors:
= K r - D~ + W
(4)
pRcpR Y and ~ A C;
and W =
(
DCin - GQ
DTRin -
)
for chemical reactors
K A y and W A DCin - G for bioreactors
It is well-known that the modelling of r is a difficult and hazardous task. In order to overcome these difficulties, we shall propose nonlinear observers for the on-line estimation of these variables. Such observers will also allow the estimation of the non measured component concentrations. For this aim, the following realistic conditions are considered : (C1) K, D, W are known and the temperature TR is supposed to be measured when chemical reactors are considered. (C2) The time derivatives of the reaction rates are bounded. (C3) There are M-1 component concentrations which are measured. Moreover, if ~ is partitioned into the sets of measured variables ~' and non-measured variables {"" ~ -
~,,
and
654
(K')
accordingly, K - -
K"
' then we have rank(K') -- rank(K) = M.
According to condition (C3), system (4) can be rewritten as follows:
{ ~' = K' r - D~' + W' ~"= K'lr- D~" + W"
(w,)
where
W"
(5)
denotes the partitions of W induced by the above partitions of ~ and K.
Now, on the basis of system (5), we propose the following nonlinear observer-based estimator for the on-line estimation of the rates rj and the non measured component concentrations r from the measured state variables r ,z!
-
+ W'-
30(
' -
- 1] - 302K'-1 ( ~ ' - ~')
-- _03Kt-l(~t_~ t)
(6)
It -
where 0 > 0 is a design parameter and r I denotes the first time derivative of r. 4. SIMULATION RESULTS In this section, the main features of the proposed approach are illustrated in simulation through an example dealing with one reactant chemical reaction held in a batch jacketed reaction calorimeter. It should be emphasized that this example has been chosen mainly for its simplicity and illustrative properties. Nevertheless, the whole procedure previously described can be applied to more sophisticated reaction scheme. The considered mathematical model is :
{ TR-- anr -
~
pRCpR
CA
+ Q
(7)
~" - - r
where r is the reaction rate, CA is the reactant concentration and the other terms keep the same meaning as before. Our control objective consists in tracking a desired profile for the reactor temperature. The plant input and output are the input jacket temperature Tji n and the reactor temperature respectively. While controlling the temperature, our aim also consists in obtaining on-line estimates of the reaction rate r and the component concentration CA from the sole temperature measurements. The obtained results have been compared with data issued from simulation, i.e. we have simulated the process model (7) by considering a first order reaction rate. The term Q and the jacket temperature Tj were obtained by considering the dynamics of the latter. Before being used by the control and estimation algorithms, the reactor temperature (issued from simulation) has been corrupted by an additive noise. It should be also noted that the heat production inside the reactor induced by the exothermic reaction is considered as an output perturbation. For comparison purposes, a PID controller has been synthesized using the relay method [ 1].
655 5
0
o -5
~'~Q" .
COMPLEMENTARY SENSITIVITY
"
I -10
g_
~-15
SENSITIVITY FUNCTION
-25
o~
"350
oo,F,eq,-,,..w (r.,d~)00,5
o~,2
o~
25
/
~
oi,,
0;,5
o~,. . . .
F,eq,~,.-y (~V=)
SENSITIVITY FUNCTION
2O
/
le
i --25
le
~,
g o
TION -lo
CONTROLLER
o~o5
--400
o;,
oo,5
0O25
OO2
F,eq,~rcy (~.d,~)
0005
001 0;15 Frequency (rad/I)
O0 .
.
.
.
Fig. 1. Sensitivity functions (m GPC;-- PID)
015
015
01
005
0
20
40
I
60 Time (ran)
I
80
I
100
120
_0
o51
20
40
60 Time (mn)
80
100
120
32 30
30
2e
28
2e
20
24
24
22
20
le
20
4O
"nm~mn)
O0
Fig. 2. Input-Output performances
IO0
120
18
-
20
40
60 "nine (mr,)
80
100
120
656 550
r (mol.m-3.s -I)
500 450
(mol.m -3)
400 350 30(]
25a 20C
15C 10C 5C
20
40
T~m6~mn)
80
'
~o
0
;o
~.
. . -rim, (mr,) eo
80 .
1o o
Fig. 3. Estimation of r and CA
Figure 1 shows the Bode plots of the usual sensitivity functions corresponding to the robust predictive control and PID control systems. Corresponding input-output temporal behaviors are reported in Figure 2. We remark that the performances of the robust predictive controller are better according to the input behaviour. Indeed, for the PID controller, the gain of the regulator times the sensitivity function is too high to perform well in the presence of measurement noise and unmodelled dynamics. Estimation results are reported in Figure 3 where the estimates of r and CA are compared to their 'true' values issued from the model simulation. We remark the good performances of the estimators in tracking the parameter variations and in dealing with noise rejection (estimated and simulated curves are almost superimposed). We recall that, as for the control scheme, the expression of the reaction rate introduced for simulation purposes is ignored by the nonlinear observer.
5. CONCLUSION An integrated approach for on-line control and monitoring of biochemical processes has been presented. The main features of the proposed approach have been highlighted in simulation through a typical chemical reaction. REFERENCES
1. K.J. ~str0m and T. H~igglund. PID Controllers : Theory, Design and Tuning. Instrument Society of America, 1995. 2. D.W. Clarke, C. Mohtadi, and P. S. Tufts. Generalized predictive control. Automatica, 23(2):137-160, 1987. 3. J.P. Gauthier, H. Hammouri, and S. Othman. A simple observer for nonlinear systemsapplication to bioreactors. IEEE Trans. on Aut. Control, 37:875-880, 1992. 4. M. M'Saad and G. Sanchez. Partial state reference model adaptive control of multivariable systems. Automatica, 28:1189-1197, 1992.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
657
Control Structure Selection for an Evaporation Process Marius S. Govatsmark a, Sigurd Skogestad a* aDepartment of Chemical Engineering, NTNU, N-7491 Trondheim A systematic procedure for control structure selection is applied to the evaporation process of Newell and Lee (1989). First, promising sets of controlled variables are selected, based on steady-state economic criteria. The objective is to find sets of controlled variables which with constant setpoints keep the process close to the economic optimum ("self-optimizing control") in face of disturbances and implementation errors. Second, stabilization and controllability analysis is performed for the most promising sets of economic controlled variables. 1. INTRODUCTION Control structure selection consists of selecting controlled variables, manipulated variables, measurements and links between them. A poor choice can give both dynamic and steady-state problems, such as instability, input saturation, operation outside constraints and non-optimal operation. This can be partly counteracted by using logic, model predictive control and on-line optimization, but the control system then becomes more complicated and costly than necessary. Selecting a good control structure is a precondition for getting a simple control system with good control behavior. Inspired by the many methods presented in the area of control structure selection, both from a steady-state economic point of view and from a dynamic analysis point of view, a procedure is proposed and applied to a simple, but illustrating example of the evaporation process of Newell and Lee (1989). The control structure selection is mainly based on a plant-wide control procedure presented by Larsson and Skogestad (2001). Back-off, Perkins (1998), from the nominal operation is sometimes needed in order to obtain feasible operation. We define backoff (or setpoint adjustment) as the difference between the nominally optimal setpoint and the actual setpoint, used e.g. to achieve feasible operation when there are disturbances. There are two cases: (a) Control of variables at active output constraints. The required back-off is given by the implementation (including measurement) error. (b) Control of unconstrained variables. We adjust the setpoints to achieve feasibility when there are disturbances which otherwise move the active constraints or include a new one. Note that the required back-off and corresponding economic loss depends on the selected controlled variables. Thus, the primary issue is to select the right control structure (variables), whereas the back-off is just a setpoint adjustment to deal with nonlinearities and in particular constraints. Optimal back-off can be determined by robust optimization, which minimizes the nominal steady-state economic criteria, given that the constraints are satisfied for all expected disturbances and implementation errors, see Glemmestad *e-mail: [email protected];phone: +47-7359-4154; fax: +47-7359-4080
658 .L ~ |
~
,
,
,
',F~---,
T ~ v ,,;T:--!P ~__~
FlOO
Cooling water
| Condensate .I ~ Separator F
~ ~3
B
%oo
EvapoP'rat~~~
F
C o n d e n s a t e ~2 -~,~" Fe.Pump ---~_..
y" L Fx T
Fig. 1. Evaporation process
:~--~'I"F
08
,
F', FF,G
,
9
F 1 [~g~min]
11
12
royct F2x2T2
Fig. 2. Loss as function of feed flowrate for different alternatives with robust setpoints and simple logic
et al. (1999). 2. CONTROL STRUCTURE SELECTION PROCEDURE A short summary of methods and rules recommended in control structure selection, Larsson and Skogestad (2001): 1) Formulate a steady-state economic objective. 2) Define control objectives, steady-state and dynamic degrees of freedom, manipulated variables with allowed variations, disturbances with expected variations and possible controlled variables with expected implementation errors. 3) Steady-state optimization should be performed at the nominal point and, if possible, for extreme values of expected disturbances and implementation errors. 4) In order to reduce the number of sets of controlled variables to be evaluated in detail, use active constraint control, see Maarleveld and Rijnsdorp (1970), and the minimum singular value rule, see Skogestad (2000). 5) For the remaining combinations of controlled variables the economic loss imposed by keeping the sets of controlled variables constant (rather than at their optimal values) should be evaluated for expected disturbances and implementation errors. 6) To avoid infeasibility, try back-off from nominally optimal setpoints. 7) For stabilization select the variables to control and manipulate based on computing the poles and their associated input and output directions, see Havre (1998). 8) Perform controllability analysis for the most promising sets of economic controlled variables. Tools are provided by Skogestad and Postlethwaite (1996). 9) Design controllers and run nonlinear simulations for the most promising alternatives. 3. EVAPORATION PROCESS CASE STUDY In the evaporation process of Newell and Lee (1989) the concentration of dilute liquor is increased by a vertical heat exchanger with recirculated liquor, see figure 1. The steady-state economic objective is to minimize the operational cost (S/h) related to steam, cooling water
659 and pump work, see Wang and Cameron (1994): Y - 600F10o + 0.6F2oo -+- 1.009 (F2 + F3 )
(1)
Process constraints related to product specification, safety and design must be met: X2 > 35%
(2)
40kPa _< P2 < 80kPa
(3)
PlO0 _< 400kPa
(4)
F2oo _< 400kg/min
(5)
0kg/min < F3 _< 100kg/min
(6)
There are four manipulated variables; steam pressure, cooling water flowrate, recirculating flowrate and product flowrate: u T = [F2oo/9100F3 F2]
(7)
One degree of freedom is purely dynamic (the condenser level which needs to be stabilized), hence there are three steady-state degrees of freedom. The major disturbances are feed flowrate, feed concentration, feed temperature and cooling water inlet temperature, with expected variations about +20%: d T = IF1 Xl I"1 T20o] = [10 + 2 kg/min
5 + 1%
40 + 8~
25 + 5~
(8)
Controlled variable candidates are all possible measurements and manipulated variables: yT = [F2 F3 F4 F5 X2 T2 T3 L2 P2 F100/91o0 Qloo F200 T201 Q200]
(9)
Expected implementation error associated with each variable is based on the following rules: Flowrates (+10%), compositions (• temperature (+I~ and pressure (• The model equations are given in appendix A. 3.1. Steady-state economic evaluation and optimization The steady-state optimal values at the nominal point for the objective function is 61625/h and corresponding to the following optimal values: T
Yopt
--
[1.4 27.7 8.6 8.6 35.0 90.9 83.4 1.0 56.2 10.0 400.0 365.6 230.2 45.5 330.0]
(10)
Steady-state optimization at the nominal point and for extreme values of the disturbances yield that two of the constraints, product composition (x2) and steam pressure(P100), are always active.Active constraint control then consumes two steady-state degrees of freedom. The last degree of freedom is optimally unconstrained for most disturbances (There is one exception, for low feed flowrate the last degree of freedom is consumed by the minimum operating pressure constraint). The best economic choice for this controlled variable is related to the selfoptimizing control properties. There are 13 candidate variables. The minimum singular value rule, see Skogestad (2000), is applied to eliminate some of them. For one single input the rule is to select the controlled variable with the largest absolute process gain (IGI), when the variables are scaled with respect to the sum of the variation in their optimal value and the expected implementation error. The seven most promising combinations of economic controlled variables are
660
Table 1 Most promising alternative sets of controlled variables based on the minimum singular value rule Ys,3 IG[ Rank Alt. Ys,1 Ys,2 1 G 22 el00 T201 0.0150 2 FF X2 el00 F2oo/F1 0.0135 3 F x2 PlO0 F2oo 0.0108 4 C x2 PlOO P2 0.0044 5 A x2 Plo0 T2 0.0042 6 H x2 P10o T3 0.0042 7 B x2 /10o F3 0.0018 -
E
X2
P2
F3
-
Table 2 Average economic loss with optimal back-off to avoid infeasibility Rank Alt. Ys,! Ys,2 Ys,3 Average (X2) (P100) loss 1 F* 36 390.0 231.2 0.55% 2 G 36 390.0 48.0 0.59% 3 FF 36 390.0 20.6 0.60% 4 E* 36 57.3 41.0 0.91% 5 H 36 390.0 90.3 1.24% 6 C 36 390.0 69.6 1.24% 7 A 36 390.0 98.7 1.24% 8 B 36 181.0 1 3 4 . 4 3.24% F Infeasible E Infeasible *=With use of logic
listed in table 1. Here we have also included a feed-forward improvement of alternative F (denoted F F ) . In addition we study alternative E, proposed and used by Newell and Lee (1989). In this alternative the recirculating flowrate (F3) is not used as a manipulated variable in the basic control layer. They do not use active constraint control, because the last available manipulated variable, cooling water outlet flow (F2oo), has too small effect on the product composition (x2). The steam pressure (Ploo) is not kept on its constraints, but is used to control the product composition (x2). The cooling water flowrate (F200) is used to control the operating pressure (P2). To achieve feasibility, we compute the optimal back-off from the nominal optimum setpoints by robust optimization(Glemmestad et al. (1999)). The losses related to keeping the operation at these robust setpoints are given in table 2 and in figure 2. There exists no feasible setpoint adjustment (back-off) for alternatives E and F. Alternative F has feasibility problems only at low feed flowrates when the operating pressure (P2) gets too low. To avoid infeasibility a simple logic scheme can be introduced: If the feed flowrate becomes lower than 8.5 k g / m i n , the cooling water flowrate is reduced to avoid that the operating pressure becomes too low. For the proposed structure of Newell and Lee (1989) (alternative E) several constraints may be violated, and the logic becomes more complicated. For large feed flowrates and a high inlet cooling water temperature the cooling water flowrate should be kept constant instead of the operating pressure (P2). 3.2.
Stabilization
and
controllability
analysis
The rest of the analysis is based on a scaled linear, dynamic model. The holdup in the separator must be stabilized. Based on computing the poles and their associated input and output directions we find as expected, that the separator level must be controlled by the product flowrate. Controllability analysis is performed for the three most economic promising alternatives F, F F and G, plus alternative E. The alternatives are largely controllable. The relative gain array (RGA) is used to select links between the controlled and manipulated variables, see table 3. Alternative F F and G are shown in figure 3 and 4. Decentralized controllers were designed, and nonlinear simulations were performed to verify the controllability of the designs.
661 Table 3 Control structure used in decentralized control Alt.
Loop 1
Loop 2
E* F* FF G
x2 ~ PlOO x2 ~ F3 x2 4-~ F3 x2 ++/73
P2 ~ / 7 2 0 0 PlOO P100 P100
Loop 3
F3
F200
Qoo/FI
T201 ~-~ F200
Loop 4** L2 L2 L2 L2
* = With use of logic, ** = Stabilizing loop
++ F2 ~-~ F2 ++ Q ~ F2
(~-~t, ...."B~ " ................................................................. 17~:,,~r .. ....
:vo..........-~--~-,FC~---,, Condenser ' Cooling water
Cooling water
T ~ 2o~
T~
-F T Condensate
P '~" i ~
Steami ,~
~
Foo~oo
~ ' Condensate A A[ Separator F L'! L~,
~i o
F3"
Eva
.
Cor~densate ~ F i i ' m - ~ Feed
~ ~ ,
E vap~176
,~ .... ::
~(21
[I II I F,.
l
i.
:x
C o n d e V n s a t e ~ ,_~ihj-purnp ...........!-.......!~ Product
F~
FxT
Fig. 3. Evaporation process with control structure FF
FxT
FxT
Fig. 4. Evaporation process with control structure G
4. C O N C L U S I O N A systematic procedure for control structure selection, based on both steady-state economic evaluation and controllability analysis, has been demonstrated. The evaporation process example has been used to illustrate how important the selection of control structure is and how rewarding it may be. By putting serious effort in selecting a good control structure, it is possible to avoid significant control problems and reduce the complexity of the control system. Both alternative F F and G have small economic loss, good control behavior and no need of logic. In the end alternative G is proposed, because it is based on pure feedback control. A. M O D E L E Q U A T I O N S
20dL/dt 20dxz/dt
- - F1 - F4 - F2 -
Fix1 - F2x2
4dPz/dt =
F4 - F5
T2 = 0.5616P2 + 0.3126x2 + 48.43 T3 -- 0.507P2 -+- 55.0
(11) (12) (]3) (14) (15)
662 T100 = 0.1538P100 + 90.0
(16)
aloo = 0.16(F1 + F3) (Tloo - T2)
(17)
Q l o o - 36.6Floo
(18)
Qloo -- 38.5F4 q- 0.07Fl (T2 - T1)
(19)
Q2oo -- 6.84(T3 - 0.5 (T2oo+ T201 ))
(2o)
Q200-- 38.5F5
(21)
Q200 -- 0.07F200(T201 - T200)
(22)
Newell and Lee (1989) assumed the time lag connected to the slave controllers to be 1.2 min. This seems too large, and we use 0.1 min.
REFERENCES Glemmestad, B., S. Skogestad and T. Gundersen (1999). Optimal operation of heat exchanger networks. Computers Chem. Engng. 23, 509-522. Havre, K. (1998). Studies on the controllability analysis and control structure design. PhD thesis. NTNU. Available from http://www.chembio.ntnu.no/users/skoge/. Larsson, T and S Skogestad (2001). Plantwide control: A review and a new design procedure. Modeling, Identification and Control 22(1), 1-32. Maarleveld, A and J.E. Rijnsdorp (1970). Constraint control on distillation columns. Automatica 6(1), 51-58. Newell, R.B. and P.E. Lee (1989). Applied Process Control - A Case Study. Prentice Hall. Perkins, J.D. (1998). Plant-wide optimization: Opportunities and challenges. FOCAPO III, Snowbird, Utah pp. 15-26. Skogestad, S. and I. Postlethwaite (1996). Multivariable Feedback Control - Analysis and Design. pp. 246-248. John Wiley & Sons. Skogestad, Sigurd (2000). Plantwide control: The search for the self-optimizing control structure. Journal of Process Control 10(5), 487-507. Wang, F.Y. and I.T. Cameron (1994). Control studies on a model evaporation process - constraint state driving with conventional and higher relative degree systems. Journal of Process Control 4(2), 59-75.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
663
MPC using Nonlinear Models Generated by Genetic Programming Benjamin Grosman and Daniel R. Lewin PSE Research Group, Dept. of Chemical Engineering, Technion I. I. T., Haifa 32000, Israel This paper describes the use of genetic programming (GP) to generate an empirical dynamic model of a process, and its use in a nonlinear, model predictive control (NMPC) strategy. GP derives both a model structure and its parameter values in such a way that the process trajectory is predicted accurately. Consequently, the performance of the NMPC strategy, based on this model, is expected to be good. The genetic programming approach and the nonlinear MPC strategy are briefly described, and demonstrated by simulation on a multivariable process. 1. I N T R O D U C T I O N All control systems are either implicitly of explicitly model-based. They can be as simple as a PI controller, which is implicitly based on a first-order lag model of the process, or as sophisticated as a set of DAEs that model the process and are solved in a nonlinear predictivecontrol algorithm. The performance to be expected from a model-based controller is directly related to the fidelity of the model, in the frequency range where performance is sought. However, generating an accurate model is sometimes a difficult task. The development of first-principles models can be expensive and often leads to disappointing results, since it is hard to capture all relevant phenomena in the model. In contrast, empirical models are relatively easy to calibrate, but rely on model structures often selected by intuition, and may require repeated updating because of changing process conditions. Genetic programming (GP) is an evolutionary method for automatically generating nonlinear input-output models. In this work an improved Genetic Program is developed for generating models. The algorithm relies on previously published work, but involves novel methods to assess model fitness accounting both for modeling and prediction error, as well as efficient population management, and control of constant assignment for each candidate model depending on its degrees-of-freedom. These features significantly improve the predictive power of steady-state models generated by our algorithm. In this contribution, the novel GP code is used to generate empirical non-linear dynamic models for control purposes. More specifically, the models so derived are to be used for nonlinear MPC. The performance of the GP-based MPC is compared with that of a conventional linear PI control strategy, when applied to a simulated multivariable process. This paper begins by describing the GP used for model derivation, and the implementation of the nonlinear MPC algorithm. Preliminary results of testing the algorithm on a relatively simple MIMO process are given.
664 2. G E N E T I C P R O G R A M M I N G Genetic programming (GP) is an evolutionary method for automatically generating nonlinear input-output models. The probability of a given model surviving into the next generation is proportional to how well it predicts the output data, and components of successful models are continuously recombined with those of others to form new models. The GP thus optimizes the model structure, with a lower level nonlinear least-squares algorithm harnessed to perform the associated parameter estimation. Gray et al. (1996), McKay et al. (1997) and Willis et al. (1997) have discussed the application of genetic programming to nonlinear modeling. Lakshminarayanan et al. (2000) used a composite GP-PCA approach to generate nonlinear models for product design applications. GP is based on simple rules that simulate biological evolution. Combining basis functions, inputs and constants creates an initial model population, whose complexity is controlled by the user. The models are structured in a tree-like fashion, with basis functions linking nodes of inputs and constants. Each individual model in the population is then fitted to the empirical data using nonlinear regression, and then graded according to how well it matches the data. In each generation (iteration) of the algorithm, relatively successful individuals are selected as "parents" for the next generation and form a reproduction pool. A new generation of solutions evolves, using one of three possible operations: crossover, mutation and permutation. In crossover, two individuals from the pool are selected, their tree structures are divided at a randomly selected crossover point, and the resulting sub-trees are recombined to form two new individuals. In mutation, a random change is performed on a selected individual by substitution - this can be a functional group, an input variable or a constant. In permutation, two branches of a selected individual are randomly switched. The parameters for each new individual in the new generation are regressed, the models are then graded by fitness as before, and the procedure is repeated until a stopping criteria is attained. In most cases, as in this application, the population size, npop, and the total number of generations, ncen, are decided in advance. The GP code used in this work is similar to that published by McKay et al. (1997), with some important differences: One. Overfitting-controlledfitness evaluation. The most important difference concerns how the fitness is computed, which should be a combination of the quality of the model to match the empirical data with some penalty to guard against over-complex models. In contrast with previously published work, our code preserves a portion of the data to test predictive power of the model, and uses the prediction error to dynamically limit model complexity during the course of evolution. This method successfully prevents over-fitting without requiring an arbitrary cut-off length to be specified a priori. Two. Smart control on new generation synthesis. The second difference is related to how a new generation is created from its parent population. Most codes adopt so-called patrimony, creating the next generation by keeping some fixed part of the parent population, usually the better solutions, and filling the balance from evolutionary operators applied to the reproduction pool. This approach is expensive, since no guarantee can be made that the evolved children are always superior to those parents they replace. Instead, our approach is to create npor new individuals as above, making temporarily a total of 2 npop individuals, and pick the best npop of these to constitute the next generation. To make this approach work well, it is also necessary to identify repeated solutions, and eliminate those from the gene pool. This requires tree-structure recognition capability.
66~ Three. Smart tree-structure recognition capability. To identify critical sub-tree structures and to eliminate repeated solutions from the population, it is necessary to be able recognize tree-structure components. For example, if n is the number of addition and multiplication nodes in a tree, then there are 2" equivalent trees possible. Thus, a check is performed on each newly generated model to ensure that it is not equivalent to one already existing in the population, and if so, is marked for deletion Four. Setting the number of constants dictated by model degrees-of-freedom. To ensure the parametric optimizer can exploit the degrees of freedom in a model, the code considers augmenting the model with additional constants. Good results have been obtained using our code to identify steady state models in a number of applications. The same principle used to generate steady-state models is also employed for identifying dynamic models. The main difference lies in the way the prediction and the optimization are realized. Instead of developing a conventional one-step-ahead predictive model, the complete trajectory is estimated assuming only the initial condition for the output. For this computation, pseudo-random sequences of steps in the inputs are used to excite the process, with two sets generated - one for modeling (used to optimize the model parameters) and one for prediction (used for verification of the optimized model). The weighted sum of the trajectory-matching performance obtained with both sets is used to compute fitness. 3. NONLINEAR MODEL PREDICTIVE CONTROL (NMPC) The nonlinear model predictive control strategy implemented here consists of several components: 1. Each MISO input-output model, created by the GP, which takes the general form: ~(k)= f ( .~(k -11 k),u_(k -11 k)),
(1)
where ~(k) is the predicted model output at instant k, ~(k- 11k) is a vector of the previous P predicted outputs, computed based on information available until instant k, and u_(k- I l k ) is a vector of P inputs at instant k. In principle, the approach also handles nonlinear state-space models. 2. Selected values for P, the prediction horizon that controls the predictive range of the model, and M the control horizon, which establishes the number of future moves to be optimized, noting that M __P. 3. A matrix of historical data, A, used by the model to predict the influence of previous outputs and inputs on the future outputs. The matrix has P rows and a number of columns equal to the total number of inputs and outputs, and is refreshed every sample instant. 4. A constrained quadratic objective function to be minimized, whose arguments are calculated on the basis of predicted values of inputs and outputs generated using the nonlinear GP models, resulting in a sequence of M optimal future inputs. The objective function used in this work is: (2)
666 where
~(k + l l k ) is a vector holding the estimated output trajectory P moves ahead com-
puted at instant k,
Ys,correctediS
a vector holding the corrected desired trajectory,
Au T (k + 11 k) is a vector of M future manipulated variable changes, optimized at instant k, and Q and S are tunable weights. Because of the inevitable model-plant mismatch, the actual set points used in Eq (2), YS,corrected,if left uncorrected, will lead to optimal control only in the open-loop sense. Thus, the set points implemented use current process outputs as feedback, which is equivalent to integral action, and allows the plant/model mismatch to be estimated: A
d(k)= y(k)- ~v(k),
(3)
where y(k)and ~(k)are the process and model outputs at instant k, respectively. This mismatch is assumed constant for all future predicted values, and leads to a correction to the desired set point values: Y s,corrected
(k)= Y S (k )- ~(k )
(4)
Naturally, there are upper and lower constraints on the manipulated variables, 0 < U _<1. To ensure a realistic control response, it is also assumed that the rate of change of manipulated variables is limited, IAU[ < 0.1. The formulation also supports linear and nonlinear constraints on the outputs, and for this application, it is assumed that both outputs are also constrained. This nonlinear MPC scheme is similar to that outlined by Henson (1998), with the main difference being the GP model used to predict the future outputs. Our implementation is carried out with MATLAB 5.3, using our GP code for empirical modeling and MATLAB' s fmincon for constrained optimization.
4. APPLICATION OF G P - N M P C ON T H E C O N T R O L OF A M I X I N G T A N K 4.1 Process description and modeling Preliminary testing of the method is carried out using a simulated cylindrical mixing tank, consisting of two feed flows: one of fresh water and the other of saturated salt water. The objective is to ensure the fluid level in the tank and the effluent salt concentration are at the desired set points. The mathematical model is based on the following two assumptions: 1. The outlet flow behaves as
Uout = ko~/-H, where
k 0 is an empirical constant.
2. The tank is assumed to be perfectly stirred. Thus, two nonlinear ODE's describe the process dynamics, derived to describe the total and salt mass balances:
A dH =Qw+Qs_ko.x[-ff dt A d(C. H)=aw "Cw +Qs" Cs -kox[-ff C dt
(5a) (5b)
In the above equations, H is the fluid level in the tank, C is the tank salt concentration, A is the cross-sectional area of the tank (=1), Qw is the sweet water flow; Qs is the salt water flow, Cw is the sweet water salt concentration, (=0); C~ is the salt water concentration (=1); ko
667 is the valve constant (=1). Since Qw and Qs are normalized to lie between zero and one, the normal operating ranges of the concentration and the height are also in this range. Thus, a level value in excess of unity implies overflow of the tank, and indicates catastrophic loss of control.
4.2 Creation of the state-space model using GP Discrete input-output models are generated to allow the prediction of level and concentration trajectories using the GP. Delayed outputs ( k - 1, k - 2, k - 3) and inputs (k, k - 1, k 2, k - 3 ) are presented to the GP for predicting y(k). Pseudo-random sequences of steps in the two inputs are used to excite the process, with two sets generated - one for modeling and one for prediction, as stated previously. The sampling time was selected to be 0.15 time units, bearing in mind the desired closed-loop response. The use of trajectory matching for both modeling and prediction means that our approach is relatively insensitive to the sampling rate selected, which is one of its advantages. The models that scored the highest are:
C(k ) - 0.818. C(k - 3)+ 0.134. Q,. (k - 1)- 0.078. Qw(k - 2)+ 0.68 H (k)= 0.890. H (k - 1)+ 0.170. Q,. (k)+ 0.167-Qw (k)- 0.061
6(a)
6(b) It is noted that both models are linear, which is not surprising, noting that the actual process is itself almost linear. This suggests that a linear MPC strategy would be expected to do well on this process, which has been confirmed by simulation.
4.3 Comparison of the GP-NMPC with PI Control A commonly used approach to multivariable control system implementation is decentralized PI control, often selected because it is easily implemented. Thus, the performance of the GP-NMPC approach is compared here to that of a decentralized PI control system, with pairings: [ H - Q w , C - Q s ] , with IMC-based tuning (Rivera et al, 1986). Here, we report two representative tests that indicate the regulatory and servo performance of the controllers. Regulatory performance: A pulse of 40% in the influent salt-water flow is invoked, when the process is at the steady state [H(0), C(0)] = [0.5, 0.5]. The GP-NMPC is set up with a prediction horizon of 10 and a control horizon of 5, with all weights in the quadratic function set to unity. The response of the GP-NMPC and the PI control configurations are shown in Figure 1. It is noted that GP-NMPC provides almost perfect disturbance rejection, whereas the response using the decentralized PI controller is oscillatory, indicative of the strong coupling in the system. Detuning the PI controllers reduces the intensity of the oscillations, but at the price of increased settling time. Servo performance: A setpoint change in both outputs is commanded, from the steady state [H(0), C(0)] - [0.2, 0.2] to values at [0.9, 0.9], noting that these are close to the upper constraints of the process. The GP-NMPC is set up as in the regulatory test, with the responses of the GP-NMPC and the PI control configurations shown in Figure 2. The GP-NMPC provides rapid acquisition of the targets, while satisfying the process hard constraints. In contrast, the decentralized PI controller, again strongly oscillatory, violates the upper level constraint. 5. C O N C L U S I O N S This paper has shown that the combination of GP and NMPC provides a means for the rapid commissioning of a nonlinear MPC strategy. The results have shown that the GP-
668 NMPC approach provides significantly better regulatory and servo performance than decentralized PI control. Rapid acquisition of an empirical nonlinear model is achieved efficiently using genetic programming. This model provides reliable prediction of future output trajectories in the NMPC scheme, which also accounts for both process interactions and constraint violations, and thus allows the computation of improved control moves. Currently, work is in prog??ress on the application of the approach on a more complex MIMO process, a simulation of a Karr liquid-liquid extraction column (Weinstein et al., 1998). Our preliminary results indicate the importance of non-linear models for improved MPC for this example. (a) G P - N M P C
(a) G P - N M P C - -
H
..........
C 0.5
i-I'P s ~," "~ ~
...........................
Qw
Qs
"':': r
~
0.3 0"1 i~" iI "~"~'~ 'i;~'i "'~ 4
0
0.7
~
,
6
(b) Decentralized PI control , , '
8
10
,
".. -.. -%...
0~ 0
2
1|
..
-~.,
.: ! .:
0.5 .
0
2
4 6 Dimensionless Time
8
Figure 1 Regulatory performance of the NMPC and PI controllers,
10
0
4
/ i
/
6
8
(b) Decentralized PI control 9 ' t " " ", '
:.. :
._.m.--
.,._.
10
' "
;...r."
. . , r-~."
2
4 6 Dimensionless Time
8
10
Figure 2: Servo performance of the NMPC and PI controllers.
ACKNOWLEDGEMENT
The authors are grateful for the support of the Mitsubishi Chemical Corporation. REFERENCES
1. Gray, G. J., D. J. Murray-Smith, Y. Li and K. C. Sharman (1996). "Nonlinear Model Structure Identification using Genetic Programming and a Block Diagram Oriented Simulation Tool", Electronic Letters, 32, 1422-1424. 2. Henson, M. A. (1998). "Nonlinear Model Predictive Control: Current Status and Future Directions", Comput. Chem. Engng., 23(2), 187-202 3. Lakshminarayanan, S., H. Fujii, B. Grosman, E. Dassau and D. R. Lewin (2000). "New Product Design via Analysis of Historical Databases", Comput. Chem. Engng., 24(2-7), 671-676 4. McKay, B., M. J. Willis and G. W. Barton (1997). "Steady-state Modeling of Chemical Processes using Genetic Programming", Comput. Chem. Engng., 21(9), 981-996. 5. Rivera, D. E., S. Skogestad and M. Morari (1986). "Internal Model Control. 4. PID Controller Design", I&EC Proc. Des. Dev., 25, 252-265. 6. Weinstein, O., R. Semiat and D. R. Lewin (1998). "Modeling, Simulation and Control of Liquid-liquid Extraction Columns", Chem. Eng. Sci., 53(2), 325-339 7. Willis, M. J., H. Hiden, M. Hinchliffe, B. McKay and G. W. Barton (1997). "Systems Modeling using Genetic Programming", Comp. Chem. Engng., 21(S), S 1161 -S 1166.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
669
Wavelet Shrinkage Based Covariance Matrix Estimation from Process Measurements Yuanjie Huang, G. V. Reklaitis 1 and Venkat Venkatasubramanian Email: (yuan jie, reklaiti, venkat) @ecn. purdue, edu School of Chemical Engineering, Purdue University, W. Lafayette, IN 47907-1283, USA
1. I N T R O D U C T I O N Measurements of process variables are subject to both measurement errors and process variabilities. Such errors may be random or biased. Thus, before the measurements can be used, the errors should be removed and the corrupted measurements corrected. This is usually done through data reconciliation and rectification. Data reconciliation is the procedure of optimally adjusting measured data so that the adjusted values satisfy the conservation laws and other process constraints, and is often implemented by minimizing a quadratic form of the difference between the measurements and the estimates subject to the process constraints, which are generally non-linear. Most of the reported data reconciliation techniques (Mah, 1987; Tamhane and Mah, 1985; Crowe, 1996; Bagajewicz and Jiang, 1998) assume that the measurement errors are Gaussian distributed, and the covariance matrix Q is known. However, in practical situations, Q is unknown or known at most approximately. In this paper we present a method based on wavelet threshold shrinkage for estimating Q from the process measurements. Let x(1),x(2),...,x(n) be the n measurements, each measurement x(i) includes m process variables. The basic measurement model is assumed to be of the following form:
x(i)=x*(i)+e(i)+8(i)
i= l,2,...,n
(1)
where x*(i) - (X*l(i),x~(i),...,Xm(i)) T is the true value, e(i) - (el (i),e2(i),...,em(i)) T is the Gaussian measurement error with zero mean and covariance matrix Q and 5(i) = (81 (i), 82(i), 9.., 8m(i))r is the gross error vector in which each element is a step function. If the process is truly at steady state during the time interval in which all n measurements are taken, and the measurements do not contain any gross errors, then the sample variance of the n repeated easurements of xi is adequate and easy to compute (direct method). In real processes, generally the steady state values are under variations during the measurements are taken, which makes the estimate from direct method misleading. In order to overcome this drawback various indirect methods which make use of the covariance matrix of the constraint residuals are proposed. In (Almasy and Mah, 1984), the authors presented an indirect method of estimating the covariance matrix of the measurement errors from the covariance matrix of the constraint residuals under the assumption that the covariance matrix Q is diagonal or diagonally dominant. However this estimate is very sensitive to the correlated easurements. (Keller et al., 1992) derived a least squares method for estimating a diagonal variance matrix from the sample covariance 1Author to w h o m all correspondence should be addressed
670 matrix of the constraint residuals when the square of the number of balances exceeds the number of measured variables. The authors also extended their method to estimate the non-diagonal Q with known locations of non-zero off-diagonal entries. Another extension of this work was given in (Chen et al., 1997), in which the authors presented a robust estimation technique based on an M-estimator, that is insensitive to the outliers. However, in real situations, the locations of the non-zero off-diagonal elements are usually unknown. In this paper we present a new method for estimating the covariance matrix. This method is based on the wavelet threshold shrinkage theory and can be used to estimate Q from both steady state and dynamic measurements or even measurements corrupted with gross errors. As a result it is possible to avoid the procedure of checking whether the process is in steady state or not when the measurements are taken. 2. WAVELET BASED ESTIMATION For the following non-parametric regression model: iid
yi= f(ti)+~Zi, zioN(O, 1), i-- 1 , 2 , . . . , n
(2)
the objective of function estimation is to recover the unknown function f(t) (process variable, signal, image, etc.) from the samples with small errors. The quality of the estimate f is defined by the mean square error (MSE):
R ( f , f ) - E II f
--
f
II22
(3)
For practical use, a good estimator should be: adaptive that is the estimator should be able to automatically adapt to the smoothness of the underlying functions; spatially adaptive that is the estimator should also capture the spatially inhomogeneity of the underlying functions; computationally efficient that is the estimator should be feasible for a large set of data; and asymptotically optimal that is the estimator should have minimum estimation errors. Wavelet estimation methods remove noises through thresholding of the empirical wavelet coefficients obtained from the Discrete Wavelet Transform (DWT) or the Stationary Wavelet Transform (SWT). The basis functions, or wavelets possess the following three properties. Localization: Local features of a function are captured by its local wavelet coefficients. Efficient Representation: Wavelet transform compacts the energy of a function into a small number of (relatively large) wavelet coefficients. Unconditional Basis" Wavelet bases are unconditional bases for a wide range of function spaces. This makes wavelets the optimal bases for data compression and statistical estimation (Donoho, 1993). As a consequence of these properties the estimation techniques for model (2) based on the thresholding of the coefficients obtained from the expansion of the noisy function in some wavelet function basis are spatially adaptive (Donoho and Johnstone, 1994), computationally efficient and asymptotically optimal. The steps involved in a wavelet based estimation method are as follows: 1. Transform the noisy measurements Yi into the wavelet domain via a DWT or SWT:
ff~=W.y where y = (yl,yz,...,y,) r, ff~ = (gjo,1,...,gjo,Zjo,djo,l,...,djo,Z~O,...,dj_l,zJ-1) empirical coefficients of the noisy data, and W is the transform matrix.
are the
671 2. De-noise the noisy empirical detail wavelet coefficients via hard or soft thresholding:
where ~ is the threshold. And the shrinkage rule: fla.(y) _ { Y" I(lyl > ~)
hard shrinkage
sgn(y). (lyl- 7~)+ soft shrinkage
3. Estimate the whole function f by J-1
L(,)- Eejo, ,jo, (t)+ E EdJ, vJ, (t) k
j=Jo k
3. WAVELET SHRINKAGE BASED VARIANCE ESTIMATION
In the measurement model (1), x* (t) and x* (t) + 5(t) are generally much smoother than e(t) even when the process is dynamic. This makes it possible to extract the random information from the measurements and then use the direct method to estimate the covariance matrix of the random errors. Or equivalently we can treat x* (t) + 8(t) as the true value of x(t), thus the measurement x(t) only contains Gaussian noise. Then appropriate de-noising/non-parametric function estimation techniques can be used to extract the random errors. Since measurement x(t) may contain gross errors which will add spatial inhomogeneity to the underlying function x* (t) traditional non-parametric function estimation techniques will not work well in this case. From the discussion in section 2 that wavelet shrinkage based estimation techniques are well fitted to this problem. Since e(t) is a multivariate normal distributed vector. Each element in vector e(t) is also a Gaussian random variable by definition. For data contaminated with Gaussian noises wavelet based de-noising methods favor the properties mentioned in section 2. To estimate the error covariance matrix Q, first the true values ofx* (t) + 5(t) are estimated using wavelet thresholding methods given in section 2, then the estimate of the random errors ~(t) are obtained by taking the difference between x(t) and x* (t) + ~5(t). Finally, Q is calculated as the sample covariance of ~(t). 4. NUMERICAL EXAMPLES
In this section, three examples are presented using the above mentioned method. In each case, 2048 measurements are used for estimating Q. To eliminate the "edge-effect" of wavelet de-noising, which is caused by using periodic wavelets for non-periodic underlying function estimation, an easy and effective way is to throw away some of the data at the two ends of the estimated ~(t). For both examples, Donoho and Johnstone's VisuShrink with hard thresholding function is used. The used thresholds in these examples are computed automatically in the VisuShrink method. Example I---Diagonal First the proposed method is applied to a diagonal case investigated by Almasy and Mah (1984). The process network is shown in figure 1. The true values of the
672 diagonal elements used are: 4, 36, 36, 4, 4 and 4. The estimated Q is: 4.24 -0.39 0.29 0.04 0.12 0.05
-0.39 36.60 0.88 0.24 0.03 0.22
0.29 0.88 35.12 0.13 0.38 -0.01
0.04 0.24 0.13 3.81 0.06 0.04
0.12 0.03 0.38 0.06 3.98 0.03
0.05 0.22 -0.01 0.04 0.03 4.10
This estimate of Q shows that the relative errors of the non-zeroes are less than 6%, and the mean and deviation of the noise in the zero entries are 0.14 and 0.26.
v
. . . . .
Fig. 1. Process network 1
f - -
.
.
.
.
.......A'o -~12
Fig. 2. Process network 2
Example 2--Non-diagonal The process network is shown
in figure 2. The diagonal elements of Q are taken to be (30,30,20,20,7.5,15,10,10,10,8.1,20,20) T, off-diagonal non-zero elements are Qa,5 - Q5,2 - 15, Q7,10 - Q10,7 - 9. The true values of the measured variables are
673
undergoing linear increase. The strong correlations between the measurement errors in stream 2 and 5 and 7 and 10 will lead to large estimation errors if the indirect method given in (Almasy and Mah, 1984) were used as shown by the authors there. Due to lack of space, the complete estimated Q is not listed here. The maximum error of the non-zero entries is 4.7%. The mean and deviation of the zero entry estimates are 0.17 and 0.33. Example 3 - - M e a s u r e m e n t s Contain Gross E r r o r s For the system of case 2, gross errors in the measured data of stream 1 and 2 are introduced. Both gross errors are step increases occuring at the middle of the sample time. For such cases, the conventional indirect methods will give misleading estimate as shown in (Chen et al., 1997). To give an idea how wavelet shrinkage method removes the noise in the measured signal. The stack plot and DWT of the measured flowrate of stream 1 are shown in figure 3. The estimated covariance matrix of the measurement (random) errors Q is very close to the true covariance matrix Q as of: the maximum error of the non-zeroes is less than 5%, and the mean and deviation of the zeroes are 0.18 and 0.33. Although the robust estimator technique (Chen et al., 1997) gives a good estimation of Q in this case, the method depends on the knowledge of the locations of the non-zeroes in Q.
Data Signal
_f
t,__.
Resid 0
idwt dl d2 d3 d4 d5 d6 s6
500
1000
..........
.
.
.
.
.
.
.
1500
2000
1
,
.
,
,
|
1
i.
.
.
.
.
.
.
.
.
.
.
.
I'
I
[
= 0
.
"' ,
,
,
. . . . . . . . . . . 500
.
.
.
i i
' 1
,
,
I
J
, , , ~ i l i l l i l l l i l l l i l 1000
1500
2000
Fig. 3. Stack and DWT plots of stream 1
5. C O N C L U S I O N In this paper a new method for estimating the covafiance matrix of the measurement errors is presented. The method extracts the random measurement errors contained in the measured data
674 using wavelet estimation techniques. The method can be used even when the measurements are sampled during dynamic transition period, or the data contains gross errors. The results of the numerical examples show that this wavelet shrinkage based method is robust to the model constraint residuals and gross errors, and in effect for both uncorrelated and correlated measurements.
REFERENCES Bro, R. (1998). Multi-way Analysis in the Food Industry. Academish Proefschrift, Universiteit van Amsterdam. Almasy, G. A. and R. S. H. Mah, Estimation of measurement error variances from process data. Ind. Eng. Chem. Process Des. Dev., 23(4), 779-784 (1984). Bagajewicz, M. J. and Q. Y. Jiang, Gross error modeling and detection in plant linear dynamic reconciliation. Comput. & Chem. Engng, 22(12), 1789-1809 (1998). Chen, J., A. Bandoni, and J. A. Romagnoli, Robust estimation of measurement error variance/covariance from process sampling data. Computers & Chem. Engng, 21(6), 593-600 (1997). Crowe, C. M., Data reconciliation- progress and challenges. J. Proc. Cont., 6(2/3), 89-98 (1996). Donoho, D. L., Unconditional bases are optimal bases for data compression and for statistical estimation. Applied & Computational Harmonic Analysis, 1 (1), 100-115 (1993). Donoho, D. L. and I. M. Johnstone, Ideal spatial adaptation via wavelet shrinkage. Biometrika, 81(3), 425-455 (1994). Keller, J. Y., M. Zasadzinski, and M. Darouach, Analytical estimator of measurement error variances in data reconciliation. Comput. & Chem. Engng, 16(3), 185-188 (1992). Mah, R. S. H., Data screening. In Reklaitis, G. V. and M. D. Spriggs, editors, Foundations of Computers - Aided Process Operation. Elsevier, Amsterdam (1987). Tamhane, A. C. and R. S. H. Mah, Data reconciliation and gross error detection in chemical process networks. Technometrics, 27, 409 (1985).
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
675
Efficient Gross Error Elimination Methods for Rigorous On-Line Optimization Cornel Jordache, David Temet and Scott Brown Simulation Sciences Inc. 601 Valencia Ave., Suite 100, Brea, CA 92823, U.S.A. Correct identification and elimination of gross errors in plant measurements is a key factor for the performance of industrial on-line optimization. For rigorous nonlinear plant models, the gross error detection problem is very challenging. This paper presents two data reconciliation and gross error detection strategies for nonlinear models: one based on a serial elimination algorithm using a linearized model and another one based on the Tjoa-Biegler contaminated normal distribution approach. The comparison is based upon the results obtained with a commercial on-line data reconciliation and optimization package using a rigorous model of two industrial refinery processes. 1. INTRODUCTION Data reconciliation is widely used to adjust the plant data and provide estimates for unmeasured variables and parameters. Data reconciliation improves the accuracy in measured variables and model parameters by exploiting redundancy in the measured data. Traditional data reconciliation assumes that only random errors exist in process data. If gross errors also occur, they need to be identified and eliminated. Otherwise, the reconciled solution will be highly biased. Since data reconciliation is often used to provide better starting points to the economic optimization, it is very important that gross errors will not significantly affect the quality of reconciled data. Statistical hypothesis testing techniques have been employed to detect persistent gross errors [1]. However, correctly identifying all gross errors is still a challenging task, even for steady state models. The existing strategies based on statistical tests sometimes wrongly report a different number or location of gross errors than in reality. This problem occurs for the following two reasons. First, a gross error may propagate in data reconciliation and contaminate the reconciled data. Second, redundancy in measured data for a chemical or refinery process may be low compared to the size of the process. Most gross error detection and identification strategies have been designed for linear data reconciliation models, such as plant mass flow balances. A linear data reconciliation problem can be written as a constrained weighted least-squares minimization problem: min F - (x- xDT y-1 ( x - ~) X
s.t. B(~ + a)- c
(x~ - xi )2 i--1
(1)
O- i
(2)
676 where ~ is an n-vector of measurements, and a is the vector of adjustments to ~ so that the balances in Equation (2) hold. B is an m x n matrix of rank m, consisting of the incidence matrix for the flow mass balances and c is a m-vector of constants. E is an n x n variancecovariance matrix of ~. Since in most applications 2; is assumed to be a diagonal matrix (independent measurement errors), the objective function F reduces to the summation expression on the right hand side of Equation (1). For simplicity, Equation (2) contains only measured variables. If unmeasured variables also exist, they can be easily eliminated by a projection matrix method [1-4]. Assuming that ~ follows a normal distribution, the optimal adjustment vector is
a = i - ~ = -ZBT (BEBT)-l (B~ - c) with the corresponding
n x n
variance-covariance matrix V = coy(a) = EBT (BEBT)-1BE. With raw measurements, the process constraints (2) may not be closed. The residuals of the process constraints resulting from the errors in the data are expressed as r = B ~ - c with the corresponding m x m nonsingular variance-covariance matrix H = c o v ( r ) - BEBT. If ~ follows a multivariate normal distribution, 2 ~ N ( x , Z ) , where x is a vector of the unknown mean values, both a and r follow multivariate normal distributions, a ~ N ( 0 , V ) , and r ~ N ( 0 , H ) , respectively. Therefore, the expectations of a and r are zero, if there are only random errors. This is the statistical foundation of gross error detection. In addition, the objective function value (1) follows a chi-square distribution with m degrees of freedom: F* = aTZ-~a = r T H - l r ~ Z2m . We can test the data globally by comparing the optimal objective function value (F*) against a threshold of a chi-square value with m degrees of freedom. If F* >_Zm2~, where ot is the chosen level of significance for the statistical test, one or more gross errors are detected. The chi-square test global test (GT) defined by F* can be used for gross error detection, but for identifying the location of gross errors additional operations, such as serial elimination, have traditionally been implemented. For a better gross error identification, univariate tests can be used instead of the global test. Since a ~ N(0, V ) , we can derive a univariate measurement test (MT) statistic
Za,j = ~ aj
N(0,1), j : 1,2,...,n
(3)
Unlike the global test, the MT processes each measurement adjustment separately and gives rise to m-univariate tests. If any of the test statistics Za,j exceeds the test criterion Zl_~/2 (the critical value for a-level of significance of the two-sided standard normal distribution test), a gross error is detected in measurement j. Although it associates a gross error with a particular measurement, the MT was found to detect many non-existent gross errors. Additional identification is also required in order to enhance the performance of the univariate tests. Other statistical tests such as the generalized likelihood ratio (GLR) test, and the principal component (PC) test, are analyzed in detail elsewhere [1 ]. Each type of statistical test has its own benefits and associated problems.
2. NONLINEAR DATA RECONCILIATION AND GROSS ERROR DETECTION Rigorous nonlinear models, which are desired for enhanced optimization and more precise process control, add another level of complexity to the data reconciliation and gross error
677 detection problems. While nonlinear data reconciliation with rigorous models enables simultaneous solution for both reconciliation of measured data and estimation of model parameters, it makes the gross error detection problem a much more challenging task. The general nonlinear data reconciliation problem can be formulated as a least-squares minimization problem (Equations 1-2), but the linear constraints (2) are replaced by nonlinear equations representing material and energy conservation relations, thermodynamic equilibrium constraints, etc. Inequality constraints such as upper and lower bounds on variables or complex feasibility constraints related to equipment operation can also be included. The minimization of objective function (1) subject to nonlinear constraints can be achieved by using a general purpose nonlinear optimization technique. Also, since the objective function is quadratic, efficient SQP techniques have been developed to solve this problem. All statistical tests and gross error identification strategies reviewed above can also be used for nonlinear models. The usual procedure is to first linearize the process model followed by a gross error identification method designed for linear equations. This strategy, however, may not be suitable for highly nonlinear processes with data corrupted by significant gross errors. An improvement can be achieved from a good data prescreening and validation before the data reconciliation run. Alternative solutions to applying statistical tests to linearized models have been proposed, but they all take a significant amount of computational time. For example, Kim et al. [5] modified the MIMT serial elimination strategy based on MT of Serth and Heenan [6] by calculating the measurement adjustments for the statistical test in Equation (3) based on the optimal nonlinear solution ~, that is, a = ~ - ~ . However, the variance-covariance matrix of the adjustment vector a is calculated from a linearized model [7] and results in a long matrix expression which is computationally prohibitive for a large-scale industrial problem with a rigorous model. If gross errors are present in data (e.g., large measurement biases), this objective function, which comes from the normal distribution, provides a biased data reconciliation solution. One way to reduce the degree of bias in the reconciliation estimates is to use a contaminated normal distribution, as suggested by Tjoa and Biegler [8]. This distribution is less sensitive to gross errors of medium size [9]. The objective function for this approach changes to F =-
(1 p~)exp i=1
+ ;--- exp - 0 . 5 ( _ _ ) 2
(4)
DiO" /
In Equation (4), p~ represents the probability of occurrence of a gross error and b~ is the ratio of standard deviation for the gross error distribution to the standard deviation of the normal random error in measurement i. This approach enables a simple statistical test (similar to MT) for gross error detection. The test simply declares a measurement i in gross error if:
Another type of objective function is based on robust statistics. Various robust distribution functions have been proposed, which are insensitive to large gross errors. One of them is based on the Fair function, which gives the following objective function [4], [9]"
678
F = ~._.c i=l
2F'a' coi ( 'ai'll + log 1 +
(6)
CO'i flJ
where c is a tuning parameter (see [4] for details). The major problem with the robust statistics is that it does not allow a straightforward statistical test for gross errors detection. But it usually provides a data reconciliation solution that is less biased than the solution obtained from the normal distribution [9]. Due to the extremely large size of the industrial problems with nonlinear rigorous models, not many gross error strategies based on statistical tests can be applied for such models. Simple, but efficient gross error methods are desired for large-scale industrial problems. A classical methodology of gross error detection based on the GT and the MT tests, similar to the IMT algorithm of Serth and Heenan [6] was first studied in this work. The original nonlinear problem was reduced to a linear data reconciliation problem involving only measured variables via model linearization followed by elimination of the unmeasured variables by a projection method. Note that a projection matrix is constructed only once in order to obtain the reduced linear model with only measured variables. This algorithm was designed and implemented for gross error detection associated with the ROMeo TM on-line optimization package. A complete description of the gross error detection algorithm, which also differs from the original IMT algorithm by some decision criteria to select the most likely gross errors, is found in [10]. We will refer to this particular algorithm as Gross Error Detection and Elimination (GEDE). The performance of this method is compared against another strategy based on a contaminated normal distribution (CND). 3. CASE STUDY: A REFINERY GAS PROCESS The GEDE algorithm presented above is illustrated on a gas separation process. The plant consists of a debutanizer, a naphtha splitter and a de-isohexanizer. The rigorous nonlinear model contains 10858 equations and 11663 variables (793 variables were fixed). This particular plant has 54 measured variables. Only two measured variables are theoretically nonredundant. The number of measurements is small in comparison with the problem size, but this is typical for a refinery process, particularly with a rigorous model. Initially, the measured values were chosen to be almost equal to the simulation values. Next, gross errors were simulated in various locations, and with various orders of magnitudes. In this example, six gross errors exist in the measured data as indicated in Table 1.
Table 1: Gross error detection results by GEDE and CNDT for the gas plant example GE location (Tag, UOM)
MS5:Flow, m3/s MS12:Flow, m3/s MS27:Flow, m3/s
Simulation Value
Measured
Standard
Detected
Detected
Value
Deviation
by GEDE
by CNDT*
0.01145 0.00184 0.00066 2.22222 2.22222 2.22222
YES YES NO YES YES NO
0.6335 0.5725 0.0999 0.0819 0.0244 0.0334 MS3 :Temp, deg.K 449.10 432.42 MS6:Temp, deg.K 426.10 409.57 MS42:Temp, deg.K 439.40 456.16 * p=0.01, b=20, for all measurements
YES YES YES YES YES YES
679 By using the GEDE algorithm with the level of significance for the statistical tests o~ = 0.05 (95% confidence interval), four measurements were found in gross error. One detection problem was measurement MS27:Flow, which is the flow rate of a smaller outlet stream of a splitter. This particular flow measurement has a very low detectability factor [1]. Another problem is with MS42:Temp, which is the shell side inlet temperature of a heat exchanger. That particular heat exchanger has only one other temperature measurement, the tube side inlet temperature. Since none of the outlet temperatures are measured, the two measurements are practically equivalent as far as gross error detection is concerned. Thus, the gross error spreads equally between the two measurements and the same MT statistic is obtained. Unless the gross error is very large, none of the measurements are found in gross error. This is a case of weak identifiability [ 1]. The test based on the CND (denoted here as CNDT) however, was able to find all gross errors. The values chosen for the tuning parameters p and b did not show significant difference in the outcome of the gross error test, as also found by Tjoa and Biegler [8]. In our testing, the following pairs (p, b): (0.1, 10), (0.01, 10), (0.01, 20), (0.001, 20) produced the same result. Allowing the p's to be independent variables in the SQP solver did not change the results either; the computational time, however was significantly increased (about four times). Apparently, the CNDT is sharper in finding gross errors for measurements with lower detectability or identifiability. Unfortunately, our study on a larger flowsheet (a crude unit atmospheric separation which also includes a vacuum distillation and a gas processing plant) showed that the CNDT behave closer to the pure MT. It reports too many gross errors. For example, out of the 359 process measurements, the MT (without any serial elimination) declared 49 measurements in gross error. The CNDT with (p=0.05, b=20) found 71 gross errors; the CNDT with (p=0.01, b=20) found 52; the CNDT with (p=0.001, b=20) found 49. The GEDE (which does serial elimination) declared only 15 in gross error. It is obvious that additional identification such as serial elimination is very important for any strategy of gross error detection. Table 2: Reconciled values for various types of objective function for the gas plant example Reconciled Value GE location (Tag, UOM)
MS5:Flow, m3/s MS12:Flow, m3/s
Simulation Measured Value Value GEDE
CND ~
Fair function (2)
0.6335 0.0999
0.5735 0.0819
0.63345* 0.1015"
0.63445 0.1012
0.63445
0.0244 449.10 426.10 439.40
0.0334 432.42 409.57 456.16
0.0264 447.72* 428.27* 446.85
0.0237 449.19 425.07 446.43
0.0244 449.07 425.52 455.57
0.1011
MS27:Flow, m3/s MS3 :Temp, deg.K MS6:Temp, deg.K MS42:Temp, deg.K
* estimated as unmeasured variables after eliminating the measurements in gross error o) p=O.O1, b=20; ~2)c = 0.04409
680 Another interesting fact is related to the solution obtained from solving the data reconciliation problem with various objective functions. Table 2 above shows the results for the gas plant example. The reconciled values coming from the contaminated normal distribution are closer to the initial simulation values than those obtained from solving the reconciliation problem after GEDE. The solution based on a Fair function is even better for most measurements, except for the MS42:Temp. Again, this result is probably due to the lack of identifiability of a gross error in this particular measurement, as described above. 4. CONCLUSIONS The gross error algorithms for nonlinear models in this study are capable of detecting all significant gross errors that do not have low detectability or identifiability. The GEDE algorithm is more computationally expensive, but it is a better gross error detection approach for large industrial problems. The simple test based on a contaminated normal distribution requires additional identification methods, since it usually predicts too many gross errors. In the presence of gross errors, the data reconciliation solution however, is usually better for the contaminated normal distribution and also for robust statistics. Therefore, these distributions can be used to provide better starting points for the economic optimization. Gross errors in measurements that are not easily identifiable are difficult to handle, and the corresponding reconciled values are generally not reliable. REFERENCES 1. S. Narasimhan, and C. Jordache, Data Reconciliation and Gross Error Detection: An Intelligent Use of Process Data, Gulf Publ. Co., Houston TX (1999). 2. C.M. Crowe, Y.A. Garcia Campos and A. Hrymak, "Reconciliation of Process Flow Rates by Matrix Projection. I. The Linear Case," AIChE J., 29, (1983) 818. 3. C.L.E. Swartz, "Data Reconciliation for Generalized Flowsheet Applications," AIChE Spring National Meeting, Dallas, TX, (1989). 4. J.S. Albuquerque and L.T. Biegler, "Data Reconciliation and Gross Error Detection for Dynamic Systems", AIChE Journal, Vol.42, No. 10, (1996), 2841. 5. I.W. Kim, M.S. Kang, S. Park, and T.F. Edgar, "Robust Data Reconciliation and Gross Error Detection: The Modified MIMT using NLP," Computers & Chem. Engng., Vol.21, No.7, (1997), 775. 6. R.W. Serth and W.A. Heenan, "Gross Error Detection and Reconciliation in Steammetering Systems", AIChE Journal, Vol.32, (1986), 733. 7. R.W. Serth, C.M. Valero, and W.A. Heenan, "Detection of Gross Errors in Nonlinearly Constrained Data: A Case Study ", Chem. Eng. Comm., Vol.51, (1987), 89. 8. I.B. Tjoa, and L.T. Biegler, "Simultaneous Strategies for Data Reconciliation and Gross Error Detection of Nonlinear Systems" Computers & Chem. Engng., Vol.15, No.10, (1991), 679. 9. X. Chen, R.W. Pike, T.A. Hertwig and J.R. Hopper," Optimal Implementation of Online Optimization", Computers & Chem. Engn. Vol.22 Suppl., (1998), $435. 10. C. Jordache, and D. Ternet, "Avoid Gross Error in Process Measurement", World Refining, Vol. 10, No.4, (2000), 34.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
681
A Combined Data And Gap Metric Approach to Nonlinear Process Control E. Kint a, Y. Samyudiaa* and P. de Jong b a Department of Chemical Technology, Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands. b DSM Services, Engineering- Stamicarbon, P.O. Box 10, 6160 MC Geleen, The Netherlands
A procedure to design a model-based controller for a non-linear process is proposed in this paper. The procedure combines the gap metric analysis to determine a control-relevant nominal model and the model updating strategy using low-order filters. The filters are identified from a set of data generated from an experiment to the closed-loop system so that the discrepancies between the design and the actual closed-loop performances are minimised. Several simulation studies illustrate the applicability of the proposed procedure.
1. I N T R O D U C T I O N Most of chemical processes are non-linear in nature. The use of linear controllers designed without considering the process non-linearity would lead to a poor closed-loop performance, particularly when we operate the process at a widen operating region. In contrast, non-linear control design methods that capture complex behaviour of the plant into the controller design, though attractive, would lead to computational difficulty and complexity as compared to linear control design. Another alternative to non-linear process control has been employing an adaptive control strategy. This strategy aims at adapting the controller parameters according to the change of process characteristics. In this attempt, the process non-linearity has been treated as the plant-model mismatch. The adaptive approach, however, could lead to instability when the process identifier did not receive sufficient excitation to identify a process model. These facts lead us to the operating regime-based control as a practically attractive approach to non-linear process control [ 1-5]. In principle, this approach was kept maintaining simple linear controllers for a set of local linear models generated from the defined operating points, and added a "co-ordinating" controller that takes into account for the non-linear (and/or transition) behaviour between operating points. Different synthesis procedures have been employed depending upon how to represent the process non-linearity. In this paper, we propose a data based control strategy for non-linear process control. The basic philosophy of this approach combines the basic ideas of the adaptive control and the operating regime-based control. In this work, our attempt is to minimise the performance deterioration due to the process non-linearity by updating the nominal model used in the To whomall correspondence shouldbe addressed. Phone. +31 15 278 6327, fax. +31 15 278 4452. Email: [email protected]
682 controller design. The model updating strategy makes use of the data generated from an experiment to a closed-loop system. Hence, the development of this approach relies upon a (to be updated) nominal model of the process, a set of closed-loop data and a (linear or nonlinear) model-based design technique. A design framework of the data based control strategy follows a four-step procedure: 1. Generate a control-relevant nominal model and design a model-based controller; 2. Perform an experiment on the closed-loop system to collect a set of data that represents the actual closed-loop performance, and compare with the designed closed-loop performance, to determine the performance deviation. 3. If the performance deviation is large, identify low-order filters using the collected data to up-date the nominal model. 4. Repeat step 2-3 until we achieve a small discrepancy between the designed and actual closed-loop performances. This paper will focus on presenting the method of generating a control-relevant linear model for a non-linear process, and the strategy of updating the nominal model using low order filters to improve the actual closed-loop performance. A high-purity distillation column of [6] is chosen as an application example to demonstrate how the developed procedure works in achieving a high closed-loop performance.
2. SELECTION OF CONTROL-RELEVANT NOMINAL M O D E L Consider the dynamics of a non-linear process as, iCo = fo(Xo,Uo,do,t, Oo) Po" (1) Yo - go (x,u,t) where x is a vector of state variables, y is a vector of controlled variables, u is a vector of manipulated variables, d is a vector of disturbance variables, t is time and 0 is a vector of process parameters. In general, f or g is a vector of non-linear (known) functional relationships. The subscript o denotes the true system. Suppose that we generate a set of linear models in the specified operating regimes .~. The ith linear model has a state space form as, ic = Aix + Biu + v P~. (2) y = C~x + D~u + d Assume that this set of linear models approximates the dynamics of non-linear process within the operating regime .~. Let us select a linear model in the set as a nominal model, denoted as P,,. Then, we calculate the gap between the nominal model P, and the rest of linear models in the set as [7]" e = max{6i (P,, P~), c~,(P~, P, }
for/an
(3)
where fi(P,, Pi ) is defined as: c~ (Pn, P~) = Qinf lIP, - QP, l[oo eH~"
(4)
Applying this gap calculation for the whole set yields a profile of the gap in the specified operating regime. Referring to [8], this profile represents how far the actual performance
683 deviates from the nominal performance. The actual performance here refers to the closed-loop performance of a controller designed using P, when applied to other operating points. According to [8], the profile of gap represents the effect of process non-linearity in terms of co-rime factor uncertainty within the operating regime. Further, for the chosen nominal model, P,, we can calculate its maximum stability margin [9] that indicates its ability to robustly stabilise the co-prime factor uncertainty: bma x = [1 + p ( Y X ) ] -~/2
(5)
where p(.) is a spectral factor; Y and X are solutions to Kalman Filter Algebraic Riccati Equation (KFARE) and Control Algebraic Riccati Equation associated with the nominal model P,. Different nominal models within the set may produce different profiles of gap and bmax . By comparing the profile of gap and bma x produced for different nominal models within the specified operating regimes, we are able to determine a control-relevant nominal model. The criterion to be used for determining a control-relevant nominal model is the nominal model that has a largest coverage in which the gap is less than bma x . To illustrate this selection procedure, let us consider a 41 tray, non-linear high purity distillation column of [6]. We assume that the distillation process will be operated within the specified range of disturbances in feed flow-rate, F and feed composition z F as follows: 0.75 < F < 1.25 0.3 < z F ~ 0 . 7
(6)
In this case, we select two fifth-order linear models, P,,~ and Pn,2 a s nominal models. These nominal models are obtained after applying a model reduction technique (e.g. b a l a n c e d truncation technique) to full-linearized models at the following operating points: P,,,1 (Operating point 1): Feed flow rate = 1 k-mole/min and Feed composition = 0.5. Pn,2 (Operating point 2): Feed flow rate = 1.25 k-mole/min and Feed composition = 0.4. To include the performance specification in the calculations of gap and stability margin, we apply a loop shaping technique, in which a weighting function: 5 W= ~ I 2 •
s(s + 0.4) is introduced to shaped the nominal models. The gap profiles for the shaped nominal models within the specified operating regime can be seen in Figures l a and lb. We observe that the nominal model P,,,1 produces the gap
profile smaller than
bma x
for the whole operating regime. In contrast, the second nominal
model P,,2 produces an intersection at the edge of operating regime. This implies that the performance of model-based controller designed using the nominal model P,,1 is better than that of based on P,,,2 within this specified region.
684 A closed-loop simulation presented in Figures 2a and 2b confirms the analysis. In this simulation, we use a loop shaping Hoo controller that is designed for the associated nominal model. The controller is then applied to a full non-linear model of the distillation column operated at F = 1.1 and Z F - - 0.65. i
Figure la. Gap metric profile for the first
nominal model, Phi
Figure 2a. Closed-loop responses for Pn.l
'
'
.~
Figure lb. Gap metric profile for the second nominal model, P..,.
Figure 2b. Closed-loop responses for P..2
The responses in Figures 2a and 2b show the comparison between the design and actual performances of loop shaping controllers produced by different nominal models. Clearly, the nominal model Pn,1 produces a controller having a small discrepancy between the design and actual performances. Hence, this case study demonstrates how the gap metric approach can be used to determine a control-relevant linear model as the basis for designing a model-based controller for a non-linear process within the specified operating regime.
685 3. M O D E L U P D A T I N G S T R A T E G Y W I T H D A T A Commonly, the process non-linearity (or in general plant-model mismatch) deteriorates the design performance significantly. When this occurs, we need to update the nominal model accordingly. In this work, we exploit the actual data collected from a closed-loop system as the information to update the nominal model. Suppose that the actual and designed responses are denoted by {u o , Yo } and {u, y}, where subscript o represents the actual response. Let us introduce two stable, invertible filters F 1(z) and F 2(z) such that
Yo(t) ~ F~(z)y(t) = y,(t)
(7)
Uo (t) ~ F, ( z ) u ( t ) = u I ,
(8)
and
Using the initial nominal model and its associated model-based controller, we evaluate its closed-loop performance using the following criterion: N
N
J = Z (Yo (t)- y(t)) r (Yo (t)- y(t))+ 2~,-" (u o (t)- u(t)) r (u o (t)- u(t)) t =1
j
L
~.
t =1
Z
.....
(9)
9
I f J is large, we need to update the nominal model. To do so, we perform an identification experiment on both the actual and designed closed loops by applying a persistently exciting signal, e(t). Then, we collect N-data of {Yo (t),Uo (t), y(t),u(t)} for t = 1.... , N . From the collected data, we identify the low-order filters F~(z) and F2(z ) using the available identification methods [10]. Once we get the filters, the updated nominal model follows the following form ~ = F~P~F[ 1(Proof is omitted here because of the space limitation). By repeating the procedure of loop shaping controller design for the updated model, we produce a new controller K = FZIKo F~, where Ko is designed for the updated nominal model. To illustrate this model updating strategy, we use again the case study of a non-linear distillation column. We chose the initial nominal model P,,~ and we apply a loop shaping technique by introducing a weighting function as: W = 5(s + 0.4) I2• S
The response of closed-loop system is shown as solid lines in Figure 4. At identification experiment, we injected a band-limited white noise with power 0.01 to actual and the desired closed-loops and then we identified the filters F~ and F 2from 1000-collected input-output data at sampling time 1 minute. For )~ = 0.33, we get following filters for updating the nominal model"
F1
"-
0.996(s + 0.0217) s + 0.0138 0
0 0.984(s+0.0958) ; s + 0.0615
3(s + 0.0202) F[l_
s+0.0193 0
0 3(s + 0.0153) s + 0.0119
the the the the
686 The closed-loop responses before and after updating the nominal models are presented as in Figure 4. It can be seen now that the closed-loop performance improves significantly and is closer to the desired performance.
l 099 0
jjjl es nAcua -50
1O0
time [min]
150
0.013
0.011
"--
001 l
0 009
___. - ~ _
............. 0
50
time [min]
1O0
150
Figure 4. Closed-loopresponses for the controller designed using the initial nominal model (dashedlines); the updated nominal model (dashed-dottedlines); and the desired response (solidline).
4. C O N C L U S I O N S This paper has addressed a design method for non-linear process control that combines the gap metric analysis and the data based model-updating strategy. The method has been successfully applied to control a non-linear, high purity distillation column. The proposed method could lead to an iterative model-based control design that integrates the identification and control design.
REFERENCES 1. K.A. Kosanovich, J.G. Charboneau and M.J. Piovoso, J. Proc. Control, 7(1):43-56, 1997. 2. T.A. Johansen and B.A. Foss, Control Eng. Prac., 6(10):1277-1286, 1998. 3. A. Kordon, P.S. Dhurjati, Y.O. Fuentes and B.A. Ogunnaike, J. of Proc. Control, 9(5):453-460, 1999. 4. Y. Chikkula, J.H. Lee and B.A. Ogunnaike, AIChE J., 44:2658-2674, 1998. 5. P.L. Lee, H. Li and I.T. Cameron, Chem Eng. Sci., 55:3743-3758, 2000. 6. S. Skogestad and I. Posleithwaite, Multivariable Feedback Control, Wiley, 1996. 7. T.T. Georgiou and M.C. Smith, IEEE Trans. Auto. Control, 35(6):673-686, 1990. 8. Y. Samyudia, M. Green, P.L. Lee and I.T. Cameron, Chem. Eng. Sci., 50(11): 1695-1706, 1995. 9. D.C. McFarlane and K. Glover, IEEE Trans. Auto. Control, 37(6):759-769, 1992. 10. L. Ljung, System Identification. Theory for the User, Prentice Hall, Inc. 1999.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
687
Dynamic study of inferential sensors (Neural Nets) in quality prediction of crude oil distillation tower side streams. Jos6 J. Macias a and Josep A. Feliu b aAdvanced Control Department, CEPSA (Tenerife Refinery) and Chemical Engineering Dept, University of La Laguna, Tenerife, Spain. Email: [email protected] ; [email protected] bCustomer Services Department, Hyprotech Europe S.L., Pg. de Gr&cia 56, 08007 Barcelona, Spain. Email: [email protected] Prediction of properties by statistical methods, and especially by neural networks, is a field that is developing extremely fast. For refinery units, there exist many examples of application that use real plant data (usually, hourly averages) to calibrate neural models. The models are able to predict interesting properties like distillation end points for the products of a crude oil distillation tower. Nevertheless, few examples are known where a dynamic study is performed to highlight the importance of variable evolution along time. The objective of this work is to show how first-principles based dynamic simulators can be used to take into account the dynamics of a crude oil distillation column. In few words, dynamic data is generated with a commercial simulator (Hysys.Plant§ this data is used to generate the neural net model and, later, the goodness of the proposed methodology is checked against dynamic plant test runs. This proposed mixed procedure combines the use of real plant data to generate the steady state gain and the dynamic simulation data to develop the finite impulse response models. The corrected dynamic models should offer better predictive characteristics than models with variables not conveniently delayed. 1. INTRODUCTION Many reasons arise to justify the use of statistical techniques to determine a process variable instead of having an on-line analyzer. A common reason is the unavailability of the analyzer of interest or its high associated cost that makes it difficult to justify economically. On the other side, having high amount of data stored in instrumentation databases and in laboratory reports should permit a cheap way of creating virtual or soft analyzers. This data mining is what current methods of multivariate calibration like Partial Least Squares (PLS) and Neural Networks (NN) are doing [1 ]. The main techniques used up to date to calibrate high amounts of data are: 9 Neural Networks, non-linear models Very good in calibrating non-linear data. High dependency to process noise. 9 PLS calibration methods, high noise immunity It is normally applied to linear structures of high collinear data and high level of noise. The inner relation can be either, linear or non-linear [2-4] With respect to data, in a chemical plant will have sources coming from:
688 9 Instrumentation hourly averages Hourly averages are the common way to store instrumentation data. This is a way to have data filtered and to store more information with less investment but process variations and instrument actions are not consolidated as stable process changes and hourly averages do not permit to evaluate dynamic interaction between variables. Additionally, variables with a dynamic constant higher than an hour are not well represented with hourly averages. 9 Lab spot values In plant databases there is lab information with very low periodicity. In this way, the number of samples used to calibrate the models is limited, and therefore, the data error can be propagated easily to the model. In front of a discrepancy between the steady state model prediction and real plant data there will always arise the doubt if the model is wrong or if the plant was not in steady state when sampling. The error between model prediction and real plant data is proportional to the distance of the plant to its steady state situation. 2. OBJECTIVES The objectives of this work are to propose a methodology able to cope with the above situations and to apply it to the prediction of final distillation points of a crude oil distillation unit. A case of predicting a 95% ASTM D86 of Straight Run Naphtha from process data is investigated using real process data with dynamic simulated tower.
2.1. Usual procedures in inferential modeling As a guideline, hereafter there is a brief description of the normal implementation phase of an inferential model. The whole picture is shown in figure 1. IM()l)!'.i.lbJ:i.
~)ATASETS, HOURLY AVERAGES I'
i'~ ii ,",r r.-!N ]
>
~ O D e L W~WHNO DYNAMIC
I
IPP,l:I)I(:rl'l()N. ~'I.S c-.,~-~qt'J [DATASETS, SPOT VALUES
[
. . . . . .
:> [DYNAMI(? MODt:~I..I.ING F.RRORS _]
Figure 1. Usual procedures on inferential modeling
Data collection: This phase is to collect hourly averages around the time where lab samples with the proposed prediction characteristic exist. Data around this time has to be evaluated to find process instability. Some data basic statistics should be calculated like data correlation matrix, standard deviation, etc. Data Calibration and Model Generation: Several methods could be used but, in any case, attention has to be paid to data prediction statistics to be sure of the capacity of the model. Runtime model prediction: Usually, spot values are used to have the best response under process upsets.
689
Long range Model Bias update: Care should be taken if it is decided to update the model bias. We have found that in most of the cases this will only propagate lab and instrumentation errors to the model.
2.2. Proposed procedure, Dynamic PLS Algorithm The proposed methodology, schematized in figure 2, is the following: Obtain Steady State Models: It is possible to use any available calibration technology. In general, if the system is highly non-linear, and if the function that correlates inferential or output variable with the input variables is not known (and if data is good enough) then NN will be the best choice. If the range of available data is small, if data have errors and the system is highly collinear, it is probably better to use a method that has a rigid data structure and is protected against noise like PLS. Data could come from plant instrumentation or from first-principles simulators. Simulators are used more and more often due to the fact that data generated with a simulation program in steady state has not noise and is reproducible, the variation range of variables is bigger than in a real plant and, finally, the time to create the data is shorter than any method. However, real data gives the inherent characteristic of a plant than can not be found in a simulation, like process discrepancies, instrument errors, etc. Probably, the best approach is to use a hybrid methodology, using a simulator to find the best-input variables and real plant data to caliber those variables with more accurate input coefficients. ,
" ,
t:~lllhlN
,', ! t:.ti)'~ S T A l l ,
M()DI'.I,,~
_
LJ~.I.L,\I.li(,)i_:Itl.Y,,,\vi'I~,,\(.il;s ] P,I~tt .1) i ) Y N k ~ l t ( '
[
S'I'I.::\[)Y S i'A I'I: t~,,\ f~',]S
,~I~,tU I.,UtI,~)N
[ ~,11:cl ~.,,,Ni<:,,\l~ I.)~:sl(:;>~ l
[ ()PERATIN(]
DATA
-..,,,,.. I DYNAMIC SIMULATION I
/
["tTNSTRI,rM c,NTATt.)*~ r " ( ,,r [
[
N t % l t : I . k't I" P I . A N t" l ' t t ~ ' l ,,~,Nt~ II~1-,~' t It <'~, D~ X,%),il(
tlO1)t,.1
[ ,<;lJ\,1t ;L,,'V[ ED SP(YI" V A L U E S ] I ' t ~ , E t ) I ( l' V , ' t T t l
I MODEL W,THNODYNAM,C I
~I~II'I.A
1 t : I ) 1)~, x , t M l (
~ t.~l) ( AI.IBIIA
I MODEL WITH DYNAMIC tM!' i
FIMP. RESPONSEC()EF. ] MODEL
WITH
DYNAMIC
IM P. Rf,,Si-~()NSI,; ('(~i,,1,
Figure 2. Proposed methodology to build inferential models
Build Dynamic Simulation: Build a dynamic model in a simulator by completing a steady state simulation adding the rest of elements like valves, (type and characteristic), check valves, and automatic instrtmaentation. This last item is very important because it is used to make plant test and depending of what elements are in auto and which ones in manual, the number of degrees of freedom will change, and so the model results. So, it will be better to
690 reproduce the normal mode of operation including advanced control application if any. Controllers should be fine-tuned to find the closest behavior to reality. Simulate Plant Test and Identify Dynamic Models: In this phase, dynamic models are identified corresponding to variable interactions in inferential modeling. Step change or PRBS can be used to obtain the simulated data. In any case, and once the dynamic data is generated, it is necessary to use a statistical tool to create the interaction model. In this case we have used Dynamic PLS to create Finite Impulse Response models (FIR). Predict with Simulated dynamics and calibrated gains: Once FIR models are generated, it is only necessary to put together both technologies, Inferential and FIR models. From the dynamic point of view we have used the FIR coefficients, normalized to unit gain. These variables now enter the inferential model with steady state gains coming from real data. Now, variables are delayed, without any impact in the model availability to predict in steady state, but improving the prediction power in transient states.
3. CASE STUDY 3.1. Unit Description The unit is an atmospheric crude oil distillation tower with capacity of 80000 bbl/d of Arabian crude. It has 47 valve trays with a diameter of 5.2 m at the top section and of 2.4 m at the bottom section. The tower has two cold pumparounds (PA), KNO, GOL and a hot PA GOP (washout). It has five sidestreams: Heavy Naphtha (HN), Kerosene (KN), Light Gasoil (LGO), Medium Gasoil (MGO) and Heavy Gasoil (HGO). This last sidestream is currently closed. It also has a vapor sidestripper for each sidestream and a bottom vapor injection. 3.2. Database The database that was considered in this study contained, from instrumentation, the temperatures of tower overhead, reflux, tower feed, HN, KN, LGO, MGO, HGO, bottoms, KN return PA and LGO return PA, the overhead pressure, the flowrates of gas, light naphtha, HN, KN, LGO, MGO, HGO, bottoms, KN PA and LGO PA streams, the density of feed stream and the levels of the reflux drum, tower bottom and water level in reflux drum. In addition to data from instrumentation, it contained lab data consisting on 5% ASTM D86 for light naphtha, HN, KN, LGO and MGO, on 85% ASTM D86 for LGO, on 95% ASTM D86 for light naphtha, HN, KN and LGO and on density for MGO. 3.3. Steady State NN modeling results To calibrate the NN model, a time period of 8 months, between April and November 2000, with 173 lab samples and their corresponding hourly averages has been used. As an example we show the NN to predict 95% ASTM D86 for HN. For the rest of lab analysis, similar results can be obtained. For this prediction the main variables are reflux and HN flowrate and HN and KN temperatures. Taking as 20% cross-reference training and for a NN of a hidden layer with eight nodes, four inputs and one output, we obtain 3.4 ~ prediction error. The relative importance of the variables in the NN model is shown in Table 1. Table 1. Relative importance (%) of the four input variables to the NN Reflux ]HNSS HN Temp ] KN Temp 36.5 35.3 13.6
14.6
691
3.4. Dynamic Simulation Tool The commercial simulation program used has been Hysys.Plant + v.2.2. The powerful capabilities of this software permit to dynamically simulate entire and complex flowsheets, allowing to represent the crude distillation unit in detail, including all controllers and related instrumentation. To simulate the tower using HYSYS, all topological information (overhead system) needed to reproduce actual plant has been included. The dynamic model includes as well the actual controllers in the installation.
Figure 3. Hysys.Plant controller face plates
model views of condensing overhead circuit, tray section and
Lately, and once the simulation model was calibrated with plant data, it was needed to identify which variables should be changed and which ones should be kept constant. It is necessary to define the ones that will vary and the ones that will give support information as they are named in the inferential model. Simulating lab samples will be the last simulation information required. Clearly, three types of variables can be identified: Manipulated variables: They are the ones directly manipulated. They are totally independent and give the degrees of freedom of the simulation. In this case they will be the furnace outlet, KN return and LGO return temperatures, the tower overhead pressure, the flowrates of refux, HN, KN, LGO, MGO, KN recycle and LGO recycle and feed density. Induced response variables: We consider the variables that depend on the manipulated variables and belong to the rest of instrumentation of the plant. If they appear in the inferential model, they can give relevant information. Therefore, it is required to know their dynamic dependence with the inferential variables so, a more precise inferential model can be built. Those are tower overhead, reflux, HN, KN, LGO, MGO, HGO and bottoms temperatures and gas and light naphtha flowrates. Dependent variables: Those are the ones to infer from the model, as they are not available from plant instrumentation. In this case they correspond exactly to the variables that came from laboratory when building the NN model (see page 4)
3.5. Dynamic simulation data Following a step test methodology, dynamic simulations were carried out using pulse variations for each one of the manipulated variables. The pulse height and duration was
692 related to the sensibility of each variable. As a general rule the pulse had a length between 90 to 120 minutes and the step was around 5-10% of the variable value.
3.6. Dynamic PLS modeling phase results All data was collected and processed simultaneously in a Dynamic PLS with the four NN inputs: reflux and HN flowrates and HN and KN temperatures. Several model lengths were tested and some of the best-input data are obtained within 80 minutes. Data file hist01 % Importance RefluxFlow d95astmnp 4.2977 Gain Reflux Flow D95astmnp 0.0094681
Model name m95bpna HN SS 6.8708 HN SS 1.4453
PLS Comps. 2 HN Temp 21.9963 HN Temp 0.34004
KN Temp 66.8352 KN Temp 0.4436
Length,min 80 80 80 80
In the following graph, the step model responses are shown for the four input variables. Reflux
HN SS
0.01
HN Temp.
2
0 /'I" 0
50
100
-2
KN Temp.
0.4
0
50
100
0
0.5
0
50
100
0
0
50
100
Figure 4. Step response models As it can be observed, HN SS and Reflux have high importance in NN model but low importance in PLS Model and its dynamics are slower than temperatures. Therefore, a model taking into account only spot values will have higher errors, because of flows, than with these two temperatures. HN flowrate will contribute the 95% ASTM almost an hour later than when the change was made. However, other variables like temperatures are much faster, and all of them should be corrected for their dynamic influence. 4. CONCLUSIONS The proposed method implies the dynamic study in simulation of the most important variables in the model to be incorporated to the on-line model predictions. In the calibration phase, data should be carefully checked for dynamic infuence in order to select variables time frame.
REFERENCES 1. Macias, J.: "Aplicaci6n de la T6cnica de Calibraci6n Multivariante PLS al Control y Modelado de Procesos", Tesis Doctoral, Universidad de La Laguna, (1998). 2. Geladi, P.; Kowalski, B.R., An example of 2 block Predictive Partial Least Squares Regression with Simulated data, Analytica Chimica Acta, 185, 19-32, 1986 3. Geladi, P.; Kowalski, B.R., Partial Least Squares Regression: a tutorial, Analytica Chimica Acta, 185, 1-17,1986 4. Kaspar, Michael H.; Ray, W.H. :"Chemometric Methods for Process Monitoring and High Performance Controller Design", AIChE J., 38(10), 1593-1608. (1992).
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
693
An Efficient MILP Continuous-Time Formulation For The Optimal Operation Of General Multipurpose Facilities C. A. M6ndez and J. Cerdfi INTEC (UNL- CONICET) Gtiemes 3450 - 3000 Santa F e - ARGENTINA E-mail: [email protected]!.edu.ar This paper presents a unified MILP mathematical formulation for the scheduling of resource-constrained multipurpose facilities involving continuous processes and intermediate storage tanks. Different sequences of processing steps can be carried out in the plant to produce a significant number of final products and required intermediates. In order to reduce equipment idle time due to unbalanced stages' capacity, storage tanks are available for temporary inventory of intermediates. The problem goal is to maximize the plant economic output while satisfying specified minimum product requirements. The proposed approach relies on a continuous time domain representation that accounts for unit-dependent processing rates, sequence-dependent changeover times and storage limitations. The approach was applied to a manufacturing facility involving a sequence of three manufacturing stages and producing seven intermediates and fifteen final products. Compared to previous continuoustime scheduling methodologies, the approach shows a drastic reduction in variables/constraints, a much higher computational efficiency and it additionally provides a better production schedule. 1. INTRODUCTION Much of the work made in the area of production scheduling has been focused on batch processing facilities. Recently, however, several publications have paid attention to the scheduling of continuous processes. Continuous time formulations for the short-term scheduling of continuous processes, all of them based on the RTN/STN representation, have been developed by Schilling and Pantelides (1996), Ierapetritou and Floudas (1998) and Xueya and Sargent (1998). In turn, Karimi and McDonald (1997) proposed a couple of MILP formulations to deal with multiple product demands at specified due dates. The present work generalizes the MILP algorithmic methodology for the short-term scheduling of batch processing facilities, introduced by M6ndez and Cerd~i (2000), to also account for continuous processing units.
2. PROBLEM DEFINITION Given: (a) a processing facility that involves a set of continuous operations producing intermediate or final products; (b) a set of available equipment items each one performing different processing tasks; (c) a set of tanks of limited capacity for temporary storage of intermediates; (d) a predefined set of runs for each required task; (e) minimum requirements of final products to be satisfied; (f) a given scheduling horizon. The problem objective is to find (i) the optimal sequence of runs to be executed in every continuous unit; (ii) the task being performed and the intermediate/final product yielded by each one and (iii) the campaign starting and completion times, in order to maximize the economic return from production
694 sales while satisfying allocation constraints, storage limitations and end-product minimum requirements. 3. M O D E L ASSUMPTIONS (a) Several intermediates can be required to run a continuous processing task but each one produces just a single intermediate/final product; (b) a sth-producing run i~I+s can supply intermediate s to one or several production tasks i'~Ys; (c) an intermediate s required by a production task i'~ I-s can be either directly provided by run i~ r s bypassing storage or taken from a tank containing s; (d) minimum run lengths and equipment processing rates can vary with the product being manufactured; (e) changeover times between campaigns are sequencedependent; (f) the time interval during which a suitable tank t~ Ts is assigned to a particular campaign i~ I+s (the duration of the storage task) begins at its starting time and ends as soon as the amount of intermediate s produced by run i has been totally consumed. Contrarily, Ierapetritou and Floudas (1998) assumed that the storage task ends at the start of the related consuming runs, e.g. a lower bound on the actual storage task duration; (g) no initial inventory of any intermediate s~ S ~is on hand; (h) unlimited storage capacity is available for the final products whereas tanks for intermediates have limited capacities; (i) the production rate of intermediate s at campaign i~ l+s is greater than the overall consumption rate of s by runs i% I-s, (j) the plant is operated on a closed-shop mode. 4. T H E M A T H E M A T I C A L M O D E L 4.1. Problem Constraints 9 Scheduling horizon Ci
(1)
Vi~l
9 Campaign~unit allocation constraints. A single unit can at most be assigned to any campaign i ~Yij < 1
(2)
Vi ~ l+~ ,s ~ S
j~Js
9 Minimum run length constraint lminv.. _ L # <_M Y i y sj I , j <
V i ~ I s , j+~ J s ,
s~S
(3)
9 Effect offinite state release times and unit ready times upon campaign starting times
c,- Z
>_Z Max [ru ,ro,] Y,,
j~J.,
Vi~ l+,s~ S
(4)
j~Js
9 Feasible campaign size Z rs7 in Lij < Qi <_ Z rs7 ax Lij j~J., j~Js
V i e I s,S~ + S
(5)
9 Minimum final product demands ,ts < ~ Q ~ i~l + s
Vs s S e
(6)
695 9 Campaign sequencing at processing units Ci'-Li' j >_Ci + "~i'j - M ( 1 - Xii')- M ( 2 - Yo" - Yi' j) C i - L{i >_Ci'+'a'iy- M Xii'-M(2 - Y { i - Y i ' j )
Vi, i' e l , i < i', j e Jii'
Vi, i'e I , i < i', j e Jir
(7) (8)
9 Material balances. Material balances ensure that enough intermediate se S I is produced to run any campaign ie Ys requiring s. (i) Sinks for intermediate s produced by campaign iEFs. Intermediate s produced by campaign ie l+s is supplied to one or several sth-consuming runs featuring Fii- > 0. Qi = E Fii,
V i e I s, + se Si
(9)
i'~l 7
(ii) Sources of intermediate s required by campaign iEls. Sources of intermediate s for a run ie Ys are those campaigns i'e I+s featuring Fi.i > 0, pisQi= E F i , i
Vie Is, Se S
(10)
i'~ I +
9 Source/sink campaign matching conditions. A run ieI+s supplying intermediate s to campaign i'e I-s (i.e. Uii' -" 1 and Fii' > 0) must not start later than campaign i' Eli,<_ n Uii,
V i e l+~ , i ' e I s , S e S
Ci - E Lo" <_Ci,- E Li, j + M ( 1 - Uii, ) j~J~
(11) V i e l+s , i' e l s , s e S
(12)
j~J~'
9 Duration of storage tasks. The storage task for campaign ieI+s begins at the starting time of run i (ITi) and finishes as soon as the latest consuming campaign i'eI-s (with Uii' =1) has been completed (CTi). ITi : Ci -
E Lij
V i e l+s , S e S I
(13)
j~Ys
CTi >_C i ' - g (1-Uii')
V i e l+s,i'e I s , S e S
(14)
9 Storage task sequencing constraints l T i ' >__C T i -i- l~i' t - g
( l - Xii' ) - g
I T i >_ C T i ' - l - ~ ' it - M
Xii'-M
(2 - W i t - W i ' t )
(2 - W i t - W i ' t )
Vi, i' e l , i < i' , t e Tii'
Vi, i' e l , i < i' , t e Tir
(15) (16)
9 Storage capacity constraints. To fulfill storage capacity constraints, the net amount of material stored in a tank can never exceed its capacity. To establish the amount of material in the tank assigned to run ie I+~ at time Ci, it is important to first identify the sth-consuming runs i'e Is starting after Ci. (i) Runs i'd'sstarting after the end of campaign iEFs (Zii' =1) Ci'-~'Li'j-Ci<M
Zii' V i e l+s,i'e l - ~ , s e S
(17)
j~Ji'
(ii) Bounds on the amount of intermediate s supplied by run iEFs to the consuming campaign i ' e l s (Vii') at the completion time Ci. As stated by constraint (18), Vii' should be equal to zero
696 whenever run i%I-s starts after Ci. Moreover, it can never exceed the total amount of intermediate supplied by run i to the matching campaign i'. In turn, inequality (20) provides a bound on the value of Vii' for consuming campaigns i' still running at Ci.
Viel+s,i'e l;,se S
Vii'<_n(1-Zir)
Vir
V i e l+s ,i'e I s , s e S /
Vii,<
(18)
srrjaXlC, -- C r + E L r j I + M Z i r + M ( I - Y i ' j ) j~ji.
) (c) Limited storage capacity assigned to run ie I+s O i - EVii'<__El~tWit i'~l~
(19)
x
Viel+s,seS
V i e I +s, ife I s, s e S, j e Ji'
(20)
(21)
t~Ts
9 Objective function. The problem objective is to maximize the economic return from production sales whereas meeting minimum production requirements. Max
2 ps E Qi i~ I+
(22)
SE S e
5. RESULTS AND DISCUSSION
The proposed MILP scheduling methodology has been applied to a couple of medium-size case studies both based on an industrial fast moving consumer goods manufacturing plant (Schilling and Pantelides, 1996). Fifteen final products are manufactured in the facility following a common production sequence: mixing, storage and packing. The mixing stage comprises three parallel mixers operating in a continuous mode in which seven intermediates are produced from three different base materials available as required. Such intermediates are then stored in three storage tanks or directly packed in five continuous packing lines. Problem data are given in Tables 1 and 2. Case I assumes an unlimited intermediate storage (UIS policy), i.e. storage capacity constraints are ignored. In turn, limited capacities of storage tanks (FIS policy) are taken into account in Case II. The optimal schedules for Cases I and II are depicted in Figures 1 and 2, respectively. They were found by using ILOG OPL Studio with the embedded CPLEX Optimizer (Ilog, 1999). Figure 2 also shows the change of the intermediate inventory with time as well as the sequence of intermediates assigned to every storage tank. Table 3 compares the size of the proposed MILP formulation, the optimal objective value and the computational requirements for Cases I and II with results reported in previous work. It is observed: (a) a one-order-of magnitude reduction in the number of binary/continuous variables and constraints; (b) an improvement in the optimal solution and (c) a significant reduction in CPU time requirements. 6. CONCLUSIONS A highly computationally efficient continuous-time MILP algorithmic approach to the short-term scheduling of multiproduct facilities involving batch and continuous processes has been presented. When applied to a medium-size fast moving consumer goods manufacturing plant, it provides a better production schedule through a small-size MILP formulation requiring significantly less CPU time than previous approaches.
697
Table 1. Equipment data units
rate/capacity (tn/h or tn) M1 17.00 Mz, M3 12.24 17.00 T1,T2 T3 60 LI
5.8333
L2
2.7083
L3
5.5714
L4
2.2410 3.3333 5.3571
L5
suitability
Table 2. Production requirements
change-over requirements
Ii, Iz 15, 16, I7 13,14 Store all intermediates P2, P3, P7 {P2, P3} and {PT}-I h P4, P5, Ps, P9 {P4, Ps} and { Ps, P9 } - 4h {P1} and P1, P6 {P6}- 1 h P12, P13 {P12, P13} and P14, PI5 {P1g, P15}- 2h Plo, Pll
intermediate/ demand intermediate/ demand product (tn) product (tn) 11/ P1 220 15/ P9 13.5 11/ P2 251 I5 / P10 11A 5 I2/P3 15 I6/Pll I3 12/P4 116 16/P12 16.5 I2/P5 7 I6/P13 8.5 I3/P6 47 I7/P14 2.5 I4 / P7 144 17/ P15 17.5 14/ P8 42.5
Fig. 1. Optimal schedule with UIS policy Table 3. Comparison of results example
binary vars, objective CPU time cont. vars, rows function
- UIS policy Schilling et al., 1042,2746,4981 1996 Ierapetritou et 280,1089,2873 al., 1998
2604
3407
2689.42
540
r h!.s..ap.p.roach.................3.8.:4.4....1.40..................269...5...3~..................14.:.?.7.......... - FIS policy Xueya et al., 1318,4555,4801 1998 This approach 60,87,361
2556
1085
2670.28
398.92
Fig. 2. Optimal schedule with FIS policy and inventory levels for intermediates
698
Acknowledgments. T h e
authors a c k n o w l e d g e financial s u p p o r t f r o m F O N C Y T u n d e r G r a n t 14-00356, and f r o m " U n i v e r s i d a d N a c i o n a l del L i t o r a l " u n d e r C A I + D s 048 and 121.
Notation
(a) Sets S SI SP I Is I+s I, J Js
Jss' T Ts Ts~.
states (intermediates or final states) intermediate states (SIc_ S) final states (S P c S) production runs campaigns producing or consuming state s (Is c I) campaigns producing state s (I+sc Is) campaigns consuming state s (I-s c Is) continuous processing units available units to run tasks producing state s (Ji = Js, for any campaign i~ I+s) available units to run tasks producing state s or state s' (Jii, = Jss', for campaigns i~ I+~and i'~ I+s.) storage tanks for intermediates available tanks to store the intermediate s~ S~(Ti = Ts for any campaign i~ I+s) available tanks to store both states s and s' (Tii' = Tss' for any pair of campaigns i~ I+s and i'~ I+~.)
(b) Parameters h ruj ros ~s'j ($ii't lminsj rminsj rmaXsj vt p,s ps ds
time horizon length ready time of unit j release time for any campaign producing state s changeover time between runs i~ I+s and i'~ I+s. at unitj (~i'j = ~ss'j for a pair of campaigns i~ I+s and i'e I+s.) changeover time between campaigns i~ I+s and i'~ I+s. , both assigned to tank t~ Tss. minimum allowed length of a campaign ie I+, producing state s and running at unit j minimum production rate of state s at any campaign i~ I+s being run at unitj maximum production rate of state s at any campaign ie I+s being run at unit j maximum capacity of tank t~ T amount of state s required per unit length of campaign i~ I-, price of final state s~ S P minimum requirement of final product s~ S P
(c) Variables Yij Wit Xii, Zi i, Uii, Lij Ci ITi CTi Qi Fii, Vii,
binary variable denoting that campaign i~ I+s is run in unit j~ Js binary variable denoting that the intermediate se SI produced by run i~ I§ has been assigned to tank t~ Ts binary variable denoting that campaign i is run/stored before (Xii'-" 1) or after campaign i' (Xii'-" 0) in some available unit/tank binary variable denoting that campaign i'~ I-s starts after campaign ie I+s has ended binary variable denoting that campaign i~ I+s supplies intermediate s to campaign i'~ I-s length of campaign ie I+s in unit je Js completion time for campaign i starting time for the storage task of intermediate s supplied by campaign i~ I+s completion time for the storage task of intermediate s provided by campaign i~ I+s production size of campaign i amount of intermediate s supplied by run i~ I+s to the consuming campaign i'~ I-s accumulated amount of state s consumed by campaign i'~ Is at the completion time of campaign ie I+s
REFERENCES Ierapetritou, M . G . ; F l o u d a s , C.A. (1998). Ind. Eng. Chem. Res., 37, 4360. I L O G O P L S t u d i o 2.1. (1999). User's Manual. K a r i m i , I.; M c D o n a l d , C. (1997). Ind. Eng. Chem. Res., 36, 2691. M 6 n d e z C.; Cerdfi J. (2000). Comput. Chem. Eng., 24, 369. Schilling, G.; Pantelides, C.C. (1996). AIChe Annual Meeting, p a p e r no. 171 d. X u e y a Z.; S a r g e n t R . W . H . (1998). Comput. Chem. Eng., 22, 1287.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
Increasing the selectivity process by periodic operation
of
699
the
Fischer
Tropsch
E Michiel Meeuse ~, Ronald M. de Deugd ~, Freek Kapteijn r Peter J.T. Verheijen t and Sjoerd M. Ypma r tprocess Systems Engineering Section, Department of Chemical Technology, Delft University of Technology Julianalaan 136, 2628 BL Delft The Netherlands *Section of Industrial Catalysis, Department of Chemical Technology, Delft University of Technology Julianalaan 136, 2628 BL Delft, The Netherlands The Fischer Tropsch synthesis converts synthesis gas to hydrocarbons. An inherent limitation of the reaction mechanism is the very wide product distribution. In this work the possibilities for narrowing this distribution by periodic operation are examined using a model based approach. The basic idea is to switch periodically between two feed streams with different feed compositions. It was found that the diesel fraction could be increased with 20%. Future research will have to verify this experimentally. Also some of the implications of this periodic operation on the rest of the process are presented. 1. INTRODUCTION The Fischer Tropsch synthesis is a process to convert synthesis gas to hydrocarbons that can for instance be used as transportation fuels. The process is heterogeneously catalyzed and can be carried out in a variety of reactors including trickle bed reactors, slurry reactors and fluidized bed reactors. A drawback of this process is the very wide product distribution, ranging from methane till heavy waxes. The reason for this can be found in the reaction mechanism. The ratio between the termination reaction rate and the propagation reaction rate is chain length independent, giving rise to the Anderson-Schulz-Flory distribution (Dry, 1996): Mn -- o~n- l ( 1 - O~)
(1)
Where Mn is the molar fraction in the product with a chain length of n, n is the chain length and o~ is the chain growth probability that is independent of the chain length. The value of 0r can be influenced by the choice of catalyst, the reactor temperature and pressure and the hydrogen carbon monoxide ratio of the synthesis gas. As a consequence the maximum achievable diesel weight fraction is 39%, corresponding with an o~ of 0.87. In the recently build Shell Middle Distillate Synthesis (SMDS) process in Bintulu Malaysia the diesel fraction is increased by first producing predominantly heavy products that are subsequently hydrocracked into the desired product spectrum (Sic, 1998). Figure 1 gives a schematic overview of the resulting process. In this work the possibilities for increasing the selectivity by dynamic operation are investigated using a model based approach.
700 The basic idea is to periodically switch between a feed with a low hydrogen/carbon monoxide ratio and a feed with a high hydrogen/carbon monoxide ratio. During the low ratio period predominantly chain growth takes place, while the high ratio is in favor of chain termination. The aim is to maximize the selectivity towards diesel (C10 - C20). In the next section the
hydrogen ,=lFischer TropschH v[ synthesis synthesisgas
Hydr~
t
=1 Distillation ]" 1 .~v I
products
Fig. 1. Simplified flowscheme of the SMDS process.
basics of dynamic operation is given. Then the modeling assumptions are discussed, followed by the implementation. Then the results are presented, including a discussion of the process implications for dynamic operation. Finally some conclusions are presented. 2. PERIODIC OPERATION Most continuous chemical processes are designed to operate under steady-state conditions. However transient conditions can sometimes provide benefits. By periodic operation the time averaged yield or selectivity sometimes can be increased to values that are not achievable in the steady-state. This is called resonance. Stankiewicz and Kuczynski (1995) give an overview of processes that can benefit from periodic operation. In the literature some indications can be found that periodic operation has an influence on the product distribution of the Fischer Tropsch process, but no work in the field of optimizing the diesel fraction, or model based studies for this process were found. Adesina et al. (1995) presents an overview of the experimental work on periodic operation of Fischer Tropsch synthesis. Van Neer (1999) has investigated the requirements for resonance in catalytic processes. Two phenomena for catalytic processes that are sufficient to obtain such resonance phenomena are presented: 9 The sorption behaviour of the forced component must be at least as fast as the sorption of the other component involved. 9 The surface of the catalyst has to be almost totally occupied at steady-state in the considered concentration range. The essence of these conditions is that strong non-linearities need to be present to obtain resonance. Since the two conditions mentioned above are satisfied in the Fischer Tropsch synthesis process it is expected that a higher selectivity can be obtained by dynamic operation.
701 3. M O D E L I N G A S S U M P T I O N S
Kinetic model In the literature there is no general consensus about the reaction mechanism. The mechanism we used in this work is shown schematically in figure 2. It is mainly based on the work of Dry (1996) and Van der Laan (1999). Hydrogen and carbon monoxide adsorb dissociatively on the catalyst surface. The oxygen reacts in two steps to water. The adsorbed Carbon reacts with the adsorbed hydrogen to the carbide intermediate that serves as building block. Chain growth occurs do to the reaction of the adsorbed chain with the adsorbed carbide building block. Termination occurs either to paraffins or olefins.
H2(g) ~
2 Hs
co (g) ~
Hs
Hs
Hs
Cs -~
CH2s " ~ Hs
Os C2H4(g) ~ OH
11
H20 (g)
Hs
C2H4s = - ~ - ~ C2Hss CH2s
Hs
H20s
Hs
C3H6(g) ~ C 3 H 6 s =
-~- -
= C2Hs(g) Hs
HTS
- =C3H8(g)
etc.
Fig. 2. Reaction mechanism.
Kinetic parameters For the kinetic model presented above, two types of parameters are required: rate constants and surface coverage fractions of the various components. In the literature no complete set of parameters for this model can be found. Therefore the estimation of parameters for this model is a combination of various literature sources. A detailed description of all parameters with their numerical value is presented in De Deugd et al. (2001). The only reaction modeled far from equilibrium is the dissociative adsorption of hydrogen.
Reactor model It is important that a distinction can be made between kinetic effects and reactor effects. Especially since a large variety of reactor types can be applied for this process. In order to be as generic as possible we have modeled the reactor as a ideal mixed reactor with a high space velocity. The reactor is assumed to be operating in a regime where mass transfer limitation
702 can be neglected. Although the Fischer Tropsch process is highly exothermal, the reactor is assumed to be isothermal. It is assumed that all the heat can be transferred to a cooling medium, e.g. boiling feed water that is flowing through cooling tubes. 4. IMPLEMENTATION The model was implemented in gPROMS (gPROMS user guide, 2000). The model consists of molar balances for paraffins and olefins up to C21. All heavier components are lumped into a heavy fraction. The diesel weight fraction is defined as the sum of the weight fractions of the paraffins and olefins from C 10 till C20. The periodic operation was implemented using a block profile for the feed composition. Figure 3 gives such a typical profile. This profile is described by four parameters. We have chosen to describe the profile with the cycle time, pulse time, base ratio and peak ratio. The total inlet flowrate is adapted such that the pressure in the reactor and the outlet flowrate are maintained constant.
pulsetime pul s e rati o ~, H2/CO ratio baserD ~cletime~
~-~~
time
Fig. 3. Feed profile.
The time average diesel fraction is given by:
f ~m,dieseldt cycle Xdiesel : f TCbm,hydrocarbonsdt~ cycle where Xdiesel is the weight fraction diesel, the total hydrocarbon massflow.
(2)
(~m,diesel is the mass flow diesel and (~m,hydrocarbons is
5. RESULTS
Steady-state analysis In the literature no consistent set of dynamic data is available to validate the dynamic model. Therefore only a steady-state validation of the model was done. Also a quantitative steady-state analysis is very hard to do since consistent data sets are lacking in the literature. The product distribution obtained by static operation was indeed the Anderson-SchulzFlory distribution. Also the maximum diesel yield under static operation was in accordance with the ASF distribution.
703 Furthermore a number of trends that are generally accepted were checked qualitatively (Van der Laan, 1999). Table 1 presents these trends.
Table 1 Steady state model verification
Cause
Effect
increase H2/CO ratio
decrease average chain length decrease olefin content increase average chain length decrease olefin content decrease conversion
increase pressure decrease space velocity introduce water in feed
Dynamic optimization The optimization was performed by varying the four variables that determine the block profile from figure 3 in order to maximize the objective function given by equation 2. The optimum feed-profile has a cycle period of 1.0,104s, a pulse time of 1.1 9 103s, a base ratio of 1.0, 10 -3 (almost pure carbon monoxide) and a pulse ratio of 19. The corresponding value of the objective function is 0.47, about 20% higher then the best achievable steady state result. However the time average ratio is far from stoichiometric (0.19 instead of 2) resulting in very low carbon monoxide conversions. Adesina et al. (1996) report time scales in the range from 102 to 103 seconds instead of the 104 from this present work. However this is not in conflict with our findings since they focus on smaller products (till C 10). Figure 4 shows the optimal product distribution that was obtained with the periodic operation, compared with the optimal distribution with static operation. This figure shows that not only the diesel fraction is increased, also the heavy fraction is increased, resulting in a decrease of the light fraction. This is also favorable from an economic point of view.
0.06
optimumperiodicoperation mass
~.action0.04
0.02 diesel range l
i
i
I
,
,
,
,
,
,
2
4
6
8
10
12
14
16
18
20
carbonnumber
Fig. 4. Optimum with static and periodic operation.
704 6. PROCESS IMPLICATIONS
The feed profile in this study is a block profile. On a lab-scale this profile is easy to generate. However on an industrial scale the profile will probably be smoothened. This can have a negative effect on the selectivity. Usually the feed to the Fischer Tropsch process is a mixture of hydrogen and carbon monoxide with a molar ratio of about 2. Sufficient technologies exists to produce this in one step. For periodic operation two feed streams are required with different compositions. So additional separation might be required to make the two streams with different composition. The reactor effluent will also have a periodic composition. This can lead to difficulties in operating the distillation section. Several parallel units that operate out of phase might solve this problem. 7. CONCLUSIONS It was demonstrated that periodic operation of the Fischer Tropsch synthesis can produce a diesel yield that is 25 % higher then the best achievable yield under steady state operation. However the model depends on number of assumptions was required in order to obtain a consistent model. The dependence of the results on the assumptions needs to be checked more rigorously. Ongoing research is aimed at the experimental validation of these results. REFERENCES
Adesina, A.A., Huidgins, R.R. and Silveston, P.L. (1995) Fischer Tropsch synthesis under periodic operation, Catalysis Today, Vol 25, 127 - 144 Deugd, R.M. de, Ypma, S.M., Kapteijn, E, Meeuse, EM., Moulijn, J.A. and Verheijen, P.J.T. (2001) Model-based optimization of the periodic operation of the Fischer Tropsch Synthesis, Presented at the 3rd International symposium on reaction kinetics and the development and operation of catalytic processes, Oostende. Dry, M.E. (1996) Practical and theoretical aspects of the catalytic Fischer Tropsch process, Applied Catalysis A, Vol 138, 319- 344 gPROMS User Guide (2000) version 1.8, Process Systems Enterprise LTD, London Laan, G.P. van der (1999) Kinetics, selectivity and scale-up of the Fischer Tropsch synthesis, PhD Thesis University of Groningen Neer, E van (1999) Forced oscillations in heterogeneous catalysis, PhD Thesis University of Amsterdam Sie, S.T. (1998) Process development and scale up: IV case history of the development of a Fischer Tropsch synthesis process, Reviews in Chemical Engineering, Vol 14, No 2 Stankiewicz, A. and Kuczynski, M. (1995) An industrial view on the dynamic operation of chemical converters, Chem Eng Process, Vol 34, 367 - 377
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. J~rgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
705
On-line optimal operating policy and control of a batch free-radical polymerization process G. Mourikas a' b, p. Seferlis a, A. J. Morris b, and C. Kiparissides a a Department of Chemical Engineering and Chemical Process Engineering Research Institute Aristotle University of Thessaloniki, P.O. Box 472, Thessaloniki, Greece 540 06 b Department of Chemical and Process Engineering, University of Newcastle-upon Tyne, NE1 7RU, U.K. An on-line optimizing control scheme that ensures the satisfaction of polymer property specifications under the influence of time-varying model parameters and unknown initial conditions is developed for the free-radical polymerization of methyl-methacrylate in a batch reactor. The control scheme combines a state/parameter estimation step based on on-line measurements of process variables that updates the values of model parameters with an optimization step that calculates the optimal operating policy. The control objectives require the satisfaction of quality specifications based on molecular weight properties (e.g. average molecular weights) of the final polymer product subject to time-varying kinetic parameters and unknown initial level of impurities in the reactor. 1. INTRODUCTION The operating objectives in many batch polymerization processes must satisfy complex property requirements for the final polymer and simultaneously achieve the greatest economic potential during the batch operation. Most mechanical and rheological properties of the polymer products are directly or indirectly linked with the molecular structural properties of polymer chains (e.g., molecular weight distribution, MWD, copolymer composition distribution, CCD, chain sequence length distribution, CSD, and so forth), which are difficult (sometimes impossible) to measure on-line. Average polymer molecular weight properties (e.g., number and weight average molecular weights), which can be indirectly inferred from the on-line measurement of the solution viscosity or melt index of the polymer, are selected as the major controlled variables that need to be maintained within well-determined limits so that desired product quality criteria can be satisfied. Control strategies require that pre-determined trajectories for key process variables (e.g., reactor temperature) are implemented during the batch operation [ 1]. However, the operation of the batch polymerization reactor is influenced by process disturbances and model parameter variations due to the inherent process-model mismatch or the changing operating conditions of the process. Unless the time-varying model parameter values and subsequently the optimal control trajectory are updated regularly during the batch operation, the control strategy would fail to meet the product quality specifications and the operating requirements [2-4].
706 The present work presents a scheme for the on-line optimizing control of the number and weight average molecular weights of the polymer product in a free-radical batch polymerization reactor under significant model parameter uncertainty. An on-line temperature profile represented as a discrete sequence of temperature set point changes is evaluated that can ensure the satisfaction of product quality and process operating objectives (e.g., product yield, temperature constraints) in an optimal sense. A regulatory control system attempts to enforce the calculated set point sequence during the batch. A state/parameter estimator utilizes on-line measurements of process variables to estimate the values of the time-varying kinetic parameters and unknown initial level of impurities in the reactor. 2. OPTIMIZING CONTROL SCHEME DESCRIPTION Figure 1 shows the schematic of the information flowsheet for the on-line optimizing controller. The measured output variables, Ymeas, is compared to the model predictions, YmThe anticipated error incorporates the unmeasured process disturbances, d, the measurement noise and the process-model mismatch. The state/parameter estimation block uses the error term to retrieve the entire state vector of the system, Yest, and further provide estimates for the stochastic time-varying model parameters. Subsequently, the updated values for the state variables and the model parameters, Yest and p, enter the optimizer block. A set of target values associated with the molecular weight properties, Yq,sp, and a performance index that specifies the overall process control objectives in a hierarchical order conclude the input stream to the optimization block. The optimizer calculates a series of optimal set point changes for the controlled variables that would ensure the satisfaction of the overall control objectives in an optimal sense. The time-optimal sequence of set point changes, Yopt,sp, are then passed to the regulatory controller that forces the process to follow the optimal trajectory as close as possible. The execution of the optimization task is performed on a regular basis that mainly depends on the rate of change of the model parameters, the frequency of the process disturbances and the ability of the process to track the optimal control trajectory. 2.1. Calculation of the time-optimal control trajectories The major goal of the optimization task is to determine a sequence of set point changes for the control variables that would satisfy a certain performance criterion. The process control variables are selected based on their impact on the product quality and the process ability for real-time implementation. For example, in the case of MMA free-radical polymerization, the polymerization temperature plays the most important role in controlling the final polymer properties (e.g., average molecular weights, Mn, Mw) and monomer conversion. Furthermore, a desired temperature trajectory can be easily followed by the regulatory control system of the reactor. Other examples of optimization variables could be the amount of initiator in the reactor or the feed pattern in semi-batch settings. The performance index usually depends on the final state of the system and the dynamic characteristics of the batch polymerization. In the free-radical polymerization of MMA the dynamic optimization problem aims to minimize the batch duration while meeting the product quality specifications and satisfying the operating constraints.
707
Min s.t.
J=w~tf+w
2
Cf
-1.0
/2 /
Mnf + w 3 Mnd - 1 . 0
)2 / +
W4
/~ = f ~ x , u , Q y : h(x,u,t)
Mwf Mw d
/2 -1.0
(P1)
f(x,u,t) and h(x,u,t) are the polymerization reactor's modeling relations with x, y, and u being the state, output and manipulated variables, respectively. The performance index, J, penalizes deviations of the final number and weight average molecular weights (Mnf, Mwf) from a set of target values (Mnd, Mwd). Furthermore, the satisfaction of operating goals such as the achieved monomer conversion per batch (cd) are incorporated in the objective function. Deviations from the product quality targets usually have a substantial impact on the process economics (e.g., product value may drop or the entire batch content may have to be discarded). On the other hand, the operating targets aim to maximize the polymer production per batch cycle and minimize the duration of each cycle. The weighting factors, wl-w4 specify the relative importance of the usually competing terms in the objective function. The solution of (P 1) is achieved through discretization of the differential equations using orthogonal collocation on finite elements techniques [5]. The resulting equation set is hence consisted of purely algebraic equations, which can be solved by conventional nonlinear programming techniques. 2.2. Nonlinear state and parameter estimation
In the present work two types of estimators are employed, namely an extended Kalman filter (EKF) for estimating the states and the time-varying model parameters and an optimization based estimator for calculating the initial values of unknown state variables. Both approaches require that additional sensible non-stationary stochastic states representing the time-varying model parameters and the unknown initial states need to be introduced to ensure the bias elimination from the state estimates. The optimization based estimator calculates the adjustments to the state variables (deterministic and stochastic) by minimizing the distance between the model prediction and the actual measurements from the process over a specified time horizon. The estimator balances the reliability of the process model predictions against the reliability of the process measurements. In the case of unknown initial conditions the time horizon extends from the beginning of the batch until the time instance the most recent process measurements become available. The optimization-based approach is more tedious than the EKF estimator but allows the use of the full nonlinear model and satisfies explicitly the process constraints. 3. OPTIMIZING CONTROL OF A BATCH FREE-RADICAL POLYMERIZATION
REACTOR The estimation/optimization scheme outlined in Figure 1 is applied to a simulated model of a free-radical polymerization of methyl-methacrylate (MMA). The mathematical model presented in [4] closely describes the dynamic behavior of an experimental pilot scale system illustrated in Figure 2. Heating and cooling of the reaction mixture is achieved by controlling the flows of two water streams (e.g. a hot and a cold water stream) through the reactor jacket. The polymerization temperature is controlled by a cascade control system consisting of a primary PID and two secondary PI controllers. The polymerization is highly exothermic and
708
Figure 1. Flow diagram of the optimizing control scheme,
Figure 2. Schematic of the MMA batch polymerization reactor.
exhibits a strong acceleration in polymerization rate due to gel-effect (e.g. the termination rate constant decreases with conversion). Usually the gel effect contribution on the termination rate of reaction cannot be determined accurately, mainly due to modeling uncertainty and difficulty in obtaining the true values of the kinetic parameters. The uncertainty is described as an additional stochastic term, gSt,corr, introduced in the functional relationship that provides the termination rate of reaction, kt.
kt = kt0 "gt'g~t,corr
(1)
The term gt describes the influence of the gel-effect on the termination rate of reaction. A random walk model was assumed for the time variation of the term gSt,corr, leading to the following relationship for the behavior of the stochastic state: dg S
t,corr
dt
= f~(x,u,t) + w sg = 0 + w sg
(2)
where WSgis a random variable that represents white Gaussian noise. In the considered scenario, the actual value of term gSt,corr varied linearly with time from a value of 1.0 at the beginning of the batch to a value of 0.67 at the end of the batch. On-line measurements of the reactor and jacket temperatures and monomer conversion were available every minute. The state/parameter estimator used an extended Kalman filter algorithm for the estimation of the state variables and the parameter values for the termination rate constant (through the stochastic term gSt,corr). Good modeling relationships allowed the estimation of the otherwise unobservable, with the current set of measured variables, dead polymer moments [6]. The specifications for the number and weight average molecular weights were set equal to 4.9.105 and 1.44.106 kg/kmol, respectively, leading to a polydispersity index of 2.94. A final monomer conversion of 0.87 was the target value after 120 minutes of operation. State variable and the model parameter estimates are calculated every minute, while a new optimal temperature trajectory was evaluated every 15 minutes using the most recent values for the
709
Figure 3. Initial 'o' and updated (solid) Figure 4. Average molecular weight temperature set point changes and properties for the final polymer product with implemented (dashed) reactor temperature, kinetic parameter variation. kinetic parameters. The optimization task evaluates a piecewise constant polymerization temperature profile for the remaining of the batch cycle (total batch duration is fixed). The initial optimal temperature trajectory (Figure 3) suggests a big temperature rise at the appearance of the gel-effect in order to suppress the increase of the weight average molecular weight and hence keep the polydispersity index at the desired level, a behavior consistent with previous studies [ 1]. The optimal polymerization temperature profile using the updated values for the termination rate of reaction attempts to shift the occurrence of the gel-effect phenomenon later in time with lower temperature levels at the initial stages of the batch, while a higher temperature peak restricts the magnitude of the increase in the weight average molecular weight due to the gel-effect. The dashed curve represents the dynamic behavior of the reactor's temperature and indicates possible dynamic limitations imposed by the regulatory control system. Figure 4 compares the evolution of the key average molecular weight property indicators for the initial and the updated profiles during the batch. The plots clearly suggest that the updated control trajectories result in a definite improvement of the final product properties, thus successfully compensating for the effects of disturbances. Unknown amounts of impurities in the monomer or in the reactor vessel may influence the effectiveness of the initiator and subsequently the overall achieved monomer conversion and product quality. The effects of impurities can be estimated during the batch operation from on-line measurements of the achieved monomer conversion. An updated effective initial concentration of the initiator was calculated, as sufficient information were available from process measurements, which was used to re-integrate the model to predict the time-optimal temperature trajectory. The MMA batch reactor was subject to a 50% reduction of the effective initial initiator's concentration. Measurements were subject to Gaussian noise with standard deviation equal to 2.10 -3 for the conversion and 0.2 K for temperature. A target polydispersity index of 5.08 (Mnd=4.9 9105, Mwd=2.03 9106 kg/kmol) was set after 80 minutes of operation. The optimization based estimator provided new estimates every minute while the optimizer evaluated a new optimal control trajectory every four minutes. The updated
710
Figure 5. Time evolution of the optimal Figure 6. Average molecular weight temperature trajectories, properties for the final polymer product with unknown initial conditions. sequence of temperature set point changes keeps the characteristic shape of the initial profile (Figure 5) but requires slightly higher temperature levels before the gel-effect becomes dominant and much higher temperature after the big rise in the monomer conversion occurs to compensate for the reduced initiator concentration. Figure 6 depicts clearly the superior performance of the optimizing controller in meeting the desired values for the final number and weight average molecular weights over the off-line alternative (initial profile). 4. CONCLUSIONS The two examples verify the ability of the proposed control framework to operate under the influence of parametric uncertainty for highly sensitive batch polymerization reactors, while satisfying stringent quality specifications. The detrimental effects of time-varying kinetic parameters and unknown initial states on the final molecular weight properties were successfully alleviated by combining a nonlinear estimator to provide updated values for the model parameters with an optimization stage to calculate an optimal operating policy for the batch reactor. Dynamic optimization methods were employed for the determination of optimal control trajectories that meet the control objectives and satisfy the operating constraints and specifications. REFERENCES
1. I. M. Thomas and C. Kiparissides, Can. J. Chem. Eng., 62 (1984) 284. 2. D. Ruppen, D. Bonvin. and D. W. T. Rippin, Comput. Chem. Eng., 22 (1997) 185. 3. T. J. Crowley, and K. Y.Choi, Ind. Eng. Chem. Res., 36 (1997) 3676. 4. G. Mourikas, P. Seferlis, A. J. Morris and C. Kiparissides, sub Ind. Eng. Chem. Res. (2000). 5. L.-B. Tjoa and L. T. Biegler, Ind. Eng. Chem. Res., 30 (1991) 376. 6. D. J. Kozub and J. F. MacGregor, Chem. Eng. Sci., 47 (1992) 1047.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
711
Using genetic algorithm in robust nonlinear model predictive control Z. Nagy a, S. Agachi a, F. Allgower b, R. Findeisen b, M. Diehl c, H. G. Bock c, J. P. Schloder r a"Babes-Bolyai" University, Arany. J. 11, 3400 Cluj-Napoca, Romania, bUniversity of Stuttgart, Pfaffenwaldring 9, 70550 Stuttgart, Germany, CUniversity of Heidelberg, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany, In this paper two NMPC algorithms are proposed which improve considerably both the computational complexity and especially the robustness of the classical NMPC algorithms. Both algorithms use the advantages of genetic algorithms (GAs), in combination with sequential quadratic programming (SQP), to solve the complex optimization problem from the NMPC. The control performance, computational burden and robustness of the proposed NMPC techniques are tested for the control of a simulated high purity distillation column. 1. INTRODUCTION One of the most widely used advanced control strategies in chemical industry is Model Predictive Control (MPC), because it can explicitly handle the multivariable interactions [ 1]. The main idea of MPC algorithm is to find the control vector trajectory that optimizes a performance objective over a future prediction horizon. Predicted values of the controlled parameters are obtained from a process model. The strongly nonlinear behavior of most of the chemical processes suggests the implementation of nonlinear model predictive control (NMPC) strategies, which use a nonlinear model for prediction. If an accurate first principle model of the process can be developed, it describes more correctly the dynamic behavior of the system, in a wide operation region, compared to other, empirical models. Consequently, the use of the first principle model in the nonlinear control algorithm (assuming the accuracy of the model) can be very benefic for the performances of the control system. In this case predicted outputs in the NMPC are obtained by integrating the process model over the prediction horizon. Consequently, solving the optimization problem for the control movement calculation in the case of complex process models might demand great computational effort and time. The optimization usually is carried out in each sampling time. There are efficient optimization algorithms for convex optimization problems. However, in NMPC a constrained non-convex optimization problem has to be solved on-line. Thus, the global solution with the conventional single shooting methods can not be guaranteed. Furthermore, the conventional iterative optimization techniques are very sensitive to the initialization of the algorithm and usually lead to poor solutions due to convergence to local optima. Consequently, the control performance and safety of the controlled system might depend dramatically on finding the global optimum. One of the optimization techniques that seem to be suitable for constrained, nonconvex, nonlinear optimization problems is based on genetic algorithms.
712 In this paper genetic algorithm (GA) is proposed to carry out the optimization in each sampling time, and its performance is compared to a sequential-quadratic-programming (SQP) technique. GAs function in a similar manner as an optimizing search routine. These algorithms are based on the mechanics of natural selection and basic natural genetics. The most impressive feature of GA is its ability to combine survival of the fittest among individuals (in a population) whilst performing an almost random search through the parameter space minimizing the freezing of the search in local minima. In this approach the population consists of the control variables from the control horizon. To combine the advantages of both algorithms (GA and SQP) two approaches are presented in which GA is combined with SQP, and the proposed control schemes are tested for a simulated high purity distillation column. 2. GA BASED NMPC
The objective of NMPC is to calculate a set of future control moves (control horizon - M) by minimization of a cost function, like the squared control error on a moving finite horizon (prediction horizon- P) [2]. The on-line control problem can be expressed as below: Solve:
minIJ(x(t),u(.))) u(.) t
(1)
x
with: P
J(x(t),u(.))
=
ZIIQ, t=l
M
(r(k + i)- yp (k + 1)11= + Zl[R, Au(k § i-1~1 ~
(2)
t=l
Subject to the constraints derived from the process model equations and restrictions concerning the process inputs and outputs. Here x = state variables, u = manipulated variables, y = output variables, r = set points, Q,R - weighting factors. To solve the on-line control problem imposed by the NMPC algorithms a sequential approach was implemented. This method uses different algorithms for the integration of differential equations and optimization. First, according to the algorithm of optimization, a sequence of control movements is considered; with this, the system of differential equations is numerically integrated to obtain the trajectory of the controlled variables. Then, the scope function is computed. Depending on its value, the method of optimization yields a new sequence of control movements and the algorithm is repeated until the optimal sequence is obtained. Only the first value of the sequence is applied to the process. Since a constrained non-convex nonlinear optimization problem has to be solved on-line, the major practical challenge associated with NMPC is the computational complexity that increases significantly with the complexity of the models used in the controller. There has been significant progress in the field of dynamic process optimization. Fast on-line optimization algorithms have been developed that exploit the specific structure of optimization problems arising in NMPC, and real-time applications have been proven to be feasible for small-scale processes. However, the global solution of the optimization cannot be guaranteed and the development of fast and stable optimization techniques is one of the major objectives in the NMPC research.
713
Fig. 1. Sequential GA-SQP optimization algorithm
Fig. 2. Parallel GA-SQP optimization algorithm based NMPC approach
In this paper genetic algorithm (GA) is proposed to be used in combination with SQP algorithm to carry out the optimization in each sampling time. Genetic algorithms are randomized search algorithms that are based on the mechanics of natural selection and genetics [3]. It uses the principles of natural selection based on the survival of the fittest in order to search the solution space of an optimization problem. GAs can be used efficiently in application in the field of optimization, because of their ability to search efficiently large state spaces, which makes them more robust with respect to the complexity of the optimization problem compared to the more conventional optimization techniques. Genetic algorithms code the candidate solutions of an optimization algorithm as a string of binary digits. GAs consider a number of solutions, called individuals, which together form a population. They modify and update the individuals in a population iteratively, searching for good solutions of the optimization problem, by evaluating afitness function. This function can be compared with an objective function in classical optimization and it indicates how good a candidate solution is. Each iteration step is called a generation. According to the survival of the fittest idea, genetic algorithms maximize the fitness value, in contrast to classical optimization, where one usually minimizes the objective function. GAs evaluate the individuals in the population by using a fitness function. This function indicates how good a candidate solution is. GAs start with the generation of an initial population, containing individuals, which represent initial estimates for the optimization problem. It should be emphasized that GAs evaluate a set of solutions in the population at each iteration step, in contrast to classical optimization methods, which evaluate a single solution at each iteration step. Thus, GAs usually find very quickly the neighborhood of the optimal solution. However, GAs present a slower convergence in finding the exact optimal value, because it does not use the derivatives of the objective function (being a kind of random search technique). To combine the advantages of both algorithms (GA and SQP) two approaches are presented. The first approach combines the two optimization methods in one, more robust
714 technique, in which GA is used for preoptimization to find the neighborhood of the global optimum. From the best solution found after a certain number of generations the SQP technique completes the optimization. As the GA operators are designed to maximize the fitness function, in the preoptimization step, the above minimization problem, has to be transformed into a maximization one. This can be done, for instance, by using the following transformation:
1
F =~ 1 + J(-)
(3)
The main idea of this approach is shown in figure 1. With this combination used to solve the optimization in the NMPC algorithm a very good robustness of the control scheme is achieved. The second approach uses SQP as the main optimization technique in the NMPC algorithm but parallel the control problem is also solved with GA. Whenever a solution from the slower GA is available it is compared to the one obtained with the SQP algorithm and the better one is implemented. This approach, presented schematically in figure 2, allows real time feasibility of the NMPC due to some special initial value embedding techniques, which can be implemented with SQP. 3. SIMULATED SYSTEM The column considered in our simulations has N=40 trays and is used for the separation of Methanol and n-Propanol. The feed flow (F) enters into the column at tray 21, and it is considered as the main source of disturbance through the feed flow composition (XF). The control system is considered in LV configuration, i.e. the reflux flow (L) and the vapor flow (V) are considered as the control inputs. As usual in distillation control, x8 and XD are not controlled directly. Instead, an inferential control scheme that controls the deviations of the concentrations on tray 14 and 28 from the set points is used. Since for the standard operating conditions the turning point positions of the waves approximately correspond to these trays, one can expect good control performance with respect to the concentrations at the bottom and top (x~ and xD). 4. SIMULATION RESULTS For comparison, simulations were performed with the first principle model based NMPC algorithm, using the above mentioned two different approaches in solving the on-line optimization problem and the classical algorithm where the SQP is only used to solved the open-loop on-line optimization from the controller. In the simulations the following form of the objective function was used:
x,4(k + i)_ xi,4] []2Q M-,
i=1
I
xz8 (k + i)- x2"~
+ Z,:0II
1) +y+ -
u(k + j)]]12n
(4)
The results obtained with Q = I; R = 0; P = 5; M = 1; T s a m p --- 30sec., for a disturbance scenario with step changes in XF and in the set points are presented in figure 3.
715 / .~o% ^
/
oot
0.85 x
0.15
0%'se'0~ + \ /
0"1 / : 0 4.5
:~--in x_l, 500
x 10 .5
,
,
1000
1500
0.75 0 x 10 s 8/r t !
v 6i o
500
looo
t[s]
~5oo
r-
x
t [s]
3.5 3
I
set point ,
__ : _ f ' ~
4
o
~oo
10'00
1500
t [s]
----. Seq. GA-SQP - - - Par. GA+SQP
~
500
looo
t[sl
15oo
Fig. 3. Comparison of the performance of the three NMPC approaches for the following disturbance scenario: -20% step in XF at t=- 90s; +20% step in XF at t=- 420s; after that setpoint changes: Xl4.sp=0.2 at t=-840s, Xl4,sp=O.1at t=-1500s, x28,sp=0.75 at t=l 180s, x28,w=0.9 at t=1500s The parameters of the GA are selected based on the characteristics of GA to quickly find the neighborhood of the global optimum. In figure 4 the best values of the fitness function found by the GA are presented for one optimization in the case of different control horizon. It can be observed that, in all cases after only 20 generations the GA found a close-to-optimum value, after which it converges slowly to the exact value of the optimum. Consequently in the sequential GA-SQP algorithm the GA is used at the beginning only for 20 generations. The best solution found after 20 generations is close enough to the global optimum so that the SQP algorithm starting from this solution finds fast, with small computation effort the exact value of the optimum. Similar conclusions can be obtained from figure 5, too, where the computational burden, expressed in the number of floating point operation (flops), of the sequential GA-SQP algorithm is compared to the situations when only the GA or SQP are used individually to solve to optimization. One can observe that GA becomes computationally more attractive when the dimension of the optimization problem increases, and that the combined GA-SQP approach decreases significantly the computational burden in all cases. In all simulations a population with 50 individuals for every optimization variable is used. The control performances obtained with the genetic algorithm based NMPC techniques are similar in this case with the results obtained using only the SQP algorithm to solve the on-line control problem. Nevertheless, the sequential GA-SQP based NMPC improved slightly the control performance. However its main advantage is the improved robustness of the on-line optimization as it can be shown in table 1 and the significant decreasing of the computational time needed for the solution of one open-loop control problem, presented in figure 5. Table 1. Comparison of the robustness of the 3 optimization algorithms SQP Sequential GA-SQP Number of optimization failures in the case of 9 2 660 open-loop optimization with 20 disturbance
Parallel GA-SQP 0
716
~0
~1-~
0.0r
|ili "m ..
i
......
ilI'| 0.4 " -J ........
i
i
i
:: ............
i: .....
!:
|
i ; ...........
! 4............
i i............
i i...........
/1
:
:
:
'~
..............
10 8
--"".
- - - M=2
Ol.~
i
0
20
L-'-" M:9 40 60 Number of generations
80
i ..............
......
...,-"
M=I
[---"
j i ....... : ....... ~ ~
GA
.....
/
: ...........
O.2 tl_lI
I- -*-
~ 100
Fig. 4. Evolution of the fitness function in one optimization with GA for different M
106
:
:
: .,,.
i .....
.Z. /
:
:
:
6
7
8
.,soSO~"*'~162 .s"
1
2
3
4
5 M
9
Fig. 5. Comparison of the computational burden of the algorithms for different M
For the parallel combined GA-SQP algorithm, both the control performance and computational burden are similar to the ones obtained with the SQP (the best results obtained when the algorithm does not fail), because in this case the control action implemented usually is the one obtained from the SQP algorithm (which is faster). However the parallel running GA can increase significantly the robustness of the control structure, because whenever the main SQP fails, a close-to-optimum control action is always available from the GA. 5. CONCLUSIONS In this study simulation results of first principle nonlinear model based predictive control (NMPC) of a high purity distillation column are presented. Two different NMPC approaches are proposed to improve the robustness of the NMPC controller, in which the advantageous properties of GAs in solving successfully, complex nonconvex nonlinear optimizatiojn problems, are exploited. The first approach uses GA for preoptimization in solving the on-line open-loop control problem. From the best solution obtained the SQP continues the solution. This approach improves both the control performance and the robustness of the NMPC. Additionaly, the computational burden in this case is decreased. The second approach uses GA to solve the optimization problem in parallel with the SQP improving thus the robustness of the NMPC, practically eliminating the controller failures due to the failure of the optimization, thus, conferring to it a great importance for practical NMPC implementations.
REFERENCES 1. C. E. Garcia, D. M. Prett and M. Morari, Automatica, 25, (1989), 335. 2. F. Allgower, T. A. Badgwell, J. S. Quin, J. B. Rawlings and S. J. Wright, Advences in Control, Highlights of ECC'99, (1999), 391. 3. D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, 1989.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
717
Inferential C o n t r o l of M i c r o b i a l Batch Culture K. Preu[3 a, M.-V. Le Lann b a Institut National Polytechnique de Toulouse, ENSIGC, CNRS UMR 5503
18, Chemin de la Loge - 31078 Toulouse Cedex 4 - France E-mail: [email protected] b LAAS/INSA- CNRS UPR 8001 - 7, Av. du Colonel Roche - 31077 Toulouse Cedex 4 E-mail: [email protected] Keywords: Brevibacterium linens, Disturbance, Estimation, Filter, Specific growth rate 1. INTRODUCTION Control of specific growth rate (referred to as growth rate in the following) requires the online estimation of this parameter. In order to obtain good closed loop performance, it is crucial that the estimation procedure provides a quick response to the applied control actions. In an industrial environment it is likely that the only available on-line measurements are components concentration in the gas phase. Under these conditions it is difficult to use observerbased [ 1] or Kalman filtering [2] estimation procedures if yield coefficients and kinetics are not known. Other approaches deduce the reaction rate of one component from gas phase measurements [3], [4]. In these cases the time derivative of the respective reaction rate must be known. This variable dealt out to be very corrupted with noise, so that smoothing of the time derivative is inevitable. Unfortunately this renders the estimation insensitive to quick changes of operation conditions. These problems can be overcome by the estimation procedure that is based on exhaust gas analysis, outlined in section 3 of this paper. This estimation procedure has been validated with experimental data from a 20 liters industrial pilot bioreactor [5]. The concerned pilot plant is described in section 2 of this paper. In order to reject disturbances form the growth rate estimate, the model based filter described in section 4 was implemented. Several manipulated variables might be used to control the specific growth rate. Therefore a model based supervisory routine (section 5) determines the action on these control variables (the setpoints of the respective slave controllers). Currently one slave control loop is implemented: culture temperature. With this approach, a time-varying setpoint profile of growth rate of Brevibacterium linens during batch culture was controlled. The experimental results (section 6) show a good performance of the proposed approach.
2. PILOT PLANT The pilot plant consists of a 20 liters stirred and aerated CHEMAP bioreactor. The considered process is batch growth of Brevibacterium linens, a bacterium used in the diary industry. It is grown on a synthetic culture medium containing lactic acid and amino acids as main substrates. Samples were taken at irregular intervals and the concentration of the following components was analyzed: biomass X (optical density), lactic acid L (enzymatic kid), amino acid A (HPLC). In fact, 16 amino acids were analyzed separately and regrouped to one
718 pseudo-component. Exhaust gas composition (CO:, 02), pit, dissolved oxygen concentration (P02), reactor temperature (7), stirrer speed (vs) and inlet aeration rate (Fin) are measured online. Data acquisition, recording and visualization are performed by the MFCS/win-soflware (I3. Braun Biotech). Its internal PID control reactor temperature and dissolved oxygen concentration. The latter is cascaded by stirrer speed control and regulation of the aeration rate if the maximum stirrer speed is reached. The entire process is operated via the MFCS/win operator interface. A macroscopic reaction scheme, describing some parts of the metabolism of Brevibacterium linens, was recently proposed by [6]. It can be summarized as follows: L + 0 2 - > X q- C 0 2 (R1) A at- 0 2 -> Y-+- C 0 2 (R2) This representation implies that oxygen consumption and production of carbon dioxide are proportional to the production of biomass. Generally both reactions take place simultaneously but R1 is predominant the beginning of a batch. The relation between reaction rates rl and r2 changes during the course of a batch run and depends on the concentration of the substrates lactic acid and amino acid. 3. G R O W T H RATE ESTIMATION At the origin of growth rate estimation is the differential equation for microbial growth with a neglected maintenance term: equation 1. dX
---~=lt
Vn,a:Yax* dX (2) , ~
*X (1)
'
Qa = ~ V n , a , d t + Q a ( t o ) : Y a , x , x to
(3)
The reaction rate vR,a of component a, which must be directly linked to growth, is defined as indicated by equation 2. Its integral Qa is equation 3. Replacing biomass X and its derivative in equation 1 by equations 2 and 3, under the assumption of a constant yield Ya, X, gives an expression, equation 4, for the growth rate:
/~(t) = vR'~
dvn,"(t)
(4)
Oa (t)
fl(t) :
at VR,a(t)
(5)
The initial quantity of component a, Qa(to), is unknown. When calculating the growth rate for a terminated batch run a posteriori, the appropriate value can be found manually. In case of starting the estimation procedure during the course of a batch run, Q~(to) can hardly be determined a priori, neither manually. Hence an additional procedure, allowing for the on-line automatic determination of Qa(to), is necessary. It relies on the alternative expression for the growth rate given by equation 5. d2X dt 2
= #*
dX --~
(6)
dvn'" Y~ x * d 2 X = dt ' dt 2
(7)
vPR
(t) = ~ a i * ti i:0
(8)
Equations 6 and 7 result from deriving equations 1 and 2 assuming that/z 4=f(t) and then introducing equations 2 and 7 in expression 6. In practice, the derivative of the reaction rate will be corrupted with noise. This problem is solved by approximating the reaction rate by a polynomial (equation 8). The data used for this approximation is collected on a gliding window of past measured values. The length of this window Atw depends on the order of the polynomial: the higher the order, the more data is required. In order to start the estimation of the growth rate quickly after the start of a batch, it is preferable to use a low order. Nevertheless the order should allow for approximating an exponential curve (microbial growth with constant growth rate). Hence n=2 appears to be a reasonable choice. Deducing the derivative
719 of the reaction rate from equation 8 is straightforward. The initialization procedure can be triggered manually or depending on a certain condition with respect to process state. It uses equations 4 and 5 to determine the initial quantity Qa(to).After initialization, only equation 4 is used for growth rate estimation. Its advantage compared to equation 5 consists in the fact that no smoothing is carried out, so that the estimation provides a quick response to variations in the process. However disturbances contained in the reaction rate are directly transferred to the estimation. For this reason the estimated growth rate must be filtered (see section 4), in order to reject disturbances, before it can be used for closed loop control. Under the assumptions that (i) the concentration of carbon dioxide in the liquid phase remains constant and (ii) the gas volume between the culture surface and the exhaust gas analyzer is zero, the reaction rate of C02 is equivalent to the exhaust rate CER. For practical implementation of the estimation procedure the reaction rate is then considered to be vR,a=CER. 4. M O D E L B A S E D F I L T E R
Disturbances contained in the reaction rate are directly transferred to the estimation. For this reason the estimated growth rate must be filtered in order to reject disturbances. The basic idea of the model based filter described in the following is that at any point of time the future evolution of the value to be filtered depends on some underlying, in the present case biological, principles. If these principles are, at least approximately, transformed into a model and if furthermore model uncertainty and not modeled effects are taken into consideration, then a dynamic range, according to the principles represented in the model, for possible evolution of the value can be calculated: ,L/fil,min (i) Crier,max(i)1 ]Llfil (i) = / ]/fil,min (i)/f r < ,L/fil,min (i) [ (10)
/u(i) else
L
J
r
(11)
at flfil,min(i)= lu~,(i-1)+(d-~-~ min(i)-ec) *At at
(12)
The bounds of the admissible range are calculated by equations 11 and 12. The positive parameter ec allows for convergence of the filtered value towards the real evolution. In these equations appears the minimum and maximum derivative of the value to be filtered. They are determined by the model mentioned above. By means of parameters e (+) and e (-) model uncertainty is taken into consideration.
+>0
d/tmax= -~
dt
l+lOOe)_)~ - - ~ -
dlt .I -~ 1- ~6-@ else
d/l.
(13)
e
.
I ( '+'/
d/~<0
-~ 1+i-0-6 if--~ d'u dt min = [ dl~,(le'-) I L --~ [,-i-O-d) else
(14)
The time derivative of growth rate d/~/dtis deduced from the filter model. Suppose a kinetic model for growth rate of the following type: tn
/~ =/t~ I-[ f ( x , ) i=1
m
1 - I-[ f(x~ i=1
(15)
720 o
with p~ being the reference growth rate resulting from the reference states x i. Its time derivative is equation 16, which can be transformed into equation 17.
d~ = l,o,
u
df,(x,)
i=1
L
dt
m
* Hf/(x')
(16)
j=l,j•i
0
-dt- =
~,o
9
&, df,(x,) i=1
dt
dx/
9
[ i f,(x,)
j=l,j~-i
(17)
Equation 15 can be transformed into equation 18, which allows for replacing, the a priori unknown, parameter p ~ in equation 17 so that the desired time derivative is obtained (equation 19). 1-If(x,
-~ = P
,=, - -
dr,
*
(19)
i=l In practice, as the true growth rate /z is unknown, its filtered estimate /~fit is used. Furthermore the derivative of the state variables dx/dt must be known. In the case of culture temperature and dissolved oxygen this value can be easily deduced from measurements. The derivative of the kinetic function f is known from the kinetic law. For example for the influence of temperature on growth rate Rosso et al. [7] proposed the following law: (T-Tmax)*(T-Tm,,,) 2 fr(T):
%,
__Zmin,), [(Top, _ rmin ), (r _ Topt )_ %t _ rmax ), (Top, +Train
_2,T)
]
(20)
5. CONTROL STRATEGY Depending on the respective process, different operation conditions can be used to control growth rate (i.e. temperature, dissolved oxygen concentration, substrate concentration). Generally these operation conditions are controlled by independent slave controllers. In order to control growth rate, its setpoint Pset has to be transformed into setpoints X_setfor the slave controllers. This is done via a supervisory control routine. This routine distributes also the total control action on the different available controlled operation conditions. Suppose a parameter c so that the growth rate setpoint is related to estimated growth rate/zfit under current operation conditions xt: p.,.e,(X,e,) =C* pt~,(X,) (21) By means of the kinetic model (equation 15) the growth rate at both relevant sets of operation conditions can be written as: m
,u~, (X,) = ,u~ * ]'-I f~ (x,.,)
m
(22)
,u~, (x.,.~,) = u~ * ]--I f (x ,.....,)
i=1
(23)
I=1
Obviously the goal of control is to minimize the difference between the growth rate setpoint and the estimated value. With equations 21 to 23 the MISO-supervisory control routine becomes the following non-linear constrained optimization problem that has to be solved by an appropriate numerical method:
Min J(x.,.e,)= H f ( x, t)-c* T-I f~(x,,,) -P(x*e"X') ....
/=1
Xi .....t,min ~ Xi ....t ~ Xi,set,max
i=1
(24)
i = 1,...m
Function P accounts for the preference among the possible manipulated variables (operation conditions). In the present case this function was not implemented because the
721 culture temperature is the only manipulated variable considered. For this reason, equation 20 is the only kinetic functionf appearing in the optimization problem (equation 24). 6. EXPERIMENTAL The experiments shown in this section were carried out on the pilot plant described in section 2 of this paper. For several data sets the following parameters dealt out to be a reasonable choice: Atw=4h, ec=0.02 h z, e(+~=0 %, e (-)=15 %. From a previous study [5] the parameters of temperature kinetics are known to be Train=0.8 ~ Tm~=34.2 ~ Topt=32.2~ All the procedures for growth rate estimation, filtering and control were programmed as independent software application in Visual Basic. This application is connected via an OPC-based software interface for on-line data exchange to the MFCS/win system in order to receive the required process measurements and to transfer the controller output to the pilot plant. During the experiment depicted in figure 1 step changes of culture temperature (figure l b) were applied to the process. As can be seen from figure 1a they accelerate or slow down growth. This effect is correctly indicated by the filtered estimated growth rate #{CER,fil}. The comparison with the unfiltered estimated growth rate/t{CER} shows that noise is removed from the estimate to a large extend. Moreover the model based filter rejects disturbances resulting from pH-control - occurring in particular between 20 and 27 hours (figure 1b) - almost completely.
Fig. 1a: Growth rate estimation with model based filter,
Fig. 1b: Evolution of culture temperature and pH.
Figure 2a shows control of growth rate using the approach described in the preceding section. The culture temperature T was used as manipulated variable to control the growth rate. By means of the initialization procedure the growth rate estimation procedure was started automatically. The setpoint profile of growth rate/u{set} was chosen by the operator prior to starting the controller. Tracking of a setpoint profile - consisting of constant, step and ramp shaped elements - is quite well achieved by the proposed control approach. Mean relative error between setpoint and estimated value of growth rate is 3.1%. When the substrate expires at the end of the batch run, the growth rate drops rapidly. The controller reacts by increasing culture temperature. At 32.2~ the optimal temperature is reached, no further increase of growth rate is possible and the setpoint can not be reached anymore. In order to validate the growth rate estimation, the estimated values were used to calculate the evolution of biomass (figure 2b). Biomass estimation was initialized at 20 hours. Mean relative error between estimated X{est} and measured biomass X{mes} is 7.5 %. As this value depends on the estimated growth rate it can be concluded that this estimation is near the real values. Nevertheless, from the evolution of the estimated biomass can be deduced that until 26 hours the estimated growth rate is below its real value. Then, due to the fact that the devia-
722 tion between estimated and measured biomass remains constant, the estimated growth rate can be considered to be close to the true value until 34 hours. In the period between 34 and 38 hours the growth rate is overestimated.
Fig. 2a: Inferential control of growth rate.
Fig. 2b: Validation of growth rate estimation by comparing estimated X{est} and measured biomass X {mes }.
7. CONCLUSION Batch culture of Brevibacterium linens is grown on lactic acid and amino acids as substrate in a 20 liters bioreactor. During the culture growth rate is controlled according to a time varying setpoint profile. Growth rate is inferred on from exhaust gas measurements. In the present case carbon exhaust rate is used. As this variable is corrupted with noise and disturbances, a model based filter, including a simple kinetic model, is implemented. Experimental evidence shows that the filter rejects noise and disturbances to a large extend and hereby renders the growth rate estimation suitable for control purposes. The setpoint for growth rate is the input to a supervisory control routine based on the simple kinetic model. This routine determines the values of the manipulated operation conditions so that the growth rate matches the desired setpoint. As shown, this routine is inherently suitable for MISO-control, although culture temperature is the only manipulated operation condition used in the present case. The controller software was connected to the pilot plant via the MFCS/win OPC-based software interface in a comfortable and reliable way. Experimental results prove that the presented approach allows for precisely tracking time-varying growth rate setpoint profiles. ACKNOWLEDGEMENT
Technical support of this research by SOREDAB S.A. is gratefully acknowledged. REFERENCES [ 1] G. Bastin and D. Dochain, Elsevier, Amsterdam, 1990. [2] H. Shimizu, T. Takamatsu, S. Shioya and K.-I. Suga, Biotech. Bioeng., 33, 354-364, 1989. [3] D. Levisuskas, R. Simutis, D. Borvitz andA. Ltibbert, Bioproc. Engng., 15, 145-150, 1996. [4] V. Ljubenova, M. Ignatova, Bioproc. Engng., 11, 107-113, 1994. [5] K. Preul3 and M.-V. Le Lann, Submitted to CAB8, 24.-27.06.2001, Quebec (Canada) [6] A. Moreau, PhD Thesis INP Toulouse/France, 1999. [7] L. Rosso, J.R. Lobry and S. Bajard, Applied Environmental Biotech., Vol. 61, No 2, 1999.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
723
Multistability and robust control of the ammonia synthesis loop Maurizio Rovaglio, Davide Manca, Francesco Cortese, Paolo Mussone Dipartimento di Chimica Industriale ed Ingegneria Chimica "G. Natta" Politecnico di Milano, Italy Auto-thermal catalytic reactors may show limit cycle behavior due to inverse response and thermal feedbacks simultaneous presence. In this study a classical ammonia reactor layout (two beds, radial flux) was considered and simulated together with the overall synthesis loop. Oscillations, caused by feed gas temperature disturbances, were confirmed, while a pressure dependent hysteresis patter appeared when the temperature was reset to the original design value. A more sensitive dynamic behavior was highlighted for the synthesis loop compared to the reactor alone because of the energy and material recycles. Finally, a control strategy extended to the all synthesis loop was tested in order to avoid dangerous dynamics or undesired reactor shutdown. 1. INTRODUCTION In a recent article Morud and Skogestad (1998) address the problem of an industrial accident in an ammonia production plant. The accident was caused by a sudden loss of stability induced by a decrease of the reactor pressure. The instability determined the steadystate to move to a dynamic regime characterized by sustained oscillations. In Morud and Skogestad work the analysis was performed through a non linear model consisting of two partial differential equations (PDE) limited to the reactor description but capable of qualitatively interpreting the accident data. The onset oscillation is correctly described in spite of the model simplicity, and, in order to reduce the risk of instability a control strategy is proposed but not verified. A minor limitation of the Morud and Skogestad's approach is the lack of a complete dynamic analysis able to prevent the possibility of other multistability scenarios that might be discovered in actual operations. In a following work by Mancusi et al. (1999), the results of a complete dynamic analysis are presented. Here, the same model proposed by Morud and Skogestad was adopted to show how to improve the model predictive capability. Reactor pressure and exchanger efficiency are considered as bifurcation parameters and a complete description of both static and dynamic attractors is made available for typical operating conditions. However, the analysis was limited only to the ammonia reactor disregarding the external heat and mass recycles that characterize the ammonia synthesis loop, without clearly addressing the control problem and the related design. Aim of this work is to develop a robust control system for the overall ammonia synthesis loop. The control system in object should be able to reject external disturbance on the main process variables and uncertainty on the adopted control settings. Moreover, a complete dynamic analysis including all heat and material recycles was performed to clearly address the multistable behavior from one side and the controllability issue from the other. A detailed
724 analysis on the most appropriate model scale description was carried out for a better and quantitative understanding of the operating conditions range where the instability can appear. Therefore, the momentum balance and an heterogeneous model was investigated as possible improvements on the previous works. Moreover, the model extension to the overall synthesis loop implies new equations related to a cascade of heat exchangers plus a two stage flash where the ammonia is condensed and separated as a liquid product while the gas is recycled to the reaction stages. It is important to underline that two of the aforementioned heat exchangers are adopted for feed preheating so determining a strong heat system integration. Consequently, the impact of such a double heat recycle plus the material one related to the unreacted gas was analyzed in terms of both dynamic behavior and stability. In particular, the results clearly show that, with this more complete layout, sustained oscillations can be induced by smaller disturbances than those pointed out by previous studies. Finally, a conventional control scheme was analyzed in terms of pairing and optimal control settings. The sensitivity analysis here performed shows that errors on the control parameters due to a poor tuning can lead to sustained oscillations with respect to pressure disturbances even in presence of closed loop conditions. 2. R E A C T O R M O D E L
The ammonia converter here studied is a radial flow fixed bed reactor. The pressure shell is assumed to be adiabatic and it hosts an inter-bed heat exchanger. Fig. 1 schematically shows the flow pattern: the gas enters the converter through two main inlets at the bottom of the pressure shell, then it passes upwards through the annular space between the catalytic baskets and the pressure shell. From the top of the converter the gas goes to the tube side of the inter-bed heat Fig. 1- Flow pattern exchanger where it reaches the reaction temperature needed for inside the ammonia the first bed. Then it flows inward through the catalytic basket converter and again passes to the exchanger (shell side) where it cools at the temperature needed for the second catalytic basket. One of the two bottom inlets makes a small part of the cold syngas to enter the first bed without pre-heating. This by-pass ("cold shot") adjusts the gas temperature at the first bed. The equations for a pseudo-homogeneous model of the reactor include energy and material non-stationary balances and a momentum stationary balance to take into account the pressure drops along the catalytic beds:
(~C, --=-v
OC, G~2C, x 9 +D+ v i -R
Ot
3x
. iOcat'Cp'cat
OT Ot
---
. --Pgas
. Cp,gas
o=_dp+G, N/ D,N dx
-
-
(1)
--~
2gp ~
c e3
aZT OT . Vx ~OX + k . --~- + (-AHr ) . R
ReIN
ln DOvr+ 1 . 7 5 . ( 1- D1N DIN Douy
(2) + 1 1-
e~
DIN DO~T
(3)
725 The last equation is based on a modified Ergun expression (Jennings (1991)) where the plus sign is chosen when the flow is moving centripetally. This type of reactor gives rise to a minimal pressure drop because of the large section area and the short length of the catalyst bed where the gas has to pass through. The simulation results obtained confirm this hypothesis and allow Eq. 3 to be neglected. The holdup terms (c~/c~) for the gas phase in Eq. 1 is also negligible because of the residence time that is clearly lower than the characteristic time of the temperature response to any disturbance (~50 s vs. 15 min). Mancusi et al. F k,i F k-l,i (2000) also tested such assumption for both stationary and periodic solutions. As far as the modeling of the catalytic reactor is concerned it can be represented by either a continuous Tk-1 P Tk Pk system or a sequence of stirred tank reactors. The latter model, also called mixing-cell model, k has the capability of transforming a PDE system Fig. 2 - The K stirred tank reactor into an ODE (Ordinary Differential Equation) system. Considering a k th cell (see Fig. 2) and the assumptions made, Eq.s 1 and 2 become: 0 -~ E , k - 1 -- E , k 3t- V , " R i 9V k
(4)
Cp.co,.pca,.Vk . d T k - ( T k _ l - T k ) . C p m ~ k 1 .F~o,k 1 + R k . ( - A H k ) . V k (5) dt " ''As shown, the system arising from this model is constituted by algebraic and differential equations because, as already underlined, the gas phase holdups are neglected. The main point to be emphasized is that while a unique asymptotic temperature and NH3-flux profiles are obtained for steady-state conditions, when increasing the cells number with NC > 30, two different dynamic evolutions are derived from transient simulations. As a matter of fact, in fig. 3, which refers to the dynamic evolution determined by a disturbance on the inlet temperature, two significantly different behaviors can be pointed out: the reactor shutdown when NC=5 or 30 and the limit cycle behavior when NC=100. The cells number is related to the energy and material dispersion in the gas flow direction: the fewer the cells the greater their depth and also the Fig. 3-Dynamic evolutions of the phase diagram characteristic length of the dispersion phenomena. If the Reynolds number for the particle is about 100 (fully developed turbulent flow): Re p = p" v~ ~ 7.102 , the characteristic length must be equal to the particles diameter ~.a.#
726 (Dp=3 mm). In order to obtain each cell of 3 mm depth, more than 100 cells are needed according to the geometry of the first catalytic basket but no significant difference in dynamic behavior is found when comparing simulations with NC >_ 100. Similar conclusions may be drown for the second basket where about 200 cells are needed since the difference between external and internal radii is greater. Details about kinetic expression and the model of the heat exchanger are reported elsewhere (Rovaglio et al., 2001). Performing the dynamic analysis of such a reactor model, an interesting result can be derived considering a temperature disturbance on the inlet flow. When the original conditions are reset, after a limit cycle is developed, the ammonia converter enters in a new limit cycle. This hysteresis pattern (Fig. 4) means that this reactor shows one stable limit cycle (the darker line) at the operating pressure of 190 bar in addition to the static attractors of high conversion (starting point) and low conversion (not reported in the figure). Therefore, it could be not possible to obtain the starting conditions by simply inverting the disturbance step on the manipulated variable as happened during the 1988 accident in Germany. A slightly higher pressure may change this dynamic pattern. As a matter of fact, at P=200 bar the converter goes back to the original steady-state con-ditions by simply switching back the inlet gas temperature to the starting values. Fig. 5 highlights the different dynamic behavior at the different pressure value. Similarly, Mancusi et al. (2000), developed a nonlinear analysis for the German reactor above described and, selecting the Fig. 4- Hysteresis pattern related to a disturbance of + 15 K pressure as a bifurcation parameter, they detected two static attractors at the design pressure. In analogy with the results here reported, they also found two static attractors and a limit cycle in a different range of the operating pressure (163 bar -166 bar). Such a different pressure range can be related to the different reactor geometry and size.
Fig. 5 - At the operating pressure of 200 bars the hysteresis disappears (both the diagrams refer to a disturbance of + 15 K)
727 3. THE SYNTHESIS LOOP Previous paragraph underlines the influence of the pre-heater on the reactor dynamics and it shows how the auto-thermal structure and the contemporary presence of an inverse response in the catalytic bed allow the presence of the limit cycle behavior to be justified. For such reason this section focuses the attention on the overall synthesis loop with the aim of addressing the following specific issues: 9 dynamic effect of more cycle integration; 9 stability impact of heat and material recycles; 9 operating conditions range affected by limit cycles and control loop scheme. The synthesis loop structure here considered was taken from literature (Nielsen, 1995) and it is sketched in fig 6. The synthesis loop examined is composed of a synthesis reactor, a cascade of heat exchangers and two separation units. As shown in the Fig. 6 there are four recycles in the adopted layout. The first one is determined by the third heat exchanger, where the effluent stream from the reactor releases thermal energy to the stream entering the reactor. This configuration is typical of the auto-thermal system where feed stream is preheated by hot outlet stream. The fifth heat exchanger is also used to preheat the recycling stream to the reactor by means of the heat exchanged with the hot stream leaving the reactor. The last recycle is determined by the flash that separates the liquid ammonia product from the unreacted gases. Indeed, hydrogen conversion per reacting-pass is approximately 30% and it is necessary to provide a recycle of the raw unconverted material to the process unit. In summary, the loop is characterized by the presence of two heat recycles used in order to configure the system as much auto-thermal as possible and by a material recycle due to the conversion needs.
Fig. 6- Layout of the ammonia synthesis loop The main model assumptions and philosophy related to the mathematical description of the loops are reported in the following while more details on the system equations are given
728 elsewhere (see Rovaglio et al., 2001). As previously mentioned, heat exchangers are used to cool the process stream exiting from the reactor and to separate, by condensation, the produced ammonia. Such exchangers are assumed instantaneously at steady-state conditions. In particular, the models proposed for the three heat exchangers encountered by the process stream, after the reactor, are written without considering the ammonia condensation since the stream temperature can be assumed to be always higher than the mixture dew point (in the range of the examined pressures and compositions). In the remaining exchangers, where the ammonia condensation takes place, the RKS equation was adopted to compute the equilibrium gas and liquid compositions (Reddy and Husain, 1982). Flash drum separators are adopted to separate the liquid ammonia from the recycle stream and the liquid ammonia from the purge gas. These units, again, are considered instantaneously at steady-state conditions because of the gas phase holdups and the corresponding inertial effects that can be assumed negligible. Data for the calculation of the equilibrium constant are taken from literature (Prausnitz et al., 1992). Purge gas and make-up are described through global mass and energy balances, neglecting the corresponding mixing enthalpy terms. The most important disturbances that have a significant impact on the plant behavior are listed in the following, namely: make-up composition, make-up temperature, variations related to the loop compressor and consequent disturbances on the loop pressure, variations on the liquid ammonia temperature in the last chiller adopted to regulate the temperature inside the flashing unit. The first two disturbances can be considered depending on the reforming section. Disturbances on the loop pressure are directly related to the working conditions of the compressor loop. Finally, disturbances on the ammonia temperature can be associated with bad operations of the refrigeration circuit. Moreover, there are also some minor disturbances that can act on the synthesis loop. In analogy with the accident reported by Morud and Skogestad (1998), the attention will be now focused on the effect of pressure disturbances on the loop behavior. In Fig. 7 it is shown the temperature transient of the outlet stream from the reactor when the considered system is the reactor alone (case a) or the overall loop (case b). While in the first case it is necessary to reduce the pressure from 195 bar to 185 bar to enter a limit cycle, in the second case a lower pressure drop down to 190 bar is enough to induce the limit cycle. This confirms the fact that integrated systems behave in a more sensitive way than the units considered separately.
Fig 7 - case (a): dynamic behavior of the reactor after two disturbance of -5 bar
Fig 7 - case (b): dynamic behavior of the w h o l e loop after a disturbance of -5 bar
Similarly, other disturbances previously mentioned can induce the same behavior. In facts, the actions modifying feed composition and/or feed temperature can induce the limit cycle
729 but, generally, even with a minor step on the forcing variable with respect to those described for the reactor alone. Therefore, since these systems are more sensitive and because of this, more potentially dangerous, it is very important to develop a robust control scheme in order to reject the disturbances mentioned above and avoid the raise of any possible oscillating condition. 4. C O N T R O L STRATEGY FOR THE SYNTHESIS LOOP From last analysis it was underlined the need of providing a robust control structure for the overall loop to avoid possible risks connected with the effects of the aforementioned disturbances. As a matter of fact, raising limit cycle behavior or undesired shutdown of the synthesis reactor can involve consistent damage of the reactor itself and/or of the other cycle units. Sustained oscillations can be dangerous due to the periodic and frequent variations of temperature that can seriously damage the catalytic material. On the other hand temperature waves that move through the heat exchangers cascade can alter the steady-state operating conditions. For example a great temperature change can induce a large variation in the boiler section and in the refrigeration circuit. Fig. 8 - A disturbance o f - 1 0 bar forces a total The main control structure for the valve closing and the raise of a limit cycle synthesis reactor and consequently for the behavior. overall synthesis loop can be based on the cold by-pass inside the reactor: a fraction of the inlet stream (cold) is separated from the main flow that passes through the heat exchanger while the by-pass flow is mixed just before entering the catalytic bed. The obtained result is a smooth thermal regulation: when the valve is progressively closed the inlet temperature becomes higher, on the other side, a valve opening involve a lower inlet temperature. The most critical condition to be maintained, for safe operations of the reactor, is the inlet temperature of the catalytic bed. Low temperatures may involve the reactor shutdown, while high temperatures may lead to the synterization of the catalyst. From the available data, the correct temperature and flow rate profiles were obtained for a valve position corresponding to a by-pass flow of 1.5 % of the total flow. Lower openings cause higher inlet temperatures while higher values lead to reactor's extinction. As a result, the control system is able to reject all the disturbances that force the valve opening, while it is evident that strong disturbances, characterized by a negative action, can easily take to the complete valve closing. This saturated condition leads to an open loop system and consequently to the possibility of reaching again a limit cycle behavior, as shown Fig. 9 - 1 ~t a n d 2 'd control l o o p
730 in Fig. 8. Therefore, it is obviously necessary to adopt another degree of freedom in order to get higher inlet temperatures. Let us now consider the part of the loop including the reactor and the first three exchangers (Fig. 9). It can be easily pointed out that manipulating the heat duty to the second heat exchanger, the inlet reactor temperature can be modified. Consequently, it is possible to help the previous controller. The by-pass valve of such exchanger is the second manipulated variable, and the corresponding value of the steady-state condition is around the 15%. The corresponding controlled variable is the inlet temperature to the second catalytic bed. In facts, varying the reactor inlet temperature does not effect the first bed inlet temperature (until the first controller works) but it modifies the inlet temperature of the second bed through the internal exchanger. A disturbance of-10 bar on the operating pressure (same as before with only one loop) may be rejected adopting the new control configuration of Fig. 9. 635
0.16
634
0.14
633
0.12 0.1 1st and 2nd 0.08 valve [%) 0.06
1st and 632 2nd bed input T 631 [K] 630
0.04
629 628
0.02 I
0
5
I
I
10 15
I
I
I
I
20 25 30 35 Time [min]
I
I
40
45
0 50
0
5
10
15 20 25 30 Time [min]
35
40
45
50
Fig.10 - Introducing a second control loop the critical disturbance is rejected
From the preliminary results it can be highlighted that the proposed control system is able to reject the main plant disturbances. Extensive simulations and analysis are still undergoing. 5. R E F E R E N C E S
Jennings J.R., "Catalytic ammonia synthesis", Plenum press, New York, (1991) Mancusi E., Merola G., Crescitelli S., Maffettone P.L., "Multistability and Hysteresis in an industrial ammonia reactor", AIChE J., 46, 824, (2000) Mizsey P., I. Kalmar, "Effect of recycle on control of chemical processes", Comp.Chem.Eng, 20, s883, (1996) Morud J., S. Skogestad, "Analysis of instability in an industrial ammonia reactor", AIChE J., 44, 888, (1998) Nielsen J.B., "Ammonia, Catalysis and manifacture ", Springer-Verlag, New York, (1995) Reddy K.V., A. Husain, "Modeling and simulation of an ammonia synthesis loop", Ind.Eng.Chem, 25,359-367 (1982) Reid R. C., J. M. Prausnitz, B. E. Poling, "The properties of gases and liquids", Mc-Graw Hill, New York, (1988) Rovaglio M., D. Manca, F. Cortese, P. Mussone "Complex dynamic behavior and control structure for the ammonia synthesis loop", submitted for publication, (2001)
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
731
Predictive functional control (PFC) applied to an unstable system. An industrial application Carlos Ruiz* a,c, Marta S. Basualdo b'c, Aldo Molina d, Laureano Jim6nez e, Benjamin Parisse f, Jacques Richalet f (a) SOTEICA S.R.L., Buenos Aires, Argentina (b) Dept. Electr6nica. FCEIyA-UNR, Rosario, Argentina (c) GIAIQ, UTN-FRR, Rosario, Argentina (d)REPSOL YPF, Refineria Plaza Huincul, Neuquen, Argentina (e) Chem. Eng. Dept. ETSEQ, Universidad Rovira i Virgili, Tarragona, Spain (0 ADERSA, Paris, France This paper describes an industrial application of Predictive Functional Control (PFC), which belongs to a family of Model Predictive Control (MBPC) developed by Richalet and coworkers during the last decades. The case presented in this work consists on controlling the bottom level and the reboiler temperature of a stripper distillation column, which belongs to the naphtha hydrotreater (Hydrobon) plant located at REPSOL-YPF Plaza Huincul refinery. The Hydrobon plant objective is to eliminate the downstream naphtha reforming (Platforming) plant catalyst. Level and temperature control by zone strategy is implemented; it involves keep the level and temperature inside a specific allowable band with respect to the set point and the corresponding controller tuning parameters are adjusted depending on that. The project consisted in three main steps, process identification, off-line controller design and on-line commissioning. The controllers were implemented in the control processor of the Distributed Control System (DCS). This paper presents the details of the identification, and off-line design, as well as the results obtained after the control strategies commissioning. 1. INTRODUCTION Several authors have published excellent reviews of MPC theoretical issues as Morari and Lee, 1999; a very good review from vendors and users of MPC point of view was presented by Qin and Badwell, 1995. In this work a combined experience between vendors, users and university researchers is presented showing industrial application results of a relative new technology of MPC philosophy such as PFC. The first industrial application of a Model Based Predictive Control (MBPC) was done in 1973 (Model Algorithmic Control). In 1975-79 the first generation appears with IDCOM (Identification and Commande, 1980-85 second generation with HIECON (Hierarchical Constraint Control), 1989 third generation with PFC (Predictive Functional Control). Since 1997, a subset of PFC called PCR (Predictive Control of Reactors) was developed for cover the gap between PID and big MBPC applications, especially focused to solve control problems where PID is not enough Author to whom all correspondence should be addressed: Alvarez Thomas 796, 3 C, 1427 Buenos Aires, Argentina. Tel.: +54 11 4555 5703, ext. 218; Fax: +54 11 4551 5701; e-mail: [email protected]
732 and Multivariable MBPC does not apply because of expensive or high computing power requirements. However, although its name suggests applications for reactors only it can be used on other chemical units such as heat exchangers, furnaces, and distillation columns. The PCR technique resides on represent the plant with a linear impulse response model, generate the control algorithm for a coincidence point of the reference trajectory, solve it and apply the calculated input action. The later can be constrained on its maximum and minimum values and its rate of variation. The target is to slim MBPC to make it simple and easily implementable, at a very low DCS or PLC level, without losing its efficiency. The application field ranges from petrochemical (continuous) to pharmaceutical (batch) processes. In this work, an industrial application of PCR on two control loops of a distillation column implemented on a Foxboro IAS DCS in HLBL language is presented. The paper is intended to be useful to control practitioners and researchers by presenting specific theory and details of control algorithms. A detailed description of this technology can be found at Richalet, 1993, 1998. 2. PROCESS DESCRIPTION
In the Hydrobon plant the gasoline is treated in order to hydrogenate those components that are poisons for the catalyst of the downstream Platforming unit. The main contaminants are S, N and heavy metals. In Figure 1 a simplified flow diagram of the Hydrobon plant is shown. This plant consists of a reactor (R-301), where the elements mentioned above reacts with hydrogen producing SH2, NH3, H20 and the metals are kept on the reactor catalyst, a heat exchanger network for heat recovery and a distillation column (T-301) to strip out the impurities. The column bottom level needs to be controlled by manipulating the Hydrobon plant feed rate because the Platforming unit feed, the natural candidate to be used as the manipulated variable, should be maintained stable to improve the Platforming plant efficiency. Due to the long delay involved and the integrative (unstable) nature of this loop, it is not suitable to be controlled properly with a PID.
Fig. 1. Hydrobon plant flow diagram. 3. P R O J E C T PLANNING This project involves the steps of system identification, controller design and implementation in the basic library of the control hardware equipment. The IEC 1131-3 norm
733 is a convenient procedure for the embedding the new control modules in proprietary and nonproprietary environments. 3.1. Plant Test A plant test was carried on for both the column bottom level and the reboiler (H-302) temperature. For the level the difference between the set points of the feed flow rate entering and product flow rate leaving the Hydrobon plant, named "Diff"(FIC-304 - FIC-311), marked at Fig. 1 with arrows, were chosen. In this way the last one flow rate is taken into account as a direct feed-forward disturbance. At the left of Figure 2 the steps on "Diff" variable is presented, at the right the bottom level evolution of the column (LT-304) is shown. The transfer function was a first order with delay and integrator which parameters are: Kp = 0.02 %/sec./m3/h, Tp = 2.73 sec, Dp = 450 sec. Similarly, the gas valve output of the H-302 was manipulated to obtain the transfer function for the temperature. In this case, a first order plus time delay transfer function, with Kp =0.8 ~ / %, T p = 2 0 0 sec, Dp=50 sec parameters were obtained.
Fig. 2. Test plant for identification step. 3.2. Controller design The main PFC controller technology elements are: independent model approach which predicts the dynamic behavior of the plant on a certain prediction horizon; decomposition principle for unstable systems such as integrative process; exponential reference trajectory of first order closed loop response; polynomial structuration of the future manipulated variables that minimize the difference between reference trajectory and model prediction at one or most coincidence points; constraints on MV and state variables; easy tuning in time and frequency domains. 3.3. Mathematical calculations for PFC/PCR design According to the identifications done in the previous step in this section is presented as illustrative examples the procedures for designing both controllers for level and temperature loops. 3.3.1. Model Since for level control the identified transfer function was of first order plus time delay and integrator with Kp as gain, Tp time constant, D p dead time. The Z-transformed is a0.z
G p (z-')=
1-
-l
+ al.z
-2
(1 + ec ).z -l + tx.z -2 "z-q'
(1)
734 T~
with: a o = K p .ITs - Tp.(1 - or)l, ot = e T~ a~ = K p . [ -
or.T, +
T p.0 - ~)],
rp = int(
-~)'
Ts" sampling time
3.3.2. Internal model G m ( Z - l ) -- a ~
+ alm'Z-I 1 - ot ,z -i m
H
(2)
Z-' "1 -- Z -1 "z-rm
I
1
T, With:
a o m = K m .[Ts
- T m.(1 -
(~m)], am
"-
e Tm, a i m
= K m . [ - (3~m " L "+" Tm
.(1 -
a m )],
rm = int( D m )
The integrator term is decompose as is shown in Figure 3.
un, u(n)
"! I
Ii~ t
I
I I
I
H~
~
Yr(n)
I
M~
] [ . M~
I
I
Ym2(n)
Fig. 3. Decomposition of the integrator term T~ M l(Z-l)
z-l
= 1-
-l
M 2(z-l)
=
(1 - a m,).z-' been Otto1 = e v.,
CZm~.Z
(3)
1--Ctm~.Z-~
Tml is the decomposition time generally considered equal to the closed loop time response. The Z-transformed for the H 2 transfer function is H2 ( z _ l ) =
.Z -1 -2 a0m + aim .z 1 - ( a m + Ctml).Z q + a m .Otml.Z -2
(4)
Transforming to the observable state canonic equation the output of M 2 is" Ym2(n) = ~ m l "Ym2( n -
being
1) + (1--(~ml).Yp( n - 1)
(5)
yp (n- 1) the output of the plant at instant n-l, hence the internal model is written as
Ym (n) = Yml (n) + Y m2 (n) if time delay is considered (6) is transformed in (7)
ym2(n)=aml.Ym2(n-1)+(1-aml).[yp(n-1) +ym(n-1)- y,,(n-l-rm)] 3.3.3 Control algorithm For the coincidence point, C - ~p (n + h) = XH.(C- yp (n))
(6) (7)
(8)
735 3.H.Ts
Where C represents the constant set point, and ;~" = e Trbr . The predicted output is given by (9) ~p (n + h) = y p (n) + Ym (n + h) - Ym (n) The future output of the model is" Ym (n + h) = Ym~(n + h) + Ym2 (n + h) (1 O) From the linearity theorem it can be written for the model y,,~" Ym~(n + h) = Ym~(n + h).fo=ea+ Ym~(n + h).free
(11)
Yml (n + h) forzed : SF(H).u(n) ; Yml(nWh)free-SI_o(H)xom(n)+Sf_l(H)Xlm(n)
(12)
here, u(n) is the input of the plant at instant n, SL is the free response of the system which describes the time evolution from the actual time without any driving force. SF is the forced generated by external driving force and starting from zero initial conditions. The subindex "0", "1", etc. are related with the base function responses to classical inputs such as steps, ramps and so on. This allows, by considering the superposition principle, to write the forced response as a sum of base responses. Therefore, u(n) results: + (1-SI~(H))'Ym'(n)+(1-t2'~)'(Ym2(n)- yp(n)) (13) u(n)= (1-2n)'(C- yp(n))-SI-~176 SF(/4) SF(H) SF(H), SL 0(H), SLt (H) are pre-calculated in an iterative way for the coincidence point H, the state equations are calculated accounting that u(n)-I for SF(H);
u(n)-0 for SLo(H ),
SL~ (H). In addition for systems with delay u(n) is: H).(Ym2(n) - yp(n)) (1-)]H ).(C -- yp ( F / ) ) - 8/.13(/-/).X0m(F/)q- (1- SZ~(H)).ym~(n) + (1- aL; SF(H) SF(H) with yp (n) = yp (n) + Ym(n) -- Ym( n - rm) u(n) =
(14) (15)
Therefore (14) is the control algorithm, which will be used for the level. For temperature control the identified process was first order plus time delay, hence the control algorithm used is given in Eq. (16) u ( n ) = ( C - yp (n))(1 - An ). 4 y- "~ (n)
K(1-a •)
(16)
K
4. R E S U L T S PCF/PCR technologies include a Computer Aided Design (CAD) tool, which makes easier to identify and tune the controller. The parameters to be fixed or tuned for these controllers are: coincidence point (H) [sec], decomposition time: [sec] for unstable systems such as integrative, low and high closed loop time response: [TRBF, sec] applied when the controlled variable is outside the allowed control zone. By using a control zone as in the application considered here the parameter TRBF values are moving linearly between the two extremes. Hence, when the controlled variable is exactly the set point high TRBF is used. Otherwise, when it is far away from the set point TRBF decreases linearly in order to drive the controlled variable inside the zone as quickly as possible. Transition zone [%] set the allowed zone for the controlled variable expressed as + n% with respect to set point value, constraints to manipulated variable are also included. The application results are presented in Figure 4. At the right, the column bottoms level after and
736 before PCR control are compared. A dramatic control improvement is shown. At the left, the reboiler temperature controlled with PCR by manipulating the fuel gas valve output is compared with the previous PID control scheme. z a s oo 95
..Ipw .Ir,~
20 ~LoYel
9O
bofore PCR
~L~vol
wll~ PCR
19
B5
io
BO
17
'~
1 7s 70
15-
65
t4
6O
13
55
i
i
i PCR
.....
Tump S P bofo~ PC~
Tsmp SP ,,-,,h PC~
244 {30
L::
-
.., i_ .
. . . . . . Temp wllh PCR
.
.
.
.
.
.
.
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. ,
r :::
-
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.
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.
i-
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.
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1955
2219
043
307
531
755
[18130
(1200
0400
061:}0
DBDO
1000
12OO
14013
Vi,.,.
Fig. 4. Left: bottom level variations after and before predictive control implementation. Right: Temperature controlled with and without predictive control 5. CONCLUSIONS The PFC/PCR technologies turn out to be a convenient option for control loops where PID is not effective enough: unstable systems, reverse response, long time delays, constraints on manipulated and state variables, etc. Because of its cost effectiveness, practical industrial applications can be done either on large-scale plants but, also, in medium size and small industries. A significant effort has been made at 2 levels: - to make the technology easy to understand and to tune with tuning parameters that have an elementary physical significance like the Closed Loop Time response. The target is to have ground floor PID users to install controllers by themselves. A generic CAD of modeling, identification and controller design helps the local instrumentist; -to make it easy to implement into PLC's or low level control boards of DCS's, in assembly language, for safety reasons and ease of connection of the control algorithm with the overall control system. The applications cases presented in this paper are a clear demonstration of the effectiveness of the approach. Very good results were obtained compared with the previous results from classical control. Producers, operators and control experts appreciated that test case and the approach is being deployed on other sites. Moreover, it is not necessary to install new special hardware nor additional process control computers to implement this kind of predictive controllers. REFERENCES
1. M. Morari and J. H. Lee, "Model Predictive Control: Past, Present and Future", Comp. & Chem. Eng., No 23, (1999), 667-682. 2. J. Qin and T. Badgwell, "An Overview of Industrial Model Predictive Control Technology", Chemical Process Control Conference V (1996), 232-256. 3. J. Richalet, Pratique de la Commande Pr6dictive, Editorial Herm6s, Paris-France (1993). 4. J. Richalet, Pratique de l'Identification, Editorial Herm6s, Paris-France (1998).
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
737
Selecting Control Configurations for Performance Henning Schmidt a, Elling W. Jacobsen a as3-Process Control, Royal Institute of Technology, S-10044 Stockholm, Sweden An important task in the design of decentralised control systems for multivariable plants is the decision of the structure of interconnections between manipulated variables and measurements. In this paper we focus on the selection of control structures with the aim of obtaining the best control performance using independent controller design. We point out the importance of the consideration of non-perfect control, contrary to the usually assumed perfect control, for this task. Furthermore we show, that not only the effect of interactions on the magnitude, but also on the phase of the subsystems should be considered. The decentralized Relative Gain Array is proposed as a simple frequency dependent tool. 1. I N T R O D U C T I O N We consider decentralized control of a general square multivariable n • n system G(s), i.e., having n inputs and n outputs. Decentralized control implies that the overall system is decomposed into a number of interacting subsystems for which individual controllers are designed, i.e., the overall controller may be written on a blockdiagonal form. Such a decomposition of the control problem is usually preferred due to the inherent robustness of such structures (both with respect to model uncertainty and sensor/actuator failures) and the ease of (re)tuning compared to the case with full multivariable controllers. However the potential cost of using a limited controller structure is reduced closed loop performance due to to the presence of interactions among the subsystems. An important task in the design of a decentralized control system is the selection of the structure of the controller. During this task, several objectives are to be considered. There is the issue of independent controller design, which is especially important, if one should be able to tune or retune a loop without having to retune the whole plant. Then there are the issues of performance of the overall system and of certain subsystems. Last, but not least, there is the issue of stability, e.g. decentralized integral controllability (DIC). The objective considered during control structure design will usually be a combination of the above mentioned objectives. Thus, there will exist a trade off between the closed loop performance, DIC and the independent tuning issue. For example we might want to choose a control structure for which independent (re)tuning of the single loops is possible and we accept to pay this with a reduced, but still acceptable, performance of the closed loop. Most results based on existing tools for the selection of control structures, like the Relative Gain Array [ 1], the Niederlinski index and the ~t-interaction measure [2] do not address performance issues, but focus merely on stability. A tool addressing the closed loop performance was introduced by Hovd and Skogestad [3]. This tool, the performance relative gain array (PRGA),
738 can aid in determining the required open loop gain (under the assumption of perfect control) in each subsystem in order to achieve a specified closed loop performance. In this paper we focus mainly on the selection of control structures with the objective to achieve the best closed loop performance using independent controller tuning. Under the assumption of independent controller design it is reasonable to assume, that a control structure resulting in a less interactive system (in terms of gain- and phase- changes) should be preferred. Therefore we introduce in the following the decentralized Relative Gain Array (dRGA), which is a useful extension of the RGA, based on the assumption of independent finite bandwidth control.
2. INTERACTIONS UNDER FINITE BANDWIDTH C O N T R O L For the case of independent controller design it is reasonable to assume, that the control structure resulting in a less interactive system will lead to a better performance of the closed loop system. Therefore we introduce in this section the decentralized RGA as an extension to the RGA for the issue of independent decentralized control. We will give examples clarifying the improved properties of the dRGA over the RGA and show, that not only the magnitude of the dRGA but also its phase information, is important in the selection of a control structure.
2.1. The RGA The RGA [1] has found widespread use as a tool for selecting control configurations. The relative gains are the ratios of the "open-loop" and "closed-loop" transfer-functions for the scalar subsystems. The common rule is to pair on subsystems with relative gains close to 1, based on the assumption that the interactions then will have a small effect on the subsystems. Since the frequency corresponding to the desired bandwidth is the most important, the focus is usually on the relative gain around this frequency. An often cited advantage of the RGA is that it is controller independent, due to the assumption of perfect control. However, this assumption is at the same time an important shortcoming of the RGA. The effect of interactions around the desired loop bandwidths is probably most important, and at these bandwidths the gain changes due to finite bandwidth control can be highly different from the changes predicted by the RGA. This is not only due to the loops having non-perfect control at this frequency, but also due to the fact that the ( n - 1) x ( n - 1) subsystems, considered perfectly controlled when calculating the RGA, in general will have a bandwidth which differs significantly from the individual loop bandwidths. This may be one reason why the RGA, while usually working well for 2 • 2 systems, is less effective for systems with n > 2. In other words: the RGA does not take into account the issue of decentralized control, while it is commonly used to decide decentralized control structures.
2.2. Example 1 Hovd and Skogestad [4] introduced the following system conventional RGA pairing rule.
1-s
G(s) -
{
-,--~-.~2 ~
(1 t J ~ }
1
-4.19
-25.96
6.19
1
-25.96
1
1
1
)
G(s)
as a counter example to the
(1)
739
(1
The RGA of the system is frequency independent and given by
A(im) =
-5 5
1 -5
5 1
(2)
Hovd and Skogestad have demonstrated, that pairing on the + 1 RGA elements results in a bad closed loop performance with a bandwidth of approx. 0.00086 rad/s and that pairing on the +5 elements leads to a much better performance with a bandwidth of approx. 0.0045 rad/s. In both cases the max. singular value of the sensitivity was restricted to be less than 2 (IISlI~ < 2). The RGA above tells us, that if we close the loops 1/11 and 2/2 perfectly, the gain in loop 3/3 will not change. While if we close the loops 1/2 and 2/3 the gain in loop 3/1 will decrease by a factor of 5. Using finite bandwidth control the gain changes are however different. For the loops mentioned above controllers were designed in an independent way, such that in each single closed loop the bandwidth was 0.001 rad/s. This is a quite low desired bandwidth compared to the RHP zero at + 1, but a slightly higher desired bandwidth (0.003 rad/s) leads to instability for the + 1 pairing, while for the +5 pairing the desired bandwidths can be increased by a factor of 100 without getting an unstable system. The resulting gain changes (open loop gain divided by the resulting gain when the other two loops are closed) in the open third loop are plotted in fig. 1-1eft. The gain in loop 3/3 doesn't remain constant, as the RGA predicts, but increases over frequency by a factor of up to 30, while the gain in loop 3/1 doesn't change very much around the desired bandwidth and above. The reason for the different gain changes for the selected pairings lies in the different performances of the controlled 2 • 2 subsystems of G(s). As mentioned above, controlling the two single loop systems up to a desired bandwidth does not imply that the 2 • 2 subsystems are controlled up to the same bandwidth. What can be seen is actually, that for the +5 pairing the performance of the 2x2 subsystems is much closer to the desired one, than for the + 1 pairings. One tool evaluating the achievable performance in subsystems, is the partial relative gain (PRG) proposed in [5]. The main shortcoming of this tool, is that the RGA of all possible subsystems down to 2 x 2 subsystems of G(s) has to be computed under the assumption of perfect control of the remaining systems. While this is still possible for 3 • 3 systems, the number of combinations to evaluate will be excessively large for n • n systems with n > 3. 2.3. Definition of the Decentralized RGA The decentralized RGA introduced below is able to predict the gain changes due to different performances of subsystems, as demonstrated above correctly. It is based on independent controller design and takes fully decentralized real controllers into account. The dRGA represents the gain changes in the diagonal elements of the transfer matrix G(s), when the remaining n - 1 diagonal elements are independently controlled using a real decentralized controller, such that the desired sensitivity in each controlled loop is given by the corresponding element in the diagonal matrix SO:
SO- ( l + 1output/input
! a ( s ) ~ ) -1
(3)
740
Fig. 1. Left: Magnitude of relative gain changes in the elements 3/3 (x) and 3/1 (o) due to finite bandwidth control of the corresponding other elements in Ex. 1. Bandwidth in single loops: f.oi = O.O01rad/s. Right: Magnitude and phase of the dRGA elements for pairings on +1 (x) and on +5 (o) elements. Bandwidth in single loops: f.l)i O.O01rad/s
=
is a diagonal matrix containing chosen desired single loop bandwidths o)i. The term A(s) results from the separation of the system (~(s) - diag (G(s)) into a diagonal allpass transfer matrix A(s) and a diagonal minimum phase system Gm(s)
(4)
G(s) =A(s)Gm(s) It can be shown, that the decentralized RGA is given by
[
[md]ii = gii gii--g re
/(1 (~)-1~-~// )-1_]_aii/-1 gCi] -1
(5)
where only the matrix f~ with the desired bandwidths has to be specified to consider different performance requirements in different subsystems of a plant (e.g. for distillation columns the performance of the composition loops is much more important than the performance of the level loops). Gm is the minimum phase term from equation (4). gii denote the diagonal elements of G, G ii the remaining ( n - 1) x ( n - 1) plant obtained by removing row i and column i in G. gri is the i-th row of G without the i-th element, gCi the i-th column of G without the i-th element. (~ is a diagonal transfer matrix containing the diagonal elements of G. K ii denotes the diagonal controller for the system ~ii. S~ is the sensitivity, when (~ is controlled using K. 2.3.1. Phase-Information from the dRGA Usually the phase information of the RGA is not used in choosing a decentralized control structure. However, the phase lag plays an equally important role for stability and performance as the change in magnitude, and it is therefore essential to consider the effect of interactions also on the phase. Because of the assumption of perfect control in the RGA its phase information around the bandwidth will not be very useful. For the example above the phase change in each element predicted by the RGA is :t:0 degrees. In terms of the dRGA, the change in phase in loop i due to interactions is given by
arg [Ad( iOl)]ii
(6)
741
Fig. 2. Left: Magnitude and phase of dRGA elements for diagonal (x) and off-diagonal (o) pairing in Example 2. Right: Diagonal elements of sensitivity for the diagonal (x) and offdiagonal pairing (o) in example 2. The dashed line shows the desired sensivitity S4.
Note that a positive phase of the dRGA implies that the phase lag increases due to interactions. Interactions may even be such, that closing loops will make zeros cross from the LHP into the RHP or vice versa (see [6]).
2.4. Example 1, continued The magnitude of the dRGA applied on the plant in Example 1 is plotted in fig. 1-right. The magnitude plot in fig. 1-right is almost equal to the one in fig. 1-1eft. It captures well the gain changes due to the behaviour of the subsystems. For low frequencies the magnitude of the dRGA is close to the standard RGA. But around and above the desired bandwidth the closed loop gains for the +1 pairing increase by approx, a factor of 25, while for the +5 pairing they increase only by a factor of approx. 1.2. Thus the pairing on the +5 RGA-elements seems to be less interactive around the desired bandwidth. The phase plot gives a similar result. The + 1 pairing seems to be especially bad, because in the frequency region, where the magnitude is significantly increased also the phase loss is large. 2.5. Example 2 Consider the 2 x 2 system
G(s) -
1 5s + 1
(
0.5 1
(s+l)2 1
)
(7)
Assume we want to design a diagonal controller which, by independent design, achieves a bandwidth of 1 for the individual loops. Fig. 2-1eft shows the dRGA elements for the diagonal and off-diagonal pairings, respectively. From the magnitude plot alone one would conclude that the interactions around the crossover is less severe for the diagonal pairing (the RGA would conclude that the pairings are equivalent). However, if one considers the phase of the dRGA one finds that the interactions will give a significant increase in the phase lag for the diagonal pairing. For the off-diagonal pairing the effect of interactions on the phase is smaller and furthermore such that the phase lag decreases. Based on this one might expect that the off-diagonal pairing
742 should be preferred, at least if independent tuning is desired. Fig. 2-right shows the diagonal elements of the sensitivity for the two pairings, when the individual open loop sensivitities are chosen with a bandwidth of 1 rad/s. Also shown is the desired Is~l. As expected from the above analysis, the interactions have by far the most severe effect on the performance if we pair on the diagonal of G. Similar results are obtained if one instead considers the overall sensitivity in terms of ~(S).
2.6. Properties of the dRGA 1. coi = o o or s -- 0 corresponds to perfect control in the subsystems and the dRGA becomes equal to the standard RGA. 2. The dRGA is pairing-dependent, which is a drawback, but probably necessary in order to get the information needed to choose the best pairing. 3. It is based on the assumption of independent controller tuning. 4. A generalization to a block dRGA is possible. 5. As an interaction based pairing rule using the dRGA one should select the control structure, for which the magnitude of the dRGA is closest to one around and above the desired bandwidth and for which the phase loss is close to 0. 6. Even if for the determination of the dRGA a certain controller is assumed, it is mainly dependent on the chosen desired closed loop bandwidths in the single loops. 3. C O N C L U S I O N S In this paper we pointed out the importance of the consideration of decentralized finite bandwidth control in the determination of the interactions between the subsystems in a decentralized control system. These interactions are especially important for the achievable closed loop performance if independent design of controllers is desired. We also showed, that not only the influences of the interactions on the magnitude of the controlled elements is important, but that the phase changes are at least equally important for the decision about the control structure.
REFERENCES 1. Edgar H. Bristol. On a new measure of interactions for multivariable process control. In IEEE Trans. Autom. Control, 1966. 2. Pierre Grosdidier and Manfred Morari. Interaction measures for systems under decentralized control. Automatica, 1986. 3. M. Hovd. Studies on control structure selection and design of robust decentralized and SVD controllers. PhD thesis, NTNU, Norway, 1992. 4. Morten Hovd and Sigurd Skogestad. Simple frequency-dependent tools for control system analysis, structure selection and design. Automatica, 1992. 5. Kurt E. H~iggblom. Partial relative gain: A new tool for control strucutre selection. In AIChE Annual Meeting, Los Angeles, 1997. 6. E.W. Jacobsen and H. Cui. Performance limitations in decentralized control. In ADCHEM 2000, Pisa, 2000.
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rights reserved.
743
Design of Controllable Batch Processes S. S. Shah a K. P. Madhavan a* aCAD Centre, Indian Institute of Technology, Bombay, Mumbai- 400 076, India Batch process design can be more challenging than continuous process design because of the unsteady nature of the batch operation. This precludes the use of conventional controllability measures in evaluating the controllability of a given batch process design. Further the operating strategy of a batch process is characterized by trajectories of manipulated variables. Integrated approach to batch process design and control needs to address the problem of controllability of the batch process during the design phase. This is best achieved by treating the problem as a dynamic optimization problem with time invariant (design) and time variant (operating) variables. The method proposed in this paper uses the decomposition feature of Generalized Benders Decomposition (GBD) to evolve a 2-level nested optimization problem (Primal and Master), one involving time variant decision (manipulated) variables and the other involving time invariant decision (design) variables. To enhance the computational efficiency, a relaxed LP formulation of the Master problem is proposed. This variant of GBD, termed as ExGBD, is guaranteed to converge to the optimum for convex problems. ExGBD with slight modification in the iteration strategy can be shown to converge to optimum for quasi-convex problems. A simple batch reactor design problem has been chosen to demonstrate ExGBD. 1. I N T R O D U C T I O N The need to have an integrated design approach which will address concurrently issues of operability and control has been advocated in recent years. The integrated approach can be evolved with greater ease for continuous process through well defined measures of controllability and flexibility. Direct extensions of this to batch processes is not that easy. Furthermore, nonlinear dynamic models characterize the behavior of batch processes over the whole domain of their operation. The presence of uncertainty is much greater in batch processes due to short time spans available for model development. Batch process control objective is to move the process towards desired targets without violation of path constraints. Explicit incorporation of a control structure in the design problem will call for determination of time variant controller parameters. A more direct approach would be to solve the integrated design and control problem as a dynamic optimization problem, the solution of which will cover all the feasible control configurations. Such a dynamic optimization problem has to contend with two types of decision variables: time invariant decision variables related to the design and time variant decision variables related to operation (manipulated variable trajectories). *Author to whom all correspondence should be addressed to. Email: [email protected]
744 In this paper we have restricted our attention towards the development of an integrated approach to design and control of a key unit in a batch process: the batch reactor. The objective is to develop an efficient and versatile optimization methodology, which can solve in a structured manner a host of dynamic optimization problems, involving both design and operating variables. 2. PROBLEM FORMULATION AND SOLUTION METHODOLOGIES
A general mathematical formulation of a batch process design and control problem (BP1) can be represented in the following functional form: rain J = G ( x ( t f ) , d , e , t f ) +
d,u(t)
F(x(t),u(t),d,O)dt+C(d)
(l)
Subject to,
Yc(t) - f(x(t),u(t),d, O) : O, x(to) : xo ... system dynamics g(x(t),u(t),d,O) < 0 ... path constraints h(x(tf ),d, O, tf ) <_0 ... end point constraints
(2) (3)
Xmin ~ x(t) < Xmax, Umin ~_ u(t) ~ Umax
In the above dynamic optimization problem for the batch process, the decision variables are grouped into two categories: time invariant variables (d) and variables that are functions of time (u(t)). d represents design variables like area of heat transfer which are constant, u(t) represents control policies which can be varied during a batch cycle. 0 represents model parameters. Approaches normally adopted to solve such optimization problems, ignore the difference in the characteristics of the decision variables. Dynamic optimization problems with decision variables of the type u(t) can be solved using classical Pontryagin's Minimum Principle (PMP). PMP can also handle constant variables d by treating them as state variables. This however adds to the computational complexity. The optimization problem can also be cast as an NLP by converting the ODE equations into algebraic equations [2] through state and control vector parameterization. This however increases the dimensionality and complexity of the problem leading to convergence problems. The approach proposed in this work is to use Generalized Benders Decomposition (GBD) to decompose the problem, so that the design variables d can be decoupled from the dynamic optimization problem. This modification is termed as Extended Generalized Benders Decomposition (ExGBD). 3. DEVELOPMENT OF EXGBD
This section discusses the extension of GBD to handle dynamic optimization problems. Rigorous development of GBD can be obtained from the original paper on GBD by Geoffrion [3]. GBD is normally applied to NLPs of the type: rain
pEP, qEQ
f (p, q), subject to G(p, q) < 0
(4)
where, P and Q are feasible sets of variables p and q respectively, q is a vector of complicating variables, in the sense that Equation 4 is a much easier optimization problem in p when q is
745 temporarily held fixed. Problem 4 is decomposed into two nested optimization problems: the primal and the master problems with decision variables p and q respectively. In the batch design problem BP1, d is chosen as the vector of complicating variables. With d fixed at d, BP1 can be solved as the primal problem for determination of optimial profiles (x*(t), u*(t)). It is advantageous to use classical PMP to solve the problem as the resulting solution will also generate optimal value for the adjoint vector k and Lagrange multipliers/2 and co associated with constraints 2 and 3. The primal solution may be feasible or infeasible. After recasting Problem BP1 into its Lagrangian form and then implementing the relaxation technique recommended by Geoffrion, the equivalent master problem (MBP1) works out to be: min y0
(5)
dED,yo
subject to A
Yo >_L * ( d , ~ , ~fi, ~)) O >_L, ( d, ~_,~fi,~) A
where,
^ L*(d,k,'fi,~3) - min{ ts
ftto
+
i)G(.)
ft:[F(')--~T f(')+~fiT g(') +
^ L,(d,~,'fi,~) -- min{ :t: ----
u
igt Jdt +
~ G ( . ) _~_~)T Oh(.):
igt
~)h(.) )2dt} + G(x(to),d, to) + ~Th(x(to),d, to) + C(d )
(6)
(_~Tf(.)+~T g(.)+~)TOh(.) ) + ~__~T+~)T ~h(.) )~dt}
(7)
Jto
--
--
Ot
--
i)x
+~Th(x(to),d, to) Definitions 6 and 7 correspondto the Lagrangian generated due to feasible and the infeasible primals respectively. Variables (~,(/), ~fi(t), ~), t c [to, tf] correspond to the Lagrange multipliers obtained while solving a feasible primal problem and variables (~(t),~(t), ~), t C [to,tf] correspond to the Lagrange multipliers obtained while solving an infeasible primal problem. Success of the ExGBD depends on efficient solution strategies for the master problem (Problem 5). For a given d, evaluation of each of the constraints in the master problem turns out to be a calculus of variations problem, resulting in Euler-Lagrange or equivalent equations. Solution of such problems is computationally very intensive. However, under certain conditions the master problem can be further relaxed so that the solution of the problem is greatly simplified. For convex functions L* (d, k,~, ~), after rearrangement of the variables, can also be represented as, A
A
A A
L * (d, ~,, "fi,~3) >__ L* (d,'~,fi, 6)) +
a K* (d , s "fi,~)) ad
I(,,a,d3 (d - d~)
+(OG(xt:,d, tf ) OC(d) Od I(~,:,d? + Od I(d3)(d-d3
(8)
Where, L* (2, t~,d,~,~, ~), is optimal solution of the feasible primal problem and,
~K*(.) ftt~.r (~)F(.) ~Ti)f(.) +.fiT~)g~.) ~T ~h(.)l Od ](~,a,d)= [ ~d ~d ) dt + ~d J ](~,a,d")
(9)
746 An equivalent equation can be obtained for the infeasible primal. The main assumptions required for this relaxation to be valid are: functions F(.), g(.), h(.), and C ( d ) are convex in d E D, and function f(.) is either convex or concave in d E D. This relaxation approach thus eliminates the need for solving a calculus of variations problem and the resultant master problem becomes an easy to solve L P p r o b l e m in Y0, d. Initial conditions, which are part of the decision variable can be easily incorporated into ExGBD by using a simple transformation of the state variables. System equations of form: 2 ( t ) = f ( x , u ) ; x ( t o ) - xo may be transformed by 2 - x / x o for x0 ~ 0 (and translated if x0 - 0 is expected) to obtain new modified system equations" 2(t) = f ( Y , u , x o ) ; Y ( t o ) -- 1.
3.1. ExGBD optimization procedure The algorithm of ExGBD is given below: STEP(l) Assume a point d c DN V (i.e., by fixing d - d), and we have a feasible primal v(d) represented by Equation 1. Solve this primal problem and obtain an optimal solution and optimal multiplier vector ~,p,~. Put p - 1, ~ p ( t ) : ~,(t), p p ( t ) :_ p(t),O~p - co, U B D = v(d), and I - 0. Select a convergence tolerance parameter e > 0. STEP(2) Solve the current relaxed master problem: min Yo
dED,yo
subject to, Yo >_ L* ( d j , k j , h j ,
(oj) +
0 3> L . ( d j , ~ _ j , p j , o 3 j ) +
3d
(d-dj); j
(~j,aj,dj)
~d " I(xj,u_j,dj) ( d - d j ) ; j -
-
1... ,
P
1,...1
using any LP solver. Let (d, f0) be an optimal solution; f0 is a lower bound on the optimal value of Equation 1, that is, the current lower bound is L B D - Yo. If U B D - L B D < e, then terminate, d obtained during the solution of the master problem may give rise to either feasible or infeasible primal. STEP(3a)- Feasible Primal For the new d solve the dynamic optimization (primal) problem and obtain v (d'). Update the upper bound U B D - min ( U B D , v(d') }. If U B D - L B D < e, then terminate. Otherwise, use the Lagrange multipliers to update the master problem, i.e. increase p by 1 and put ~p(t) ~ ~ ( t ) . pp(t) ~(t), top - (o. Return to STEP(2). -
-
STEP(3b)- Infeasible Primal The primal does not have a feasible solution for d = d. Solve a feasibility problem to determine the multiplier vectors ~,~, 6oand the function L, (d, ~,~, (o). Set, l = l + 1, ~-t - ~-'P-I - ~ ' - ~ - 63. U B D value remains unchanged. Return to STEP(2). Assumption about convexity of v(d) in d made to simplify the inner relaxed problem reduces the applicability of ExGBD. For problems non-convex in d, the solution using this technique may get stuck at a point which is not a local optimum. However, for certain type of non-convex problems viz. quasi-convex problems this limitation can be overcome by modifying the iteration strategy of ExGBD. The essential features of this modification are: repetitive application of
747 ExGBD (similar to the algorithm suggested by Bagajewicz and Manousiouthakis [ 1] for GBD), and its further modification developed by us. This modification introduces new constraints on Problem BP1 that gradually eliminate the non-convex portions of the function v(d). Convergence to the global optimum for quasi-convex functions has been rigorously proved [6]. 4. APPLICATION OF EXGBD FOR A BATCH REACTOR PROBLEM
ExGBD has been used to solve different types of problems, varying from simple LQR problems embedded with time invariant decision variables to a realistic case study on batch reactor design in the presence of uncertain model parameters. For the sake of brevity only a simple batch reactor problem, taken from Ray [5] (Example 3.2.2), is presented in this paper to demonstrate ExGBD as an effective tool for solving integrated design and control problems for batch processes. The problem is to obtain optimal temperature profile T*(t),for a batch reactor carrying out the reaction A --+ B --+ C, with the objective function being to maximize the concentration of B ([B]). This problem has been modified to include a time invariant decision variable; the batch time, (tf). The modified objective is to have a trade off between the two objectives namely maximize [B] and minimize the batch time, tf and is accomplished by weighted objective function involving [B] and tf.The value of these weighting factors will be dictated by the economics of the plant. The objective function is similar to that used by Ramirez [4], combining performance and minimum time requirement. The objective function of the optimization problem is given below: min [M-[B](tf) +C(tf)]
T(t),tf
where, M is some positive number ( - 1) to ensure positiveness of the objective function and C(tf) - 0.01t}. Model details are available in Ray [5]. In this problem tf is the complicating variable.For a given tf, the primal problem is solved using control vector iteration to provide optimal temperature profile T (t). The primal and master problems are solved iteratively till convergence is reached.The iteration summary of ExGBD is given Table 1. Columns 3 and 4 indicate the current upper and the lower bounds of the optimal. The problem converges to the optimum in 8 iterations. Table 1 Iteration summary for the batch reactor problem No. 1
2 3 4 5 6 7 8
fl 1.000 10.000 4.646 2.421 1.559 1.944 2.168 2.056
yo 0.399 0.399 0.399 0.378 0.377 0.373 0.373 0.373
0.000 0.162 0.307 0.363 0.369 0.373 0.373
748 5. CONCLUSIONS In this paper we have demonstrated that the integrated design and control problem for batch processes can be cast as a constrained dynamic optimization problem and the 2-level approach is a viable approach to the solution of the problem. The complexity of the 2-level design problem can be reduced through a decomposed optimization strategy based on the modification of GBD. ExGBD can solve a variety of design problems involving design parameters, initial conditions for a batch and operating variables which can be varied during a batch run. Some of the merits of ExGBD have been identified as: (a) employs a natural decomposition of the decision variables: time varying and time invariant, (b) algorithm guaranteed to converge for convex and quasiconvex functions, (c) drastic reduction in the computation of the inner relaxed problem through the formulation of LP, (d) can handle problems with uncertain model parameters 0, (e) complete iteration history is stored in the master problem, the algorithm can be stopped and restarted without loss of important information. REFERENCES
1. M.J. Bagajewicz and V. Manousiouthakis. On the generalized benders decomposition. Comput. Chem. Eng., 15(10):691-700, Oct. 1991. 2. T.K. Bhatia and L. T. Biegler. Dynamic optimization for batch design and scheduling with process model uncertainty. Ind. Eng. Chem. Res., 36(9):3708-3717, Sept. 1997. 3. A. M. Geoffrion. Generalized benders decomposition. J. of Optim. Theory Appl., 10(4):237-260, 1972. 4. W. E Ramirez. Process Control And Identification. Academic Press, Inc., London, 1994. 5. H.W. Ray. Advanced Process Control. McGraw-Hill, New-York, 1981. 6. Sunil S. Shah. Integrated approach to process design and control. PhD dissertation, Indian Institute of Technology Bombay, Systems and Control Engineering, 1999.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
749
Optimization and Nonlinear Model Predictive Control of Batch Polymerization Systems Dulce C.M. Silva and Nuno M.C. Oliveira a aDepartamento de Engenharia Qufmica, Universidade de Coimbra P61o II, Pinhal de Marrocos, 3030-290 Coimbra, Portugal. Tel: +351-239-798700. Fax: +351-239-798703. E-mail: {dulce,nuno}@eq.uc.pt.
This paper addresses the optimization of batch polymerization systems, using a feasible path approach, with roots on Model Predictive Control (MPC) theory. The approach allows the reuse of many concepts previously developed for nonlinear MPC of continuous plants. It also provides an efficient and well integrated methodology for the optimal supervision of discontinuous chemical processes. The application of this technique in the optimization of the batch suspension polymerization of vinyl chloride is also described.
1. INTRODUCTION Due to their non-stationary nature, batch processes present interesting challenges in their control and online optimization, and create unique opportunities for the development of advanced supervision strategies. In the case of batch polymerization processes, optimal operation involves computing and accurately maintaining the optimal temperature and initiator (or coreactants) addition policies that can lead to a product with desired properties (such as molecular weight average, polidispersivity, chain length distribution) and final conversion, while minimizing the total operation time. In many cases, batch operations are still carried according to recipes based on heuristics and past experience. However, the recent availability of detailed mechanistic models, experimentally validated, provide a significant incentive for a wider use of newly developed optimization algorithms. Previous research on the determination of optimal policies for batch processes concentrated on techniques for the solution of optimization problems subject to algebraic and differential constraints. To avoid the numerical difficulties associated with the solution of a nonlinear twopoint boundary value problem, various methods based on the discretization of the differential equations have been proposed, using a simultaneous solution and optimization approach [ 1]. On the other hand, nonlinear MPC algorithms using a feasible path approach have been tested with success on continuous processes, in the presence of general constraints [2,3]. While considering different objectives, these algorithms constitute in fact general NLP optimizers, and are able to deal efficiently with differential constraints. Their use for general optimization of discontinuous processes is therefore investigated in this paper.
750 2. PROBLEM FORMULATION AND SOLUTION STRATEGY The dynamic optimization problems to be solved can be formulated as min
W(x(t),u(t))
u(t)E Y{ik
s.t.
:~ = fp (x, u, d; O)
y - gp(x; O)
(1)
Ul ~ U ~_~ Uu XI ~ _ X ~ Xu
Yl <_ Y < Yu, where fp and gp are usually assumed differentiable and continuous, except perhaps at a finite number of switching points. Depending on the task at hand, different forms of the objective function can be considered. For instance, the Newton Control formulation for nonlinear MPC uses a quadratic objective of the form
w(,) -- f~+,o~(y-Ysp)TQy(t)(y-Ysp)+(U-ur)TQu(t)(U-Ur)dZ,
(2)
It 9k
where Ur and Ysp are reference trajectories for both the inputs and outputs. Other common objectives can be related to the properties of the final product, and be formulated as soft constraints of the form W(o) -- (y(tF) --Ysp)TQ1 (y(tF) -- Ysp), (3) where Ysp represents the desired final value. Important product specifications might require their formulation as hard equality or inequality constraints. In this case the objective can coincide directly with the minimization of the operation time
V(o) =tv, subject to several product constraints, which can be expressed in terms of restrictions involving the u, x and y variables of formulation (1). Piecewise constant input profiles are assumed. The resulting nonlinear programming problem can be solved using a successive quadratic programming (SQP) approach, requiting the linear and quadratic terms of the constraints and objective. To simplify the notation, augmented vectors U, X and Y are defined, containing all values of the corresponding variables inside an operating horizon. An exact linearization of the model around a nominal trajectory can be written as
~Y
AU
- - Y -Jr-S m A U ,
where Sm represents the dynamic matrix of the model, containing the first order information for the system relative to the input variables. This matrix can be efficiently computed from the original differential model through the use of appropriate sensitivity equations. When the objective has the form of (2) or (3) the algorithms described in [2,3] can be directly used. However, some modifications are required in this formulation to treat minimum-time problems. In these problems, the final time is usually defined by a certain output variable, which
751 reaches a predefined value YF at the end of the operation. We assume that this happens during the nth discretization interval, from tn to tn+l, inside a larger horizon defined as a maximum bound on tv. Given the previous assumptions about the model, it is possible to write tF as an implicit function of the initial condition and input variables during this interval
(4)
tF -- h(xn, Un).
The first order information for tF can then be obtained by writing a Taylor series in this interval: tF - - t-F +
The derivatives
x=x
and
~tF
~tF x=~ 9
. u=~
UUn u=~
9(btn--Un ) .
are, in some cases, difficult to obtain directly, by integration
of the sensitivity coefficients, since (4) is usually not available in explicit form. However, since these coefficients are only needed in the last time interval, they can also be approximated by finite differences, without a great penalty. Applying the previous concepts, the linearization of tF with respect to the input variables can be written as tF -- ~ + S*AU.
This allows formulation of the optimization problem as the SQP iteration of minJ2 - t~ + S*AU + A U T H A U AU s.t. UId <_ AU <_ Uud Yld <_ Sm AU <_ Yud Sm.,jAU - AyF,
where AyF = Ysp -- y(tF), and H represents an approximation of the Hessian of the Lagrangian of (1). This formulation is closely similar to the one used in the nonlinear Newton control law, making the algorithms developed for its solution applicable for minimum-time problems as well. 3. APPLICATION TO THE PVC SUSPENSION POLYMERIZATION
In this section we consider the application of the previous strategy to the optimization and nonlinear control of the batch suspension polymerization of vinyl chloride (VCM). This system involves four phases (monomer, polymer, aqueous and gas), and an heterogeneous reaction. Various kinetic models have been proposed to describe the process, with significant differences at the level of complexity and detail given to various chemical and physical phenomena taking place. In order to compare the optimization results, and to better assess their sensitivity, two mechanistic models for this process were built, based on the kinetics information provided by Xie and co-workers [4-6] and Kiparissides and co-workers [7]. Both of these models consider diffusion controlled reactions. The monomer distribution in the different phases is computed as a function of the conversion and reactor operation conditions. In the Xie model the rate constants are
752 modelled using the free volume theory, while in Kiparissides's model the termination and propagation rates are expressed in terms of reaction limiting and diffusion limiting terms; the later term depends on an effective reaction radius and on the diffusion coefficients, calculated from the extended free volume theory [7]. The state variables in the Xie's model are the monomer conversion, the first and second moments of the dead polymer distribution in the polymer phase, number and weight accumulated molecular weight average and quantities of the initiator. In the Kiparissides's model the state variables are: conversion, zeroth, first and second moments of the dead polymer distribution, and quantity of initiators. The predictions from both models are compared in Figure 1, using a constant polymerization temperature of 55~ As can be observed the conversion profiles are close until a conversion of 70% is reached. Their divergence after this point can be attributed to the fact that in Kiparissides's model the initiation efficiency after the critical conversion is not considered to be diffusionaly controlled, as in the Xie's model. With respect to average molecular weights, we can observe in Figure 2 that for the same temperature the models predict polymers with slightly different properties at the end of the operation. These differences can be due to the values of the kinetic parameters used in each model, especially the chain transfer to monomer that controls the molecular weight of the polymer, as well as other considerations made in their development.
. . . . . . . . . . . . . .
175
0.8
~. 150
0.6
0.4
~
0.2 0
k
200
1 ..........................
~
0
I00
200 300 t (rain)
400
500
Fig. 1. Conversion profile for isothermal operation ( - Xie's model; - 99Kiparissides's model).
PMw
125
.
I00
I~....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75 : < PMn 5o ...................................................... PMn 25 0
0.2
0.4
X
0.6
0.8
Fig. 2. Conversion dependence of number and weight average molecular weights ( Xie's model . . . . Kiparissides's model).
4. OPTIMIZATION RESULTS The main variables available for manipulation in this example are the reaction temperature and the initiator (or co-reactants) concentration. We studied the independent effect of each of these variables separately.
4.1. Effect of the reaction temperature: Figure 3 shows the results obtained for the optimal temperature profile that minimizes the total batch time. This optimal profile was obtained subject to restrictions in the polidispersivity and weight average molecular weight that guaranteed that the same product was obtained. Also, due to operational constraints, bounds of i 5 ~ relatively to the nominal temperature were imposed. When compared with isothermal operation (the current industrial practice), the profiles
753 obtained with Kiparissides's and Xie's models are able to achieve reductions of 8.7% and 23% in the reaction time, respectively. Xie's model shows more sensitivity to the reactor temperature and therefore has lower deviations relatively to the nominal trajectory.
1.03
.
.
.
.
.
.
.
1.02
.
.
.
.
.
.
.
.
.
.
.
~--~..~ ..~
1.01
"'~'-%
1
.
.
.
.
.
i "'%..
.
.
.
.....
I
1:
~..~.. -~___X .......
0.99 0.98 0
I00
200 300 t (min)
400
500
Fig. 3. Normalized optimal reactor temperature policy ( - Xie's model; - . model).
Kiparissides's
The on-line implementation of the optimal trajectory obtained in the Kiparissides's model was considered, using the nonlinear Newton-type model predictive control formulation. This is illustrated in Figures 4 and 5, where these results are also compared with a linear control strategy. For the predictive controller the tuning parameters used w e r e Qa = I, Q2 = 10-31, with a sampling time of 200 seconds. The discrete PI controller used a sampling time of 100 seconds. Since the process was found to be open-loop unstable, the linear controller was tuned using the Ziegler-Nichols criteria, and the resulting parameters were subsequently fine-tuned, in order to avoid the appearance of oscillation in some operating zones; the final settings obtained were kc - 20 and I:/-- 1500 seconds. Besides being easier to tune, smaller amplitude changes are observed in the MPC profile, with a closer tracking of the optimal trajectory.
1.03 1
9
0.99
0.98
1.35
. . . . . . . . . . . . . . . . . . . . . . . . .
02
1.3
5,
1.25
ii!iiiii
~
1.2
1.15 I.i
6 . . . . 160. . . . 260. . . . 360. . . . 460. . . . t (min)
1.05 s~0
Fig. 4. Output profile for the optimal policy of the reactor temperature (... PI; - MPC).
. . . .
1+o . . . .
2+o . . . . t (rain)
3~o . . . .
4+o
Fig. 5. Normalized inlet temperature jacket profile for the optimal profile of the reactor temperature (... PI; - MPC).
754 4.2. Effect of the initiator quantities: The composition of the initiator, and the best amount that should be added at the beginning of the operation, in order to produce a polymer with desired properties in minimum-time, were also the subject of optimization. The results are described in Table 1. In this case, both models also allow for important reductions in the cycle time, with a similar optimal initiator composition. The reduction obtained with the Xie model is smaller, because the initiation efficiency becomes diffusionaly controlled after the critical conversion. Significantly, no important reductions in the reaction time were found when the continuous addition of initiators was considered with this system. Additional optimization results, such as optimal profiles for polymers with improved properties, that cannot be formed with simple isothermal operation, can be found in [8].
Table 1 Optimization of the initial quantities of the initiators. Isothermal case Xie's model Total amount (mol) 18.0 19.4 Initiator A (%) 50 53 Initiator B (%) 50 47 Reduction in the cycle time (%) 7.2
Kiparissides's model 22.1 53 47 14.4
5. CONCLUSIONS The application of a nonlinear feasible path optimization strategy for the determination of optimal policies for batch processes was considered. The results obtained with a batch suspension polymerization reactor clearly illustrates the advantages and possible improvements in the operation of discontinuous processes, associated with a more generalized use of theses methodologies. REFERENCES
1. J.E. Cuthrell, L.T. Biegler, AIChE Journal, 33(8) (1987) 1257. 2. N.M.C. Oliveira, L.T. Biegler, Automatica, 31 (1995) 281. 3. L.O. Santos, N.M.C. Oliveira, L.T. Biegler, Proc. Dynamics and Control of Chemical Reactors, Distillation Columns and Batch Processes (DYCORD+'95), (1995) 33. 4. T.Y. Xie, A.E. Hamielec, EE. Wood, D.R. Woods, J. Applied Polymer Science, 34 (1987) 1749. 5. T.Y. Xie, A.E. Hamielec, EE. Wood, D.R. Woods, Polymer, 32 (1991) 537. 6. T.Y. Xie, A.E. Hamielec, EE. Wood, D.R. Woods, Polymer, 32 (1991) 1098. 7. C. Kiparissides, G. Daskalatis, D.S. Achilias, E. Sidiropoulou, Ind. Eng. Chem. Res., 36(4) (1997) 1253. 8. D.C.M. Silva, N.M.C Oliveira, Proc. Controlo'2000, 4th Portuguese Conference on Automatic Control, (2000) 49.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
755
Interval Matrix Robust Control of a M M A Polymerization Reactor Andrea Silva B., Antonio Flores T. * and Guillermo Fern~dez A. Universidad Iberoamericana, Prolongaci6n Paseo de la Reforma 880 M6xico DF, 01210, M6xico In this work results on the robust stability of interval matrices are applied to the robust closedloop control of a MMA polymerization reactor. Reactor operation is analyzed around an optimal design point where the system exhibits output multiplicities, besides the operation point is practically a limit point. A robust linear MIMO control law is synthesized where plant and input disturbances are captured as interval matrices. The closed-loop system performance was analyzed in presence of input disturbances and model uncertainties.
1. INTRODUCTION The closed-loop control of polymerization reactors is an important task in the chemical processing industry. Such control is sometimes complicated due to the fact that strong nonlinearities are commonly exhibited. Among these nonlinearities the presence of input/output multiplicities, isolas and disjoint bifurcations can be mentioned. In addition one of the most important end use polymer properties, molecular weight distribution, normally cannot be on-line measured for closed-loop control purposes. Because most of the polymerization reactions kinetic parameters are seldom known with enought precision, and they can be time variant, the impact of modeling errors on the closed-loop performance must be taken into account. In the control literature there has been some ways of taking into account robustness issues. One of such approaches consists in representing model uncertainty as upper and lower bounds on the nominal plant description. Such upper and lower bounds define and interval system where the plant to be controlled can be any plant located between such bounds. In this work we propose a control synthesis technique to address robust stability issues by using a constant gain linear multivariable control law. The synthesis technique is partially based on previous work [ 1] based on capturing modeling errors as upper and lower bounds on the state-space representation of a model, and from this plant representation to derive theoretical conditions to guarantee closedloop robust stability for linear systems. However, no synthesis technique was proposed in [1]. Hence, the main contribution of this work consists in the design of the control law for systems described in terms of multivariable linear interval plants.
2. MATHEMATICAL MODEL The polymerization system addressed was the free-radical bulk methyl methacrylate (MMA) polymerization using AIBN as initiator and toluene as solvent. The set of polymerization re*Author to whom correspondence should be addressed. E-mail: [email protected],phone/fax: +52 5 267 42 79, http://kaos.dci.uia.mxJ'aflores
756 actions takes place in a CSTR, the mathematical model of the MMA polymerization is given by:
dCm at = dCl = dt dT dt = dDo = dt dD1 = dt dTj dt =
F(Cmin -Cm) v
-(kp + ~:~) C~Po +
(1)
-ktCt + (FICtin- FC1) V F(r,. - ~) (-AH)kpCm UA pc~ P o - pcov~ ( r - rj) + (0.5ktc + ktd)Po2 + k f m C m P o
-
(2) (3)
(4)
vo-----Z~ V
Mm(kp + k:=) C~eo- FO----11
(5)
V
Fcw(Two- Tj) UA (~ Vo + pwCpw-----~o
~)
(6)
where,
/2:*C,k, Po = Vk-~d~tc,
kr--Are
, r = p, fm, I, td, tc
the polymer average molecular weight is defined as the ratio D1/Do. Under nominal conditions, the system model exhibits three steady-states (see [2] for model notation and for a complete description of parameter values and steady-state solutions). The low conversion (7.8 %) steadystate is used as the nominal operating point throughout this study. It is important to mention that even though the operation point is stable, it is practically a limit point; that is, it almost corresponds to the point in which the slope sign changes. This fact may imply control problems.
3. CONTROL TECHNIQUE Through a non-linear behavior analysis of the polymerization system [2] it was expected that the control problem would not be simple. In [1 ] the authors provided a series of theoretical conditions that have to be met in order to assure robust stability of interval matrices and established that Hurwitz stability is guaranteed if condition 1 is met and at least one of the three conditions described in condition 2: 9 condition 1.
9 condition 2.
757 see [1] for notation. In this work the above results were applied to the robust closed-loop control of the MMA polymerization system. Using a proportional controller u = kx to control the plant represented by the linear description :~ = Ax + Bu, the following closed-loop expression is obtained Jc = (A + Bk)x. In order to use the stability conditions in such system, it is necessary that the (A + Bk) matrix be represented as an interval matrix. Defining the uncertainty level of matrices A and B as A: .4=A(I+A),
A_=A(1-A),
B=B(I+A),
B=B(1-A)
the interval and nominal matrices (At,Ao), respectively are given by:
a~= [a+B_k, ~+~k],
ao=a+Bk
using the above expressions it becomes possible to apply the conditions for robust stability as stated in [ 1] to design an interval robust control system. In order to use the proposed technique it is necessary to have a square system. Therefore, the selected controlled variables were monomer and initiator concentrations (Cm and Ct), and reactor and cooling system temperatures (T and Tj). The manipulated variables selected were the system inputs: monomer, initiator and cooling system flow rates (F,/~ and Fc~) and the inlet temperature (Tin). The central matrix was obtained by linearizing the plant around the optimal operating point, and the interval matrices were derived assuming a 4- 99% uncertainty in the central matrix. The controller k was synthesized, using the optimization toolbox, by solving any of the following unconstrained optimization programs:
ein{n k
( Iel
, i,j=l
I i
the closed-loop operability of the linear plant was analyzed and the results proved that the controller provides robust and performance stability characteristics. Nevertheless, we decided to prove the performance of the linear MIMO controller on the original highly non-linear plant representation. 4. RESULTS AND DISCUSSION
The controller performance was analyzed through closed-loop dynamic simulations of the original non-linear system in presence of input disturbances, in particular changes in the monomer inlet concentration. The dynamic simulation of the closed-loop system when the inlet monomer concentration increases by 10% is shown in figure 1. Figure 1(a) shows the states deviation. In figure 1(b) the manipulated variables relative percentage deviations are shown (the notation used in the results figures is as follows. Outputs'. ACre .... ACIin -.-., A T .... ATj - - , A M W ~ . Inputs'. AF w , A F t , AFcw - - , ATin ...) . It can be observed that there is a large control action. The monomer and initiator flowrates had to be reduced by more than 40%, and the system is stable after 0.5 hours. The largest deviations in states are observed in monomer and initiator concentrations (ACre +4% and ACt -6%), while the molecular weight had a smaller deviation (+2%). Both temperatures remained at the optimal operation point (see figure 1(a)). In figure 2 closed-loop system simulation results when the propagation activation energy increases by 0.3% are shown. The output variables were always under control, but the behaviour
758
o.
-5
.......................................
-IO
~_-lS
i:
~.
'.
~o . . . . . . . . . . . . . . . . . . . . . . .
o~
0'1
0~5
0'2
0~5
r~n8 (h)
0'3
o~s
014 0.~,5
-45 -15(
os
(a)
o~
o',
o',~ o'~ o~
rwne (h)
o'~ o~
o'4 o~
o~
(b)
Fig. 1. Closed-loop deviations in controlled (a) and manipulated variables (b) using a 10% disturbance.
0.3%Epunoert,~r~ . . . .
Cmi,
2O0
~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14~ 12C
g
10C
c ac
. . . . . . . . . . . . . .
Time(h)
r r n e (h)
(a)
(b)
Fig. 2. Closed-loop deviations in controled (a) and manipulated variables (b) using a 0.3% disturbance.
Ep
of the manipulated variables shows that potential saturation problems are likely to occur in the initiator flowrate control valve. This is so because the nominal initiator flowrate is small and therefore any slight modification of this flowrate might lead to saturation problems. No saturation problems were observed for the remaining manipulated variables. In order to test the interval controller under more demanding operating conditions, the monomer inlet concentration was increased by 10% and a 10 minute monomer concentration measurement delay was also considered. In figure 3 closed-loop simulation results for this case are shown. Output variables were controlled around the desired nominal operating region, however
759 lo'~C~n,'qltutt)ancamcllOrmemr,entr=endew
,
,
0
g
1o
i
J
I
.
-300
,
Ot'~ Or2 0=3 014 Or5 0=5 017 ~=8 O=g
o*~ ol2 o'a 0'4 o'5 o's ol7 o s ~me (h)
"n rne (h)
(a)
o s/
(b)
Fig. 3. Closed-loop deviations in controlled (a) and manipulated variables (b) using a 10% Cmin disturbance and 10 min. concentration delay.
5
0 3 % Ep
uncertaintyen~1 0 rranconc~ntrenonaelay
loo c
!o
(a)
(b)
Fig. 4. Closed-loop deviations in controlled (a) and manipulated variables (b) using a 0.3% Ep uncertainty and 10 min. concentration delay.
wide variations in the monomer and initiator flowrates were observed. Comparing figures 1 and 3 we observe that the introduction of the concentration delay makes infeasible the use of the interval controller since saturation problems could emerge. In figure 4 the closed-loop system simulation when the propagation activation energy increases by 0.3% and there is a 10 minute monomer concentration delay is shown. Comparing figures 2 and 4 we notice that closed-loop results look similar, i.e. the introduction of the concentration delay did not make worse the controller performance. Again, potential initiator flowrate saturation problems are observed.
760 From the analyzed cases we observe that, due to the quite small values of the initiator flowrate, almost any disturbance might cause saturation problems on this input variable. Because the interval controller is a pure gain controller, offset problems are likely to occur. The problem could in principle be corrected by introducing integral action into the control system. Because most of the saturation problems occur in the initiator flowrate one might question about the utility of using the initiator flowrate as manipulated variable. In some cases product composition could be obtained by using the remaining 3 input variables. Besides, when they were obtained, acceptable closed-loop responses were observed in around 4 times the reactor residence time (0.1 h), which implies good closed-loop disturbance rejection characteristics. 5. CONCLUSIONS Polymerization systems, like the one in study, have highly non-linear behaviour specially when they operate around multiplicity regions. The nonlinear behaviour may imply control dificulties. In this case, the system was controlled around an operating point that is practically a limit point. Using the results on stability of interval matrices [1] it was posible to obtain a controller that provides robust stability on the controlled system. It is clear that some deviations were observed, but since the average molecular weight is the most important variable in a polymerization system, it can be concluded that the system was succesfully controlled since, under the considered upsets, not very large deviations in this variable were observed under the considered upsets. Besides, being the interval controller a pure gain controller, it would be easier to implement than other control structures.
REFERENCES 1. Wang, S., S.B. Lin and L.S. Shieh. Proceedings of the lASTED International Conference CONTROL' 97 (1997), 31-34. 2. Silva, A. and A. Flores Ind.Eng.Chem.Res. 38 (1999) 4790-4804.
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rights reserved.
761
Model predictive control of apoptosis in mammalian cell cultures L. Simon a*, N. M. Karim a aDepartment of Chemical and Bioresource Engineering, Colorado State University, Fort Collins, CO 80523
Model predictive control (MPC) was implemented to control starvation-induced apoptosis in CHO cells. Apoptosis or Programmed Cell Death (PCD) is characterized by an enzymatic and cellular machinery in which the cells engage in an orderly pattern of events that lead to death by suicide. In this work, classical kinetic models were extended to account for apoptotic cell death. A series of batch type fermentations were run in order to obtain a specific apoptotic death rate. The amino acids that played a key role in the onset of apoptosis were identified by performing a Neural Network based sensitivity analysis. By estimation of the number of viable cells using extended Kalman filter (EKF), the concentration of apoptotic dead cells (PADC) could be monitored on-line. The estimated PADC was used for on-line model predictive control. Based on the prediction of PACD two-sample intervals ahead, corrective control actions were taken to minimize apoptotic death in the bioreactor. The manipulated variables were the flow rates of glutamine, cysteine and tyrosine. The output variable was the concentration of apoptotic dead cells in the bioreactor.
1. I N T R O D U C T I O N The two mechanisms of cell death reported in the literature are necrosis and apoptosis (PCD) (Wyllie et al., 1980). Necrosis is a passive process that occurs when the cells are exposed to extreme environmental or physiologic stresses. Apoptotic cells are induced to commit suicide under normal physiological conditions. Researchers, working with CHO cells established a connection between amino acid starvation and the protein synthetic machinery of CHO cells (Stanners et al., 1978). Sensitivity analysis should to be conducted in order to reduce the set of amino acids responsible for PCD. Controlling apoptosis by addition of key amino acids to the medium is a well developed method. The novelty of the paper is the use of an amino acid dependent death rate to account for apoptotic death and potentials for real-time control of apoptosis. Classical descriptions of the dynamics of mammalian cells rarely include death by necrosis and apoptosis. The control of PCD is therefore difficult in the current framework. *Supported by the National Science Foundation (BES-9622526), the UNCF-Merck Graduate Science Research Dissertation Fellowship, the Shrake Culler scholarship, and the Colorado State Experiment Station
762
2. CULTIVATION CONDITIONS AND SAMPLE ANALYSIS 2.1. Cultivation conditions The CHO cell line, CHO 1-155500, obtained from ATCC was used in this research. The cells produce recombinant tissue-type plasminogen activator (tPA) that has clinical applications as a thrombolytic agent. The cells were initially thawed and diluted ten-fold (1 ml of culture and 9 ml of 90% HAMSF-12 and 10% FBS). These were then transferred to a 75 ml T-flask containing 95 % selection medium and 5% dialyzed fetal bovine serum. The flask was placed in a incubator set at 37~ and 8% carbon dioxide overlay. 2.2. Sample analysis The concentration of total cells was measured using a hemacytometer. The samples were initially diluted to 2,000-20,000 cells per ml. Cell viability was assessed by trypan blue exclusion. Flow cytometry was the methodology employed to detect and quantify apoptotic cells. Ammonia was assayed by enzymatic reactions. The concentrations of lactate and glucose were determined using a YSI analyzer. The compositions of amino acids were determined by HPLC method. The product, tPA, was analyzed via ELISA. 3. KINETIC MODEL OF CELL GROWTH, PRODUCTION FORMATION AND SUBSTRATE CONSUMPTION The kinetic model is described by the following equations (see Table 1 for definitions of variables and parameters)"
dXr[t] = l~v[t] dt
(1)
dXv[t] = (11- ka)Xv[t] dt
(2)
dXda[t] = kdaXv[t] dt
(3)
dXdn[t] = kanXv[t] dt
(4)
dGln[t] = -kdegGln[t] - qGlnXv[t] dt
(5)
dAm[t] = kdegGln[t] + qAmXv[t] dt
(6)
dP[t] = ctdXv~t~[1 dt dt dCys[t] dt
1 dXv[t] Yxdcys dt
dTyr[t] = 1 dXv[t] dt --YXv/Tyr dt
(7)
(8) (9)
763
dGlc[t] dt
1 dXv[t] YXv/Glc dt
dAsn[t] dt
YXv/Asn
rac[t] -
1
c[O]
Glc[O] -Glc[t]
(lO)
dXv[t] dt
(11)
= YLac/Glc
(12)
PmaxGlc[t].Gin[t] .Asn[t] ].1 -- (kGl c + Glc[t]) (kGln -k- Gln[t]) (kAsn + Asn[t])Kini
ki~c
+1 [ kiAm +1
[Glc[t] 1] kiGlc +
(13)
(14)
Solving Eq. 3 is a difficult task since the dependence of kda on Tyr and Cys is unknown. To model the concentration of apoptotic cells (Eq. 3), feedforward neural networks (NNs) were combined with physical knowledge of the real system. The inputs to the networks are the concentrations of tyrosine, cysteine, and viable cells and the output is the concentration of apoptotic cells, kda c a n then be obtained discretely from Eq. 3. Tyrosine and cysteine were selected among the set of 20 amino acids after performing a sensitivity analysis (see Sections 7 and 8.1). Eq. 3 is replaced by
Xda[t]- fNN(Tyr[t],Asn[t],Xv[t])
(15)
where fNN is the NN mapping. 4. M O D E L I N G OF O X Y G E N IN THE B I O R E A C T O R
With CHO cells, the liquid phase oxygen balance is
VLdCo2,L dt =VLKLaB(Co2,u _Co2,L) +VLKLaH(Co2,LH_Co2L) -OUR
(16)
where CO2,L is the concentration of dissolved oxygen in the liquid phase. CO2,LH and CO2,LB represent the liquid phase concentration of oxygen in equilibrium with the headspace and bubble phase respectively. KLan and KLaH are the overall mass transfer coefficient in the bubble and head phase respectively. OUR is the oxygen uptake rate by the cells. The oxygen uptake rate is obtained using the following equation
OUR - qo2,gXV+ qo2,mXV
(17)
where qo2,g and qo2,m are the specific uptake rates for growth requirement and maintenance respectively.
764 Table 1 Definition of variables and parameters Variable X T [(103) cells/mL] Xv [105 cells/mL]
Glc [g/L] Asn [pmole/mL] Gin [pmole/mL] Tyr [pmole/mL] Cys [pmole/mL] Lac [g/L] Am [t.tg/mL] P [pg/mL] /-/max [h-1 ] kdn [h -1 ]
kGlc
[g/L]
kAsn [pmole/mL] kGln[pmole/mL] kiLac [(g/L) 2] kiGlc [ g/L] kiAm [lug/mL] kdeg [h -1 ]
qGtn [(pmole Gln)/(105 cell.h)] QAm [(]tlg Am)~(105 cell.h)] Yxv/Clc [(105 cells)/(pmole Glc)] Yl_ac/Glc[(g of Lac)/(g Glc)] Yxv/Cys [(105 cells)/(pmole Cys)] Yxv/asn [ ( 105 cells)/(pmole Asn)]
Yxv/ryr [( 105 cells)/(pmole c~ [(p g tPA)/(105 cells)]
Tyr)]
Definition Total cell concentration Viable cell concentration Glucose concentration Asparagine concentration Glutamine concentration Tyrosine concentration Cysteine concentration Lactate concentration Ammonia concentration tPA concentration maximum specific growth rate necrotic death rate Monod constant for Glc Monod constant for Asn Monod const, for Gin Inhibition constant for Lac Inhibition constant for Glc Inhibition constant for Am Decomposition rate of Gin Specific Gin uptake rate Specific Am production rate Glc yield coefficient Lac to Glc yield coefficient Cys yield coefficient Asn yield coefficient Tyr yield coefficient Growth associated coefficient
Value
0.0577 0.0262 4.3707 4.4859 4.4997 0.9774 480.1097 266.3237 0.OO25 1.6406 0.0170 5.5772 0.5106 0.2144 0.1065 0.1253 0.5616
5. EXTENDED KALMAN FILTERING TECHNIQUE A Kalman filter or Kalman estimator is an linear optimal observer that estimates the states of a dynamic system from measurements with random errors. Kalman filters extract the best estimate of a state vector at time k based on noisy output vectors observed from prior time and information about the statistical properties of the noise (Sargantanis and Karim, 1994). The state variable vector x(t) is represented by [ XT[t], Xv[t], Xcln[t], Glc[t], Gln[t], Am[t], Asn[t], Tyr[t], Cys[t], P[t]] 7" is the 10 x 1 state vector, f(x(t)) a 10 • 1 state function vector given by Eqs. 1, 2, 4, 5, 6, 7, 8, 9, 10, 11. The output z(tk) = OUR[tk] is a scalar output at discrete time tg, g(x(tk)) is the observer equation given by 17, w(t) is the state model error vector (10 x 1) and v(tk) is the measurement noise. In the EKF version, the state and output equations are linearized. The initial conditions are perturbed by 3% of their true values.
765 6. M O D E L P R E D I C T I V E C O N T R O L (MPC) MPC is a multistep predictor method in which both input and output constraints are explicitly integrated in the controller design. The model predictive control law follows from the minimization of an objective function over a control horizon. For details on MPCs, see (Eaton and Rawlings, 1992; Pr~311and Karim, 1994). 7. N E U R A L N E T W O R K S AND INPUT SENSITIVITY ANALYSIS Neural network was used in order to help with the sensitivity analysis. Different architectures of neural networks have been proposed in the literature (Eikens and Karim, 1999). The development of a neural network requires a training set made up of known input and output patterns. The network tries to learn the underlying relationship based on the given set of data using an iterative error-minimization procedure. By performing sensitivity analysis on a trained neural network, one can often find and eliminate irrelevant inputs. A sensitivity analysis about the mean was implemented to extract relevant inputs. 8. RESULTS AND D I S C U S S I O N S
8.1. Sensitivity analysis results The amino acid candidates for apoptosis were chosen from the set of 20 amino acids. The structure of the neural network is : 11 input nodes, 20 nodes in the hidden layer, 1 node in the output layer (apoptotic cells) (11 • 20-20 x 1). The neural network was trained using only 11 amino acids (Asn, Gln, Arg, Pro, Tyr, Val, Met, Cys, lie, Leu, Phe) since the concentrations of the remaining amino acids increased during the fermentation. The neural network accurately predicts the concentration of apoptotic cells. Apoptosis was found to be more sensitive to Tyr and Cys. Glutamine and proline had no effect on PCD in the experiment.
8.2. Kalman filter results An extended Kalman filter was used for estimation of state variables. Total, viable, necrotic and apoptotic cell profiles are given in Fig. 1. The error in the estimation of the apoptotic cells is due to the calculation of the specific rate of PCD (Kda). This number is obtained from the neural network prediction of the concentration of apoptotic cells.
o
20
40
T|r,,e(h)
80
80
1
oo
.
.
.
.
%,,,ov.]' .
.
.
.
.
.
o~_- I ..... O.2
u ~
~'o
4"o
8'o
8'o
~ oo
~
20
40
T|mo(h)
eo
..,0
I
oo
Fig. 1. Noise-containing and EKF estimation of total, viable, necrotic and apoptotic cells
766 8.3. MPC results A model predictive controller was implemented once the concentration of apoptotic cells reached 1.5 • 104 cells/mL. More details on the use of MPC can be found in the works of Prtill and Karim (Pr611 and Karim, 1994).The concentration of apoptotic cells was controlled using the flow rates of cysteine Fcys, tyrosine FTyr and glutamine FGtn. The results are shown in Fig. 2. The length of the control and prediction horizons are 1 and 2, respectively. o.2
....
o.ls
o.os
o o
20
40
oo
8o
-I o o
20
40
60
80
I O0
o
=o
40
20
40
Time(h)
eo
-,,o
1 oo
80
so
1 oo
Fig. 2. Control of apoptotic cells using the flow rates of Fcys, tyrosine FTy r and glutamine Fctn
9. CONCLUSIONS A model was proposed to account for apoptotic death. Tyrosine and Cysteine were selected based on a neural network sensitivity analysis. A neural network was developed to express the concentration of apoptotic cells as a function of viable cells, tyrosine and cysteine concentrations. The specific apoptotic rate constant, kda, was obtained discretely from the concentration of apoptotic cells. An extended Kalman filter was used for state estimation. The concentration of apoptotic cells was controlled at 1.5 • 104 cells/mL. REFERENCES
Eaton, J.W. and J.B. Rawlings (1992). Model predictive contol of chemical processes. Chem. Eng. Sci. 47, 705-720. Eikens, B. and M.N. Karim (1999). Process identification with multiple neural network models. Int. J. Control 72, 576-590. Pr611, T. and M.N. Karim (1994). Model-predictive pH control using real-time NARX approach. AIChE J. 40, 269-282. Sargantanis, J. and M.N. Karim (1994). Multivariable iterative extended kalman filter based adaptive control: Case study of solid substrate fermentation. Ind. Eng. Chem. Res. 33, 878888. Stanners, C.P., T.M. Wightman and J.L. Harkins (1978). Effect of extreme amino acid starvation on the protein synthetic machinery of cho cells. J. Cell. Physiol. 95, 125-138. Wyllie, A.H., J.ER. Kerr and A.R. Currie (1980). Cell death: The significance of apoptosis. Int. Rev. Cytol. 68, 251-306.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
767
Convex Underestimators for Variational and Optimal Control Problems Adam B. Singer, Jin-Kwang Bok, and Paul I. Barton a aDepartment of Chemical Engineering Massachusetts Institute of Technology Cambridge, MA 02139 An overview of convexity for variational problems is presented. These notions of convexity are then used to develop a theoretical framework for convex underestimators for variational problems. Necessary and sufficient conditions for a minimum of the convex underestimating problem are developed. 1. I N T R O D U C T I O N Beginning with the work of Falk and Soland [2], much interest has been generated in the field of optimization pertaining to the construction of convex underestimators to be used by algorithms for determining global solutions to nonlinear, nonconvex, algebraic optimization problems. More recently, Ryoo and Sahinidis [5] have developed a branch and reduce method for the global optimization of NLPs and MINLPs using convex underestimators. Moreover, Adjiman, Dallwig, Floudas, and Neumaier [ 1] have developed a method that enables the construction of convex underestimators for any twice continuously differentiable functions. In conventional optimization, the objective is a function that maps from a finite dimensional real space to a real value. However, in variational problems, the objective is actually a functional that maps an infinite dimensional function space to a real value. Of particular interest are integral functionals on linear spaces of functions. Integral functionals are functionals that map a linear space of functions to a real value via a definite integral. This exposition describes our recent developments that utilize convexifications of integral functionals to compute rigorous lower bounds for variational problems. Additionally, we believe that these methods may eventually lead to the development of algorithms for generating global solutions to nonconvex, nonlinear variational and optimal control problems. 2. C O N V E X I T Y At the root of developing convex underestimators for variational problems is the notion of convexity for real valued functions on Na. For the purposes of this discussion, only continuously differentiable functions will be considered. This restriction implies the existence of the gradient of the function and permits convexity to be defined in the following manner. Definition 2.1. The real valued function f will be convex on D C R a if it has continuous partial derivatives on D and satisfies the inequality
f ( x ) _> f(xo) + Vf(xo)" ( x - xo),
V x, xo C D
Moreover, f is strictly convex on D when the above inequality holds at each xo C D with equality if and only if x - xo.
768 Remark. An equivalent definition for convexity on D C R d is f ( x + v) - f(x) >__V f ( x ) . v ,
VvER a
for which x, x + v E D.
Moreover, when v is taken as an arbitrary vector, 5 f ( x ; v ) = Vf(x) 9v, where 8f(x;v) is the G~teaux variation of f . The purpose of expressing convexity as in the above Remark is merely to illustrate a direct analogy between convexity in an Euclidean space R d and convexity in a linear space of functions 2C, the definition of which is given below.
Definition 2.2. A functional J on a set 9 C 2C is said to be [strictly] convex on 9 provided that when x and x + v E 9 then ~)J(x; v) is defined and J(x + v) - J(x) > ~)J(x; v) [with equality if and only if v=O, where (9 is the unique vector such that cO = 0x = (9, Vx E 2C, c E R]. Convexity is an important notion in conventional optimization schemes because convexity implies that any stationary point found is a global minimum (the unique global minimum for strict convexity). As expected, an analogous result exists for the minimization of convex functionals, as stated in the following proposition.
Proposition 2.3. If J is [strictly] convex on ~D C T~ then each xo E 9 for which 5J(x0; v) = 0, ~/ xo + v E ~D minimizes J on ~D [uniquely]. The proof of Proposition 2.3 is immediately evident from Definition 2.2 and is thus omitted. The interested reader is referred to Troutman [7, pp. 54-55] for the details of the proof. Note that the above Proposition illustrates the fundamental principle that minimization of convex functionals occurs for the function that makes the G~teaux variation equal to zero. The following defines convexity of functions on a subset S C R3. Specifically, functions of this form comprise the class of functions from which the integrands of functionals will be selected. Note that underlined variables are held fixed in the inequality hence only requiting the partial derivatives of f of the remaining variables.
Definition 2.4. f ( t , x , x ~) is said to be [strongly] convex on a set S C R3 if f = f ( t , x , x ~) and its partial derivatives fx and fx' are defined and continuous on this set and satisfy the inequality:
f ( t , x + v,x ~+ v') - f ( t , x , x ~) > fx(t,x,x~)v + fx' (t,x,x~)v ', V(t,x,x !) and (t,x + v,x ~+ v') E S [with equality at (t,x,x ~) only if v = 0 or v' = 0]. 3. C O N V E X U N D E R E S T I M A T O R S The existence of tight convex underestimators for real functions of special form such as univariate concave, bilinear, and trilinear functions has long been established in the field of optimization. Recently, Adjiman, Dallwig, Floudas, and Neumaier [ 1] have developed a method for deriving convex underestimators for twice continuously differentiable real functions. This section develops a rigorous analytical method for extending these known theories of convex underestimators for real functions to variational problems. At the heart of this analysis is the following fundamental theorem, which links convexity and underestimation of real functions to convexity and underestimation of integral functionals.
769 T h e o r e m 3.1. Let the functionals
F(x) --
/a
f(t,x(t),x'(t))dt
and
G(x) -
/a
g(t,x(t),x'(t))dt
be defined on the set 9 = {x E Cl[a,b]; (x(t),x'(t)) E D C R2}. /fg(t,x,x') is convex on [a,b] • D in the sense of Definition 2.4 and
g(t,x,x !) < f(t,x,x'),
V fixed t C [a,b] andx(t),x'(t) C D
then G(x) is convex on I) and G(x) <_ F(x). That is, if g(t,x,x t) is a convex underestimatorfor f ( t , x , x t) on [a,b] • D, then G(x) is a convex underestimatorfor F(x) on I). Proof When x, x + v C 9 then Definition 2.4 shows that at each t C (a, b), the convexity of g yields
g(t,x + v,x' + v') - g(t,x,x') > gx(t,x,x')v + gx,(t,x,x')v'. The above equation is integrated to yield
f f g ( t , x ( t ) + v(t),xl(t) + v'(t)) - g(t,x(t),x~(t))dt 3> fabgx(t,x(t),x' (t) )v(t) + gx,(t,x(t),x' (t) )v' (t) dt. However, this is equivalent to
+ v) - C(x) _>
v),
which by Definition 2.2 shows that G(x) is convex. It remains to show that G(x) <_ F(x), but this is evident from the comparison theorem for integrals because by assumption, we have that g(t,x,x ~) <_f(t,x,x~), V fixed t C [a,b], andx(t),x~(t) E D. I-1 R e m a r k . This theorem enables any known method for convex underestimation of real functions to be harnessed in the convex underestimation of integral functionals. At this point, a short example is in order to demonstrate the construction of a convex underestimator for an integral functional. Example 3.2. Suppose we wish to minimize the following functional (with respect to x(t)):
F(x) - f01 [x'(t)] 2 - [x(t)]2dt on the set
O - {x(t)E CI[0,1]" x(0)-x0, x(1)- x1, 0 ~ x(t) ( 1 Vt C [0, 1]} Clearly, F(x) is nonconvex; therefore, a convex underestimator will be derived by finding a convex underestimator for the integrand. The following formula [ 1] is appropriate for underestimating the integrand. Note that [x~]2 is already convex. Since the sum of convex functions is still convex, only the nonconvex term, -[X] 2 needs to be convexified. That x and x ~ are functions of t has been dropped to emphasize that for convexification, x and x ~ are treated as elements of
770
R rather than as elements of cn[o, 1] (where n = 1 for x(t) and n = 0 for xt(t)). In the following formula, x c and x U respectively represent the lower and upper bounds on x.
f ( x L) + f ( x g ) - - f(xg) (x--x L) -- f ( O ) + f ( 1 ) - - f ( O ) (x--O) -- --x. xU - x L 1 -0 Hence, a valid convex underestimator, denoted as G(x), for this function is 1
G(x) -
f0
[x'(t)] 2 - x ( t ) d t
Theorem 3.1 provides a method for constructing a convex underestimating integral functional by constructing a convex underestimator for the integrand. However, Theorem 3.1 does not address the problem of determining a minimum for this convex underestimator. At first glance, Proposition 2.3 appears to offer a method by which to solve the underestimating integral; however, Proposition 2.3 is applicable only to unconstrained variational problems. The generation of convex underestimators requires constraints on the nonconvex variables [ 1], as illustrated by Example 3.2. Necessary and sufficient conditions for optimizing these constrained variational problems are discussed in the following section. 4. NECESSARY AND SUFFICIENT CONDITIONS FOR CONSTRAINED VARIATIONAL PROBLEMS
Many algorithms designed for solving variational problems focus exclusively on satisfying necessary conditions for optimality. For variational problems, this technique is synonymous with finding stationary functions, or those functions which satisfy the Euler-Lagrange equations. The downfall of this approach is that stationarity does not necessarily even imply local minimality. However, using the convexity inherent to the underestimators, a complementary sufficiency condition has been discovered thus guaranteeing that any function satisfying the Euler-Lagrange equation is a minimum [a unique minimum under strong convexity]. For this discussion, existence conditions are not discussed. Rather, it is assumed that a feasible minimum for the problem exists. The sufficiency condition is developed in two stages. First, a lemma is presented that illustrates a method for transforming a constrained problem into an unconstrained problem, the minimum of which implies the minimum of the original constrained problem. Second, the optimality condition for a constrained, convex variational problem is stated. The proofs for the lemma and for sufficiency [6] are both omitted due to space limitations. However, the reader is directed to Hestenes [3] for the proof of necessity. Note that the "hat" symbol is used to denote piecewise continuity or piecewise continuous differentiability. L e m m a 4.1. Suppose f -
f ( t , x ( t ) , ~ ( t ) ) and ~, - ~,(t,x(t),~(t)) are continuous on [a,b] • ][{2 and there exists a function ~(t) E C[a,b],for which xo minimizes F(x) - f f f(t,x(t),x~(t)) dt on
~) C_ ~,1 [a,b] where f d e f f following constraints:
7t -
1. ~(t,x(t),x~(t)) < O, 2. ~(t) >_ O,
~ . Then xo minimizes F(x) - fb f(t,x(t),x~(t))dt on ~) under the
t E (a,b)
t E (a,b)
3. ~,(t)~,(t,xo(t),Xlo(t)) - 0
771 Remark. Although the product ~,(t)~(t,x(t),x~(t)) may only be piecewise continuous, this poses no difficulty to integration provided there exist only finitely many points of discontinuity (cf. Rudin [4]). The following Theorem states the conditions of optimality for minimizing a bounded convex functional. Theorem 4.2. For a domain D o f R 2 suppose that f ( t , x ( t ) , x ~ ( t ) ) C Cl([a,b] • D) is convex, and we wish to minimize F(2) --
/aa
f(t,2(t),~'(t))dt
on
~) -- {2(t) C Cl[a,b]'2(a) - al,2(b) - bl}, subject to the inequality constraint g(t,2(t)) <_ O,
t E (a,b),
where g(t,2) is also convex. For any 2o C 9 satisfying the inequality, the following conditions are necessary and sufficient to guarantee a minimum: There exists a ~,(t) C C[a,b] such that ~(t) >_ 0 and ~(t)g(t, xo) - O. Additionally, f o r all intervals excluding corner points the following equation must be satisfied: d -d~fx'(t,2o(t),21o(t)) - fx(t,2o(t),21o(t) -- Z(t)gx(t,2o(t)). At any corner point c, the following condition must hold: f x ' ( C - , 2 0 ( c - ) , 2 1 0 ( c - ) ) - fx,(C+,20(c+),2~o(C+)) -- kgx(t,2o(t)) f o r a constant k (for necessity, k <_ 0 and f o r sufficiency, k - 0). Currently, a small gap exists between the necessary and sufficient conditions.
Remark. It should be noted that the differential equation defining optimality is the first EulerLagrange equation for the modified function P(2). For a detailed discussion of stationarity, the reader is referred to Troutman [7]. Comer conditions for constrained variational problems are discussed in detail in both Troutman [7] and Hestenes [3]. Additionally, while the above theorem is stated only for one constraint, it should be obvious that the same theorem can trivially be extended to multiple constraints provided the constraints are nonintersecting. This follows because the minimum 20(t) cannot exist at multiple distinct points simultaneously. Thus, when 20(t) lies on one constraint, the multiplier for any other constraint is simply 0. 5. CONCLUSIONS Convex underestimators for variational problems are useful for determining rigorous lower bounds and potentially for developing global optimization schemes for nonconvex problems. In this paper, the fundamentals of convexity for functionals, specifically integral functionals, has been presented. Additionally, the theoretical groundwork has been developed enabling the derivation of convex underestimators for variational problems. Moreover, optimality conditions for constrained convex problems of special form have been developed and proven. While currently limited only to variational problems, the future of this work will be to extend this theory to encompass general problems of optimal control.
772 REFERENCES
1. C.S. Adjiman, S. Dallwig, C.A. Floudas, and A. Neumaier. A global optimization method, ctBB, for general twice-differentiable constrained N L P s - I. theoretical advances. Computers and Chemical Engineering, 22(9): 1137-1158, 1998. 2. James E. Falk and Richard M. Soland. An algorithm for separable nonconvex programming problems. Management Science, 15(9):550-569, 1969. 3. Magnus R. Hestenes. Calculus of Variations and Optimal Control Theory. John Wiley & Sons, Inc., New York, 1966. 4. Walter Rudin. Principles of Mathematical Analysis. McGraw-Hill, Inc., New York, third edition, 1976. 5. H.S. Ryoo and N.V. Sahinidis. Global optimization of nonconvex NLPs and MINLPs with application in process design. Computers and Chemical Engineering, 19(5):551-566, 1995. 6. Adam B. Singer and Paul I. Barton. Convex underestimators for variational and optimal control problems. In preparation, 2000. 7. John L. Troutman. Variational Calculus and Optimal Control: Optimization with Elementary Convexity. Springer-Verlag, New York, second edition, 1996.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
773
Modeling, Simulation and Control of an Industrial Electrochemical Process R. Skotte 1, W. An 2, D.H. Lenz 2, D.R. Baptiste 2, D.S. Lapham 2, C.J. Lymburner 2, J.M. Kaylor 2, M. Pinsky 3, R. Gani 1 and S.B. Jorgensen 1 1 CAPEC, Department of Chemical Engineering, Technical University of Denmark, DK-2800 Lyngby 2 FMC Corporation, Active Oxidants Division, Tonawanda, NY 3 FMC Corporation, R&D Center, Princeton, NJ CAPE methods and tools were successfully used to develop a simulation of an electrochemical process. Analysis of the developed models showed immediate improvements that could be made on the process. The existing control scheme was evaluated to identify any operational problems. An on-line process application was implemented to provide the operators and engineers with solubility predictions for the process solutions. 1. INTRODUCTION Many industrial processes today are continuous systems with multiple mass recycle streams and energy integration. However the complexity of these processes often lead to operability problems where PID controllers are implemented on the integrated processes. Problems arise when little time is spent on tuning the controllers and certainly not enough time is spent on optimizing the control structure for the integrated process. CAPEC recently collaborated with FMC Corporation to improve and optimize the control of one of FMC's industrial processes with the above characteristics. In addition to poorly tuned controllers, this particular process also had manually controlled loops, which lead to quite different plant operation depending on the operator running the plant. Improved operation and control was anticipated to provide increased capacity and lower operating costs. This case study illustrates the application of methods and tools for modeling, simulation, control structure analysis, and controller tuning in order to understand the static and dynamic behavior of the process, to identify the sensitive control loops, to tune the existing controllers and to suggest improvements in the existing control loops. The main unit-operations in the plant studied are electrolytic cells and a continuously operated crystallizer linked together by several recycle-loops with mixing operations. The first step considers model development [ 1,2,3] in order to represent the behavior of the plant with a combination of dynamic models and in-house correlations representing the electrolyte cell chemistry. After the modeling (steady state and dynamic) step, the existing control scheme was successfully implemented and simulated. The control loops were analyzed and the controllers were tuned using the simulator. Finally, control structure changes and online applications were implemented and validated on the actual industrial process.
774
2. PROBLEM DEFINITION 2.1 Process Description The electrolyte process contains water and 4 inorganic salts, which in solution dissociates into 7 ions. The main unit operations in the plant are the electrolytic cells and the crystallizer. In addition a series of mixing tanks (A-Feed, A-Collect & Neutral Tank) are used to maintain the correct chemistry. On the anodic side of the electrolyte cells, anions react to form the anions needed in the final salt while the raw material is added to the cathodic side, which is transferred to the anodic side through the membrane and through the Neutral Tank. In the crystallizer, water evaporation occurs under vacuum, which saturates the solution and precipitates the product salt. Salts formed in the crystallizer are treated in a special system that ensures the formation of a pure final salt. A simplified layout of the process is given in Figure 1. I
t
'
,
Acid ~
-I
-]
,--
c k ~ Alkah --i:~- ~ H20
H20
Ba
,
--~
Catho
Fir ,=s
--~ I
_1 ~11Neutral Tank -
Alkali
(
Flow measurement
(~
Temperature measurement
•
(~
Pressure measurement
(~
Density measurement
I
6
I
I
Product Purification& 4 Concentratton
~
Alkali
H=O
Salt
(~-- Levelmeasurement ( ~ p H measurement - - - PID signal
Setpoint signal
Figure 1" Simplified P&I diagram with original control structure.
775 2.2 Operation Operators and engineers try to keep the process efficiency as high as possible by operating the different unit-operations at specified chemical compositions. The current efficiency in the cells specifies the theoretical production rate, which is based on specific evaporation and extraction rates in the crystallizer. If the evaporation and extraction rates in the crystallizer are below the specifications, the dissolved product is returned to the cells adversely affecting the current efficiency and the production rate. When operated separately, the cells and the crystallizer have large operating windows, but when they are linked together the combined operating window decreases significantly, and therefore, good control is needed to keep the integrated process working close to optimal productivity. 2.3 Control System The process has 46 control loops, out of which 9 were run in manual and 4 have saturated actuators. The critical control loops are highlighted in Figure 1 and described briefly below. 9 Neutral tank control loop: Prepares the solution that is sent to the cells by controlling pH
and density and by specifying a K-Out flow. 9 A-Feed control loop: Keeps the feed flowrate to the cells constant to maintain required
electrochemical cell operation. 9 Crystallizer control loop: Maintains constant evaporation and extraction rates in order to
precipitate and grow the salt crystals. 9 Electrolyte cell control loop: Constant amperage is maintained to achieve a constant
production rate. 9 Product purity control loop: Determines the purity specification of the final product/salt.
2.4 Computer Aided Control System The main components of the computer aided system being developed for this electrochemical process consists of a number of methods and tools:
9 Honeywell- Distributed Control System. 9 Fourier Transform Near Infrared (FTNIR) spectrometry- On-line measurement of compositions. 9 Honeywell PHD (Plant History Data) - Data acquisition. 9 Simulation Engine- Steady state and dynamic (open loop + closed loop) simulation and optimization. 9 Control Toolbox- Analysis of controller structure [4] and tuning of controller parameters. 9 On-line analysis - On-line monitoring of solution chemistry based on measured data and "phenomena" (first engineering principle) models. The objective of the computer aided control system is to improve production efficiency through improved control of the process operation, improve operator productivity through operator training and better information (data) flow, and improved process operability through better controller structure and controller design (set-points, controller parameters, controller interaction, etc.).
776 3. PROCESS CONTROL OPTIMIZATION 3.1 Modeling & Simulation The temperature in most operations is either controlled or assumed to have a negligible effect. The pressure, except for the crystallizer, is atmospheric. Therefore, the mass balance equations are solved together with the constrained phase equilibrium and "phenomena" model equations (solution chemistry, cell chemistry) and decoupled from the energy balance equations. In this paper, only the specially developed crystallizer model is briefly described together with the phenomena (on-line) models for pH and solubility index*. The growth-type crystallizer [5] used in the process can be modeled as illustrated in Figure 2. Fvl, PvI
I
!
Fv2, Pv2
~
FDc
FML ni,BOT
~
FsALT
Figure 2: Crystallizer model.
Since no evaporation occurs in the lower chamber (BOT), only the liquid phase molar component balances are formulated for the i'th component: dn i ,BOT
dt
(1)
__--G c K i , T O P -- FsXi,BO T -- FMLXi,ML -- FSALTXi,sALT
Assuming that only water is evaporated in the upper chamber (TOP), the component molar balance equations are written separately for water and the compounds in the system. dni,TOP
dt
-- F I x i,i q" Fs x i,BOT -- G C X i,TOP
i r H20
(2)
--- F i x i , I q- FsXi,BO T -- FDcXi,TO p -- FVAP
i=H20
(3)
dni,TOP
dt
The holdup (accumulation) of water vapor in the upper chamber is also considered. dn vnv dt
~=Vv~-Vv,-Vv2
(4) K Ct //
Def'med as the activity product of a salt divided by its solubility product:
SI =
aKaAaw
K K,,A~nH20
777 The molar flows of water vapor leaving the upper chamber is described as a function of valve constants (VCvi) and the pressure difference: nvApRT Fvl = g C v l " (Pv~ - Pvl ) ; , where PVAP= Fv2 = VCv2 "(Pv~ - Pv2) J ( V T o P -- Vni,ror ' )
(5)
The molar flow of liquid in the down-comer is determined from the pressure in the upper chamber and the pressure head of the solution in the upper (PHs,TOP) and lower (PHs,BOT) chamber: FDc = VCDc "(PvAP +
PHS,TOP -- (X" PHS,BOT) ,
where PHs = P" g" h
(6)
The density p is determined from a correlation developed specifically for this process, because it is not available in the phenomena models, c~ is a parameter taking the shape of the crystallizer into account i.e. not all of the pressure head in the lower chamber is affecting the flow in the down-comer. The total flow of mother liquor and salt leaving the lower chamber is determined from the liquid level (which is a function of the liquid holdup) and a valve constant. The "phenomena" model then determines the distribution of the flows by calculating the degree of saturation. FML "4- FSALT = V C B o T 9LevelBo
(7)
T
The rigorous model of the crystallizer was successfully implemented in ICAS [6] and dynamically simulated to reach a steady state. 3.2 Process Control
The performance of the control loops shown in Figure 1 has been investigated through closedloop dynamic simulation of the process. In order to improve the performance of the process, several operational (control) tasks were identified. Table 1 lists the controller performances measured in terms of variability, v, where X is defined as the average deviation from the mean process variable (not the setpoint) and R is the average of the increments in the process variable between consecutive samples [7]. A recent review of control loop monitoring using minimum variance based measures is given by Harris ad Seppala [8]. Controller Type I No. of Controllers Density Flow Level Pressure PH Temperature Table 1: Controllerperformances.
3 12 5 3 3 4
v=R/X 0.015 % 0.53 % 1.36 % 0.11% 2.47 % 0.9%
The analysis of the dynamic behavior identified several operational difficulties and in this paper three of the operational difficulties related to level control, density control and online tools are highlighted together with their solution through better control.
778
From Figure 1 it can be noted that changing the crystallizer "overflow" controls the AFeed level. In the real process this loop was run in manual because the level controller did not perform adequately. The operational problems were very significant when filters were washed because a lot of the solution from the Neutral Tank was used to wash the filters. This also affected the A-Feed level and because of a large time delay and interference of the Neutral Tank level controller, the A-Feed level controller often became unstable. An analysis of degrees of freedom for control [9] indicated that controlling the A-Feed level by the inlet flow to the A-Feed tank and controlling the Neutral Tank level by the inlet flow to the Neutral Tank might improve the control of the levels in these tanks. From Figure 1 it can also be noted that changing the water addition to the K-Feed tank controls the density, but the multiplier (shown as MULT in Figure 1) was used to maintain a certain ratio between acid and water. A simple degrees of freedom analysis [9] showed that it is not possible to maintain both a ratio and a density. It turned out that the ratio controller was installed before density measurements were available but it was never removed when the density measurement was added to the process. The ratio controller was removed and the density was controlled by the water addition directly. Figure 3 and Figure 4 show the closedloop response on the real plant before and after the change in control structure based on the degrees of freedom analysis over two randomly picked three-day periods. It can be noted that the control of the operation is quite good. 1,3 1,295 o ..9o v
1,29
(3rl
.~ 1,285
.m
1,28 1,275 1,27 00:00
12100 00':00 12100 00':00 12100 00:00 Time (hours)
Figure 3" Closed-loop response with original control structure. 1,3 1,295 1,29
.~ 1,285
. m ffl
ag
1,28 1,275 1,27 00:00
, 12:00
~ ~ 00:00 12:00 00:00 Time (hours)
12:00
00:00
Figure 4: Closed-loop response of density controller after the change in control structure.
779
In electrochemical processes it is very important to have a "perfect" control of the solution chemistry before the mixture is sent to the cells because it determines the efficiency of the cells and thereby the production rate. Implementation of on-line concentration measurements was therefore determined to be of the greatest importance. As a result of this addition to the process instrumentation, CAPEC developed an on-line tool that uses the online concentration measurements (obtained by FTNIR) to calculate the solubility indices (degree of saturation) and the saturation temperatures of the different salts using the "phenomena" models from ICAS. Figure 5 and Figure 6 show two responses from a dynamic simulation of the process. Figure 5 shows the response of a pH-controller in a single tank. It can be seen that the controller oscillates around the neutral point but finally reaches the setpoint of 6.5, which is a typical behavior of pH-controllers because of the non-linearities in the titration curve. Figure 6 shows the response in the solubility index of the product salt in the cell anode-feed after a 5% increase in the extraction efficiency in the crystallizer. The solubility index of the product salt decreases because more salt is removed in the crystallizer. 0.010
35
0.009
lkah
-
0.008
30
9
0.007
25
'~ 0.006
2O gl.
~ o.oo5
8
15~
~ e 0.003
o
j
L
0.001
~'""
10
5
pH
0.000
0 0
10
20
30
40
50
60
70
80
90
100
T i m e (rain)
Figure 5: Simulated response of a pH-controller in the Neutral Tank. 0.335 0.334 0.333 x
0.332
.--_~ 0.331
0.33 0.329 0.328 0.327 0
100
200
300
400
500 T i m e (min)
Figure 6: Response of solubility index 9
600
700
800
900
1000
780 4. CONCLUSIONS Modeling and simulation of the process in question were performed successfully and CAPE tools were used to provide the operators and engineers with information about solution properties. It has been possible to perform steady state and dynamic simulations of the process and use these to analyze the operation of the process. The modeling and simulation tools have been integrated with a control toolbox and on-line calculation options that serve as the framework for an integrated computer aided system for the plant. A few changes were made to the existing control scheme after identifying some operational difficulties and CAPE tools were used to show the improvement obtainable by these changes. The next step is to add an optimization toolbox to optimize process operation. REFERENCES
1. Jensen, A.K. "Generation of Problem Specific Simulation Models within an Integrated Computer Aided System", Ph.D. Thesis 1998, Department of Chemical Engineering, Technical University of Denmark. 2. Thomsen, K. "Aqueous electrolytes: model parameters and process simulation", Ph.D. Thesis 1997, Department of Chemical Engineering, Technical University of Denmark. 3. Thomsen, K., Rasmussen, P., Gani, R., "Simulation and optimization of fractional crystallization processes", Chemical Engineering Science, vol. 53, No. 8, pp. 1551-1564, 1998. 4. Seider, W.D., Seader, J.D., Lewin, D.R., "Process Design Principles: Synthesis, Analysis, and Evaluation", Wiley & Sons, Inc., 1999. 5. Myerson, A.S. "Handbook of Industrial Crystallization", Butterworth-Heinemann Series, 1993. 6. Gani, R., Hytoft, G., Jaksland, C., Jensen, A.K. "An integrated computer aided system for integrated design of chemical processes", Computers Chemical Engineering, vol. 21, No. 10, pp. 1135-1146, 1997. 7. "Statistical Quality Control Handbook", AT&T by Western Electric Co., Inc., 1956. 8. Harris, T.J. and Seppala, C.T.: "Recent Developments in Controller Performance Monitoring and Assessment Techniques", Preprints from Chemical Process Control 6, Tucson, Arizona, pp. 220-254, January 2001. 9. Russel, B.M., Henriksen, J.P., Jorgensen, S.B., Gani, R., "Integration of Design and Control Through Model Analysis", Computers and Chemical Engineering. 2000, 24 (2-7), pp. 967- 973.
European Symposium on Computer Aided Process Engineering - 11 R. Gain and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
781
Generic properties of time and space dependent optimal control of (bio-)chemical processes Ilse Y.M. Smets a and Jan EM. Van Impe a*
aBioTeC- Bioprocess Technology & Control, Katholieke Universiteit Leuven Kasteelpark Arenberg 22, B-3001 Leuven (Belgium) This paper deals with the optimal control in the space domain of (bio-)chemical reactors. As an example, the determination of optimal heat exchanger temperature profiles of exothermic tubular reactors under the assumption of steady-state and plug flow characteristics is investigated. To enable a trade-off between process performance and energy loss a weighted cost criterion is defined. Application of the minimum principle of Pontryagin leads to extremal control profile structures which are very similar to those obtained during optimization (in the time domain) of biotechnical production processes in well mixed bioreactors. The latter problem has been analyzed in detail over the last 2 decades. The analogy is reflected at various levels during analytical optimal control calculations. 1. INTRODUCTION Nowadays the advantages of optimal control theory are well established [1 ]. Indeed, given a user defined performance criterion optimal control theory (OC) provides the optimal solution to various specifications. It provides a systematic and direct approach to a large variety of control design problems and even constrained optimization (where the manipulated variables are interrelated) is possible. The last decades OC has also found its application in bioprocess optimization. Optimal yield or productivity solutions for the control-in-time of biotechnical growth or production processes in well mixed bioreactors have been analyzed in detail over the past few years (see, e.g., [3], [4] and the references therein). In contrast with these time domain problems, this paper is concerned with the optimal control in the space domain of (bio-)chemical reactors. More specifically, we focus on the determination of optimal heat exchanger temperature profiles of exothermic tubular reactors under the assumption of steady-state and plug flow characteristics. The problem results in an optimal control problem where the usual independent variable (the time t) is replaced by the spatial variable z. To enable a trade-off between process performance and energy loss a weighted cost criterion is defined. The similarity between the analytical calculations and the obtained results in the two abovementioned domains (time and space) is striking and will be emphasized throughout this paper. The organization of this paper is as follows. In the first section optimal control theory in the sense of the Minimum Principle of Pontryagin is introduced in a genetic way for a class of *Corresponding author; Fax: +32-16-32.19.60and e-mail: [email protected]
782 models and performance indices. Afterwards, the theory is applied to a specific time domain and space domain case study in the subsequent sections. The final section summarizes the general conclusions. 2. G E N E R A L STATEMENT OF THE O P T I M A L CONTROL P R O B L E M The two case studies considered further in this paper (control in time of well mixed reactors and control in space of tubular reactors) can be summarized as follows. dx(~)
-- f[x(~)]+bu
with
~0 < ~ < ~f
(1)
with x(~) the state vector and u the scalar control input (both possibly restricted by physical constraints such as upper and lower bounds), and ~ the independent variable t (time) or z (spatial coordinate). Subscript 0 and f denote the initial and final value respectively. Observe that the nonlinear model (1) is linear in the input u. The performance criterion to be minimized is a sum of a terminal and an integral cost:
J[u]-
h[x(~f)]
+ fo ~s g[x(~)]d~
Terminal cost
(2)
Integral cost
Observe that the performance measure (2) is assumed not to depend explicitly on the control input u. The problem statement is then the following. Find an admissible control which causes the given system to follow an admissible trajectory while at the same time minimizing the performance criterion.
According to the minimum principle of Pontryagin [ 1] minimization of performance criterion (2) is achieved by minimizing the Hamiltonian H which is equal to the sum of (i) the integrand of the integral cost and (ii) the product of the costate vector ~, with the fight hand side of the model equations (1). H
-- g[x(~)] +~Tf[x(~)J.+~'~'u''-"'~" v Y
The costate vector X satisfies the following system of differential equations: d)~ _ d~ -
OH 0x
_ -
-
3g[x(~,)_______~]_ xrOf[x(~,)] 0x 0x
with
~(~f)~y
(3)
2.1. Extremal control determination Both optimization problems considered in this study give rise to a Hamiltonian function which is linear in the control input (i.e., the limiting substrate feed rate for the well mixed reactor and the heat exchanger temperature for the plug flow reactor). As a result, the optimal control is of the bang-bang type, with the possibility of singular arcs depending on the value of
if ~ > 0, then u* (~)
-
UMIN
783 if V = 0 for z E [~i, ~i+1], then u* (~) = using(~) if ~ < 0, then u* (~) = UMAX UMIN and UMAX represent the lower and upper bound on the control input respectively. Since the Hamiltonian does not explicitly depend on the time coordinate t in the well mixed case or on the spatial coordinate z in the plug flow case, the Hamiltonian is constant when evaluated on an extremal trajectory.
2.2. Singular arc analysis On a singular interval [~i,~i+l], we have that ~ : 0. The singular control Using(~) is then obtained by repeatedly differentiating ~ with respect to ~ until u appears explicitly. Following algebraic manipulations: 9
~:0
r
d~ ~-~-0
0g[x(~)] + ~T 3f[x(~)] /1 b = 0 3x 3x J
dk T r162
r 3g[x(~)]b + ~Td -- 0 with d A Of[x(~)] b Ox Ox
d2v
02g[x( )] dx( )
9 -d-~2=0 r
~X2
dnr
d~ b + d ~ d +
n 0adx( ) ~x d~
r162[ 02g[x(~)] b ~2) x + ~ r ~ ] d d ~ ~ ) - [[3g[x(~)] +3x r
=0
~'T 0f[x(~)] d = 0 3]x
[02g[x(~)] b + ~ T ~ ] [f[x(~)] + bu] -[0g[x(~)] + ~T0f[x(~)]] d --0 ~X2 bX bX
yield the general expression for the control on the singular arc:
__
Using(~)
[0g[x(~)] ~T0f[x(~)] 02g[x(~)]b+ T3d]f[x(~)] L ~xx q" ~xx ] d - [ ~x 2 ~ ~xJ [32g[x(~)]b + b 3x 2 3xJ
(4)
In the following case studies this general relationship will be analyzed in more detail. 3. CASE STUDY #1. OPTIMIZATION OF FED-BATCH MIXED BIOREACTORS -
r)
In this first case study a fed-batch growth process is considered in which a micro-organism X [g DW] grows on one limiting substrate S [g] in a well mixed bioreactor. The input u is the volumetric flow rate F [L/h]. V [L] is the volume of the reactor, and Cs, in [g/L] is the substrate
'1~4
Cin = C(0) Tin = T(0)
C(L)
C T
/: p "~-q~
T(L)
Fig. 1. Non-isothermal tubular reactor with heat exchanger.
concentration in the feed. Following mass balance equations of the form (1) apply:
d
X
dt
V
-
+
0
u
(5)
1
x(t)
f[x(t)]
b
where cr -- P/Yx/s + m is the specific substrate consumption rate with Yx/s [g DW/g] the yield coefficient of biomass on substrate and m [g/g DW.h] the maintenance constant. The shape of the specific growth rate p [l/h] is arbitrary (i.e., monotonically increasing or non-monotonic). If the objective is to maximize the final amount of biomass (i.e., a terminal cost of the yield type)"
J[u] = -X(tf)
(6)
then Equation (4) reduces to
~CxV
t)f[x(t)] ] d - [~'T~ ] f[x(t)]
Using(t) =
[~T
~X
::=k Using(t) = [~r ~_~dx]b
Cs,in-fs
(7)
with Cx A_ X/V and Cs ~ S/V the biomass and substrate concentration respectively. It can be easily seen that singular control (7) keeps the substrate concentration Cs constant along the singular arc. A comprehensive treatment of this case can be found in, e.g., [3].
4. CASE STUDY #2. OPTIMIZATION OF CONTINUOUS PLUG FLOW REACTORS ( ~ - z) In the steady-state tubular reactor under study a chemical C is converted into a desired product in an exothermic way. The input u is the heat exchanger temperature Tw and the controlled variable is the reactor temperature T (see Figure 4). Since dispersion phenomena are neglected, the steady-state equations for the tubular exothermic reactor reduce to ordinary differential plug
785
flow equations with as independent variable the spatial coordinate z (which are of the form (1))" -
d lTlrq dz L J C
_
~-
~
T
+ ko
x(z)
-
PeP vk~ ~
-E Ce-~
4h
pCpdv
with
u
T(O)
C(O) --fin
0
v f[x(z)]
l)n
~ b
T and C are the temperature and the reactant concentration, v is the fluid superficial velocity, AH is the heat of reaction (AH < 0 for an exothermic reaction), p, Cp, ko, E, R, h, d and Tw are the density, the specific heat, the kinetic constant, the activation energy, the ideal gas constant, the heat transfer coefficient, the reactor diameter, and the heat exchanger temperature respectively. If the objective is to minimize a cost of the form (2), following expression for Using(Z) is obtained from Equation (4): /)2g[x(z)] dC ~)g[x(z)] ko ce Re_r E
Using(Z)- T +
AHdk~ c e - ~ -4- pCpdv [ 4h 4h
t)C
v
O2g[x(z)]
~)T -------T~ -
RT 2
t)Tt)C dz ]
/)g[x(z)]( E
~)-------~, R T 2
2
T/
As an illustration following cost criterion is proposed (for which a detailed motivation and analysis can be found in [2]):
J[u] --
(1 -A)[C(L)] +A f ~ [ T ( z ) - Tin]2dz
(8)
with A [-] the trade-off coefficient between terminal and integral cost and 7)n [K] the fluid inlet temperature. The terminal cost part is a measure for the process efficiency while the integral cost part accounts for the total heat loss. This results in following singular control: AHdk0... Using(Z) -- T +
4h
e
t~e Rr
(9)
It can be easily verified that this control keeps the reactor temperature constant along the singular arc. 5.
DISCUSSION
AND
CONCLUSIONS
In this paper generic properties of time and space dependent optimal control problems are discussed and illustrated with two case studies. The first case study analyzes the optimal control in time of a well mixed fed-batch reactor and the second case study focuses on the optimal control in space of a continuous tubular reactor. Both optimization problems give rise to a Hamiltonian function which is linear in the control input (i.e., the limiting substrate feed rate for the well mixed reactor and the heat exchanger temperature for the plug flow reactor) introducing the possibility of singular control. This singular control can be explicitly computed after differentiating twice the Hamiltonian. In other words, both are singular problems of order 2. In addition, the so-called costate variables can be completely eliminated from the singular control expression, i.e., a (nonlinear) law of the state variables only is obtained (see Equations (7) and (9)). Finally, depending on the specific choice of the performance index, the singular
786 control reduces to setpoint control, for which robust (adaptive) feedback algorithms can be easily designed. More specifically, if no integral cost is specified in the case of a completely mixed reactor, setpoint control of the substrate concentration in the reactor is obtained. For the plug flow reactor case, if there is no integral cost or the integral cost is no function of the chemical's concentration (i.e., bg[x(z)]/OC is equal to zero), a setpoint control of the reactor temperature results (see again Equations (7) and (9)). At the level of optimal control in time of well mixed reactors, numerous problems have been solved thus far. At the level of optimal control in (time and) space of tubular reactors, a rather limited number of solutions to control problems has been reported in literature. In this contribution, it is clearly illustrated that many problems can be casted within the same general control framework. As a result, reported solutions and their properties for mixed reactors can be transferred/adapted to solve tubular reactor control problems.
ACKNOWLEDGMENTS Author Ilse Smets is a research assistant with the Fund for Scientific Research Flanders (FWO). Work supported in part by Project OT/99/24 of the Research Council of the Katholieke Universiteit Leuven and the Belgian Program on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister's Office for Science, Technology and Culture. The scientific responsibility is assumed by its authors.
REFERENCES 1. D.E. Kirk. Optimal Control Theory: an Introduction. Prentice-Hall, Englewood Cliffs, New Jersey, 1970. 2. I.Y. Smets, D. Dochain, and J.E Van Impe. Optimal spatial temperature control of a steadystate exothermic plug flow reactor. Part I: bang-bang control & Part II: singular control. Technical Report BioTeC, 2000. 3. J. Van Impe. Optimal control of fed-batch fermentation processes. In J. Van Impe, E Vanrolleghem, and D. Iserentant, editors, Advanced Instrumentation, Data Interpretation, and Control of Biotechnological Processes, pages 319-346. Kluwer Academic Publishers, Dordrecht- Boston- London, 1998. 4. J. Van Impe and G. Bastin. Optimal adaptive control of biotechnological processes. In J. Van Impe, E Vanrolleghem, and D. Iserentant, editors, Advanced Instrumentation, Data Interpretation, and Control of Biotechnological Processes, pages 401--436. Kluwer Academic Publishers, Dordrecht - Boston - London, 1998.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
787
Nonlinear Analysis and Control of a Continuous Fermentation Process G. Szederk6nyi a, N. R. Kristensen b, K. M. Hangos a, S. B. JCrgensen b aSystems and Control Laboratory, Computer and Automation Research Institute HAS, H-1518 Budapest, Hungary, PO box 63 bDepartment of Chemical Engineering, Technical University of Denmark, DK-2800 Lyngby, Denmark Nonlinear controllers of different type: feedback linearization with pole-placement controller and two versions of Hamiltonian controllers, together with an LQR controller are designed and compared for a simple fermentor near an optimal production operating point which is close to its fold bifurcation point. The performance analysis is based on simulation studies and on investigating the stability region of the controllers. The partial Hamiltonian controller is recommended for this case which has wide and predictable operating region, robustness and uses the measurable state variable only. 1. INTRODUCTION The analysis and control of nonlinear process systems is a challenging and emerging interdisciplinary field of major practical importance. Fermentation processes in particular exhibit strong nonlinear characteristics and are known to be difficult to control [3]. The investigated simple fermentation process is used as a benchmark problem for advanced nonlinear analysis and control techniques. In this paper different approaches are investigated for the stabilizing control of a continuous fermentation process. 2. MODEL ANALYSIS In order to be able to focus on the key issues in controller design and performance analysis of bio-reactors, the simplest possible bio-reactor, a perfectly stirred continuous fermentor is chosen as a test case. Despite of its simplicity, it exhibits the key properties which make bio-reactors difficult to control. 2.1. Nonlinear state-space model of the fermentor The dynamics of the process is given by the state space model dX
~-7 = u ( s ) x
aS
_
dt--
where
XF
v
_~(s)x + (S,~-S)F Y
V S
la(S) = lamax K2S2+S+r~
(1) (2) (3)
The variables and parameters of the model together with their units and parameter values are given in Table 1. The above model is in standard input-affine form with the centered state vector
788 Table 1 Variables and parameters of the fermentation process model X biomass concentration S substrate concentration F feed flow rate V volume SF substrate feed concentration Y yield coefficient Pmax, kinetic parameter K1 kinetic parameter K2 kinetic parameter
4 10 0.5 1 0.03 0.5
[{] [g-]t [~-] [/] [~] [-~] [g-]t [~]
x = [J? S-]r = IX - Xo S - So]r and manipulable input u = P = F - Fo
= f(x) + g(x)u
f(x) =
[ _t~(g+So)(X+Xo)+Xo) _ ] _,_ (SF-(S+So.))Fo Y
--
, g(x)=
[ (SF-(~+So))P v ]
V
with (Xo,So,Fo) being a steady-stateoperatingpoint.
(4) (5)
V
2.2. Stabilityand controllabilityanalysis
Stability Optimal production is achieved, when the outlet biomass mass flowrate (X. F) is at a maximum, which occurs at the point (Xo, So, F o) = (4.8907, 0.2187, 3.2088). In a neighborhood of this point the system is stable, but because the point is very close to the fold bifurcation point (X*,S*,F*) = (4.8775,0.2449,3.2128), this stability region is small. Stability analysis based on local linearization shows that the process is stable around the operating point but it does not give any information on the stability region. Controllability After calculating the Kalman-controllability matrix of the linearized model we find that the system is not fully controllable (in the linear sense) around the required operating point, because the controllability matrix has rank 1. Nonlinear controllability analysis based on the generation of controllability distributions is then used in identifying the singular points of the state space around which control of the system is problematic or even impossible. The local controllability distribution is generated incrementally in an algorithmic way [2] in two steps as follows. The initial distribution is Ao = span{g}
(6)
This is extended by Lie-bracket of f and g in the next step A1 =
Ao + span{[f, gl} = span{g, If, g]}
Ogf (x) - O ~f g (x) [f,g] (x) = ~-~
(7) (8)
The second step then gives the following A2 = A1 + [f, A1] + [g, Ax] = span{g,
If, g], [f, [f,g]], [g, If, g]]}
(9)
789
Singular points
At the point IX S-]T=[-Xo S F - So]r (X = 0 [~], S = SF) all the elements of A2 (and, of course, all the elements of Ao and A1) are equally zero. It means, that the controllability distribution has rank 0 at this point. Moreover, this singular point is a steady state point in the state-space. From this it follows, that if the system reaches this (undesired) point, it's impossible to drive the process out of it by manipulating the input feed flow rate. If 3~ = -Xo (X = 0 [~]) and S ~ SF -- SO (S ~ SF), A2 has rank 1. From a practical point of view it means that if the biomass concentration decreases to 0 [g] then it can't be increased by changing the input flow rate.
Non-singular points
At any other point in the state space including the desired operating point [3( S-]7"=[0 0] 7" the controllability distribution has rank 2, which means, that the system is controllable in a neighborhood of these points and we can apply state feedback controllers to stabilize the process. 3. CONTROLLER DESIGN AND PERFORMANCE ANALYSIS
Various controllers of different type: LQ controllers, feedback linearization based controllers and Hamiltonian controllers are compared here on the example of the simple fermentation process. The performance analysis of the controllers is performed by extensive simulation studies and by investigating the stability region of the controllers. For this purpose a simple Lyapunov function in the positive definite form of L(x) = lxrx is defined and the time derivative of this function is computed and analyzed as the function of the state variables. 3.1. LQ control The starting controller is the well-known LQ-controller. This case is used as a reference controller for comparison and tuning the nonlinear controllers. This type of controller is designed for the locally linearized model of the process and minimizes the cost function
J(x(t),u(t)) =
j~o~
(xT"(t)Qx(t) +ur(t)Ru(t))dt
(10)
where Q and R (the design parameters) are positive definite weighting matrices of appropriate dimensions. The optimal input that minimizes the above functional is in the form of a linear full state feedback controller u = -Kx. The results for two different weighting matrix selections are investigated.
Cheap control In this case the design parameters Q and R are selected to be Q = 12xe and R = 1. The results and the time derivative of the Lyapunov function for this case are shown in Fig. 1 (the desired operating point is at (x, s) = (0, 0) in the right subplot). As it is visible, the linear feedback stabilizes the system quite well. Expensive control The weighting matrices in this case were Q = 10.12x2 and R = 1. There were no significant differences in terms of controller performance compared to the previous case. 3.2. Local asymptotic stabilization via feedback linearization A nonlinear technique, feedback linearization is applied next for changing the system dynamics into linear one. Then, different linear controllers can be employed on the linearized system.
790 5
'
1 - ~*~'-~'~.'.':"..
I
0 a ~2S
. -
.
.
.
.
.
.
"
loi
" I= .lqlr~[h] 112
114
1,
118
2 " ~ ~ 4
Fig. 1. LQ control: simulation run and the time derivative of the Lyapunov function
We chose the substrate concentration S as output variable i.e. y = h(x)
(11)
-- x2 = S
To linearize the input-output behavior of the system, the following state feedback can be applied (see e.g. [2] )
u = o~(x) + ~(x)v ~(x) =
(12)
Lfh(x) 1 Lgh(x) ' ~(x) = Lgh(x)
(13)
where v denotes the new reference input and
Lfh(x) = ~x = i ) h(xf)
E0 1]f(x) = f2(x)
(14)
Lgh(x) = -~xxg(X i)h ) = [0 1]g(x) = gz(x)
(15)
It's easy to check that with this full state feedback the system becomes an integrator from an input-output point of view, i.e. 3) = v. From this point on, we can apply any well-known linear controller for stabilizing the system. To show the most important dynamic properties of the closed loop system we choose the simple pole placement controller v = -ky. The simulation results for feedback linearization control corresponding to k = I can be seen in Fig 2. As it is visible, the chosen output (the substrate concentration) shows a simple first order response with a time constant determined by k, and the biomass concentration also remains stable. Here we can arbitrarily prescribe the dynamics of the input-output behavior of the closed loop system but a major drawback of the method is the lack of robustness (because of the sensitivity of the functions ct and [3 [2] ).
791
5
35
,
,
0
i
-
-
-10
2S
1
-2 2
. ~
2
4
.
e
. e
.
.
"n'O[h]me
. ,2
.
.
,4
,e
2o
5"~
4
Fig. 2. Feedback linearization control: simulation results and the time derivative of the Lyapunov function
3.3. Nonlinear feedback based on the Hamiltonian model of the system The third approach is the design of a nonlinear full or partial state feedback that shapes the generalized energy function of the process.
Hamiltonian model of the fermentor Based on [1] it is easy to obtain the Hamiltonian equations of the fermentation process. For this, let us introduce the following state q and costate p variables p
_
_
q=
P2
] = [mx-mxo ] = [ox ]
Eqa]= q2 -
ms -- mso
16,
ms
[*]
(17)
where p = - V q denotes the component masses which are conserved. The Hamiltonian form of the model is written as [91 --
- V ( X o - ql)lJ(So - q2) Jr" (Xo - ql)F0 -k- (Xo - ql)F
(18)
P2
V(Xo-ql)lJ(So-q2) Y
(19)
--~
_ (SF-- ( S o - q 2 ) ) F o -
(SF-- ( S o - q 2 ) ) F
Then the coupling Hamiltonian Hi(q) used for feedback is calculated by simple integration. Since c-Jill
()H1
./)ql =. X0 . ql . , /)q2
(20)
(Se-(So-q2))
the natural output of the system Yl is written as Yl = Ha (q) = Xoqx - l q 2 - ( S F q 2 Z,
(Soq2- lq2)) Z,
(21)
792
4
, i
-50 -100
Fig. 3. Full state Hamiltonian control: simulation results and the time derivative of the Lyapunov function
Full state feedback using the whole natural output In this case the feedback consists of the entire calculated natural output, that is u = P = -klYl
(22)
where kl is an appropriately chosen positive constant gain. The simulation results corresponding to kl = 0.5 are visible in Fig. 3 indicating a good and balanced performance.
State feedback using only a part of the natural output It is known from the theory of Hamiltonian systems that the definiteness of the time derivative of the storage function (i.e. the stability of the closed loop system) may be influenced by using only some part of the natural output for feedback. Since the biomass concentration is quite difficult to measure in practice we chose the substrate concentration only for feedback 1 2 P = -k2"--(SFq2- (Soq2- ~q2))
(23)
This controller gives a similarly good performance as its full state counterpart. Moreover, the performance does not depend heavily on the controller gain either. A novel tuning method of the feedback gain for both cases is presented elsewhere [4].
Acknowledgement This research is partially supported by the Hungarian Research Fund through grant No. T032 479. REFERENCES 1. K.M. Hangos, J. Bokor, and G. Szederk6nyi. Hamiltonian view on process systems. AIChE Journal, accepted, 2001. 2. A. Isidofi. Nonlinear Control Systems. Springer, Berlin, 1995. 3. Ch. Kuhlmann, I.D.L. Bogle, and Z.S. Chalabi. Robust operation of fed bartch fermenters. Bioprocess Engineering, 19, 53-59. 1998. 4. G. Szederk6nyi and K.M. Hangos. Hamiltonian control of a nonlinear fermentation process. In: European Control Conference, submitted, 2001.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
793
Nonlinear process model-based self-optimizing control of complex crude distillation column A.Yu. Torgashov Process Control Laboratory, Vladivostok State University of Economics and Service, 690600, 41 Gogolya street, Vladivostok, Russia, E-mail: [email protected]
The control system based upon a nonlinear process model is presented in this paper. The self-optimizing control strategy is used for the maintenance of the optimal steady-state of a complex crude distillation column. The optimal distillation operation under unmeasurable feed composition disturbances has been considered. It has been also suggested that the stabilization of the certain relationship between pumparound streams in the sense of vector criterion provides the optimal operation of the crude distillation tower within the minimum and acceptable losses. 1. INTRODUCTION The complex crude distillation column (crude distillation tower) is a basic technological unit in the petrochemical industry for benzine, kerosene, diesel oil and mazut production. The maintaining of its optimal steady-state is very important task because at that the main economical benefit can be reached. The product quality control is the difficult problem due to the multivariable process nature and interactions between the control loops. At present, more emphasis is placed on this subject in the several papers [1-3]. Usually the principal obstacle attendant to the process control problem is the discrepancy between usable model and real process. The significant nonlinear distillation properties also hamper successful control performance using linear controllers (PI, PID etc.). In additional, the large dimension of the dynamic distillation model eliminates an application of such modem nonlinear control methods as feedback linearization, integrator backstepping and passivity based-control [4]. Thus, for the crude distillation tower the overall number of the nonlinear algebraic-differential equations is approximately equal to 3000. The using of such dynamic model is impossible under analytical nonlinear controller design. It is the author's opinion that the nonlinear process model-based (NPMB) control law [5] is more efficient method for complex distillation columns regulation. In that case the physical-chemical process nature (phase equilibrium, material and enthalpy balances equations) is taken into account in the control scheme by means of rigorous steady-state distillation model. In the previous investigation, the advantages of NPMB control strategy, as compared with another modem control modes, were found [3,5-6]. It is often necessary to provide the optimal process operation if the some degrees of freedom may be detected for optimization procedure. Among the recent techniques for solving such problem the self-optimizing control might be noted [7]. The method is simple from the practical point of view. The basic idea consists of the finding the feedback variable
794 characterizing the optimal process operation with minimum and acceptable criterion losses. In the previous work in the area of self-optimizing control the single criterion analysis was considered, whereas the complex process is often described by the vector criterion. It may be that the feedback variable does not reflect the optimal process operation in the sense of another criterion 9 This paper deals with the feedback variable reflecting the optimal steadystate of a crude distillation tower according to vector criterion. It was found that the losses of vector criterion are in several times lower compared with the losses of the individual criteria optimal values 9 2. P R O B L E M FORMULATION
2.1. Complexcrudedistillationcolumn Figure 1 shows the studied crude distillation column like the pipestill examined in the work [3]. The main purpose of this splitter is the crude separation on the products as benzine (B), kerosene (K), diesel oil (D). The crude is represented by 32 pseudo-components. The physical-chemical properties of these pseudo-components, including the coefficients for phase equilibrium, equation, enthalpies of the pure components in liquid and vapor phases with the nominal regime parameters of the column are presented in [8]. The products quality is characterized by the withdrawal temperatures of the key fractions. These temperatures are stabilized on the values located at the true boiling curve of the feed on 85 % before the end point temperature of the corresponding fraction.
2.2. The NPMBcontrollaw for crudedistillationcolumn In this work the NPMB control strategy is realized in the generic model control framework - ~ p(w)ss [5] 9 According to this control method the withdrawal temperatures of the key fractions T
are calculated from the following equation
= Tp + Kp, 1
-Tp
Kp, 2
•7.•_
i ~ ~V
R
Qvl~x-~ QP2~ ,
,
584, 582, ~" 580 ~. 578 576~ 0
-~K $2 "~
~
(I)
,
Slst ~e~
F steam
-Tp
0
N
am
M "r" ~ D
Fig. 1. Complex crude distillation column.
800 6 0 0 ~ W1(F,h'nol/h)
200
800
V V V
B~ (Kmol/h)
Fig.2(a).Surface of the first criterion.
795 Table 1 Product quality control system configuration for crude distillation tower Manipulated variable Controlled variable B T1w K D
T2 w Ta w
where 7' .p(w)ss
- set point of the withdrawal temperature; K p,1, Kp,2 "tuning coefficients; t-
current time; p=1,2,3 9 By the calculated values of -Tp(w)ss the manipulated variables are obtained with the help of inverse nonlinear steady-state distillation model. It has its origins in the Newton-Raphson method application and will be described below. The control system configuration is illustrated in the Table 1. 2.3. Criteria a n d d e g r e e s o f f r e e d o m s e l e c t i o n The crude distillation tower has two degrees of freedom in the form of flow rates WI and W2 (fig. 1) under the products purity specification as
265 1
380
800" ~ 0 0 4 0 Wl(Kmol/h) 200
0 W2 800 (Kmol/h)
Fig.2(b). Surface of the second criterion 9 2800 2600
I~ (Kmol/h)
200 ~
600 800 ~ (KmoVh) Fig.2(c) 9 Surface of the third criterion. X 109
~"2400
~ 2200 2000
1800 1600
~
~
800W16 400
(Kmol/h)
200 200
800
600~ (Kmol/h)
Fig.2(d). Surface of the fourth criterion.
800 6
~
W1 400 ~ (Kmol/h) 200
800
x,~
200
400 I~600 (Kmo]/b)
Fig.2(e) 9 Surface of the fifth criterion.
796
Tiw(sp) = (TiW~ef , i=1,2,3
(2)
where ref= reference value of the key fraction withdrawal temperature. The temperatures of the pumparound streams are constant. The flow rates W1 and W2 lie in the range from 100 Kmol/h to 800 Kmol/h. Thus, five optimization problems (five criteria) can be considered as 1. To maximize the benzine flow rate: Ji(W1,W2)=B --->max 2. To maximize the kerosene flow rate: J2(W1,W2)=K --->max 3. To maximize the diesel oil flow rate: J3(W1,W2)=D --->max 4. Minimization of the reflux flow rate: Ja(W1,W2)=R --->min 5. Minimization of the all condensers duty: Js(W1,W2)=Qc+Qpl+Qp2 --->min Fig.2 illustrates the surfaces of the selected criteria. It is obvious that the optimal criterion values will drift under the action of the unmeasurable feed composition disturbances. Therefore, it is necessary to find the structure of the control system which is insensitive to such uncertainty. 3. DEVELOPMENT OF THE SELF-OPTIMIZING CONTROL SCHEME
3.1. Control design under unmeasurable feed composition disturbances Table 2 shows the six basic cases of the unmeasurable (uncontrolled) disturbances in the form of the feed composition redistribution. The composition of each key fraction has a deviation of 20% from the nominal value. The optimal magnitudes of the criteria and manipulated variables are listed in Table 3 for these cases. The last column in Table 3 corresponds to the optimal vector criterion values in the relative units calculated in a similar manner as in the work [9]. Fig. 3 demonstrates the multiobjective optimization problem solution for one of the cases from Table 3. Finally, the intersection of the two contradictory criteria surfaces defines that solution. Notice that the optimal manipulated variables for 1, 2, 5 criteria are insensitive to uncontrolled feed composition disturbances action by virtue of the non-convexity of such criteria surfaces. Though, the optimal values of the control actions drift for third and fourth criteria (Table 3). It was found that the least changes of the optimal control actions are observed under the stabilization of the ratio W~/W2 on the optimal value in the sense of vector (fig.3). The maintaining of ratio W1/W2 as on the fig.3 provides the criterion at the ~~ least losses for the vector criterion as compared with the single optimization problem. This suggests the use of the self-optimizing control scheme integrated with the NPMB control strategy (fig.4). 3.2. Inverse steady-state distillation model for control algorithm The important element of the proposed control system (fig.4) is the inverse process model (IPM). In this section the fundamentals of the steady-state distillation calculation will be considered under specified withdrawal temperatures of the key fractions (2). The vector of manipulated variables u is determined by using the measurable outputs vector y in one procedure of the steady-state computing. Otherwise, it is necessary to solve the inverse problem y-->u without using optimization procedure as it was fulfilled in the previous work (see for example [10]). Here the Newton-Raphson method application is used as the basis in IPM and was originally discussed in [11 ]. The major modifications of that algorithm will be presented below.
797 Table 2 Feed composition redistribution Case Benzine fraction Kerosene fraction
Diesel oil fraction
1
+20%
-10%
-10%
2 3 4 5 6
-20% -10% +10% -10% +10%
+10% +20% -20% -10% +10%
+10% -10% +10% +20% -20%
Table 3 Optimization results under uncontrolled feed composition disturbances influence
J1, J2, J5 Case
W1 100 100 100 100 100 100
1 2 3 4 5 6
W2 100 100 100 100 100 100
Jl Opt 586 582.2 583.1 584 583 584
J2~ 264.3 266.2 267.1 262.1 264.1 262.4
J3 Js~ 9 5.5266 5.5267 5.5266 5.527 5.5268 5.5267
WI 105 100 100 105 100 105
W2 800 800 795 790 795 795
~opt
J4 J3~ 384 387.2 386.1 387.1 387.5 384.1
W1 295 290 305 300 295 300
W2 800 800 800 795 795 795
J4~ 1436.5 1433 1433 1434.9 1433.8 1435.3
0.572 0.571 0.568 0.57 0.571 0.574
For the studied distillation column (fig. 1) the input and output vectors are determined as u = [B K D W1 W2 ]; y=[Tl w T2TM 7"3w]. (3) uc uf The vector u in (3) consists from the two sub-vectors Uc and uf. In accordance with the work [10] the matrix component balance equations are formulated as well as the discrepancy functions by phase equilibrium (Fj) and enthalpy balances (@) forj-th separation stage. 1
~
W2)n~ g~
0.8
.dl norin
a0o W2(Kmol/h) Fig.3. Multiobjective optimization problem solution representation in the relative units.
1 (~/W2[)~
I
Fig.4. Self-optimizing system structure with NPMB control law. R(s) - transfer matrix of the robust controller.
798 But in the present paper the functions Gj are given by using the constant composition method as it was pointed by the author in the previous work [12] because the stable convergence of the computational algorithm was observed. In order to take into account the products purity specification (2), the additional discrepancy functions are proposed in the following form M~:Ti w- TiW(sP)---~O. i=1,2,3 (4) The introduction of the equations (4) is responsible for the appearance of an augmented matrices equations for every trial by the Newton-Raphson method JAX:-W, (5) where AX =[A(L/V)I...A(L/V)N,...ATI...ATN, Auc,1, Auc,2, AUc,3]T;W=[F1...FN.... G1...GN, M1, M2, M3]T; J - the augmented Jacobian matrix; Lj, Vs.- total liquid and vapor internal flow rates, respectively; N- total number of trays. With regard to the control system illustrated in the fig.4, the robust controller with transfer matrix R(s) is present before IPM. Its structure was determined using the known method of robust control design (~t - synthesis) because the block IPM+Process may be analyzed as the linear part with the nonlinear gains compensation by IPM. 4. CONCLUSION In the present paper the self-optimizing control system based upon the IPM has been proposed. Its main distinction consists in the indirect provision of the optimal crude distillation tower operation with the minimum vector criterion losses as compared to single criterion optimization. The proposed modifications of the standard Newton-Raphson method application to a distillation calculation give the possibility to reduce the computational efforts required for determination the vector of the manipulated variables under products purity specification. REFERENCES
1. L.H. Veland, J. Hoyland, C.R. Aronson, D.C. White, Hydrocarb. Proc., 73 (1987) 117. 2. K. Muske, J. Young, P. Grosdidier, S. Tani, Comput. Chem. Eng., 15 (1991) 629. 3. C.-B. Chung and J.B. Riggs, AIChE Journal, 41 (1995) 122. 4. M. Torres and R. Ortega, Proc. 14th IFAC Congress, Vol. E (1999) 403. 5. P.L. Lee (Ed.), Nonlinear Process Control: Applications of Generic Model Control, Springer-Verlag, Berlin, 1993. 6. H. Subawalla, V.P. Paruchuri, A. Gupta, H.G. Pandit, R.R. Rhinehart, Ind. Eng. Chem. Res., 35 (1996) 3547. 7. S. Skogestad, Journal of Process Control, 10 (2000) 487. 8. A.Yu. Torgashov, PhD Dissertation, Vladivostok State University of Economics and Service, (2000). (in Russian) 9. Yu.K. Mashunin, Journal of Computer and System Sciences International, 38 (1999) 421. 10. M. Fikar, A.M. Latifi, F. Foumier, Y. Creff, Canadian J. of Chem. Eng. 73 (1998) 1110. 11. C.D. Holland, Fundamentals of Multicomponent Distillation, McGraw-Hill, New York, 1981. 12. A.Yu. Torgashov and V.P. Krivosheev, Proc. 14th International Congress CHISA'2000, Set 2 (2000) 96.
European Symposium on Computer Aided Process Engineermg - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
Closed loop controllability analysis Application to distillation column design
799
of
process
designs:
Rob L. Tousain t and E Michiel Meeuse * t Delft University of Technology, Mechanical Engineering Systems and Control Group, Mekelweg 2, 2628 CD Delft, [email protected] Delft University of Technology, Department of Chemical Technology, Julianalaan 136, 2628 BL Delft The Netherlands, [email protected] This paper presents a new approach towards the integration of process design and control. The approach compares alternative process designs based on the optimal closed-loop performance in the presence of stochastic disturbances. The most important contribution is that a clear relation is established between the degrees of freedom in the process design and a closed loop performance measure. The approach is illustrated with a case-study of a distillation column. 1. INTRODUCTION Traditionally, the design of a process control system is postponed until the process design is completed. Nowadays it is broadly accepted that this is not a desirable situation since this sequential design approach can lead to processes that are difficult to control. As a consequence, different ways to take controllability issues into account in the process design stage have been developed and are described in literature. These methods can be classified according to Lewin (1999) in methods which (i) enable to screen alternative designs for controllability, and methods which (ii) integrate the design of the process and the control system. In the first approach, the controllability of alternative designs is tested such that alternatives that might have acceptable steady-state economics but poor control performance can be rejected in an early stage of the design. The controllability is quantified using indices like the Relative Gain Array (RGA) and singular value decomposition based indices. A brief overview of available indices is presented by Lewin (1999). This method is rather easy to integrate in existing design procedures, however the indices are often calculated based on steady-state data only, and the relation between the indices and the closed loop performance is often unclear. The second class of methods is based on the simultaneous optimization of both the process and the controller, which are parameterized by means of a so-called superstructure. Alternative designs can now be compared based on, for example, the Integrated Squared Error (ISE) for specific disturbance scenarios (see e.g. Schweiger and Floudas, (1997)). Because of the computational complexity the application of this approach is limited to small case-studies with specific assumptions on the control system to be applied. Schweiger and Floudas (1997) consider for instance only SISO PI-controllers with an a priori specified pairing. An additional limitation is that the insight gained from such an approach is limited. Finally, stochastic distur-
800 bances are often not considered. To overcome the limitations of these existing approaches we propose a new method which compares alternative designs based on the optimal closed loop performance, taking into account disturbances and measurement noise. The method is based on the approach presented by Tousain et al. (2000) for the optimal sensor selection in a High Density Polyethylene reactor. This paper is organized as follows. The next section describes the modeling of the plant and the disturbances. Then Section 3 presents the proposed controllability index which is derived using Linear Quadratic Gaussian control. The use of this index will be demonstrated in the design of a distillation column as a case study in Section 4. Finally some conclusions will be drawn. 2. PLANT AND DISTURBANCE MODELS The process model In this study we will assume that a dynamic model of the new process is available in the design stage. For the sake of generality, we consider the broad class of processes of which the behavior can be described by a system of Differential Algebraic Equations (DAE):
.iCp= f (Xp, U,y, p), O=g(Xp,U,y,p),
(1)
where Xp E Rnxp, y E Nny, and u E R n" are respectively the state, algebraic and input variables, p E P = P1 x P2 x ... x Pnp are the np design parameters, where Pi, i = 1,... ,np are the parameter sets. These sets can contain real-valued or integer numbers. An example of a real-valued parameter is a column diameter, whereas an example of an integer parameter is the number of stages in a column. The operation of the plant is subject to operating constraints, 0 _ S(xp, u, p), which are assumed to be chosen such that the steady state constraints (0 = f (xp, u, y, p), 0 = g(xp, u, y, p) , 0 <_ S(xp, u, p) ) define an unique steady state for each p E P. Process disturbances and measurement noise Often, some a priori knowledge on the sources and characteristics of disturbances and measurement noise will be available. To assess their effect on the process operation (and more important: how process design influences this !) we should include this knowledge in the design stage. We deal with disturbances and noise in a stochastic framework. This is in contrast to e.g. Seferlis and Grievink (1999) where only static disturbances are considered. The plant model can be extended with general noise models as follows
0 = Ym =
g(xp,xa,u,y, wa,p), YmY + Wn,
(2)
where [Ad, Bali is the realization filter for the disturbances, Xd are the states of this filter, Wd and Wn are Gaussian white noise stochastic variables with respectively covariance matrices Rd and R n and Ym are the measurements. The behavior of the plant in or close to its steady state operating point(s) will be described using linearized models. In the remainder of this paper we will use the standard state-space
801 notation for these models: AYe, = a(p)Ax + B(p)Au + Bw(p)wd, Aym -- C(p)Ax + Wn,
(3)
where Ax = [Axp,Axd]r, Au and Aym are respectively the deviations of the states, the inputs and the measured outputs from their steady state values, and the system matrices [A,B/Bw, C] are obtained through linearization of (2). They are functions of p only because each choice of p was assumed to uniquely define a steady state. In the remainder the dependence of p will be omitted from the notation. 3. A CLOSED LOOP PERFORMANCE MEASURE The general idea of all design methods which take process control into consideration is that the plant should be designed such that, using some kind of controller, we can achieve or optimize some kind of operating performance. The two main problems are (i) what are the assumptions for the controller, (ii) how is the operating performance quantified? To circumvent these problems we base our analysis on optimal control. The big advantage of optimal control is that, given a model of the plant and the control objective function, the controller itself is unambiguously defined, so we need not make any restrictive assumptions with respect to the controller design. Further, using optimal control guarantees that an upper limit of the practically achievable control performance is attained. Because we want to account for the effect of stochastic disturbances and measurement noise on the closed loop behavior of the process, the control objective we consider is the weighted (closed loop) variance of the process outputs and inputs. Note that this control objective ignores the process economics: ideally, the analysis of the closed loop variance as proposed here should be combined with an investigation of the steady state economics of the process design. How this can be done in a pragmatic fashion is demonstrated for the case study in the next section. Systematic ways of doing this are subject to our current research. How we can compute the closed loop variance becomes clear from the theory of Linear Quadratic Gaussian Control.
Linear Quadratic Gaussian Control In the LQG control problem we seek the control u(t) which we should apply to systems (3) to minimize the following performance measure: J = E ( rlim - ~ Tl f t r (Ax r QAx + AurRAu)dt } ,
(4)
where E denotes expectation, and Q and R are weighting matrices of appropriate dimensions. The solution of the LQG control problem is very elegant. The optimal controller is found be combining the optimal state observer (Kalman filter) with the optimal state feedback controller for the deterministic Linear Quadratic Regulator problem (with weightings Q and R), we refer to Kwakernaak and Sivan (1972) for the details. The Kalman filter gain is given by L - PC~mRn 1, where P is the solution of the estimator Algebraic Riccati Equation (ARE): 0 -- A P + P A _ pCTRnlCm P + BwRdB T w.
(5)
802 The optimal state feedback gain is given by G = R-1BTM where M is the solution of the regulator ARE: O= MA +ATM-MBR-1BTM+Q.
(6)
The optimal cost is then given as follows T r (p) : tr {BwRdBwM + O ~RoP }.
(7)
Closed loop controllability index Expression (7) is a closed loop performance measure for the plant's operation. We can compute it for different choices of p and it has thus an unambiguous interpretation as a controllability index. Clear insight in the dependence of the closed loop variance on the design parameters can be obtained via a computation of J* over a grid in the parameter space. The dimension of the search space P will generally be not very large, hence there is little danger of the number of grid points blowing up unacceptably. The function evaluations are rather expensive (in terms of calculation costs): they involve the solution of the model equations to find a steady state, and the solution of two algebraic Riccati equations, so a detailed picture of the closed loop properties of the designs may be attainable only at a significant computation cost.
4. EXAMPLE: BINARY DISTILLATION We now illustrate our approach with a binary distillation column as case study.
Process model The process model is based on the well known Column A, introduced by Skogestad (1997). The feed and product specifications for all alternative designs were based on those described by Skogestad (1997). The following modifications were made to the model: 9 the feed is introduced above the feed stage instead of on the feed stage, 9 perfect control instead of PI control is applied to the reboiler and condensor hold-up, 9 the nominal hold-up on the stages is modeled proportional to the liquid flow to take the effect of the diameter of the column into account. The design parameters that are varied are the number of stages and the feed stage location. The number of stages was varied from 31 to 51. The feed stage location was varied from the centre stage minus 5 stages to the centre stage plus 5 stages. The base case design has 41 trays and the feed tray is in the centre (tray 21). This is a traditional design with the reflux ratio 1.35 times the minimum reflux ratio. The feed is saturated liquid. The controlled variables are the top and bottom composition. The manipulated variables are the vapor boilup and liquid reflux stream. The total annual costs of all designs are calculated with the equations and coefficients presented by Luyben and Floudas (1994). This economic model computes the total annual costs as the sum of the capital and operating costs.
803
Disturbance model
Simultaneous disturbances in the feed flow rate and feed composition were considered with a variance of 0.044 and 0.011 respectively. The shaping filters for both disturbances were chosen low-pass second order with cut-off frequencies 0.016 Hz. The measurement noise on the composition measurements was assumed to be white noise with a variance of 0.0001. Results
Figures 1 and 2 show respectively the closed-loop optimal variance and the total annual costs for all designs considered. From these figures it can be concluded that the optimum design from a closed loop controllability point of view has less stages then the nominal design. Note that this is in line with the results presented by Meeuse et al. (2000), where an analysis is done based on irreversible thermodynamics. The optimum from a economic point of view lies around the base-case design. Obviously there is a trade-off between closed-loop performance and total annual costs. In order to obtain more insight in this trade-off, Figure 3 shows the closed-loop variance against the total annual costs for all designs. The solid lines in the figure are lines with the same feed stage location. The dotted line is the Pareto optimum. Apparently the lines with a feed stage around the centre lie very close to the Pareto curve. One can hence conclude that the feed stage position that is optimal with respect to both steady state economics and closed loop variance is in the center of the column. The trade-off between steady state economics and closed loop variance hence remains to be made by a choice of the number of stages only.
Fig. 1. Closed-loop variance as a function of design parameters.
Fig. 2. Total annual costs as a function of design parameters (the right picutre is a zoomed version of the left picture).
5. C O N C L U S I O N S A new approach has been introduced that can assist in the design of controllable processes. The most important features of this approach are that it can deal with stochastic disturbances and that the analysis is based on the optimal closed-loop behavior of the system. An inherent limitation of the method is that it can only deal with linear models, however the behavior of process systems around an operating point can often be described accurately with a linear model.
804 1 . 0 E - O 4
--
9.OE-O5
-
8.OE-O5
-
5 trays
below
2 trays
below
1 tray
below
centre centre centre
centre o
3 trays
above
centre
5 trays
above
centre
o:I
" 7.OE-O5
-
6.OE-O5
pareto ,
90000
1 OOOOO Total
, annual
11 OOOO costs
optimum
, 120000
Fig. 3. Total annual costs versus the closed-loop variance.
REFERENCES
Lewin, D.R. (1999) Interaction of design and control, presented at the 7th IEEE Mediterranean conference on control and automation, Haifa. Luyben. M.L. and Floudas, C.A. (1994) Analysing the interaction of design and control -1. A multiobjective framework and application to binary distillation synthesis. Comp. Chem. Eng. 18(10):933-969. Kwakernaak, H. and Sivan, R. (1972) Linear optimal control systems, Wiley New-York. Meeuse, F.M., Samyudia, Y. and Grievink, J. (2000) Design of controllable processes using irreversible thermodynamic structures, Presented at the 2000 Annual AIChE Meeting, Los Angeles. Schweiger, C.A. and Floudas, C.A. (1997) Interaction of design and control: optimisation with dynamic models. In: Optimal control: theory, algorithms and applications, eds: Hager, W.W. and Pardalos, P.M.Kluwer Academic Publishers B.V.,p. 388-435. Seferlis, P. and Grievink, J. (1999) Plant design based on economic and static controllability criteria, Proceedings of the 5th FOCAPD conference, Breckenridge. Skogestad, S. (1997) Dynamics and control of distillation columns. Trans IChemE, 75, Part A, Sept. 539-562. Tousain R.L., Hol, C.W.J. and Bosgra, O.H. (2000) Measured variable selection and nonlinear state estimation for MPC, with application to a high-density poly-ethylene reactor, Proceedings of PCI 2000 conference, Glasgow, Scotland.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
805
Online optimization integrated with online analyzers and multivariable predictive controller in industrial airlift reactors Alexandre Tresmondi a, Aimar Domingues b, Rubens Maciel Filho c Rhodia Brasil, Centro de Pesquisas de Paulinia, CP 07, 13140-000, Paulfnia, SP, Brazil e-mail: alexandre.tresmondi @br.rhodia.com b Rhodia Recherches, 85, Av. des Fr6res Perret, 69192 Saint-Fons, France c Faculty of Chemical Engineering, State University of Campinas, CP 6066, 13081-970, Campinas, SP, Brazil a
An online optimizer was developed to industrial airlift reactors in a phenol plant. The optimizer is integrated with a multivariable controller and online composition analyzers. The objective is to reduce the by-products formation during the cumene oxidation. Using an online fit model, a SQP subroutine calculates optimal composition values to the reactors and these values are sent to the multivariable controller as new setpoints. The average by-products reduction resulted in 4.7% (mass fraction), which means an important contribution to the annual plant productivity. 1. INTRODUCTION This work presents the results of an online optimization in cumene oxidation reactors in a phenol unit of Rhodia Brazil, working since September 1999. The reaction of cumene oxidation is carried out in a series of four airlift reactors. The optimizer works in closed loop with a multivariable controller that keeps product concentrations and temperatures in the optimum setpoint values, in order to reduce by-products formation. The typical production of phenol from cumene involves the following main steps: a) cumene oxidation with oxygen (air) to form cumene hydroperoxide (CHP); b) cumene hydroperoxide concentration; c) cleavage (decomposition) of cumeme hydroperoxide in an acidic medium into phenol and acetone; d) neutralization of the products of the acid cleavage; e) phenol distillation and purification. Wong et al. (1998) present a simplified block flow diagram of a typical cumene to phenol plant. Some impurities and by-products are formed during the oxidation. Major by-products formed in the oxidation reactors include dimethylphenylcarbinol (DMPC), acetophenone (ACPH) and Dicumyl Peroxide (DCP). For a given total reactor volume and feed rate, increases in cumene conversion result in increased by-products generation. Messsina et al. (1983) described the mechanism of by-products formation in cumene oxidation. According to this mechanism it is almost impossible to prevent their formation, but by-products can be reduced by optimized choices of reactors temperature (or CHP concentrations). Many authors have presented the benefits and difficulties of using real-time optimization (Korchinski, 1995; Friedman, 1995; White, 1998). In this work, many aspects of these references were considered to supply to the plant operators a robust system, easy in operation and maintenance.
806 2. OBJECTIVE FUNCTION The cumene oxidation optimizer uses a rigorous non-linear model (Camarasa et. al, 2000) of the process where the kinetics of the main reactions are represented. The following objective function was defined in order to obtain a minimum of by-products, subject to the process model and constraints: objective function: fix) = E [%DMPC, %ACPH, %DCP] --> min.
(1)
constraints: g(x) = [TI, T2, T3, T4, CHP production] (Ti, i=l ..... 4 are the reactors highest temperatures)
(2)
By-products concentrations (DMPC, ACPH and DCP) are not measured in the process, but they are model results. CHP concentrations are both estimated by the model and measured online by Near Infrared analyzers (NIR). CHP online measurements are used in the online model fit step. CHP production is represented by the CHP mass fraction at the last reactor. The optimization algorithm used to solve the problem defined by equations (1) and (2) is the "Successive Quadratic Programming (SQP)". 3. OPTIMIZATION STRATEGY The integration among the online optimizer, online analyzers and multivariable controller is shown in Figure 1 (Tresmondi, 2001).
Fig. 1. The control room architecture and steps executed by the online optimizer. The implementation of the stages described in Figure 1 is sequential, and if one of the items is not satisfied, the optimizer enters in stand-by. Considering the dynamics of the cumene oxidation process, if the optimization is not performed in a certain number of
807 intervals, the plant doesn't shown great deviations from the optimum values. In cases where it is necessary to supply continuous setpoints Dormer and Raynor, 1998 present an alternative approach where these steps could be done simultaneously. However, the results of this work will show that the proposed procedure, based on the sequential approach, appears to be very robust and stable, which are important requirements for industrial implementation. Also, it has to be pointed out that most of the published industrial case studies make use of simple linear models coupled to linear methods to find out the desired optimal conditions. In the proposed approach, both the model and the optimizer are non-linear. The optimizer performance is monitored through the registration of new optimum setpoints (generated by the full deterministic reactors model and SQP algorithm) and by estimates of by-products formation, before and after optimization. These registrations allow data exploration along the time and they are useful for the confirmation of the gains evaluated in the project stage. Optimizer availability is also monitored, through the number of successful actions of the optimizer algorithm. 4. THE STEADY STATE CRITERIA The optimizer was conceived to work in two situations: steady state or programmed production change (ramp). In the programmed production change, steady state tests are inhibited. In these cases, the optimizer will follow the production change, creating an optimum trajectory of "quasi-steady states" and the process will be conduced, after some hours, to a new optimum steady state. In constant production, it is expected that the multivariable controller will keep the unit in stable operation. In these cases, increase or decrease tendencies of the controlled variables are considered not suitable situations. The optimizer will remain in stand-by until the effects of these disturbances are annulled by the multivariable controller. To determine if the process is in steady state and ready to receive the optimization data, is used the inversion test of values in a sequence (Himmelblau, 1970). According to this test, if in a series of n measures a certain number is followed by a smaller number, it is said there is an inversion. In the next step, it is established a criteria which, depending on the value of the sum of inversions, the data set is considered in steady state or not (showing tendencies). In Figure 2 an example is presented for one of the composition variables that define the process. Some associations are made between inversion test values and situations of non-stationary state. The horizontal lines represent the range inside of which the variable is considered stationary. Small variations in the controlled variable are considered normal in the controller range. 5. THE O P T I M U M T R A J E C T O R Y After a set of optimum values (CHP mass fraction optimum setpoints) has been obtained, it is necessary to interpret these data and to judge if they can be transferred as new setpoints for the DCS. First the optima are compared within an expected operational range and also compared with the current online analyzer values. If the optima are found to be correct, it is necessary to define how they will be send to the DCS and to the multivariable controller. Because of safety considerations, it is not permitted that the new setpoints be very different from the current values of the process variables. This is made by defining an optimization trajectory, where the optimum solution is implemented in small steps, so that the restrictions
808 of the unit are not violated. It is worthwhile to be mentioned that this safety procedure developed in this work allows the system to run in a robust way in spite of possible numerical failure. 1500 1400 t 1300 1 1200 1100 m looo 9 .=_, r 9 900 I...800 ,-o ~ 700 600 50O ...,,,11. 400 300
q~o ooo
o~
9 Inversion Test
2OO
C H P _ P V reactor 4
100 0 18:00
9
i
i
u
22:00
02:00
06:00
,
10:00
i
14:00
!
18:00
time
Fig. 2. The steady state criteria test in CHP concentration. Associations between inversion test values and non-stationary conditions. After obtaining the optimum values (OPT), the current controlled variables setpoints (SP) are compared with the current values of the process variables (PV). Considering these differences and using a maximum allowed variation of setpoints (Dmax), the new setpoint (NSP) for each reactor will be correspondent to one of the following cases: NSP = SP (no modification is made) NSP = SP + or- Dmax (incremental change, in the direction of the optimum trajectory) NSP = OPT (optimal solution is admitted) 6. INTERFACE W I T H THE MULTIVARIABLE C O N T R O L L E R The Paulinia Phenol Plant cumene oxidation optimizer was conceived to work in closed loop, sending setpoints automatically to the level of the multivariable control without operator intervention. The operator can turn on or off the closed loop between automatic optimum setpoints and the multivariable controller. In the case of being disabled, the optimizer program continues running, registering the process variables and calculating optimums values that can be used as open loop setpoints. The multivariable controller doesn't interpret the new setpoints, but just implement them through changes in the manipulated variables. To avoid violations of the plant restrictions, the optimizer interprets its results (the optimum trajectory). The controller has restrictions in the values of the manipulated variables (limits and maximum rate of change) as well as limits for setpoints acceptance. In Figure 3, optimizations results (CHP_SP) and the answers of the controlled variable (CHP_PV) are presented for one (the third) of the four reactors.
809
7. R E S U L T S The gains obtained with the online optimization were verified through the optimizer historical, which registers the production of by-products evaluated by the model. In Figure 4, is shown the difference between operation with and without online optimization for the case of DMPC formation. It is shown that there was a reduction in the DMPC average production (x2 < xl) and also a variability reduction (~l < ~2). Cumene hydroperoxide production was constant during this period. The same behavior was verified to the other two by-products (DCP and ACPH), in terms of reduction in production and variability.
CHP_PV reactor 3 - + - C H P _ S P reactor 3
r
00:00
i
12:00
~
r
00:00
i
!
12:00
i
i
00:00
r
i
12:00
i
00:00
time
Fig. 3. Setpoints changes after optimization intervals and multivariable controller results. The average optimizer availability, registered after one year in operation, is 80%, for 98% of multivariable controller availability. The events that cause the 20% of non-availability are distributed in 10% due to non convergence in the parameters of the online model fit, 8% due to non convergence of SQP and 2% due to incompatibilities in data acquisition. However, when the optimizer doesn't send new setpoints to DCS, it remains in stand-by for some minutes and, if there are not hardware problems, it is never necessary to initialize again. 8. C O N C L U S I O N S The proposed system for the online optimization of cumene oxidation reactors shown to be robust and it was observed a reduction and stabilization in the by-products formation. The variability reduction can result in benefits for the unit operations after the reaction step (cleavage and purification). The total reduction of by-products was 4.7% (mass fraction). The theoretical by-products reductions were confirmed by plant annual cumene consumption control. The stabilization of CHP and by-products concentrations also can permit a production increase in the oxidation reactors, by operating closer to the restrictions. An important consideration in any control project is the previous evaluation of the economic gains. After,
810 these gains should be continuously monitored, so that system availability stays high along the time. The proposed procedure for on-line optimization makes use of a rigorous and non-linear model, with real time parameter fittings, as well as allows the evaluation of process data quality, and the consideration of the plant restrictions. This was considered essential to the success of a real time optimization system in an industrial environment.
_
With optimization
[.
(o2, x21
0 12. t~
.,q,
v
Without optimization (~1, xl)
[ t~l / ~2 = 1.846 X 1 / X 2 -" 1 . 0 1 2
,
J
,
,
;
;
,
,
(kg/h)
----DMPC
,
|
,
,
,
,
07/07 08107 09/07 10/07 11/07 12/07 13/07 14/07 15107 16/07 17/07 18/07 19/07 20/07 21107 22/07
time
Fig. 4. Stabilization and reduction of DMPC formation after on-line optimization. 9. REFERENCES
1. Wong, E.W., J.E. Wallace and T.R. Wilks (1998). "Cumene to phenol technology". Hydrocarbon Processing, December, pp 42-47. 2. Messina, G., L. Lorenzoni, O. Cappellazzo, A. Gamba (1983). "Side reactions and related by-products in the phenol/acetone process". La Chimica e l'industria, V.65, N. 1, pp 10-16. 3. Korchinski, W.J. (1995). "Otimiza9~o em grande escala bem-sucedida de refinarias". P&Q/C&I, August, pp 36-39. 4. Friedman, Y.Z. (1995). "What's wrong with unit closed loop optimization?" Hydrocarbon Processing, October, pp 107-116. 5. White, D.C. (1998). "Online optimization: what have we learned?" Hydrocarbon Processing, June, pp 55-59. 6. Camarasa, E., Meleiro, L.A.C., Carvalho, E., Domingues A., Maciel Filho, R., Wild, G., Poncin, S., Midoux, N. et Bouillard, J., "A Complete Model for Oxidation air-lift Reactors", Chem. Eng. Sci., accepted, (2000). 7. Tresmondi, A. (2001). PhD Thesis. State University of Campinas, Unicamp, LOPCA/FEQ. In preparation 8. Dormer, A. and S. Raynor (1998). "Real time optimization: an 'alternative' approach". Hydrocarbon Processing, September, pp 81-89. 9. Himmelblau, D.M. (1970). "Process analysis by statistical methods". John Wiley & Sons, 463 pp. N.Y.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
811
A Control Methodology for Product Quality Control in climate controlled operations involving Agro-materials With an application to Controlled Atmosphere container transport of agro-materials G.J.C. Verdijck a+, L.J.S. Lukasse a, H.A. Preisig b aproduction & Control Systems Department, ATO, P.O. Box 17, 6700 AA Wageningen, The Netherlands, G.J.C.Verdij [email protected] bSystems and Control Group, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands, [email protected] In this paper a control methodology for direct product quality control is presented for climate controlled operations, such as storage, transport and drying. This methodology is based on differences in time scales and the property of controlling the slowly reacting product with a fast-reacting environment. It directly drives the quality of the product to the desired setpoint. The methodology is illustrated with a full-scale industrial case study on Controlled Atmosphere container transport of agro-materials that focuses on quality monitoring and control design. 1. INTRODUCTION Presently implemented process control for climate controlled operations, such as storage, transport and drying puts tight control on individual process variables such as the temperature. These setpoints are determined beforehand and are constant, or, at best, manually adjusted. These operations are characterised by the presence of different time scales, uncertain information (both for state and objective), unidirectional/irreversible dynamics and the property of controlling the slowly reacting product with a fast-reacting environment. The dynamically changing product characteristics are not directly controlled. Tighter demands on the efficiency of post-harvest processes as well as quality requirements ask for introducing integral control. On the top level the controller is directly driving product quality, or any other relevant product characteristics, to the desired setpoint. The goal of the new control design is to optimally utilise the available process knowledge such as dynamics and product quality with the additional objective of improved safety, energy efficiency, reduction in quality variation and optimal product quality. In this paper a methodology is presented that deals with the specific properties of these processes. The starting point for this research is the opinion that to improve the processing of agromaterials in this class of processes the control methodology must directly incorporate (dynamic) product behaviour. This requires that quality (evolution) can be measured or estimated. This is an important restriction. Because of this direct control it is possible to +to whomcorrespondence shouldbe addressed
812 operate these processes closer to the operation limits, leading to improved process performance with respect to both quality and cost. In Section 2 the control methodology will be discussed that leads to the direct product control. The focus in the full-scale industrial case study, in Section 3, will be on the state estimator and the algorithm used in the selected controller. 2. CONTROL METHODOLOGY Design of a process controller that directly controls the product's quality evolution involves: 9 formulation of the control objective and selection of relevant quality attributes, 9 building the models and, if necessary, a state estimator for the quality attributes, 9 selection of the control structure based on a time scale analysis, 9 design of the selected controllers, 9 validation of the control system and 9 implementation of the controller. The objective (step 1) in most processes and also in climate controlled operations is to maximise the financial yield of the operation. This involves the selection of the relevant quality attributes. This leads to the following objective function l final
J =-P(Q)M + ~L(x,u)dt,
(1)
t
where P is the product price that depends on the end-quality Q, M is the product end-mass and the integral represents the (economic) costs, x, the state variables, and u, the control inputs. The objective function in Equation (1) should be optimised on the top level of the control structure, in Figure 1 shown as long-term controller. For the class of systems that is discussed in this paper the process state can be separated into three parts reflecting the different typical time scales, as shown in Figure 1. One time scale is associated with the quality of the product (those product components that determine the product quality attributes, such as colour, shape, taste and smell). The state variables associated with this time scale are referred to as primary state variables. The two other time scales are associated with the direct (part of the process that directly interacts with the primary state variables) and indirect environment (part of the process that does not affect the product directly, but only through the direct environment). Details on this separation are discussed in [1 ]. Most difficult is the modelling (step 2) of the primary state for which first the relevant quality attributes must be determined. On-line measurement of this substate is often not possible and a state estimator must be developed as will be illustrated in Section 3.1. The selection of a control structure (step 3) for direct control of the relevant quality attributes is an important aspect in improving the control performance of a process. The control structure determines the inputs and outputs, the objectives of the different control components and eventually the possible control performance. In [2] a control structure is selected and motivated that is fitted to a special class of climate controlled post-harvest processes and is shown in Figure 1. This class is characterised by the presence of different time scales (as most post-harvest processes), both disturbance and control inputs that only drive the fast dynamics of the process, and absence of direct measurement of product quality. In general, each
813
Figure 1: Control structure
substate is controlled with a separate control component as its dynamics can be decoupled from the other substates. The local controllers manipulate the indirect environment with their control actions to reach and maintain the setpoints from the supervisory control components. The supervisory controller of interest consists of two components, a short-term and a long-term controller. The main motivations for this separation are the different time scales and the frequencies of input signals, together with the information density, as discussed in [1] and [2].
The design of the controllers (step 4) should deal with the relevant nonlinearities that occur in model parameters and (controlled) variables. Nonlinearities that are located in model parameters depend on climate conditions, e.g. temperature dependency of reaction constants. The effect of these nonlinearities are relatively small, as climate conditions do often not change dramatically. Stronger nonlinearities in (controlled) variables occur in modelling the quality attributes. As will be shown in Section 3.2 a significant energy reduction is only possible if the airflow is controlled. This leads to a nonlinear control nonaffine problem as both airflow and incoming air temperature are controlled as discussed in [3]. It is our goal to derive algorithms that are sufficiently generic to enable their use in a large class of processes, thereby significantly reducing development cost of model-based (supervisory) controllers that are dedicated to the product and its quality. The designed control system must be tested in experiments (step 5). The results should be compared with results using the current controllers. This is a laborious step in the development of the controller, due to the slow time scale of the primary substate (product quality attributes). Furthermore, the large product variation requires numerous replications of the experiments. As for all control systems, requirements are on stability and robustness while performing sufficiently. Implementation of the designed controller (step 6) requires close co-operation with endusers to assure feedback on controller functioning in the real-life application. It is important to guarantee a certain degree of robustness as these processes operate with a large variation in the product, and/or to develop an adaptive mechanism that deals with the large variations. This will not be discussed in this paper. Due to space limitations only step 2 and 4 will be illustrated in more detail in the case study. 3. APPLICATION: CONTROLLED ATMOSPHERE CONTAINER TRANSPORT Climate-controlled container transport is a common way to get agro-materials to the desired location. Product and its quality change depending on transport conditions like temperature, relative humidity, 02, CO2 and ethylene concentrations. To minimise quality decline, usually high rates of ventilation with outside air and/or circulation of internal air are
814 applied leading to unnecessary high cost, and a high evaporation rate and weight-loss. Therefore, new controllers are currently developed, directly controlling product quality that yield a higher end-quality with lower cost.
3.1. Monitoring product quality evolution (respiration/fermentation) The models that are used consist of the three substates as mentioned in the introduction. Equations for the direct and indirect environment are deduced from the conservation laws. The direct environment consists of the product state variables temperature and moisture content, and the climate state variables air temperature, humidity, 02, CO2. The indirect environment consists of e.g. the air conditions in the headspace. The model consists of three parts refecting the bulk conditions and the minimum/maximum conditions caused by the airflow distribution inside the stowage space of the container to deal with the spatial distribution of the quality attributes. Respiration and also fermentation are the two metabolic pathways that provide fruits and vegetables in the post-harvest stage with energy. This energy is used for maintenance or in other words: to stay alive. It is known that the respiration rate is closely correlated to the rate of ripening (for more details on respiration of apples is referred to [4]). This ripening may lead to a quality decrease (softening, rotting), although on the other hand quality may improve (ripening of bananas during transport). Therefore monitoring and control of respiration/fermentation is related to control of product quality evolution and may have practical value. Of course, we are fully aware that product quality evolution has many more aspects that have nothing to do with respiration/fermentation. Control of these other quality aspects is primarily hampered by lack of reliable measurement techniques. The biochemical reaction equations of aerobic respiration and anaerobic fermentation are respectively C6H1206 +602 C6H1206
r/
ro~ >6CQ +6/-/20+ E, (2)
)2CO2 +2CzHsOH+E: (3)
where Er = oxidation energy of glucose (ATP+heat) = 2.816 MJ/mol glucose and Effermentation energy of glucose (ATP+heat) = 0.0820 MJ/mol glucose. The respiration rate, ro2, and fermentation rate, rf, are terms in the differential equations for O2 and CO/. The other terms are the dilution (flow divided by air volume), the loss of air from the container due to fermentation induced pressure build- up [5] with the underlying assumption Figure 2: estimated respiration level that the pressure build-up is negligible. A large number of papers is available on estimating ro2 and rf from Oz/COz-measurements in pilot scale facilities operated in either steady-state flowthrough or batch mode ([5],[6],[7]). These papers do not discuss (recursive) on-line estimation of ro2 and rf from O2/CO2-measurements under normal operating conditions. Therefore in this case study a Kalman filter based recursive estimator is developed. Figure 2 shows some of the promising initial simulation results with additive white noise.
815 3.2. Design of the controllers The research project focuses on the design of the short-term controller with given desired trajectories for the product respiration. These desired trajectories were deduced from extensive product experiments. The objective of the short-term controller is to reach and maintain the process at the desired trajectories with minimum cost. Undesired disturbances must be rejected. The controller optimises a trade-off between achieving the setpoint, inhomogeneity (distribution in both properties and spatial co-ordinates) inside the container and cost. This leads to an objective function that is written as
J = ~+I~((X -- X~e:)r W~ (x - x,e ) + AurW, Au)dt,
(4)
where W are the weighing factors that relate differences between actual and desired behaviour, including inhomogeneous conditions, to changes in the manipulable variables. The time horizon of this controller is denoted with H. Equation (4) is a quadratic objective where constraints can be added. This allows the formulation of a control problem in standard notation, e.g. a MPC type controller. A simulation study was performed using a linear control algorithm in Matlab for the short-term controller. In Figure 3 an indicative energy usage comparison is presented for four different cases resulting from a simulation study. The y-axis represents the energy usage in terms of hours of ventilation with Case 1 (a simulation with the current controllers) as reference. Case 2 represents a situation with the current controllers where airflow is controlled. Cases 3 and 4 represent situations with the supervisory controller where airflow is respectively not included as manipulable variable and is included. From these results can be concluded that for a reduction in energy usage manipulating the airflow can be interesting (as compared to current practice with continuous airflow). However, varying airflow leads to larger temperature variation inside the container and energy reduction possibilities are limited by the acceptable temperature variation. A special situation arises when, besides climate conditions, also airflow is controlled. In the simulation study this nonlinear control nonaffine problem is solved by linearisation. As such a control nonaffine problem is a typical control problem for the climate controlled processes discussed in this paper possibilities to improve the controller are investigated. The approach used for this control problem is based on the algorithm described in [3]. The steps followed to enable the use of this Figure 3: energy reduction algorithm are linearising the minor nonlinearities, the formulation of the nonlinear control nonaffine problem, calculation of the relevant matrices and to perform (iterative) control. The main idea is to use a qth order Taylor series approximation for the step response matrix, S, in the prediction equation l) = ~-~ S<~i!
I.o (AU)' + YP"~' + D,
(5)
i=1
with
I? = [.~(k + I I k)....~(k + p lk)] r, I?p~' = [33p~ (k + 11 k)...pp,,s, (k + p lk)] r,
AU = [Au(k) Au(k + 1)...au(k + M - 1)]r, 15 = [d(k + 11 k)...d(k + p lk)] r.
816 The step response matrix needs to be updated every iteration to reach the optimal solution for the control action. In the simulation study a first order approximation is used in cases with controllable airflow. Higher order terms are currently added. 4. CONCLUSIONS Product quality is directly incorporated in the controller to enable process settings that are more appropriate for the product that is processed. In the case study quality loss will be reduced, as it seems to be possible to directly monitor (Figure 2) and control the quality evolution. Furthermore, energy usage may decrease by controlling airflow (Figure 3), as this reduces over-circulation/-ventilation. In the oncoming year the proposed estimator will be coupled with the higher order controller designed according to this paper's control algorithm with the objective to control apple quality evolution in pilot scale and full-scale experiments. REFERENCES
1. G.J.C. Verdijck, G. van Straten. "A modelling and control structure for product quality control in climate controlled processing of agro-material." submitted to Control engineering practice. 2. G.J.C. Verdijck, L.J.S. Lukasse, J.J.M. Sillekens (2000). "Aspects of Control Structure Selection in Post-Harvest Processes", Proceedings of AgriControl2000, Wageningen, The Netherlands, to be printed. 3. R.K. Mutha, W.R. Cluett, A. Penlidis (1997). "Nonlinear Model-based Predictive Control of Control Nonaffine Systems." Automatica 33 (5): 907-913. 4. H.W. Peppelenbos (1996). "The use of gas exchange characteristics to optimize CA storage and MA packaging of fruits and vegetables." PhD. Thesis. ISBN 90-5485-606-8. 5. F. Forcier, G. S. V. Raghavan, et al. (1987). "Electronic sensor for the determination of fruit and vegetable respiration." International Journal of Refrigeration 10(6): 353-356. 6. S. Lakshminarayana, M. Muthu, et al. (1973). "Modified continuous gas stream method for measuring rates of respiration in fruits and vegetables." Laboratory Practice 23( 12): 709710. 7. M. Janssens, A. Thompson, et al. (1995). Practical determination of the respiration rate in controlled atmosphere (CA) stores, DG XIII-A, Commission of the European Communities; Luxembourg.
European Symposium on Computer Aided Process Engineering - 11 R. Gain and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
A m o d e l - b a s e d s u p e r v i s o r y c o n t r o l r o u t i n e for t e m p e r a t u r e b a t c h reactors" e x p e r i m e n t a l results
817
control of
F. Xaumier a'b, E. Ettedgui a, M.-V. Le Lann c, M. Cabassud ~b and G. Casamatta a a Laboratoire de G6nie Chimique-UMR CNRS 5503, I.N.PT./U.P.S bI.U.T Paul Sabatier-D6partement G6nie Chimique-BP4065 18, chemin de La Loge-31029 Toulouse Cedex 4-France T61.: 05.62.25.23.62, Fax: 05.62.25.23.18 E-Mail: [email protected] Institut National Polytechnique de Toulouse CLAAS-CNRS UPR 8001 7, avenue du Colonel Roche-31077 Toulouse Cedex 4-France E-mail: [email protected]
This paper describes the application of a model-based strategy for the supervisory temperature control of a 12 litres industrial pilot batch reactor. The reactor is equipped with a multi-fluid heating/cooling system. The automation of this type of heating/cooling system requires a supervisory control to handle automatically the utility fluids fed in to the reactor jacket as well as the intermediate air purge and the refilling after the changeover of the fluid. The objective is to obtain good control of the process even during transient operating conditions such as refilling of the jacket, which can not be neglected on industrial scale. The supervisory control is based on dynamic model of the reactor pilot-plant. Once the utility fluid being chosen by the supervisory routine, its flowrate is computed by a controller which is a Non-linear Model Predictive Control algorithm. Different experiments are presented in order to demonstrate the performances of the supervisory control of the pilot-plant reactor. 1. INTRODUCTION Temperature control of semi-batch or batch reactors is of increasing interest for fine chemical or pharmaceutical industry that uses essentially batch processes. Among the main thermal systems used in industry, the multi-fluid system remains the most encountered [1]. Batch operation is carried out according to a scheme, which is often decomposed in: - a heating phase, which allows the preheating of the reaction mixture up to the desired temperature at which the chemical reaction takes place. - a reaction phase during which the temperature is maintained constant. - a cooling phase to avoid by-product formation. According to these three phases and the resulting heating or cooling demand, changeovers of utility fluid occur. The main difficulty in operating such a system lies in its supervisory control. Moreover, in industry any changeover implies an air purge of the jacket. This paper presents a new methodology which simultaneously solves the problem of control and supervision of batch reactors.
818 This supervisory control procedure is based on the on-line computation of the maximum and minimum temperatures over a future horizon, which could be reached for every utility fluid available on the batch reactor pilot-plant. These predictions are made by integration of a dynamic model based on energy and mass balances of the reactor and its jacket. According to these predictions, the fight utility fluid is chosen. Once this choice is taken, the determination of the utility fluid flowrate is achieved by a Non-Linear Model Predictive Control algorithm [2]. The non-linear model predictive control [3,4,5] uses the same dynamic model of the batch reactor as in the supervisory controller [6]. This supervisory technique has been applied to the temperature control of a 12 litres industrial glass-lined jacketed reactor equipped with an multi-fluid system. 2. A M E T H O D O L O G Y F O R SUPERVISIORY C O N T R O L OF B A T C H R E A C T O R The supervisory control procedure chooses automatically the utility fluid to be injected into the jacket. The supervisory routine is linked to the other routines as follows:
Acquisition of measurement on the pilot-plant q, Estimation of unmeasured variables q/ Supervisory Algorithm: Choice of the utility fluid q/ NMPC Algorithm: Calculation of the manipulated variable (flowrate)
Fig. 1. Description of the links between the different algorithms This supervisory approach has been proposed by Ettedgui [2]. It is based on the on-line computation of maximal and minimal reactor temperature trajectories Trm~x.,(t), T, min.i(t) for every available fluid i on a future horizon [t~tk+nen] where tk represents the present time (Fig.2). HPR is the prediction horizon on which will be decided to change to an other fluid or not. The computation is performed for every available utility fluid by fixing the flowrate of the thermal fluid feeding the jacket at time tk. The model is then integrated by a Gear method on
819 interval [tlotk+HPR]. The flowrate of the thermal fluid is fixed according to the conditions corresponding to the following different scenarios: If "i" represents the fluid present in the jacket at time tk and its available temperature is greater than Tr(&) then the maximum temperature Trmax,i(O trajectory (on interval [& tk+HPR] ) is determined by taking the maximum flowrate. The minimum temperature Trm,,,i(t) trajectory corresponds to an air purge phase only. In the opposite case where the temperature of the feeding utility fluid is less than Tr(&): T~m~x,i(t) corresponds to an air purge only; T~m,n,i(t) to the maximal flowrate. - If "i" represents any other fluid except the one present in the jacket and if utility temperature is greater than Tr(tk), the Trm=,i(t) trajectory corresponds to the conditions of an air purge followed by a filling up phase with the maximal flowrate, Trm,,,~(t) corresponds to an air purge step only. Similarly, the opposite case is found when the utility fluid temperature is below the reactor temperature T~(&). -
It can be noticed that the determination of such trajectories requires the computation of the reactor and jacket temperatures for the different utilities and states of the jacket, especially in case of an air purge or refilling of the jacket. This is the reason why a lumped model of the jacket is used in order to represent the behaviour of the fluid in these specific conditions in a detailed way. An example of how the supervisory procedure works is given in Figure 2. (case of hot water present in the jacket). ~T
emr
)eratule
i I
/
9 _ I ~ -1 Tc
"~"
Trmax,st
Trmax,hw i
~1 I I I
Tr
Trmin,st Trmax,cw Trmax,mg
Trmln,cw
Past
Future
tk
I
I tk g H P R
Trmin,mg I ~ - Time
Fig.2. Principle of the supervisory procedure At every sampling period, the supervisory control procedure starts with the computation of two criteria, with regard to the deviation between the set point Tc of the reaction mixture temperature and the maximal and minimal temperatures which can be reached at tk+I-IPR for the fluid i present in the jacket. The two criteria are the following:
Chlgh = TC(t=tk+HPR)--ZIoW--Tr ..... (t=tk+HPR)--d(t--'tk+HPR)
(1)
Clo w -- rc(, -- 'k+HPR)§ lhigh- Tr ..... ('--'k+HPR)--d( ` "--'k+HPR)
(2)
820 When the two criteria are satisfied for the utility fluid i, the search is stopped and this utility fluid is kept. If the criterion noted Chighis not satisfied, this means that the set point can not be reached with the thermal fluid currently in use, then the criterion is recalculated with a warmer utility fluid. If the criterion noted Crowis not satisfied then the criterion is recalculated with a colder utility fluid. By this technique the number of fluid changeovers is minimised and less discontinuities within the process are generated (the preference is given to the current fluid). In the equations (1) and (2), the terms noticed llow and lhigh represent the limits of tolerance around the actual set point that are accepted before a switch to the utility that could exactly reach the set point is initiated. These parameters permit to reduce the number of changeovers of fluid and to obtain less discontinuities. It can be said that these two parameters represent a compromise between precise control of the process and the number of required changeovers of utility fluid. The plant model mismatch correction d is computed as a function of past values of observed errors (Trmes-Tr) on a past horizon. The parameters of this function are identified on-line. This function is then used to predict the future error d(tk+He~) [6]. 3. EXPERIMENTAL APPLICATION 3.1 D e s c r i p t i o n o f the p r o c e s s
Fig.3. Industrial glass-lined batch reactor and its heating/cooling system This pilot plant [7] consists of a stirred batch reactor. Its maximum operating volume is 12 litres. Air is used to empty the jacket when changing the utility fluid. Four utility fluids are available at a given temperature: steam, cold water, hot water and mixture of monopropylene glycol and water (50/50% weight). The hot water stream is obtained by mixing steam and cold water to produce hot water at 70~ In practice, hot water is produced by fixing the cold water flowrate and by computing, via an energy balance on the mixer, the necessary steam flowrate to produce water at 70~ The pilot-plant reactor is equipped with temperature, pressure and flowrate sensors for every fluid. The actuators on the pilot plant include two types of valves: on-off valves and proportional valves. Four proportional valves are implemented to control the flowrates of thermal fluids available on the pilot plant as follows: - cold water flowrate (A), - steam flowrate (C), - steam to be mixed with cold water flowrate (B), - glycol/water flowrate (D).
821 Valves A and B are moved simultaneously to change the hot water flow whilst the valve positions are determined to satisfy the mass and energy balances for the desired flow rate at 70~ It can be noticed that the valves A and B exhibit dead zones and remain closed between 0 and 0.4 opening degree (the valves cannot be opened at a value within this interval). A particular procedure has been implemented in the control algorithm to choose the best value among these two (0 or 0.4 opening degree) [2].
3.2 Experimental results In this part, different experiments are presented in order to demonstrate the feasibility and the performance of this model-based supervisory control. The objective of the experiments is to track a pre-defined set point profile by handling automatically the thermal fluids. This set point profile has three phases as described in the introduction. In these experiments the reactor was charged with 10 litres of water. As far as the supervisory control parameters HPR is concerned, the choice of an adequate value can be made according to time constant of the process. A too long prediction horizon results in too late changeovers. In contrary, a very short horizon leads to excessive changeovers. In our case, it has been found that 3 sampling periods are sufficient; this represents one third of the time constant. Similarly, lhigh and llow will be chosen by doing a compromise between too late changeovers and excessive changeovers on the other case. In our case, a compromise has been found for lhigh=llow=2~ Experiments have been made with 1~ which leads to more frequent changeovers [5]. Both experiments use the same controller design parameters as given previously.
Fig.4. NMPC temperature control (constant temperature phase at 60~
Fig.5. NMPC temperature control (constant temperature phase at 45~
Each experiment corresponds to a different temperature set point profile with as desired constant temperature phase: 60~ for the first experiment (Fig.4) and 45~ (Fig.5) for the second one. In both cases the reaction is simulated by electric resistors between 1200 and 2400 seconds. The notations hw, cw, mg mean that the fluid used is respectively hot water, cold water and monopropylene glycol.
822 For the example presented in Fig.4, the temperature tracking is sufficiently precise during the three phases. Slow oscillations are observed during the preheating. This behaviour is typical when steam is directly injected into the jacket: when the steam valve opens, the reactor temperature increases quickly and then oscillations occur. The supervisory control procedure handles the changeovers of the utility fluid in a way that gives satisfying quality of control. During this experiment, every available utility fluid was used. In the second experiment (Fig.5.), the temperature tracking is satisfactory. Indeed, for the heating and maintaining phases, the supervisory control procedure imposes hot water as the fluid injected into the jacket. Consequently, no oscillation is noticed as steam is not used. During the cooling phase, successively cold water and monopropylene glycol are used. Only a small undershoot appears during the cooling phase but it remains less than 1~ For both experiments, during the use of hot water, regular oscillations of the manipulated variable are observed (between 0 and 0.4). As previously mentioned, this is due to the valves A and B used to obtain hot water at 70~ that present dead zones between 0 and 0.4 opening degree. 4. CONCLUSION A methodology for thermal supervision of batch or semi-batch reactors was presented. It is based on computing the maximum and minimum temperature trajectories of every utility fluid available on the pilot reactor by means of an on-line integration of a DAE system modelling the reactor and its jacket. If the set point is between the maximum and minimum temperature of a certain fluid, this fluid will be selected. The implementation of this strategy has been performed on a glass-lined reactor equipped with a multi-fluid system. Results exhibit satisfactory reactor temperature tracking due in particular to adequate utility changeovers. This strategy is not limited to the heating/cooling system presented in this work, but may be extended to any heating/cooling system involving different fluid feeds (example: mono-fluid with mixing of different constant temperature fluid sources) which demands a supervisory control to choose what fluid has to be fed into the thermal loop. REFERENCES 1. Bonvin D., Optimal operation of batch reactors: a personal view, Journal of Process Control 8 (5-6) (1998) 355-368. 2. Ettedgui B., Commande pr6dictive non-lin6aire des r6acteurs discontinus de Chimie Fine, Th6se de Docteur I.N.P de Toulouse, France, Toulouse, 1999. 3. Bequette B. W., Ind. Eng. Chem. Res., 30 (1991) 1391. 4. Eaton J. W., Rawlings J. B., Computers Chem. Engng., 14 (1990) 469. 5. Wright G. T., Edgar T. F., Computers chem. Engng., 14 (1990) 469. 6. Xaumier F., Application exp6rimentale de la commande non-lin6aire sur un r6acteur chimique discontinu avec estimation en ligne de la chaleur r6actionnelle, Th6se de Docteur I.N.P de Toulouse, France, Toulouse, 1999. 7. Cabassud M., Chamayou A., Pollini L., Louleh Z., Le Lann M.-V., Casamatta G., Proc6d6 de contr61e thermique d'un r6acteur discontinu polyvalent ~ partir d'une pluralit6 de sources de fluides thermiques, et dispositif de mise en oeuvre de ce proc6d6, Patent N ~ 95.03753, 1995.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
823
Strategies for optimisation and control of molecular weight and particle size distributions in emulsion polymerisation J. Zeaiter, V.G. Gomes, G.W. Barton, J.A Romagnoli a and R.G. Gilbert b aProcess Systems Engineering Group, Department of Chemical Engineering J01, University of Sydney, NSW 2006, Australia bKey Centre for Polymer Colloids, School of Chemistry F11, University of Sydney, NSW 2006, Australia Emulsion polymerisation reactor dynamics were modelled with a focus on the computation of molecular weight (MWD) and particle size distributions (PSD). The evolution model equations included the detailed kinetics of styrene polymerisation in a semibatch reactor via free-radical mechanism. The effects of variations in temperature, feed-rate and the monomer and surfactant concentrations on the molecular weight, particle size and polydispersity indices were investigated. The operating temperature and the monomer and surfactant feedrates were found to have significant effects on the MWD and PSD. Based on our findings, strategies for controlling the MWD and PSD were formulated. 1. INTRODUCTION Emulsion polymerisation is widely used in industry to produce products ranging from adhesives through to surfboards. The process is preferred because the reaction medium (usually water) facilitates agitation, heat and mass transfer and provides an inherently safe process. However, the multiphase reaction medium that generates preference, renders the kinetic mechanisms and the process a complex and difficult one to model accurately. Kinetic modelling of emulsion polymerisation has been extensively researched. The recent model (Coen et al, 1998), adopted for this work, includes detailed phenomena such as, competing mechanisms, aqueous phase kinetics of radicals, particle formation by micellar and homogeneous nucleation and coagulation of unstable particles. Previously developed reactor models (Broadhead et al, 1985; Dougherty, 1986; Penlidis, 1987; Richards et al 1989; Li et al, 1993; Gugliotta et al., 1995; Liotta et al., 1998) either adopt a simple kinetic formalism or do not estimate important product attributes such as MWD and PSD. Yet it is the particle size and the molecular weight distributions that determine some of the most important properties of polymers, such as the mechanical strength, opacity, processability, etc. Consequently, the ability to predict (and ultimately control) the MWD and PSD is crucial to optimising polymerisation reactor performance. The reason why the control of product quality (e.g., PSD and MWD) is lacking in the literature due to the complexity in implementing control of distributed parameters. Recently, Asua et al. (1998) used a nonlinear model-based controller to calculate the feed rates of chain
824 transfer agent and monomer required to produce a polymer with desired MWD in styrene emulsion homopolymerisation. The control strategy was assessed by simulation and verified experimentally where a bimodal MWD was obtained. However the effects of reaction temperature among other important variables were not investigated. Similar work on emulsion copolymerisation of methyl methacrylate/butyl acrylate was attempted (Asua et al., 2000). Doyle (1999) investigated simple open-loop optimal trajectories for styrene polymerisation PSD via a numerical simulation. This paper demonstrates the advantages of a comprehensive reactor model with detailed kinetics and product attribute calculations for the purpose of optimising the process and formulating control strategies.
2. M A T H E M A T I C A L M O D E L
The polymerisation reaction under consideration in this study is a semi-batch emulsion polymerisation reactor, where styrene is polymerised to form polystyrene latex via the free radical addition polymerisation mechanism. Combining the effects of radical transfer, growth and coagulation, population balance equations were developed for particles containing one polymeric radical, zero radical and one monomeric radical respectively, shown as follows:
0---7--
k.c~..
+ p.~
- p~( - k,.c~.~ - o ( x . f
+ 6(V - Vo) C. ....,~ ~[k; ..... ,e[lM.]] - kpJ~';~'Cw[IMj~r..-~] i=~
+ ~ B(V,V v')[~o(v')~((v - v')+~,~(v')~o(v v')]dV' -
-
- ~((v) f ~(v, v')[~o(v')+,ff(V')ldV' c3n~ Ot
(1)
= p[n( + ny -no]+ k ~ n ~ + f B(V,V-V')[no(V')no(V-V') (2) + ~((v')~. ~ (v - v ' ) ] a v ' - ~o (v) ~ B(V.
V')[,.o (V') +,.." (V')laV'
any(V,t) = k , , c ~ , . ( + k.~[e],.o - (k',c,. + k ~ + p),.~' Ot
(3)
These sets of coupled partial integro-differential equations comprise the evolution equations for the latex particles in an emulsion polymerisation process. They describe the volumebased distribution according to the mechanistic model developed by Coen et al (1998). Based on the above, we estimated the average number of radicals per particle, fi, from the following equation: n= ~
~,,,(~) i=1
,_-,
rhp (V,) +~-] rh" (V,) ,=1
(4)
825 The total number of particle per unit volume, Ntot, was estimated from: Nto , = N A
(5)
f n(V)dV
The instantaneous MWD (Gilbert 1995) was calculated as follows"
OP(M) = P(M) = (k,,Cv + p)n exp - (p + ktrCp) M Ot keCe Mo
(6)
where M is the molecular weight of the polymeric chain. The number average radius was computed from: G
~n(i)r(i) < r >= ,=1
(7)
G
Zn(i) t=l
The first moment estimated above is insufficient to characterise the particle radius. Therefore, the polydispersity index was computed to indicate the spread of the distribution. The particle size polydispersity index (PSPI) is estimated as follows:
PSPI = ~
(8)
2
Similarly the number average molecular and the weight average molecular weights were calculated as follows:
IMP(M) dM <M.>=
'
IP(M)dM
[M 2P(M) dM and
<M w>="
IMP(M)dM
(9)
The molecular weight polydispersity index (MWPI) was obtained from:
M W P I = <~M W > <M n >
(10)
The reactor model includes the estimation of surface tension of the continuous phase using the Szyszkowski equation (Paine et al., 1995).. For our numerical simulations, the population balance equations were discretised using the stable backward finite difference method. The model was then written in gPROMS, a general process modelling software from the Centre for Process Systems Engineering, Imperial College, London.
826 3. DYNAMIC OPTIMISATION To illustrate an application of the model, an optimisation of the surfactant feed trajectory was conducted in gPROMS. The objective of the optimisation was to obtain a bimodal distribution in the PSD of the final product in addition to maintaining colloidal stability during the entire period of the reaction: The objective function to be maximised was defined as,
MFaX[
l,.~, ) = < r2 > ]
(11)
2J
The surfactant feed to the reactor (Fs) was chosen as the control variable and was specified as piecewise constant. Upper and lower bounds for the feed were set to ensure that the optimisation focused on the feasible range of operation. The time horizon was fixed to correspond with the completion of monomer feed to the reaction vessel (tf). To enable a stable emulsion, the surface tension (7) at the end of the polymerisation (an endpoint inequality constraint) was specified as indicated by the following equation: 34 dyn.cm -1 < 7 < 36 dyn.cm -1
(12)
4. RESULTS As shown in figures 1-2, the changes in temperature and monomer feedrates had major effects on the MWD. Higher temperatures resulted in shorter chains. This happens because at high temperature, chain termination becomes more important, which results in a low degree of polymerisation. On the other hand, at high monomer feedrates, the monomer concentration within a particle increases, which causes an increase in the polymerisation reaction rate. The propagation of the polymer chains becomes more important and a higher degree of polymerisation is obtained.
1:t ~
10],
8
~8
~ 7
~ 7 4
.-. a
.
.
. O
01 r
.
0.0~)0
5.0~1~
.
.
1.0E~7
. 1.5E~7
Molecular
. 2.0E~
2 1 0
. ZSE~)7
3.0E~ffr
weight
Fig. 1. Effects of monomer feedrate on MWD. temperature on MWD.
0.0E~
S.I~I~
1.0E~7
1.5E~7
,
,
Z0~lr/
Zf~t~'
Molecularweight
Fig. 2. Effects of reaction
827 As illustrated in Figures 3-4, the reaction temperature and the monomer feedrate affects the PSD. However the effect of the latter was more significant as the distribution was broader with a lower average particle size for a higher flowrate. Thus feedrate control is an important variable for obtaining desired PSD indicating the need to operate in a semi-batch mode for PSD control.
Fig. 3. PSD at different reaction temperature.
Fig. 4. PSD at different monomer feedrates.
Finally, the effects of the surfactant feed on the PSD were studied using the optimisation strategy defined previously. An optimum surfactant feed profile was developed for obtaining a maximum PSPI within the operating constraints. As a result, a bimodal distribution was obtained which is reasonable, because with the increase in surfactant feed, new micelles were formed and additional nucleation sites were created. Thus a new generation of smaller size particles was obtained as illustrated in Figure 5.
Fig. 5. Bimodal distribution of PSD under optimal surfactant feed. The surfactant feed optimal profile showed an initial increase in the surfactant feedrate after the particle nucleation stage. As particles grow in size, the interfacial area increases and hence more surfactant molecules are required to maintain colloidal stability. A subsequent
828 increase in surfactant feedrate caused secondary nucleation and a new generation of particles evolved. On completion of the nucleation stage, the feedrate rate was decreased to a new value, which ensures the stability of the emulsion. Thus surfactant feed can be manipulated in order to tailor the distribution modality.
5. CONCLUSIONS The investigation identified the important variables for the control of MWD and PSD in emulsion polymerisation. In particular, operation in semi-batch mode and manipulation of temperature, monomer and surfactant feedrates are important. An optimisation study illustrated the indicated that the PSD modality can be manipulated through surfactant feedrate control. Multi-variable optimisation can be conducted to optimise product quality and process operability by determining the required monomer and surfactant feed trajectories as well as the operating conditions (such as temperature) to obtain the desired PSD and/or MWD. REFERENCES
1. Broadhead T.O., Hamielec A.E., MacGregor J.F., Makromolekular chemie suppl, v 10/11, 1985, pp. 105. 2. Dougherty E. P., Journal of Applied Polymer Science, vol. 32,1986, pp.3051. 3. Penlidis A., Hamielec A.E., MacGregor J.F., Makromolekular chemie, Macromolecular Symposia, v 10/11, 1987, pp. 521. 4. Richards, J.R., and Congalidis, J.P., Journal of Applied Polymer Science, v37, 1989, pp.2727. 5. Li B., Brooks B.W., Journal of Applied Polymer Science, v48, 1993, pp. 1811. 6. Gugliotta L.M., Brandolini M.C., Vega J.R., Iturralde E.O., Azum J.L., Micra G.R., Polymer Reaction Engineering, v3, 1995, pp.201. 7. Liotta V., Sudol E.D., E1-Asser M.S., Georgakis C., Journal of Polymer Science, Part A: Polymer Chemistry, v36, 1998, pp.1553. 8. Arzamendi G., Asua J.M., Makromolekular chemic, Macromolecular Symposia,,v 35/36, 1989, pp. 249. 9. Sayer C., Arzamendi G., Asua J.M., Lima E.L., Pinto J.C., European Symposia On Computer Aided Process Engineering-10, 2000, pp.457. 10. Doyle F.J., Crowley T., Meadows E.S., Control of Particulate Processes VI, An International Conference, September 1999, Qld., Australia. 11. Coen E. M., Gilbert R.G., Morrison B. R., Leube H., Peach S., Polymer, vol. 39, 1998, No 26, 7099. 12. Gilbert R.G., Emulsion polymerization : A Mechanistic Approach, London : Academic Press, c 1995. 13. Paine, A.J., Wang, Z., and Rudin, A., Journal of Colloid and Interface Science, 1995, No 173,376. 14. Process Systems Enterprise Ltd. (PSE), gPROMS Advanced User Guide, Ver. 1.8, United Kingdom.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
829
Process Safety Management for Batch Process Operation Atsushi Aoyama, Rafael Batres and Yuji Naka Tokyo Institute of Technology, Frontier Collaborative Research Center Nagatsuta Midori-ku Yokohama 226-8503 Japan, Tel: +81-45-924-5137, Fax: +81-45-924-5144 Email: [email protected], [email protected], [email protected] The batch process is becoming increasingly important due to a greater emphasis on low-volume, higher added value products, and the need for flexibility in a market driven environment. In the meantime, the management of process safety has become a mandatory as the public concern on environmental impacts and a safety of chemical industries heightened. In this research, a scenario oriented method of abnormal situation handling procedure design and a multi-agent based abnormal situation handling operation management scheme are proposed for batch processes. In the proposed scheme, the hazard evaluation and the abnormal situation handling procedure design are structured in three layers, inter-Unit, intra-Unit and intra-CGU levels. The proposed scheme makes it easy to design and express an abnormal situation handling procedures and simplify the task of safety operation management and seamlessly integrates those two tasks. 1. INTRODUCTION The chemical and biochemical industries face an intense pressure to improve the efficiency and the product quality. Under these circumstances, the batch process is becoming increasingly important due to a greater emphasis on low-volume, higher added value products, and the need for flexibility in a market driven environment. In the meantime, the management of process safety becomes a very important issue and its implementation has become a mandatory as the public concern on environmental impacts and safety of chemical industries heightened. The process safety management has been traditionally handled by bottom-up approaches such as the TPM (Total Preventive Maintenance) where existing process safety technologies are simply bunched together. Although these approaches are necessary, they cannot explicitly show the way to systematically integrate elements such as a process safety analysis, a process safety design and a process safety operation and handle the issue of management of changes. We have proposed a batch process operation management platform [2] in accordance with ANSI/ISA88 [ 1]. In this paper, the functions of batch process operation management platform are extended for the systematic process safety design and operation management to achieve a high level of process safety. The next section briefly revisits the batch process operation management platform. Section 3 explains a process safety design method, especially a design method of abnormal situation handling procedure. Section 4
830 explains a multi-agent based batch process management structure which executes abnormal situation handling procedure designed as described in Section 3. Section 5 briefly summarizes the advantages of the proposed method. 2. BATCH PROCESS MANAGEMNET P L A T F O R M ANSI/ISA-S88 [1] has been proposed to achieve a better batch process operation management through establishing the standard models, terminology, data structure and guidelines for language used by batch process control. The batch process operation management platform [2] is also compliant to ANSI/ISA-S88 One weak point of ANSI/ISA-S88 is that the separation of the plant information reconfiguration management, the process management and the unit management is incomplete. The operation management platform has been proposed in which those three kinds of management are clearly defined. The operation management platform is modelled as a multiple layered structure of process management, unit supervision and phase execution from the upper layer to the lower layer. "process management" executes multi-batches through the management of procedure execution and executes procedures through the management of unit procedures. "process management" also has a function to generate control recipe by reconfiguring the information contained in master recipe, schedule and plant structure. This function is corresponding to the plant information reconfiguration management. "unit supervision" is responsible for the execution of entire operation in the specific unit. "unit supervision" achieves the function of unit management. Finally, "phase execution" executes phases by sending commands to DCS. The batch process operation is executed based on a control recipe. The control recipe contains the information to produce the specific batch size of product after each operation is assigned to specific equipment. It also contains the information about when each operation should be started. The procedure defined in each recipe has layered structure, Procedure, Unit procedure, Operation and Phase. Procedure is corresponding to all the production steps to produces a batch. Unit procedure is corresponding to the production steps carried out at and around a main unit such as a reactor and a distillation column. Operation is the operation carried out at a main unit and its peripherals and Phase is a individual step to complete an operation. The activation of procedure changes the state of the structure (such as opening a valve) and the behavior comes out from such changes. The plant structure belongs to the physical dimension and refers to the description of the components of which the plant is built as well as their topological representation. The most basic component of the cell is equipment (e.g. pipes, valves). Instances of elemental-controlled-group-unit (ECGU) can be generated automatically from the equipment topology. An ECGU can be identified as an assembly of equipment that has control valves at its connection ports. On the other hand, the controlled-group-unit (CGU) and Unit could not be derived from the equipment topology only. A Unit describes an aggregation of ECGU that has a common unit such as a reactor or distillation column. A CGU is an aggregation of equipment that is actually handled as an isolated region from other parts of the cell during a operation. A procedure is an entire process necessary to manufacture a
831 batch corresponding to the Cell. A unit procedure is a process executed at a unit. An operation is a process executed at a CGU that is dynamically configured from ECGU. And phases are execution steps carried out in a CGU. 3. SAFETY DESIGN The entire safety design process is divided into three steps according to the concept of independent protection layer (IPL) [3]. At the first step, IPL1 (process design), IPL2 (basic control, process alarms, and operator supervision) and a mandatory part of IPL5 (relief devices) are designed using a rule-based approach where no specific abnormal situation is assumed. At the second step, IPL3 (critical alarms, operation supervision, and manual intervention), IPL4 (Automatic Action SIS or ESD) and IPL5 (relief devices) are designed using a scenario-based approach. Finally at the third step, the safety management activities to minimize effects to outsides of plant (IPL6, 7 and 8) are designed. This paper focuses on the design of IPL3, after the design of IPL 1, 2 and a mandatory part of IPL5 are completed.
3.1. Abnormal situation handling procedure design Abnormal situation handling procedures, requirements of the plant design modification, sensor locations are simultaneously determined at the design of IPL3 in order to satisfy regulations and social requirements and various requirements from the plant owner and the society. Although these tasks are currently done using heuristics and experiences of process designers and safety engineers, such a method is error prone and above all design rationales are not clearly recorded. It is very difficult to use design rationales in a real-time operation and an operation analysis. A cure for these problems are a scenario oriented approach where various abnormal situation scenarios are generated, risks of those scenarios are estimated and a design of IPL3 is done to contain those risks into acceptable levels. The design of IPL3 is divided into two steps, a categorization of abnormal situation by a preliminary hazard assessment and IPL3 design for each abnormal situation category (recovery, partial shutdown and total shutdown). At the first step, an initial event, which is categorized into the equipment failure, the abnormal material behavior or the mal-operation, leading to abnormal situations is identified then a preliminary hazard analysis is qualitatively and quantitatively carried out and a probability, a severity, a propagation path and an affected area are estimated. Further a propagation speed is estimated using a dynamic simulator. The risk level is computed from these estimations and whether that risk is acceptable or not is determined according to the predefined criterion. If the risk is not acceptable, abnormal situation category (recovery, partial shutdown and total shutdown) is determined and an abnormal situation handling operation procedure is designed. The abnormal situation handling procedure is evaluated using dynamic simulator. If the risk is still unacceptably high, the abnormal situation handling procedure is redesigned. This cycle is repeated until the risk is reduced to be acceptable. For each successful procedure, a plant design modification and a sensor location are designed to enable the corresponding operation procedure. However they are not immediately employed.
832 Instead, the proposals of plant design modification and sensor locations for all initial events are combined to make a final decision in order to avoid redundant design modification and sensors.
3.2. Unit and CGU based abnormal situation handling procedure expression A scenario-based method has a number of advantages such as a use of design rationale. However, it has a serious drawback, a massive number of scenarios and abnormal situation handling procedures have to be considered. In order to handle this problem, this paper proposes to divide a plant into several regions and design an abnormal situation handling procedure for each region, and construct an abnormal situation handling procedure for an entire plant as a combination of these procedures. The batch process operation platform introduces the concepts of Unit and CGU (Controlled Group Unit) and the correlation between Units, CGUs and management and operation layers. The proposed scheme uses the Unit and the CGU concept as a unit for the design and execution of abnormal situation handling. There are three ways for the Units and the CGUs to receive the effects of the initial events; from the initial event within the CGU and the Unit; through physically connected Units and CGUs, through the execution of abnormal situation handling procedures. The initial event and the abnormal situation can be categorized into the following three scenarios. 9 The effects of initial event can be contained within the CGU in which the event occurs (the initial CGU). 9 The effects of initial event exceed to the outside of the initial CGU but can be contained in the Unit in which the event occurs (the initial Unit). 9 The effects of initial event exceed to the outside of the initial Unit. In the first case, only the effects of initial events within the initial CGU are evaluated and an operation-level abnormal situation handling procedure is designed. In the second case, an operation-level abnormal situation handling procedure for the initial CGU is designed. In addition, an operation-level abnormal situation handling procedure for the other CGUs belonging to the same Unit are designed. A unit procedure level abnormal situation handling procedure also has to be designed to coordinate operation-level abnormal situation handling procedures. It is worth to point out that the deigned abnormal situation handling procedure for the other CGUs and the coordination Unit can be reused for other initial events if the effects of initial event to the outside of the initial CGU are identical. Abnormal situation handling operation procedures are designed both for the intra-CGU level (operation level) and the intra-Unit level (unit procedure level). An intra-Unit level procedure (unit procedure) coordinates all CGUs and decides the expected state of CGUs (recovery, holding, partial shut down and total shut down). The intra-CGU operation procedure brings the CGU to the expected state. For the third case, an inter-Unit level abnormal situation handling procedures are designed in addition to intra-Unit and intra-CGU level procedures. An inter-Unit level procedure (multi-task management) coordinates all Units and decides the expected state of Units (recovery, holding, partial shut down and total shut down). The three parts of procedures, inter-Unit, intra-Unit and intra-CGU level, can be independently combined to generate a new
833 abnormal situation handling procedure. It makes possible to construct a massive number of operation procedures from relatively few parts and achieve an easy maintenance of abnormal situation handling procedures. Units and CGUs are convenient regions for the purpose as these regions are isolated most of the time by valves. 4. OPERATION MANAGEMNET As described in Section 2, the dynamic reconfiguration of management structure as well as the separate handling of process management and unit management are required for the batch process operation. The multi-agent system represents a new way of analysing, designing and implementing complex software systems and are suitable to realize functions of batch process operation management because they can be generated and deleted individually. The following agents are defined for the batch process operation management. 9 Model server agent: 9 Cell (plant) agent 9 Multi-batch agent 9 Process agents 9 Unit procedure agents 9 Operation agents In addition, a model server agent, which manages databases of plant structure, manufacturing schedule and recipes and sends those data to other agents by request, is defined. How these agents work in the normal operation and in the abnormal situation handling is described in the following paragraphs. 4.1. Normal operation A cell-agent and a multi-task agent are generated when a cell enters into the operation for the first time. A cell manager gives the information about which equipments are defined as main units and which are peripherals. A cell agent obtains the information of plant physical structure from the model server agent and compares it to the inputs from the cell manager and determines the regions of Units. A cell agent also generates the control recipe and determines the CGU. A cell agent reconfigures the Units information every time the structure of cell is modified. A cell agent keeps the maintenance history of equipments and has an inventory of ongoing and previously executed processes. A multi-task agent obtains the information of manufacturing schedule and generates process agents based on it. A multi-task agent keeps track of the states of processes (non-active, active, normal, abnormal, holding etc.) and sends messages to process agents to execute, terminate and hold processes. After a process is executed, a multi-task agent deletes the corresponding process agent and sends the information held by it to the database. A process agent in turn generates unit procedure and sends them messages to execute, terminate and hold processes and keeps track of their states. A process agent pass its own status information and information sent by unit procedure agents to the multi-task agent. A unit procedure agent generates necessary operation agents and keeps track
834 of their states. A unit procedure agent also manages the regions of the Unit currently not used and the valves separating the CGU region from the non-CGU regions. An operation agent sends phase commands to DCS to execute the operation and observes the differences between the phase commands and the feedbacks from DCS. 4.2. Abnormal situation handling The operation agent searches the abnormal situation scenario database for a matching initial event when the differences between phase commands and the feedbacks from DCS exceed predefined criteria. If an initial event is not found in the abnormal situation scenarios, the operation agent alarms a cell manager or an operator for further actions. If an initial event is identified, the operation agent executes an operation (intra-CGU) level abnormal situation handling procedure and sends the identification result to the unit procedure agent. The unit procedure agent executes a unit procedure (intra-Unit) level abnormal situation handling when it receives a report of abnormality from a operation agent or identified an abnormal situation in the non-CGU region. The unit procedure agent sends a report of abnormality to the process agent, the multi-task agent, and commands the other operation agents under it to execute an operation (intra-CGU) level abnormal situation handling procedure. The multi-task agent executes an inter-Unit level abnormal situation handling procedure and triggers a rescheduling if necessary. The multi-task agent sends a report of abnormality to other process agents to execute appropriate intra-Unit and intra-CGU level abnormal situation handling procedures. An entire abnormal situation handling procedure is divided into simple procedures and executed by introducing a layered structure of the cell and the operation management. 5. CONCLUSION In this paper, a scenario oriented method of abnormal situation handling procedure design and a multi-agent based abnormal situation handling operation management scheme are proposed. The proposed scheme seamlessly integrates the safety design and the safety operation management where the abnormal situation handling procedure is constructed three levels, inter-Unit, intra-Unit and intra-CGU level. It simplifies the design and expression of abnormal situation scenarios and handling procedures increases the tractability of operation. REFERENCES 1. ANSI/ISA-S88.01-1995 Batch Control Part 1: Models and Terminology 2. Atsushi Aoyama, Isao Yamada, Rafael Batres and Yuji Naka, Development of Batch Process Operation Management Platform, European Symposium on Computer Aided Process Engineering 10 (2000) 1117 3. Center for Chemical Process Safety (CCPS), Guidelines for safety Automation of Chemical Processes, American Institute of Chemical Engineers, New York, 1993
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
835
Dynamic Cross-Functional Factory-to-Business Links in the Batch Industry Mariana Badell, Diego Ruiz, and Luis Puigjaner Chemical Engineering Department, Universitat Polit6cnica de Catalunya, Av. Diagonal 647, E-08028 Barcelona, Spain The lack of a cross-functional factory-to-business link between the shop floor and the necessary supply chain relation to e-business creates a gap. In order to bridge this gap a webbusiness-plant route is developed in a pilot plant using the TicTacToe sequencing algorithm. A web-based order management system is created to generate optimal plans taking into account the factory logistic status and detailed information of the real plant through a fault diagnosis system (FDS) with re-scheduling capabilities. 1. INTRODUCTION Organizations could no longer ensure profit improvement by cutting costs or filling plant capacities. A better coordination in the use of corporate resources, where cash must be prevalently considered, could be the way to improve production economics. The paradigm is not the same as during Taylor's era (1940-1970). Now corporations must be aware of saving and recovering the money invested during de competitive race. The new paradigm requires to use the information expensively collected to derive economic benefit by make superior business decisions, especially in manufacturing and supply functions. Integration requires the financial-production integration at the planning level in enterprise resource management (ERM) systems [1] and also precise and synchronized information on the state of the enterprise. Not considering beforehand optimal sequences in order management activities is a severe shortcoming punished by important profit discounts during the enterprise lifecycle. During the present decade the enterprise resource management systems achieved a forward development. More than 350 types of MRP-based systems are now available (Enterprise Resource Planning, ERP; Manufacturing Execution Systems, MES, etc.). From its start in the 1990s ERP business arrived approx, to a 50 billion dollars figure monopolised by leading vendors. However, limitations still remain, some of them linked to its dynamical evolution. ERP developers paid more attention outward from their customer's firms to e-business and supply chain management and less concern with connections to the shop floor layer leaving this work to MES developers and plant control systems providers. On the other hand MES developers center its focus downward to the information necessities of the plant floor including the vertical integration functionality, but without a prioritized focus on the link to ebusiness external relations and the supply chain management improvement required in these circumstances (see Figure 1).There is a special gap between the shop floor and the supply chain due to the absence of an efficient and cross-functional factory-to-business link. A disadvantage is that no standard software package can substitute a tailor-made system with appropriate cross-functional links to aid the supply chain's management. In this work a dynamic cross-functional factory-to-business link is developed in a web based order
836 management system. The created transaction-oriented approach supports realistic optimal delivery dates with price-time trade-off during the marketing management. 2. O R D E R M A N A G E M E N T SYSTEMS
Requirements of accurate on time optimal quotations during the electronic commerce are strictly placed in the new paradigm. The repertoire of optimisation techniques used to locate optimal delivery times during the firm's planning activities are frequently laborious, not always arriving to an optimum. Many practical problems were too large to the computer and hence not physically realistic. The TicTacToe algorithm is a proposed solution to those old problems and could be used for the sequence optimisation in on line web-based order entry systems. However, Internet commerce also has changed the demand on pricing from fixed to dynamic. The costing must be online estimated during the scheduling time considering, at least, updated prices of the material supplies. ERM systems are potentially capable of managing dynamically realistic delivery dates and prices during the marketing activities. The production costing data are available simultaneously with the execution of the scheduled plan making possible trade off solutions. The web-based order entry system is developed within the Common Gateway Interface (CGI). A CGI-bin or a servlet program implements a request response control structure using the manufacturing information system. Figure 2 shows the proposed structure [ 1].
Figure 1.Enterprise systems
Figure 2. A web-based autonomous order entry system.
3. SCHEDULING PROBLEM FORMULATION The timing assignment of the initial/final times to units/operations requires numerous calculations. In engineering a detailed timing is certainly necessary but during enumerative search of sequences in order management systems these calculations can be simplified.
837 Figure 3. Timing with N O T i j . Objective function formulation / result, recipes in Table 1. The goal of the scheduling problem is to find a sequence that minimizes the makespan or total production time. This can be seen as a nested decision. First, identifying the 'head' or first element of the sequence. Then, for each selection of the first position or head, finding the associated production sequence that minimizes the not-overlapping times which, in turn, guarantees the strict fulfilment of the SPT - shortest processing time - priority rule. The value of any multiproduct production sequence is given by the processing time of the head of the sequence plus the sum of the not-overlapping times of the elements of the sequence in the generated order. The minimization of the overall not-overlapping time can be formulated as a variation of the Asymmetric Travelling Salesman Problem (ATSP). The ATSP is a classical combinatorial optimisation problem that can be stated as follows: Given a directed graph G = (V, A, W) where V represents the set of nodes and A denotes the set of arcs and W 9A -> R is a cost function on the set of arcs, find a subset of arcs C c A that defines a simple directed cycle through all the nodes of the graph of minimal total cost. A variation of the problem is when a simple directed path through n-1 nodes of overall minimum cost is required. The TicTacToe algorithm is a solution method for ATSP problems [ 1]. The algorithm is an iterative heuristic procedure where each iteration consists of two phases: the constructive phase that generates feasible production plans and the reservation process, where an arc outside the current solution is selected and reserved to be part of all the solutions generated in the subsequent iterations. The Difference Matrix DM is a matrix widely used to solve ATSP with this approach: it is a tool to evaluate in a concise way the deviation from its "optimal" placement for the different tasks of the schedule. For each task, there are two minimum notoverlapping times that must be incurred: one relative to its predecessor and the other one relative to its successor in the sequence. Let Mir=Min{Mij, jr and MjC=Min{Mij, i~j} denote the minimum value in row i, respectively column j, of the MNOT matrix. Four different types of labels are used to classify qualitatively the elements of MNOX: "MM" is the label assigned to elements whose value is minimum both per row and column; i.e. Mij = Mi r and Mij = Mj c. "Mm"/"mM" are the labels assigned to elements with minimum value in their row/column but not in their column/row; and "mm" are not minimum neither for their row nor for their column. These qualitative labels constitute the r o a d m a p matrix that can be used as a p r i o r i t y d r i v e r in the construction phase (priority first to "MM", "Mm" or "mM", and "mm" last). Among the elements with the same type of label, higher priority is given to those with smaller matrix value MNOTij. The DM is a quantitative road map matrix since for each cell (i,j) its value d m v represents the increase in "time cost" of producing a product i before a product j instead of the product located in position k (minimum value in row i) before the product located in k' (minimum value in column j). The sequence is constructed in a greedy fashion. Arcs are included in the solution one at a time using an ordered list that reflects the priority relative to the considered criteria. In the first version of the algorithm the qualitative road map matrix was used as a pointer for the construction. Improved versions use the DM matrix. Let L denote the list of elements of MNOr ordered by increasing values of dmo. The construction phase can be formally represented by: PATH=k~,, Enter(i) =false i= 1..... n Leave(i)=false, i=O, .... n While L # ~ do Let (a, b) be the next element of L I f P A T H ~ (a,b) contains no subtours and Leave(a)=false and Enter(b)=false then PATH: =PA TH t~ (a, b)
838 Leave(a) =true Enter(b) =true Endif L:=Ll(a,b) End
The elemems of the ordered list are included in the sequence provided that: a) no sub-tours are created, b) each product is "preceded" by, at most, one differem product (at most, one cell per column is selected), and c) each product "precedes", at most, one different product (at most, one cell per row is selected). However, at the last step of the construction, the step, the n-1 th last element to emer the solution is not optimised since this element is pre-determined by the only position left free. For this reason it is referred as to the obliged element. The reservation phase is focused to substitute the obliged elemem by some arc (outside the currem solution) with a "higher" quality. The elemem that will substitute the obliged elemem is called reserved elemem, since in the subsequem iterations of the algorithm the elemem is preserved. Candidates to be selected as reserved elemems follow a selection process: Let (ao, bo) denote the obliged element of the currem solution. If (ao, bo) were "substituted" by an arc (ao, b), in the process to recover the feasibility of the solution the arc (Pb, b) will necessarily have to be removed (Pb denotes the predecessor of product b in the currem solution). Thus, the variation of the objective function, relative to the difference matrix, in order to recover feasibility if (ao, bo) were "substituted" by (ao, b), would be at least A (ao, b) = dmaobo+ dmpbb- 2dmaob. This variation refers to inserting the arc (ao, b) and removing the arcs (ao, bo) and (Pb, b). Similarly, if (ao, bo) were "substituted" by (a, bo), would be at least A (a, bo) = dm~obo +dm~,s~- 2dm~.bo. Now, the variation occurs for inserting the arc (a, bo) and removing the arcs (ao, bo) and (a, Sa). The list of elements that are candidates to be the reserved elemem is then defined as: CL(ao, bo) = {(ao, b) : A (ao, b) > 0 } {(a, bo) : A (a, bo) > 0}, and the reserved elemem is the elemem with a higher value of A (e.g. a positive figure means a possible contribution to the minimization of the function). That is, CL comains all the elemems with the same head or tail than (ao, bo) that, if interchanged with the obliged elemem, insure local improvemem, while the reserved elemem is the one with the largest local improvemem. The best result of the differem iterations is taken as the solution to the problem. RES=~ While not end do Apply Construction phase Apply the reservation phase and Identify the reserved element (an, be,) RES := {RES} ~ {(an, b~)] End The complexity of the construction phase is O(n 3) plus the time needed to construct DM and sort its elements (which is dominated by O(n3)). The complexity of the reservation phase is linear on the number of elements of the candidate list CL that is at most 2(n-1). Variable
depth search has been successfully used in the solution of different types of optimisation problems. This gives the procedure a higher level of flexibility. The ordered list L derived from the DM matrix provides an efficient way to solve the trouble of recovering feasibility when the set of reserved cells has increased. The inner iterations of the reservation loop are: Let Sk be the solution obtained at the constructive phase of iteration k, Order elements of Sk by decreasing values of DM and include them in an ordered list 0 t=O (t: counter of inner iterations) Let (ao, bo) be the first element of O. BuiM CL(ao, bo) stop =false (the inner iterations stop when removing more cells does not lead to an improved solution)
839 While CL(at, b~) ~ ~ a n d stop=false do Select from CL(at, b~) the element with higher value AA (aRy, bRy) Rt := Rt ~ {(aRy, bRy)} Apply constructive phase with set of reserved cells Rt Let Skt denote the new solution I f value Skt < value S/-1 t := t+l Let (at, b~) denote the next element of O else stop "= true endif Endwhile R "= Rt
Some preliminary testing has been done using the TicTacToe algorithm. Programs have been coded in C++ and run on a 400 Mhz PC, 128 RAM. A series of 36 problems has been randomly generated following a uniform distribution in [1,25]. Exact optimums have been obtained initially by a MILP formulation. The percent deviation with respect to the optimal value (value-opt) / opt for all the problems searching candidates is 1.004833 and only for 12 instances the optimal value is not found, deviation is around 1%. Also, the effort required to obtain these results is acceptable. However, the reservation phase must be refined making intelligent exploration using the inner results obtained to bound local search. 4. INTEGRATION OF THE PLANNING AND SUPERVISORY CONTROL LEVELS The proposed Fault Diagnosis System consists in a combination of a pattern recognition approach based on neural networks and an inference system based on fuzzy logic. A set of suspected faults has been identified for the multiproduct plant described in Section 5. They correspond to different faults in each of the pieces of equipment that implies a major processing time. Details of FDS implementation have been recently reported [2]. When a deviation from the predicted plan in a multiproduct batch plant is diagnosed, the FDS activates the planning module (using the TicTacToe algorithm) to minimise the effect of this deviation on the remaining plan. According to the client orders, an optimal plan is generated with the TicTacToe algorithm. The co-ordination level converts the plan in the set of control actions. Once the plan is running on the real or simulated plant, the control and supervisory system communicates to the planning system the deviations detected from the proposed plan. With this information, the planning system generates new information that is sent back to the control system. This feed-back closes the loop between the planning level and control system. 5. CASE STUDY In this case study a w e b - b u s i n e s s - p l a n t - b u s i n e s s - w e b route is developed using the TicTacToe non combinatorial sequencing algorithm to generate optimal solutions taking into account the factory logistic online status and updated financial, production and plant information. Figure 3 shows the flowsheet of the pilot plant utilized for the case study. The plant consists of three tank reactors, three heat exchangers and the necessary pumps and valves to allow changes of process configuration. Equipment of this plant is fully interconnected and the instrumentation allows configuration changes by software. The case study contemplates the use of two reactors and the respective pumps, in order to obtain ten similar products. Table 1 shows the recipes of the products and the asymmetric TSP schedule
840 formulation by the matrix of not-overlapping times MNOTof the schedules of all the possible pairs of products considered in the plant shown in Figure 4. Table 1. A batch process" Matrix MNOTobtained by using ten recipes. 0 26 26 MNOT= 26 26 26 26 26 26 26
26 0 26 26 26 26 26 26 26 26
27 27 0 27 27 27 27 27 27 27
27 27 27 0 27 27 27 27 27 27
28 28 28 28 0 28 28 28 28 28
29 29 28 28 28 0 28 28 28 28
31 31 30 30 29 29 0 29 29 29
32 32 31 31 30 30 29 0 29 29
33 33 32 32 31 31 30 30 0 30
35 35 34 34 33 33 32 32 31 0
Recipes of products I to 10. Products
i=l
i=2
i=3
i=4
i=5
i--6 i=7
i=8
i=9 i=10
Pump P0,j=I
10
10
10
10
10
10
10
10
10
10
Reactor R l, j=2
3
4
4
5
6
7
8
9
9
10
Pump PI,j=3
10
10
10
10
10
10
10
10
I0
10
Reactor R3, j=4
6
6
7
7
8
8
9
9
10
11
Pump P2, j=5
10
10
10
10
10
10
10
10
10
10
Process time, min
39
40
41
42
44
45
47
48
49
51
Figure 4. The flowsheet and optimal plan (1,4,3,2,5,6,7,9,10,8, makespan 294 min, 252 iter.) 6. CONCLUSIONS A simple strategy for a web-business-plant connection between the shop floor and a sequence optimiser in a multiproduct batch plant has been presented. When a new plan is developed to fulfil the customer orders received through the web, it is considered the actual status of the ongoing plan, which also includes the FDS amendment actions taken into account at previous time. This allows realistic and efficient plant performances to support the dynamic of the manufacturing market. Real time optimisation (RTO) problems and integrated systems are waiting for and must foster improved solutions to old, complex, and unsolved problems. New approaches with trade-off scopes must be developed to fill the gap between technology and science. The TicTacToe algorithm can be suitable for a wide range of RTO problems. Financial support from the European Community is gratefully acknowledged (projects IC18-CT98-0271, WATERMAN, and IMS 26691, Global Cape Open). REFERENCES 1. M. Badell, E. Fem/mdez and L. Puigjaner, 1st World Conf. on Production and Operations Management, Sevilla, Spain, 2000. 2. D. Ruiz, J.M. Nougu6s, J. Cant6n, A. Espufia and L. Puigjaner, Computer Aided Chemical Engineering, 8 (2000) 745.
European Symposiumon ComputerAidedProcessEngineering- 11 R. Ganiand S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rightsreserved.
841
Ad-hoc Scheduling/Planning Strategies in Distributed Computing Systems: An Application to Pipeless Batch Plants Jordi Cant6n, Alexandre Afonso, Mois6s Graells, Antonio Espufia and Luis Puigjaner Chemical Engineering Department, Universitat Polit6cnica de Catalunya E.T.S.E.I.B., Avda. Diagonal 647, 08028-Barcelona, Spain*. The present work introduces a flexible scheduling framework based on a previously consolidated scheduling package. The core of the framework consists in a system co-ordinator that enables the proper actuation and communication of the different operating modules. The basic set of components should include a data module based on a hierarchical recipe representation, a timing module, a decision-making module and a series of tools for user and system interface. Over this framework, generic and/or specialised components may be incorporated, mainly in order to solve the basic decision-making associated to the scheduling problem. In the presented application, a dedicated component has been implemented and integrated in order to deal with the specific situations appearing in the scheduling of pipeless batch chemical plants, where the unloading, transport and loading of intermediates become the basic bottlenecks and their associated constraints are complex -even infeasible- to introduce in generic scheduling systems. 1. INTRODUCTION A general scheduling tool is an extremely ambitious objective very distant of the capabilities of present technology. Production facilities tend to be very particular because company policies, objectives and constraints are very specific and of different nature. Specific constraints in batch chemical processing made this case even harder. For these particular aspects, scheduling methods need to be adapted and user interfaces rebuilt each time a new application is needed. However, the modelling of batch processes may follow a common framework and share the same basic simulation procedures, user interfaces and data organisation and management. Consequently, most of the computer code developed for the general case should not be rewritten each time a new application is required, but reused. This is one of the main advantages claimed by Component Technology. Distributed components and distributed computing have the recognition of being a way to reduce software complexity and cost as well as to increase software flexibility and extensibility. * The financial support from the the European Community (project Global-Cape-Open-IMS 26691) and the Generalitat de Catalunya (through CeRTAP - Centre de Refer6ncia en Tecnologies de Producci6") are thankfully appreciated.
842 The present work introduces a flexible scheduling system based on this idea and built from previously consolidated scheduling package (Multipurpose plants Optimisation and Production Planning - MOPP) [1]. An effort has been required to migrate from the original monolithic application to a system of integrated components. However, the benefits have been readily harvest when addressing new scheduling problems. 2. STARTING POINT: THE MOPP PACKAGE In the study presented in this paper, the MOPP generic scheduling tool has been used as the basis to develop a flexible scheduling framework. The original tool included, among others, the following facilities: 9 An open modelling framework for batch / semicontinuous processes, using a continuous time representation system. 9 Optimisation algorithms to be used to calculate the optimum timing associated to a given sequence of interrelated tasks. 9 Support for tracing the use of common utilities, and alternative heuristics to detect and solve conflicts in the use of such utilities. 9 Intuitive GUI and tools for introducing manual changes based on the extensive use of the Gantt Chart representation. 3. THE RESULTING OPEN SCHEDULING F R A M E W O R K From a detailed analysis of the day-to-day operation of any scheduling system, the basic information flows have been identified, allowing the isolation of its basic functionalities. This has deal to a new architecture design, which can be identified as "generic open scheduling framework". The framework is based in a system co-ordinating the different modules: the MOPP Open Scheduling Optimisation Executive (MOSOE). This system centralises all the messages and orders and generates all decisions affecting the way of working of the different scheduler components. The basic set of components include a data module based on a hierarchical recipe representation following the ISA $88 specifications and a timing module based on the Event Operation Network (EON) model [1 ]. A graphical user interface (GUI) including an Electronic Gantt Chart allows the user to edit all the data and to use all the timing capabilities. Additional communication modules allow interaction with other pieces of software, either via ActiveX/COM interfaces, or using TCP/IP protocol. Consequently, the basic system has a great capability for being customised and for accepting new calculation features. Although the functionalities of most of these components were already incorporated in the original application, in most cases they have been completely rewritten, since the new open approach, operation procedures and framework, jointly with the new standardisation requirements, introduced radical changes in the way these functionalities should be fulfilled. 4. PIPELESS BATCH PLANTS: SPECIFIC P R O B L E M ANALYSIS Although they have received minor attention in the literature, pipeless batch plants constitute an interesting solution to minimise the amount of fixed piping, increasing production flexibility and reducing the complexity of cleaning requirements. The main idea is to execute material transportation tasks with movable vessels conveyed by AGVs (Automatic Guided Vehicles).
843 The production scheduling is a central part of pipeless batch plants operation and it is significantly different from conventional production scheduling which usually only considers the time, the lines requirements and costs. In a pipeless batch plant, the combinatorial nature of scheduling is increased by the additional problems related to AGVs assignment and routing. The plant configuration and the equipment installation become inseparable factors of the production scheduling problem. Although in principle, transport tasks can be considered in a similar way to the rest of the production tasks, they present some characteristics, which would require specific adaptation of the modelling system in order to be correctly considered during timing and scheduling: 9 Transport time: although the time required to carry out a transport task depends only on the task characteristics and the used processor, the "preparation time" (time required for arriving to the departing point) varies in a continuous way with the Processor State. Adequate formulation of this time dependency is required in order to reduce the information to be supplied to the scheduling system. 9 Special treatment should be considered to other vehicle characteristics. Among others, it deserves special consideration the existence of different vehicles, which might exhibit different characteristics in terms of speed, degree of action, manoeuvrability, materials admissible for transportation, etc. 9 Additionally, the use of Automatic Guided Vehicles imposes specific constraints that should be conveniently introduced. The most important of these are related with vehicles autonomy: Assuming that the vehicles are powered by batteries, a model of battery consumption in terms of the operation time of each vehicle should be considered in order to control battery load. Whenever low battery levels are detected, the proper charging task should be introduced in the vehicle schedule. Obviously, the models used can be easily improved to take into account, for example, the effect of the vehicle load on the battery consumption, or other effects that might be considered significant. 5. SPECIFIC SOLUTIONS FOR SPECIFIC PROBLEMS To deal with this problem, the production scheduling has been divided in two different stages, the production tasks scheduling and the transportation tasks scheduling. The production task scheduling determines the best allocation to equipment of each product and the timing of all production activities involved at each station. Once a proper solution has been devised for the production activities, the second step consists on checking the feasibility of this solution when imposing the AGVs constraints. A specific module has been developed to handle the additional constraints given by transportation tasks. Therefore, the generic system is used as the simulator module supporting the ad/hoc scheduling strategies and optimisation algorithms developed for AGV assignment and routing.
5.1. Transport task scheduling The specific module consists of four main steps, with can be integrated to form the basic strategies to find a transport-feasible schedule: 1- Identification of material/station transportation tasks; 2- AGV/stations assignment; 3- Routing; 4- Optimisation. Once a list of all the production operations are obtained from an initial (probably unfeasible) schedule, all the input and output operations can be identified as well as the
844 corresponding amounts of materials to be transported. Then, the number of transportation tasks (MOTT- MObile Transportation Tasks) needed to perform all the material transfers N 1L , ) being N needed can be easily found using the following expression: M O T T = ~ " ( M i + i=l 2 ' the total number of transfer operations, M i the number of material transfers to storage of operation i, and Li the number of material transfers to another operation of operation i. The AGV/stations assignment (step 2) fits each material transport task to an available AGV using different rules: 9 Simple charge constraints forcing to assign for one MOTT only AGV with a nominal capacity big enough to transport the amount of material needed. Qn > Qi
9 i = l ... n.~ o f M O T T s
In the limit, where all the nominal capacities are bigger than the charges of material to be transported, the problem is reduced to a Sequential Vehicle Rule -SVR [2]. 9 A greedy adaptative algorithm based on dispatch rules used with the SAGVs [2]; in this work three different dispatch rules were used: - Next Work Same Vehicle Unit (NWSVU) rule - Shortest Travel Time/Distance (STT/D) rule - Least Utilised Vehicle (LUV) rule The next step (step 3) is the routing of the AGVs. The time needed for the transportation tasks is determined using a distance matrix as well as the nominal speed for each AGV. A transport task is defined consisting in the travel operations (to reach the charge station and then to go to the discharge station), the charge and discharge operations and waiting times. Once all the assignment and time information for the transportation tasks is defined, the following step is to send this information back to the main schedule package and recalculate the whole timing involving the transport tasks. The information sent to the generic scheduling system includes the tasks to be performed with the times and assignments and the temporal constraints between the transport tasks and the production tasks. Each charge of a transport task is related to a discharge operation of a production task and vive-versa. Temporal constraints should be defined between these operations to assure the simultaneity of the beginning times (BT) and ending times (ET) of these operations" op
agv
These equations assure the simultaneity in the time of each pair of operations. And provides a new feasible schedule including the transport tasks. Finally, the resulting schedule can be optimised using a simple simulated annealing algorithm (SA) which explores other assignment combinations to find a best AGVs assignment that reduces the whole makespan or any other objective function defined.
5.2 Implementation A specific module has been developed to communicate to MOPP the additional constraints given by AGV assignment and routing 9 Therefore, MOPP is used as the simulator module supporting the ad/hoc scheduling strategies and optimisation algorithms developed for AGV assignment and routing 9 The module developed is written as a set of spreadsheet macros connected to the framework via a TCP/IP Client (ActiveX DLL). An initial production plan is
845 obtained from the standard decision-making tools available from the original optimisation system, and interactively processed by the client module. 6. CASE STUDY In this example two batches of two products (A and B) are produced following the flowsheets shown in Fig. 1 and Fig. 2.
y
II A
~.._ _ _
I
IIEA1
Rs
t
ORvl
l~XCl
I [-_.i-,:i-o.=, V
.
.
.
.
r___j
OllVl
7, o,~,
__..J Esquema de elaboraciOn del productoB
Fig. 1: Product A.
Fig. 3: Schedule without transport tasks.
Esquema de elaboraciOn del producto A,
l
Fig. 2: Product B.
The first step of the algorithm involves the creation of a schedule without the transportation tasks (Fig. 3). Once the assignment is performed a second feasible schedule including the transportation tasks is obtained (Fig. 4). Finally, after the application of the optimisation procedure, a best AGV tasks assignment is found improving the makespan from the initial value of 21,9 hours to 17,7 hours (Fig. 5).
7. CONCLUSIONS This work shows the potential of Component Technology in the area of Batch scheduling. The time for developing a new application has been significantly reduced by reusing general components providing basic calculation modules and focusing in the development of the code for including specific constraints and ad-hoc strategies. General components have been always kept as black boxes (complied DLLs), which seems to indicate a very interesting future to both proprietary and open component systems. The communication between all these tools is established by means of COM technology, thus constituting a hybrid distributed model of the decision-making system.
846
Fig. 4: First feasible complete schedule
The specific situation used as case study (Pipeless Batch Plant) demonstrates how a production scheduling system able to consider the most relevant characteristics associated to transport tasks has been easily developed. Once the main restrictions associated to these tasks (number, location and characteristics of available AGV, etc.) have been introduced, the system simultaneously optimises production and transport tasks schedule, taking into account the overall production policies and a common objective function. Decision making related with the transport system is then integrated and co-ordinated with the overall plant decision-making. The system is easily integrated with the rest of information, communication and control systems required to the management of the AGV system, so in case of plan deviations due to unexpected incidences, a new schedule, optimised according to the newest resulting scenario, can be on-line calculated and transferred to the plant-floor.
Fig. 5: Optimized schedule The use of this system can meaningfully improve the operation of both transport and production resources especially if both of them can be limiting. It can be also used to improve/optimise the design of the transport system. Additionally, the system can be used for planning and scheduling of actions in other cases where productive tasks are associated to moving processors. This is the case, for example, of automatic cleaning devices (including "Cleaning In Place" systems) and other maintenance services, as well as robot based collection of fruits and other products in the agricultural sector. REFERENCES
1. M. Graells, J. Cant6n, B. Peschaud and L. Puigjaner, Comp. Chem. Eng., 22S (1998) 395. 2. P.J. Egbelu and J.M.A. Tanchoco, Int. J. Prod. Res., 22 (1984), 359.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
847
Dynamic Modelling and Scheduling of an Industrial Batch Digester Cooking System P. Castro a, A. P. F. D. Barbosa-P6voa *b, H. Matos a B. Duarte c a Departamento de Engenharia Quimica, Inst. Sup. Trcnico, 1049-001 Lisboa, Portugal bCentro de Estudos de Gestao, Instituto Superior Trcnico, 1049-001 Lisboa, Portugal cCompanhia de Celulose do CAIMA - Const~ncia, Portugal
The optimal scheduling of a resource constrained four-batch digester system of an industrial acid sulphite pulp mill is addressed. This involves the development of two different models, one to model the scheduling operational problem and the other the batch digester operation - process model. The first model uses the Resource Task Network (RTN) representation leading to a Mixed Integer Linear Program (MILP), formulation. In this, the main operational limitation, steam availability, is modelled through the definition of a superstructure including all possible heating alternatives. The duration of these heating tasks was estimated through the use of the process model - a distributed heterogeneous dynamic model in gPROMS- validated with experimental plant data. The optimal schedule obtained, features a digester sequence different from the one in use in the plant and allows a higher level of productivity. 1. P R O B L E M A N A L Y S I S
The pulp cooking process considered consists of a set of four parallel batch digesters with different capacities. These process wood chips and chemicals and produce pulp in aqueous suspension. A batch on a particular digester consists of an ordered sequence of various operations: i) chip filling; ii) acid filling; iii) heating; iv) cooking; v) high pressure degassing; vi) low pressure degassing and vii) blowing the contents into a storage reservoir, blow-tank. At the plant the heating task is the process bottleneck. There is limited steam availability and as a consequence waiting periods during this task are frequent. To achieve high quality pulp, the temperature rise in the heating task is subject to some constraints, regarding the use of steam. As the amounts to use are dependent on the digester size, the final dil~ester sequence will affect the makespan. The digesters share other resources besides steam, such as the chip conveyer; acid-filling pumps; high pressure and low pressure degas tanks and the blow-tank. Thus, two different digesters cannot use one of these resources simultaneously. The scheduling model must be able to consider these common resource limitations, and determine the start and completion times of all tasks of the cooking process that minimise the makespan. 2. P R O B L E M
FORMULATION
The optimal scheduling of the cooking system in study involves the definition of two different types of models: a scheduling model and a dynamic simulation model. The latter *Corresponding author. Tel: +351-218419014. Fax: +351-218417638. E-mail: [email protected]
848 estimates the necessary parameters to be introduced in the first model so as to obtain the optimal system scheduling.
2.1. Scheduling model The general Resource Task Network, RTN, representation (Pantelides ~) is used to model the scheduling problem. The mathematical formulation is based on a discrete representation of time (similar to the one of Kondili et al. 2) where the time horizon is divided into a number of intervals of equal and fixed duration. All system events are forced to coincide with one of the interval boundaries. The formulation gives rise to a MILP problem. 2.1.1. RTN representation The RTN representation regards all processes as bipartite graphs comprising two types of nodes, resources and tasks. Each task is an operation that transforms a certain set of resources into another set. On the other hand, the relevant resources are classified within two different types based on their functional equivalence. The common resources, shared amongst the digesters, set R; and the individual resources, representing the possible material states in the process, set S. The latter resources and the process tasks, set K, must be referred to a given digester, an element of set D. The RTN of the cooking process is represented in Figure 1. The first task (chip filling) consumes the first state (digester empty) at its beginning and produces the second state (digester filled with chips) at its end. This state is then consumed by the second task (acid filling) and so on, until the cycle is concluded by the last task (blowing), which produces the first state at its end. In Figure 1 the heating task is simplified. To describe the complex interaction between the digesters in this stage a superstructure was derived which includes all possible heating alternatives (Figure 2). Thus, task K3, from Figure l, was replaced by 4 tasks HO(i,j) and 4 tasks Hl(i,j) with ij~D. Tasks HO precede the tasks HI. Both use all the available steam, and so this resource can be considered as a discrete resource. The first stage of the heating task for the first digester in the sequence is represented by the task HO(i, i). Digester i, can then finish its heating stage, with one of the three H1 (i,j) tasks, with i,j. These tasks implicitly consider the heating of digester j with the remaining steam of digester i changing the states of both digesters i and j. As the digesters have different steam consumption, different states in digester j are produced from different digesters. Three new states are defined - $5 to $7 - representing different temperatures for the pulp suspension. After task H1 (i,j) digester j concludes the first stage of heating with task HO(j, i), producing state $4. This process continues until the last digester (/) of the sequence ends the heating requirements (in the last production cycle) with task Hl(l, l).
849
Figure 1. RTN representation of the cooking process (heating task, K3, simplified)
Figure 2. RTN representation of the heating stage. 2.1.2. Mathematical formulation
The short term scheduling of the cooking system is achieved by the following constraints: Excess resource balances: the excess amount of a resource at a given time interval t is equal
to that at the previous interval adjusted by the amount consumed by all tasks starting at the
850 beginning of t and by the amount produced by all tasks starting at a previous interval and ending at t. One set of constraints is required for each type of resource. l'k,d Rr, t = Rr, t-I + Z Z Z ] ' l k , d , r , o N k , d , keK deD O=O
'-0 V r E R , t E r
(1)
~'k,d
S ......, = S ......,-' + Z Z ~_~vk,a .....i,oNk,,,,_o Vs ~ S,i ~ D,t ~ T
(2)
keK deD 0=0
In the above equations the parameters Pk, d,r,o and Vk, d,s,i,o represent the amount of resource r (s in digester i) produced or consumed by task k of digester d at a time 0 relative to the start of task k, while the parameter rk, d represents the fixed duration of task k in digester d in time units. The excess variables Rr, t and Ss, i,t are continuous positive variables while the variables Nk, d,t are binary ones. The latter represent the extent of task k of digester d at interval t, an element of set T. Extent constraints: The number of times a given task is to be executed is dependent of the number of production cycles to be studied, Ncycles. Although the blowing task is the last task of the cooking process, we consider that each cycle ends with the heating stage of the last digester in the sequence. The following constraints apply: Nk,a, , = Ncycles Vd e D , k e K , k < 2 (3) teT
~ Nk,a, , = N c y c l e s - 1Vd ~ D,k ~ K , k __4
(4)
t~T
~.,
~ Nk,,, , = Ncycles Vd ~ D
(5)
keK,k=HO t~T
~ Nk,a, , = Ncycles Vd ~ D
(6)
k~K,k=H1 teT
Z
Z Z Nk,a., =1
(7)
k~K,k=HO(d,d) d~D t~T
Z
ZZNk,a, ,=1
(8)
k~K,k=Hl(d,d) deD t~T
Objective .function." the goal is to minimise the make-span of the given number of batch cycles. As the last cycle ends with one of the HI (d, d), diD tasks, and only one of these tasks is executed (equation 8), the objective function can be defined as: min
E
EE
Nk,a,, (T, + rk,a)
(9)
kaK,k=Hl(d,d) deD teT
where Tt is a parameter representing the absolute starting time of interval t. 2.2. Process dynamic simulation model For the scheduling model described above the duration of all tasks needs to be known before a solution can be reached. For all tasks except heating, the necessary values were collected from plant data. Since the real plant only works with one digester sequence, D 1-D4D2-D3, the heating stage parameters for the other possible sequences needed to be estimated. This was accomplished through the development of a distributed heterogeneous dynamic model for each digester/heat exchanger system, solved in gPROMS. Each model was
851 validated with real plant data, and all possible sequences were simulated, providing the heating tasks duration. Due to the lack of space this model description is omitted. 3. RESULTS The model was solved for two different scenarios, one constrained to the current sequence used in the plant and the other without this constraint. For the first scenario only the path corresponding to the actual sequence was allowed. This was accomplished by fixing the extent variables of all other heating tasks to zero, in every time interval. The solvers OSL and CPLEX solved the MILP models in a Pentium III-450 MHz for two and three production cycles. Some statistics are shown in Table 1. The results of Table 1 clearly show that the constrained model (Scenario l) is solved faster. This is due to the lower number of integer variables of the model and to its structural simplicity caused by the use of a fixed operational sequence. As the objective is to minimise the makespan, the number of time intervals to use is not known a priori. A small number of intervals turns the model infeasible while an excessive number makes it very difficult to solve. The solving strategy used was to address first the constrained model with a reasonable number of time intervals, adjust the number based on the value of the objective and then solve the unconstrained model. Scenario 2 has a better objective value than Scenario 1. In the optimum digester sequence, D 1-D2-D3-D4, the available steam is shared more efficiently amongst digesters, which allows for a faster completion of the number of production cycles studied, when compared to the actual sequence, D1-D4-D2-D3. This allows an increase of production of 1.7 %. The associated optimal schedule (Scenario 2, Ncycles=3) is presented in Figure 3. Table 1. Computational statistics and results for the scheduling model scenarios Ncycles=2 Ncycles=3 Scenario 1 Scenario 2 Scenario 1 Scenario 2 Time intervals 252 235 374 352 Integer variables 14372 17146 20968 25314 Continuous variables 26819 25017 39751 37419 Constraints 12631 11781 18731 17631 Obj. relaxed MILP (min) 1260 1255 1860 1845 Objective MILP (min) 1260 1255 1860 1845 CPU (s) (OSL/CPLEX) 19/834 93/2284 35/2263 1966/6550
852
Figure 3. Optimal schedule for the unconstrained model (Scenario 2, Ncycles=3). 4. CONCLUSIONS The problem of finding the optimal schedule of a batch digester system with an insufficient availability of steam as a heating medium was addressed. A discrete time RTN based mathematical model was developed to model the scheduling cooking process and a superstructure was created to accomplish the several heating altematives. This was coupled with a distributed heterogeneous dynamic model of the process operation that allows the determination of all the required process data describing each possible heating altemative. As final result, the optimal schedule obtained was characterised by a different digester sequence than the one in use at the real plant representing a higher production rate. This paper shows the importance of combining different types of models to describe the plant reality. This was a first approach to solve a real problem and further improvements to the scheduling model have been undertaken by the authors so as to generalise the presented model by allowing a higher flexibility to other real plant aspects. REFERENCES
1. C.C. Pantelides, Proc. 2 nd Conf. Foundations Comp. Aided Operations, (1994) 253. 2. E. Kondili and C. C. Pantelides, Comput. Chem. Eng. 17 (1993) 211.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
853
Application and Evaluation of Linear/Restricted Nonlinear Observers to a Nonlinear CSTR S. Dash a, S. Kantharao b, R. Rengaswamy b and V. Venkatasubramanian a aSchool of Chemical Engineering, Purdue University, West Lafayette, IN 47907, USA bDepartment of Chemical Engineering, liT Bombay, Powai, India Analytical model-based diagnosis exploits the functional redundancy inherent in a system model for the design of fault-sensitive systems robust to noise/disturbances. Here, the theory of two unknown input observers, one linear and the other, a restricted non-linear are outlined and built for a non-linear exothermic reactor. A scaling matrix is used to make the residuals reflect actual fault severities. We present the performances of the observers in terms of diagnostic accuracy for single/multiple fault scenarios as well as their robustness to noise. 1. INTRODUCTION Process fault diagnosis is the difficult problem of real-time diagnosis of malfunctions in a process given sensor data and process knowledge. It is important not only because of its safety impact but also process economics [1]. The increasing complexity of plants and the development of sophisticated control systems have necessitated the parallel development of fault detection and identification (FDI) systems with superior performance. Diagnostic methods vary according to the kind and form of knowledge used [2]. Redundancy-based diagnostic methods utilize hardware or functional/analytical redundancy. Hardware redundancy uses redundant sensors which may prove to be expensive or infeasible. Analytical redundancy (AR) refers to the inherent redundancy contained in the set of static/dynamic (algebraic or temporal) relationships among system inputs and measured outputs. There are a variety of methods [3] proposed to take advantage of the AR such as observers, parity-space techniques, parameteridentification approaches. While the theory of model-based diagnosis is quite well-developed for linear systems [4], application to chemical processes has been limited because of unavailability/complexity (non-linearity) of most of the models [5]. One way to use AR is to design observers that are robust with respect to disturbance/unknown inputs. This paper deals with two observers - a linear and a restricted non-linear. Section 2 provides a brief review of the key concepts in AR and also develops the theory of the two observers. In Section 3 we evaluate
854
Fig. 1. Basic Observer Scheme
the performances of both these observers on a CSTR case study. We end with conclusions in Section 4. 2. ANALYTICAL REDUNDANCY (AR) - OBSERVERS The basic idea in AR-methods is to compare the actual behavior of the system with a nominal fault-free model driven by the same inputs, using residuals. Residuals are functions accentuated by faults i.e., they represent the inconsistency between actual plant variables and the model. The two main stages in an observer design are (1) Residual Generation and (2) Residual evaluation. The first task concerns generating the residual vector under unknown fault modes, uncertain nominal model and system/measurement noise. The second stage analyzes residuals (statistical testing) in order to arrive at a diagnostic decision (fault isolation). Observers: The technique consists of reconstruction of the outputs with the aid of observers/ kalman filters and using the estimation error/innovation or a function of it as the residual. The standard structure of a linear diagnostic observer of full order is shown in Figure 1. The actual output vector y is compared with the output vector 3~of the nominal model and the difference r is fedback with the feedback gain matrix H to compensate unmatched initial conditions and provide design freedom. The residuals should (1) detect and uniquely identify different faults and (2) be robust i.e., invariant to unstructured uncertainties such as process/measurement noise and modeling uncertainties. Next we present a very simplified design of a set of unknown input observers (UIO) each sensitive to a set of faults while being insensitive to the other faults/unknown inputs. Our main aim here is to evaluate their applicability. Frank [4] presents the necessary and sufficient conditions for observer design in the general case. Linear Observer(LO): The linear observer model [4] assumes the mathematical model of the system to be given in the discrete form as Xk+1 = A x k + B U k + E d k + K f k ;
yk=CXk+Fdk+Gfk;
Xk=O=XO
(1)
where, x is the state vector, u the input vector, y the vector of measured outputs, d the unknown inputs/disturbance vector, f the fault vector and A,B,C,E,F,K,G are known matrices of appro-
855 priate dimensions. Ed models the unknown inputs to the actuators and to the dynamic process, Kf: component and actuator faults, Fd: unknown inputs to the sensors and Gf: sensor faults.
One form of the observer is given by the equations Zk+l : Rzk + Syk + Juk;
rk -- LlZk + L2Yk;
Zk=0 = z0
(2)
where Zk = TXk, iff fk : 0 after the system transients have died out and steady-state has been achieved. This observer is valid for fault detection if the residual rk has the following properties lim rk : 0 iff fk : 0 and rk ~ 0 iff fk ~ 0 'V'U,d, x0, z0
(3)
k-+oo
For the residual to have the required properties we need [4] TA-RT:SC
J-TB
LIT+LzC=0
LzF - 0
SF = 0
TE = 0
LzG ~ 0
SG ~ 0
TI~ # 0
(4)
Also, if fk ~- 0, then it can be shown that fk/rk ----L1 ( S I - R ) - I ( S G -
T K ) d- L2G. For the
residuals due to an initial mismatch in the states of the system and the observer (Xk:0 ~ Zk=0) to go tO zero, we require R to have eigenvalues in the left half of the s-plane. One way to ensure complete decoupling of the fault effects is to have (1) R diagonal, (2) L1 = ( S G - TK) -1 and (3) LzG diagonal. Restricted N o n - L i n e a r Observer(RNLO): A partially non-linear [4] observer can be devel-
oped for a certain class of non-linear systems which can be cast in the form Xk+l : Axk + B(yk, Uk) + Edk + K(Xk)fk;
Yk : CXk + G(Xk)fk
(5)
From these equations, we see that the extraneous inputs may not appear in the output equation, the dynamics equation may be non-linear only in output and input variables, and the input distribution matrix of the fault vector may only be a function of the system states. The observer is given by the equations Zk+l : Rzk + J(Yk, Uk) -~- Syk;
rk : LlZk + L2Yk;
Zk-0 = z0
(6)
where Zk = Txkiff fk = 0. As for the linear case (Equations 4), we get the same conditions except that now J(Yk, Uk) : TB(yk,Uk), TK(xk) =/= 0. Further, the transfer function fk/rk is also same except that now L1 = (SG - T K ( x ) ) - I implying it must be computed at each instant. A better performance can be expected if K(x) in Equation 5 is replaced with K(x,u) i.e., terms such as
5uiSfj
are then included in design.
S c a l e d / N o r m a l i z e d Residuals: Nonzero values of the raw residuals in the observers
signal
faults. If a suitable scaling matrix is used in the residual equations for r~ (Equations 2, 6) the scaled residuals can be made to reflect the actual fault percentage levels. Such a scaling matrix
856
can be constructed with the no-fault values of the fault variables Foi ,
as
a diagonal matrix Sc
where the Sei = -lO0/Fo,. Thus Equations (2,6) are modified to rk = Se(LlZ + L2y) where rk = % fault levels. Also, for fault detection, ri are normalized by the maximum of the values (max/
ri)
to generate normalized residuals
Ri. Ideally we want the faults to trigger the corre-
sponding Ri (:k:l) while restricting others to 0. Perfect isolation is seldom achieved in reality so we use a threshold Rth to detect faults. Thus rk estimates fault levels and Ri (along with Rth) detects faults. 3. CSTR CASE STUDY In this section we design and evaluate the observers for a CSTR case study [6]. A first-order exothermic reaction takes place in the jacketed-reactor with proportional controllers regulating the volume and temperature of the reactants. The design of the following observers exploits the large degrees of freedom, simplifying the task, thus we provide a specific design of a more general solution. Design of LO: Linearising the model equations about the steady-state values, the matrix form obtained is as follows: ~k-Ax+Bu+Kf;
y-Cx;
u-Hy
(7)
where x = [V Ca T Tj], y = x, u = [Fj F], and f = [Fo To Tjo Cao]. Note that E, F, G are all 0. Also d - 0 i.e., there are no unknown inputs. In view of these simplifications we need - L I T K to be diagonal. There are sufficient degrees of freedom to satisfy all the Equations 4. One sufficient choice would be T -- I4, R -- - I 4 . All four eigenvalues of R are chosen to be the same so that for the same fault level in the four fault variables during transients, no fault residual eclipses others. To keep L I T K diagonal, choose L1 -- K - 1 =~ J - B; S = A + I4; L 2 - - - L 1 . Thus, the linear observer with scaling matrix Sc is
r
--
- I z + Bu + (A § I4)y
-
SeK - l ( z - y )
(8)
Design of RNLO: The non-linear model of the system is to be cast in the form shown in Equations 5. The system model is found to contain products of a fault variables with other fault variables (FoCao,FoTo). Hence to separate these terms, limited linearisation about the steadystate values of the relevant variables is required. We use K(x,u) (instead of K(x,)), thus requiring linearisation of only these two terms. As in the case of LO, E - 0 G(x, u) - 0. The design parallels that of the linear observer of the previous section. As before we choose T - I4, R - - I 4 thus giving L1 -- [ - K ( x , U)] - 1 , S - A + I4, J(y, u) - B(y, u), L2 -- - L 1 and L1 - - K ( x , u) -1. Thus, the RNLO observer designed with scaling matrix Sc is
J~ --
--14z + B(y, u) + (A + 14)u
r
Sc[-K(x,u)-l(z -y)]
-
(9)
857
Fig. 2. ri for (a) LO Case 4: Quadruple Fault (b) RNLO Case 4 (filtered): Quadruple Fault
Table 1 Normalised Residuals Ri for LO and RNLO Case I No.
Faults Simulated
1
to+
2 3 4
T+T;
ror~C2o Fo+ro+rfiC2o
1
Fo-
2
To+ T)+o
3 4
ToT,loCao FoToT~C+o
R1 R2 R3 LO (5% faults, 10% noise) 0.000 1.000 -0.006 0.000 -0.170 1.000 0.000 -0.446 -0.434 0.833 1.000 -0.674 RNLO (15% faults,15% noise) -1.000 0.000 0.000 0.000 0.869 1.000 0.000 -0.723 -1.000 -1.000 -0.892 0.999
R4 0.010 0.194 -1.000
Faults Identified T+
-0.910
Tj+ [T+1 Cao [TO Tj-o] Fo+ To+ Tfo C~o
-0.004 0.095
FoTo+ T/+
-0.725 0.852
Fo TO T~+ C+
To T/o Cao
Performance Evaluation: The CSTR along with the observers were simulated in SIMULINK. A total of 24 (8 single, 8 double, 4 triple and 4 quadruple) faults are simulated with 10% (LO) and 15 % (RNLO) noises. The faults were chosen from a combination of
F~, C~o, T~, T~.
The
scaling matrix Sc (Section 2) is used to generate the fault magnitudes directly. The threshold
(Rth) is
set at •
Due to space limitations the results from all the fault simulations could not
be presented here, so only a sample of the faults are dealt with. Results for LO: 5% fault scenarios were simulated with noise level
N/S
- 0.1. No noise
filter was used so as to test the performance in the presence of noise. An example of a quadruple fault (Case 4) is shown in Figure 2(a) with good fault isolation and estimation, and the estimated magnitudes are close to the simulated values of 5%. A sample of the residuals Ri is shown in Table 1. 7 out of the 24 cases showed either a missed fault (false negative) or an absent fault (false positive) resulting in 70% diagnostic accuracy. In many of these cases the Ri
were
close
to the threshold of +0.5. The fault-related residuals are expected to have values :J:1, while all
858 the other residuals should ideally be 0. Only residual R1 shows the values expected, because the system equations include one linear equation exclusively in the fault variable Fo. In general, it is seen that we get sufficient decoupling of fault effects for faults Fo~ Ciao,while there is plenty of interaction among the fault effects for To:k Tj~. This brings to light the non-linear nature of the CSTR which cannot be taken into account completely by a linear observer. Results for RNLO: Faults with 15% magnitude and N/S = 0.15 were simulated. The increased severity of the faults is to assess the performance of this 'better' designed observer. A sample of the normalized steady state residuals Ri (no noise filter) is shown in Table 1. A perfect isolation of faults at steady state is seen for all 24 cases achieving perfect diagnostic resolution. The
Toi Tjio fault detection problem encountered in the LO case is not seen here. However, because of the increased severity of noise in this case, deterioration in transient performance (noisy residuals) may be expected. To mitigate the noise effect, a moving average discrete filter for the observer inputs is tested and an example of a quadruple fault (Case 4) is shown in Figure 2(b). 4. CONCLUSIONS Exploiting the redundancy in a process model is an useful technique for fault diagnosis. The issues of modeling errors, robustness to noise/uncertainty and perfect fault decoupling were discussed. A linear and a restricted non-linear observer were outlined, designed and evaluated for a non-linear CSTR plant. While the linear observer is simple to build, it can't compensate for the nonlinearities resulting in fault interactions manifesting in the residuals. The restricted non-linear UIO compensates partially for the plant non-linearities, and thus has superior performance. REFERENCES [1] Nimmo, I., Adequately address abnormal situation operations. Chem. Eng. Prog. 1995, 91 (9), 36 [2] Dash, S.; Venkatasubramanian, V., Challenges in the industrial applications of fault diagnostic systems. Comput. and Chem. Engng. 2000, 24(2-7), 785 [3] Gertler, J., Fault detection and diagnosis in Engineering systems. Marcel Dekker. 1998 [4] Frank, P. M., Fault diagnosis in dynamic systems using analytical and knowledge-based redundancy - - A survey and some new results. Automatica. 1990, 26(3), 459 [5] Huang, Y.; Dash, S.; Reklaitis, G. V.; Venkatasubramanian, V., EKF based estimator for FDI in Model IV FCCU. IFAC Proceedings SAFEPROCESS2000. 2000 [6] Luyben, W. L., Process Modeling, Simulation and Control for chemical engineers. McGraw-Hill. 1990
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
859
Design considerations of computer-aided RCM-based plant maintenance management system Hossarn A.Gabbar, Kazuhiko Suzuki, and Yukiyasu Shimada Department of Systems Engineering, Okayama University 3-1-1 Tsushima-Naka, 700-8530 Okayama City, Okayama, Japan In most of the industries, Reliability-Centered Maintenance (RCM) process hasn't been fully integrated with the computerized maintenance management system (CMMS) where they run in isolation leading to outdated maintenance strategies. In this paper, the idea of integrating RCM process with the CMMS has been studied so that the maintenance strategies are dynamically changing throughout the plant lifecycle. The integrated solution has been discussed in two-folds: (1) Information, where plant design model is obtained from the plant design environment, while plant operational information is acquired from the operational systems; (2) Process logic where HAZOP, FMECA, and FTA are utilized to analyze and assess all failure types in quantitative manner, while combined Monte Carlo, Genetic algorithm, and Weibull probability distribution functions (pdf) are employed to optimize the maintenance tasks. The activity and functional models of the enhanced RCM process are developed. Modifications to the different modules of CMMS (i.e. MAXIMO TM)are proposed. 1. INTRODUCTION RCM process is intended to determine the most realistic and optimized maintenance requirements of any physical asset to continue its stated operating condition [1]. Many industries are adopting RCM technique to solve many confronted maintenance problems. Unfortunately, it didn't work as expected for many reasons: (1) RCM is a time consuming process and requires considerable amount of resources, especially for large number of assets; (2) information is not adequate to decide the suitable maintenance strategy and to optimize its cost; (3) there are non-engineering factors involved in the maintenance problems i.e. management. To achieve the expected results, it is essential to consider an integrated solution using the plant lifecycle's accumulated information. By considering quantitative asset and failure assessment techniques accurate results can be achieved. Combined HAZOP, FTA, and FMECA are used to analyze the root cause of each failure. Combined Monte Carlo simulation,
860 Genetic algorithm, and Weibull pdf are used to optimize maintenance tasks. Previously, attempts are made to integrate RCM within the design environment using fuzzy reasoning algorithms [2]. This research proposes wider vision of integration that includes plant design and operational systems, and suggests the necessary design changes to CMMS. 2. PROPOSED SYSTEM ARCHITECTURE
Fig. 1 - RCM-based integrated system architecture. Fig. 1 shows the system flowchart of the proposed integrated solution, which includes: plant design environment, operational systems, CMMS, and RCM process. The cycle starts from the plant design by developing the plant model, which includes the plant static (structure and topology), behavior, and function views. Plant asset information is extracted from the plant static model into CMMS and passed to RCM process. Failure model is developed within the plant design using HAZOP, FTA, and FMECA techniques. The failure model in the plant design domain is translated into failure hierarchies within CMMS, which is passed to RCM process for failure assessment. Assets are represented within CMMS as templates (or class definitions) and physical asset records. For example, Pumps and Valves are represented as class definitions, while the physical plant assets for pumps and valves are represented as asset records, which are associated with the corresponding asset templates. Asset information is passed to the RCM process for asset assessment practice. Maintenance strategies are initially decided during the design stage and gradually tuned throughout the plant lifecycle using feedback information from operational systems and reliability data. The data flow "D8 ") P2" is used to express the information flow from D8 (the maintenance history data within CMMS) to P2 (RCM), which
861 fortifies the utilization of the reliability data from the running CMMS. 3. RCM PROCESS MODELING
Object-oriented modeling approach has been adopted to analyze and enhance RCM process. The activity and functional models have been developed for this purpose, as below. 3.1.
RCM Activity Model
Fig. 2 - RCM process first-level activity model. The activity model has been developed using IDEFO where the RCM process has been enhanced from both business and technical views (from "As-Is" to "To-Be"). IDEFO is useful in establishing the scope of the analysis, especially for functional analysis. Fig. 2 shows the first-level activity model, which shows four major activities: (1) manage the RCM process where scope and resources are identified, (2) identify assets, functions, and performance standards where plant structure information is mapped from the plant design model, (3) analyze failure, cause, consequences, and evaluate associated risk, and (4) decide and optimize maintenance strategies and tasks. The integration on the activity model level is applied to augment the corresponding activities from CMMS and to understand the required input / output information and its originality i.e. plant design or operational systems. For example, asset identification activity has been consolidated to read the plant model from plant design environment. It creates the asset templates as the base classes for the associated assets.
862
Fig. 3 - RCM Process Function Decomposition. 3.2.
RCM Functional Modeling
Using the activity model, the function decomposition of the proposed RCM process can be constructed (as in Fig. 3). RCM can be decomposed into the following functions: assess assets, assess failures, decide maintenance strategy, decide maintenance tasks, optimize maintenance tasks, and check & validate the results. Maintenance tasks are defined as: lubrication, cleaning, inspection, replacement, repair, fabrication, and overhaul, which are used within the expert system (decision engine) to decide the suitable maintenance task based on the failure, consequence,
reliability data, accumulated
maintenance
history, and design model
requirements (including safety requirements). Maintenance task level is classified as: good-as-new (GAN), imperfect, and bad-as-old (BAO). Soft and hard life factors [3] are optimized using combined Monte Carlo simulation and Genetic algorithm (explained in details in reference [4]). Weibull function is used to model failure behavior and classify failures, which is used to construct the reliability data. Checker report using RCM knowledgebase and maintenance data is used to verify the correctness and accuracy of the obtained results (i.e. optimized maintenance tasks).
863 4. INTEGRATED SYSTEM DESIGN
Fig. 4 shows the detailed design of the integrated solution where plant model is represented in three dimensions: structure (static information about the plant i.e. Pump-1 and Valve-I), dynamic (behavioral information about the fluid, process variables changes with time i.e. fluid-pressure-out-Pump-l, which is represented by set of equations), and functional (operations performed by plant objects i.e. Valve-1 --) Open). HAZOP is used to report the possible deviations, causes, and consequences for Pump-l, and FTA reflects all the possible paths for each top-failure with the probability of failure for each minimal cut set, while FMECA defines the different failures with the different levels of details along with the criticality of each failure. Combination of these failure assessment techniques facilitates the analysis of the root cause. Assessing the root cause instead of the actual cause reduces the probability of failure and hence minimizes the associated risks.
~l
A
PlantOO ,'" u// /
, ',]
D~ Functional I
r ~ ....... r...... } I c.... q..... l / . ~ FallureModel~ ~
I
J
CMMS ~
AssetData (
I ' " , - . , ~ / ~ / ~ 7 '~'~-/~ / (Fa,lureData ( /
l
\/ / X/
.............
Fig. 4 - System flowchart of the proposed integrated solution. 5. CMMS DESIGN MODIFICATIONS RCM engine is developed using C++ as a module within MAXIMO TM (as in Fig. 5). The database structure of the failure module is modified to include failure-asset relationship and to maintain the link with the reliability data. Failure date is maintained using combined FTA/HAZOP/FMECA. RCM engine is invoked to validate changed or newly created maintenance tasks (or for new equipment), and to decide and optimize maintenance strategies. An intelligent translator is used to translate the plant design information into Asset module.
864
Fig. 5 - CMMS Design Modifications 6. CONCLUSION RCM-based CMMS is used to optimize maintenance for critical plants. The developed activity and function models are useful to understand such solution and to analyze other plants with minor modifications saving analysis time. The design modifications proposed to the adopted CMMS can be realized within MAXIMO while RCM engine can be developed as a shell integrated with the different modules of MAXIMO. The integration with the plant design and operational systems is essential to share and utilize plant design model and plant operational information. Combined HAZOP, FMECA, and FTA are used to assess failure comprehensively and quantitatively. REFERENCES
1. John Moubray. Reliability-centered maintenance, Butterworth Heinemann, ISBN 0 7506 3358 1 (1997). 2. D.J. Fonseca and G.M. Knapp. Expert Systems with Applications, No. 19 (2000), 45-57. 3. J. Crocher and U.D. Kumar. Reliability Engineering & System Safety, No. 67 (2000), 113-118. 4. M. Marseguerra and E. Zio. Reliability Engineering & System Safety, No. 68 (2000), 69-83.
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rightsreserved.
865
Capacity Planning under Clinical Trials Uncertainty for the Pharmaceutical Industry Gabriel Gatica a, Nilay Shah a'* and Lazaros G. Papageorgiou a' b a
Centre for Process Systems Engineering, Imperial College, London SW7 2BY, U.K.
b Dept. of Chemical Engineering, University College London, London WC 1E 7JE, U.K. Market globalisation, increased competitiveness and tightness of capital are some of the factors in the modem economy that affect every stage of the business value chain, especially in the pharmaceutical industry where the typical life-cycles of new drugs are becoming shorter. It may take eight years to develop a new product and the investment in it must be recovered quickly, as competitor products can appear in the market reducing its profitability. This paper presents a mixed integer optimisation model, which selects simultaneously the optimal capacity planning and investment strategy subject to the uncertainty of phased clinical trials for a given product portfolio. The applicability of the model is demonstrated by a case study. 1. I N T R O D U C T I O N Market pressures are forcing pharmaceutical companies to take a more holistic view of their product portfolio. The typical life-cycle of new drugs is increasingly coming under pressure. This is a high risk industrial sector but rewards can be substantial. It may take eight years to develop a new product and the investment on it must be recovered fast, since ten years after the approval of a new product, generic equivalents or branded competitor drugs can appear in the market reducing drastically the profitability of the drug. Consider the following situation. A corporation has four different products (CI .. Ca) in different stages of development at the current time. C1 is an existing mature product while the others are in clinical trials and it is not certain whether they will be successful products. However, based on market and epidemiology studies, the nominal predicted demands for the four products (provided that C2, C3 and Ca are successful in clinical trials) have been forecasted. The manufacturing route is common to all products (produced through a biochemical process) and is abstracted as a production line which produces crude product and a purification line which produces high purity active ingredient (AI). The secondary manufacturing process is not considered further in this paper. Initially, there is a single line of each. Therefore, the only way to increase capacity is to purchase a new line of either type.
* To whom correspondence should be addressed: n.
s h a h @ i c , ac. uk,
Fax: +44 20 7594 6606
866 The company then faces the decision of how to structure its future product portfolio optimally. One option is to invest simultaneously in manufacturing capacity and research and development to build up a large portfolio with all four products being manufactured simultaneously in anticipation of successful trials. Alternatively, it could plan to wait for the outcomes of each trial and then decide on the best course of action. Of course, several other options may exist each of which may differ with respect to the investment policy required and the potential return of the resulting portfolio. The challenge is then to maximize the expected net present value of the portfolio while avoiding unnecessary capital commitments. There are two main issues, which need to be considered during the optimisation of the product portfolio of a typical pharmaceutical company: 9 Product Management: this is concerned with the main features of each product considered as a suitable candidate for manufacturing and commercialisation. Such features are R&D cost associated with the development of each new product, potential outcomes of ongoing clinical trials, commercial characteristics of each product (e.g. demand forecast, price, marketing expenses, etc.). 9 Capacity Management: this is concerned with the allocation of existing capacity for the selected product portfolio and decisions concerning additional investments that may be required to satisfy future demands. Changeovers should also be considered as complex portfolios (i.e. with many products) at a single site often result in considerable reduction of manufacturing efficiency. In academia, the problem has never been directly addressed. However, some relevant work has been published in the area of design and planning under uncertainty (e.g. [1-4]) and product development (e.g. [5-6]). In the pharmaceutical industry, this problem has typically been addressed in a relatively simple fashion. The value of the product portfolio is estimated, for two or three orthogonal, forecasted scenarios, based on R&D costs and the potential sales value estimated for the products in the market. In this work, an optimisation-based approach is described capturing the above issues so as to select simultaneously the optimal capacity planning and investment strategy subject to uncertainty of clinical trials for each potential drug. This paper constitutes an extension of our previous work [7] where a two-stage mathematical model was proposed by considering two potential clinical trial outcomes (Success, Failure) being available simultaneously at the end of the first stage. Here, a more realistic problem representation is presented by considering four different clinical trial outcomes (High success, Target success, Low success, Failure) for each manufactured product, which is typical in the industry. The fact that these outcomes have different probabilities of occurrence and that the information from different clinical trials will become available at different times means that the investment problem becomes a large-scale, multistage, multiperiod stochastic optimisation problem. 2. P R O B L E M DESCRIPTION
For the example described above, we need to find the optimal production and capacity plan to maximise the expected nett present value (eNPV) of the project, taking into account the different possible demands for each product. For product C1 we assume that we know its future demand perfectly, so we can consider this product as a deterministic one. For the other three products, we must wait some time to know the outcomes of their clinical trials. Each
867 considered outcome has a probability associated with it, which makes this problem a stochastic one for each of these products. After the clinical trial tests, each product has four different possibilities; High, Target or Low Success or Failure. An example clinical trial structure is illustrated below.
/
/o,
Success
/
(.~
Success
Hi0.10 gh (-~ ~.J~~O.95 .
..........
1 .........
~''~Faflure --0.-0-5-
Tar~let ~
.
.
.
~Launchl
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...................................
0.1
Phase II (--1 year) Stage I
Phase III 1-2 years Stage 2 & 3
Registration (1 year)
Figure 1: Clinical trial structure 3. M U L T I S T A G E PLANNING M O D E L S Models for optimisation under uncertainty usually distinguish two types of decision: those that must be made immediately in the presence of significant uncertainty (the "here-and-now" decisions) and those that can be made upon some or all of the resolution of the uncertainty at some later date or dates (the "wait-and-see" decisions). The first set of decisions must be made in a fashion that results in satisfactory performance across all reasonably probable outcomes, while the latter decisions are implicit functions of the uncertainties. These latter decisions reflect operational adjustments made to take account of new information. We utilise this concept, and in particular formulate our problem as a multi-stage stochastic programming problem. The first stage reflects the decisions that must be made immediately: 9 product selection; 9 initial capacity investment; and 9 initial allocation of manufacturing resources to products. The subsequent stages reflect decisions made upon the completions of critical clinical trials: 9 additional capacity investment in the case of favorable outcomes; 9 abandonment/sale of capacity for unfavorable outcomes; and 9 re-allocation of manufacturing resources to products; 9 production plans. The second and subsequent stages comprise a large number of scenarios, each of which reflects a different combination of outcomes of the independent clinical trials for different drugs. In the example considered here, we assume that: 9 Product C1 is an existing product in manufacture. 9 The clinical trials for C2 and C3 will be complete in 2 years (i.e. 8 90-day periods).
868 9 The clinical trial for C4 will be complete in 3.5 years (i.e. 14 90-day periods). This gives rise to a problem with three stages. The first stage is a "deterministic" one where the only product whose demand is known is C l. However, it may be a good idea to produce other products in anticipation of them being successful, and even investing in new capacity so that it is available later on. The second stage is generated by the resolution of the uncertainty around C2 and C3. This gives rise to 16 scenarios due to the combination of the 4 different outcomes of these two products. Finally, the third stage is generated by the resolution of the uncertainty around C4 and this results in four scenarios for each of the 16 of the second stage and therefore a total of 64 scenarios are present here. The overall scenario structure is illustrated in Figure 2. ..-
1-4
1--q
C l i n i c a l Trial O u t c o m e s H T L F -
High Success Target Low Failure
A w
o
9 9
I I I I I I
-A V
~------
1 -- H
| |
1 Deterministic Product C l
" | : |
Period
HTL ~ 7 +2 Stochastic Products C2, C 3
9
1
.~
~
.... .....
"2_ -
I I I I
~
28
"x ~ 29-32
O
"
~
-
r.r
9
O
+
A
16---t Stage 2
Sen
H ~ L F
,
| Stage
.
61-64 -~
Stage 3
Figure 2" Scenario structure The proposed multiperiod, multistage, multiscenario problem is described below. 4. BASIS OF M A T H E M A T I C A L M O D E L To develop this model, the following assumptions are made: 9 There is unlimited availability of raw materials. 9 The nominal demand and price forecasts of each final product are known. 9 There is limited storage capacity for intermediates and final products. 9 The range of time considered for the planning problem corresponds to 60 time periods and each period is equivalent to 90 days. 9 After the decision to invest in a new line, the actual capacity becomes available a year later. 9 The time that each line is actually in service is not the same as the duration of each period, but reduced by maintenance, scaleup and detailed scheduling considerations. 9 Each clinical trial outcome will dictate how much of the forecasted nominal demand will be realised. In this case, if the clinical trial is a High success, then the realised demand will
869 be 100% of the forecasted nominal demand; at Target the figure is 95% and for a Low Success the figure is 65%. Based on this data, a model can be developed to determine the optimal production and capacity planning that maximises the expected plant profit. The key variables are: Binary variables: These represent decisions such as product selection, investments in lines, product-line-time period assignment and line availability. Continuous variables: These represent the amounts of each drug produced, stored, sold and wasted (due to expiry of shelf-life), and production time allocated to each drug. The key mathematical components of our model represent: 9 construction lead time of manufacturing units; 9 production timing and capacity; 9 inventory and product shelf-lives; 9 sales/marketing constraints; and 9 objective function. The objective function is the maximisation of the eNPV, which is calculated as the weighted sum of the NPV of each scenario, where the weighting factors are the scenario probabilities. 5. CASE STUDY RESULTS The problem above was modelled using the GAMS package[8]. A basic case was investigated as well as some parametric studies. For the data provided, it was found that all products are worth pursuing in stage 1. Since stage 1 we have at least one line of each type and only deterministic product C~ is manufactured. There is enough capacity initially in the system for no capacity expansion to take place on either line. The purification line is the bottleneck in the process, and although there is no capacity expansion in the first stage, there is a capacity expansion in the second and third stages in some scenarios (e.g. scenario 6) and just in the second stage in others (e.g. scenario 7). This is illustrated below, where the available time for each resource is illustrated as a function of time as is the utilisation, which is scenario-dependent.
Figure 3: Production line availability and use
Figure 4: Purification line availability and use
Additionally, production profiles are also generated and a sample is shown in Figure 5. For all periods, sales met a 95% of the forecasted demand as it was agreed for the Target outcome.
870 The inventory levels are kept to a minimum during all the time. Finally, we have only so far dealt with the eNPV. An outcome of the optimisation is the actual distribution of NPV (see Figure 6). It may be seen that the distribution is quite different from the commonly-assured normal distribution, with several eNPV clusters.
Figure 5: Product C 3 production profiles
Figure 6: Distribution of NPV
6. CONCLUDING REMARKS In this paper, we have described an optimisation-based approach in order to determine both a product development and introduction strategy, and a capacity planning and investment strategy. The overall problem has been formulated as a multistage stochastic programming problem, taking account of points in time where new information becomes available. The applicability of the proposed procedure has been demonstrated by a case study from the pharmaceutical sector. From an industrial perspective the optimisation approach was found to be extremely powerful and supported "what-if" type scenario analysis and mid/long term capacity planning. However, a key weakness with the current approach is the use only of expected NPV and not of any risk indicators (see, for example, [9]). This will form the basis of future work. REFERENCES 1. M.G. Ierapetritou, E.N. Pistikopoulos and C.A. Floudas, Comp. Chem. Eng., 12(1996)1499. 2. M.L. Liu and N.V. Sahinidis, Ind. Eng. Chem. Res., 35 (1996) 4154. 3. S.B. Petkov and C.D. Maranas, Ind. Eng. Chem. Res., 36(1997) 4864. 4. A. Gupta and C.D. Maranas, Ind. Eng. Chem. Res., 39(2000) 3799. 5. C.W. Schmidt and I.E. Grossmann, Ind. Eng. Chem. Res., 35(1996) 3498. 6. C.W. Schmidt, I.E. Grossmann and G.E. Blau, Comp. Chem. Eng., 22S(1998) S 1027. 7. G.E. Rotstein, L.G. Papageorgiou, N. Shah, R. Mustafa and D. Murphy, Comp. Chem. Eng., 23S(1999) $883. 8. A. Brooke, D. Kendrick, A. Meeraus and R. Raman, GAMS: A User's Guide, 1998. 9. G.E. Applequist, J.F. Pekny and G.V. Reklaitis, Comp. Chem. Eng., 24(2000) 2211.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
871
Multiperiod Planning of Multisite Supply Chains Under Demand Uncertainty Anshuman Gupta and Costas D. Maranas
a
aDepartment of Chemical Engineering, The Pennsylvania State University, University Park, PA 16802, USA In this work, the multiperiod planning of multisite supply chains under demand uncertainty is addressed. The presence of nested optimization problems (one for each period) coupled with the NP-hard nature of the original deterministic problem makes the multiperiod problem computationally intractable. Consequently, a modeling framework which incorporates partial information about the future demand evolution process is proposed for reducing the computational complexity of the multiperiod model. Specifically, the multiperiod model is reduced to an augmented two-stage model by considering the demand as uncertain in only the upcoming period in which the planning decisions need to be implemented immediately while treating it as deterministic in the remaining planning horizon. The coupling between the upcoming period and the rest of the planning horizon, which exists due to the transfer of inventory over time, is modeled by utilizing analytically derived expressions for the expected inventory. In addition, various rolling horizon planning policies incorporating different levels of future information are studied within a simulation environment. 1. INTRODUCTION Uncertainty in product demand has been extensively studied in the process systems literature [ 1-3]. The variability in product demand can be traced back to the basic need of planning models which is to optimally allocate resources for the future using currently available information [4]. However, one of the key component not studied in great detail is the effectiveness of various rolling horizon planning policies, which would have to be implemented in a real multiperiod planning setting. In view of this, a model formulation which utilizes partial information about future uncertain demand is proposed for the deterministic planning model of McDonald and Karimi [5]. This methodology is based on the previous works of the authors for the single period case [3,6]. The main part of the paper investigates the quantitative impact of various alternative rolling horizon planning policies through a supply chain planning case study. 2. AUGMENTED TWO-STAGE FORMULATION The proposed augmented two-stage formulation for the multiperiod planning model of McDonald and Karimi [5] (MPa2s) is as follows. min f , j ,s ,t
i, j ,s ,t
i,s ,t
i,s ,s ~,t
872
+ E EOit[O'it(Aist'Oit)] i,t=l
-k- E tistSist'+" E histlist+ E ;istI~sAt-k- E llitlit i,s,t>2 i,s,t>_2 i,s,t>_2 i,s,t>2
subject to Pijst -- RijstRZijst ; fist -- E ~i, is E Pi, jst ; fist - E Wis'st it j st
(1)
f R L f jst -
(2)
E RLijst ; E F R L f jst < Hjst i:~lf=l f
M R L f jst Yf jst ~ V R L f jst <_ Hjst Yf jst
(3)
Ais(t= l ) -- Ip -+-E eijs(t= l ) - E Wiss, (t= l )
(4)
Iis(t=2 ) -- EOi(t=l ) [Iis(t=l)] "71-EPijs(t=2) - E Wisst(t=2) - E Sis(t=2) j sI s
(5)
l i s t - Iis(t-1) + ~_~Pijst- ~_~Wiss,t - ~_~Sist
(6)
E Sist ~ Oit ; {)it- E Sist ~ lit ~_ Oit " IiLt- list ~ Iis~ ~ liLt s s
(7)
j
sI
j
st
s
min F_,tistSist + F.,histIist ~, ~istI~s~ + ~,llitli7 i,s
i,s
i,s
t
S.t. O,it(Aist, Oit) --
~.,Sist <_ Oit " list = Aist - Sist s
(8)
Oit -- f__,Sist <_ I~t <_ Oit ; I~Lt-- list <_ I~s~ <_ I~Lst s
The objective function of model M P A2S is composed of three distinct components. The first component comprising of the first four terms accounts for the production costs incurred in the entire planning horizon. The corresponding production decisions, Pijst (production amount), RLijst (runlength), Cist (raw material consumption), Wis,st (intersite shipment), Yfjst (setup) and Aist (availability), are constrained through the production constraints given by Equations 1 through 4. The second component (fifth term in objective function) captures the expected recourse costs for the first period as determined by the solution of the embedded inventory management optimization problem given by Equation 8. Finally, the last four terms model the supply chain costs incurred for t _> 2. The demand is modeled as deterministic for t >_ 2 and the supply chain decisions consisting of Sist (supply), list (inventory), I~sAt(safety stock deficit) and Iit (customer shortage) are subject to Equation 7 which are the supply chain constraints. Note that in these constraints, the expected demand (Oit) is used. The linking between the first time period and the remaining planning horizon is modeled through Equation 5 with the expected inventory level at the end of the period 1 providing the initial inventory for period 2. Solution of model M P AES requires estimation of (i) the expected recourse function and (ii) the expected inventory levels for the first period. Exact deterministic equivalents for these two
873 quantities are obtained by utilizing the analysis previously proposed by the authors [3,6]. The basic idea of the solution methodology consists of solving the recourse problem in Equation 8 analytically using linear programming duality followed by analytical integration for expectation evaluation [3]. The resulting optimal supply policies thus uncovered are subsequently used for determining the expected inventory level [6]. 3. MULTIPERIOD PLANNING POLICIES Model M P A2s is anticipative in nature as it determines the production plan for the entire planning horizon prior to the actual demand realization for even the first period. The optimal production decisions for t > 2 are of no practical significance as they do not have to be implemented immediately. This inherent flexibility of modifying future production plans in response to the demand evolution process is studied within a simulation environment which captures the evolution of time and the corresponding resolution of uncertainty through various rolling horizon planning policies. These policies incorporate varying levels of information about the demand process while considering different future timespans for making current decisions. The policies studied are: (i) Single Period Deterministic (SPD) (ii) Single Period Stochastic (SPS) (iii) Multiperiod Deterministic (MPD) and (iv) Multiperiod Stochastic (MPS) planning. In the SPD planning policy a planner solves single period deterministic planning problems within a rolling horizon setting. Aside from taking a very myopic view of the future by only considering the upcoming period, the planner also fails to recognize the uncertainty in the demand as captured by its standard deviation. After solving the appropriate model and fixing the resulting optimal production plan in the supply chain, random demand realizations are revealed to the planner. Based on these realizations the planner determines the optimal supply policies for the various production sites and the resulting total cost incurred. Even though the demand is considered to be deterministic by the SPD planner, the two-stage decision making framework is still recognized within which the supply chain variables can be optimally set to optimize in the face of uncertainty. The optimal supply policies translate into the post-demand satisfaction inventory levels and provide the planner with the initial conditions for the second period. The planner carries out this planning procedure in a rolling horizon manner for the entire planning horizon. This sequence of steps is then repeated to average over the randomly generated demand realizations. The SPS policy is similar to the SPD policy with respect to the restrictive view taken of the future timespan. Unlike the SPD planner, however, the SPS planner has information about both the mean and the standard deviation of the demand. The SPS planner, thus, solves the single period stochastic formulation for determining the production setting for the upcoming period. Subsequently, demand realizations (same as those revealed for the SPD planner) are revealed to the SPS planner and the optimal supply chain decisions are made. Decisions are then made sequentially in a rolling horizon manner in the same spirit as the SPD case. The third planning protocol considered is the MPD planning policy. In this policy, the view of the future is expanded to include the entire future timespan starting with the upcoming period while considering the demand to be deterministically known. The inventory transfer between time periods is described by deterministic inventory balance constraints of the form given by Equation 6. The optimal production plan is implemented in only the upcoming period thus retaining the flexibility to alter the production settings in the future in response to unfolding
874 events. Finally, the MPS planner extends the MPD planning framework by characterizing the upcoming period demand by both its mean and standard deviation. The demand in the future periods, however, is still considered to be deterministic. The expected inventory level is used to link the stochastic period to the future deterministic periods through Equation 5. To benchmark the quantitative performance of each of these planning strategies, the hypothetical Perfect Future Information (PFI) planning policy is used. The PFI planner is assumed to have complete information about the sequence of demands realized in the future. The PFI planner, thus, plans for the entire time horizon simultaneously by solving the deterministic multiperiod problem based on the randomly generated demand scenarios for each of the periods. This planning procedure results in minimum total costs in the supply chain as it is equivalent to considering all the decisions (both production and supply chain) as second stage, control decisions. Note that this policy, though, does not result in an implementable plan as the production decisions cannot be postponed to after demand realization. The optimal cost obtained, however, can be utilized to assess and compare the effectiveness of each of the planning policies.
4. EXAMPLE The proposed solution methodology is applied to the supply chain planning example originally studied in Gupta, Maranas and McDonald [6]. The supply chain network consists of three production sites manufacturing ten products grouped into five product families. The demand for the products exists at a single customer over a planning horizon of six months corresponding to six planning periods of one month duration each. Each product family is manufactured on a single, limited capacity processing equipment and fixed setup charges are incurred for each production campaign. The product demands are assumed to be normally distributed with specified means and standard deviations [6]. First, the multiperiod deterministic MILP problem is solved assuming mean product demand using CPLEX accessed via GAMS resulting in an optimal cost of $23,485. Next, the PFI policy is implemented within the example setting. The expected cost incurred through this policy is $23,599. This provides the lower bound on the costs of the other planning policies. Note that the cost incurred by the PFI policy is only marginally higher than the deterministic cost. This is a result of the (unattainable) assumption of perfect future knowledge which is the basis of the PFI policy. Subsequently, the other four planning policies are simulated. The resulting expected multiperiod costs obtained are $26,079 (SPD); $25,341 (SPS); $25,269 (MPD) and $24,725 (MPS). To quantify the performance of the proposed methodology, the uncertainty gap reduction (UGR) metric is defined. This is given by ( z (MPD) - z (MPS)) %UGR = \ z~P--D)2--z-~-i) x 100
(9)
where and z (.) represents the expected cost incurred through a particular planning policy. The denominator in Equation 9 represents the uncertainty gap which arises due to the failure of the MPD planner to account for uncertainty in the planning decisions. Equivalently, it is the amount the MPD planner would be willing to pay in return for complete information about the future demand realizations for the entire planning horizon. Similarly, the numerator in Equation 9 is the value allocated to information regarding the standard deviation of the upcoming period by
875 the MPD planner. The fractional savings in cost achieved by switching from the MPD policy to the MPS policy are hence captured through the UGR metric. For the current example setting, a UGR of 32.6% is obtained. This implies that the uncertainty gap can be reduced by almost a third by just incorporating an uncertain description of demand for the upcoming period. To gain further insights into the cost savings achieved through the MPS policy, the expected multiperiod costs incurred through the MPS and the MPD policies are analyzed in terms of their constitutive components. The results of this activity-based cost analysis are shown in Figure 1. As Figure 1 indicates, the MPS and the MPD policies result in almost comparable setup, transportation and customer shortage charges. Relatively insignificant difference in the fixed production charges implies that no additional setups are enforced by the MPS planner. Comparable transportation and customer shortage charges translate to comparable customer service levels achieved through the two planning policies.
Figure 1. Activity-based cost analysis The MPS policy, however, outperforms the MPD policy in terms of the inventory holding charges while the MPD policy results in lower variable production and safety stock violation charges. These observed cost trends may be intuitively interpreted as follows. The upcoming period demand can be satisfied by either (i) production in the upcoming period or (ii) inventory carryover. The MPS planner favors (relatively) the former strategy while the MPD planner relies primarily on the later. The production plans generated by the MPS policy can be expected to be more cost effective than the corresponding MPD policy plans for meeting the demand in the upcoming period. This can be attributed to the explicit incorporation of the variability in the upcoming period demand, in terms of its standard deviation, into the planning decisions through the augmented two-stage model in the MPS policy. Thus, higher production charges are incurred in return for lower inventory holding charges. Due to less reliance on inventory stock to meet demand, which translates into lower inventory levels, safety stock violations are frequent leading to high violation penalties. The MPD planner, on the other hand, does not incorporate any demand variability information while determining the production plan for the upcoming period. Consequently, use of inventory stock is preferred to meet customer demand
876 resulting in lower production and safety stock violation charges with correspondingly high inventory holding charges. On the whole, the additional production and safety stock violation charges incurred are offset by the savings in inventory charges through the MPS policy leading to overall savings in the supply chain. 5. S U M M A R Y
In this work, the multiperiod planning of multisite supply chains was addressed. An augmented two-stage model, which utilizes partial information about the future uncertain demand, was formulated and subsequently embedded within a rolling horizon simulation framework. The fact that significant cost savings could be realized with the proposed methodology over deterministic planning policies was highlighted by a supply chain planning case study. REFERENCES
1. S. Subrahmanyam, J. F. Pekny, and G. V. Reklaitis. Design of batch chemical plants under market uncertainty, l& EC Research, 33:2688, 1994. 2. M.G. Ierapetritou and E. N. Pistikopoulos. Novel Optimization Approach of Stochastic Planning Models. I& EC Research, 33:1930, 1994. 3. A. Gupta and C. D. Maranas. A Two-Stage Modeling and Solution Framework for Multisite Midterm Planning Under Demand Uncertainty. l& EC Research, 39:3799, 2000. 4. R.L. Clay and I. E. Grossmann. A disaggregation algorithm for the optimization of stochastic planning models. Computers and Chemical Engineering, 21:751, 1997. 5. C.M. McDonald and I. A. Karimi. Planning and Scheduling of Parallel Semicontinuous Processes. 1. Production Planning. I& EC Research, 36:2691, 1997. 6. A. Gupta, C. D. Maranas, and C. M. McDonald. Midterm Supply Chain Planning Under Demand Uncertainty: Customer Demand Satisfaction and Inventory Management. Computers and Chemical Engineering, 24:2613, 2000.
European Symposium on Computer Aided Process Engineering - 11 R. Gain and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
877
Combined MILP-Constraint Programming Approach for the Optimal Scheduling of Multistage Batch Processes Iiro Harjunkoski a and Ignacio E. Grossmann a* aDepartment of Chemical Engineering,Carnegie Mellon University 5000 Forbes Avenue, Pittsburgh, PA 15213, USA This paper presents an approach for combining constraint programming (CP) with mixed integer linear programming (MILP) when solving a multistage scheduling problem. The proposed hybrid approach relies on assigning orders to equipment with an MILP subproblem and sequencing with CP. Special cuts are proposed for the MILP subproblem. Results show that order of magnitude reductions can be achieved in the computation. 1. INTRODUCTION Many Mixed-Integer Linear Programming (MILP) models have been proposed for scheduling problems in chemical engineering [5,9,7]. While the advantage of this approach is that it provides a general framework for modeling a large variety of problems, the major difficulty that is faced is computational expense for large problems, which can make this approach prohibitive for industrial applications unless they are simplified through the use of heuristics. To overcome the combinatorial explosion in scheduling problems, logic based optimization methods such as Generalized Disjunctive Programming (GDP) [8] and Constraint Programming (CP) [2] have emerged. In GDP discrete and continuous optimization problems are represented through equations, disjunctions and logic propositions, and in terms of Boolean and continuous variables. CP also involves equations, logic statements and disjunctions, and has more constructs supporting thus a compact language that is largely procedural and implicit in nature. While GDP problems can be solved through branch and bound, or reformulated as MILPs, CPs are solved through implicit enumeration coupled with domain reduction of variables, which in turn makes use of constraint propagation techniques. These techniques for the case of scheduling, known as edge finding methods [ 1], have proved to be very effective for solving special types of jobshop scheduling problems. In this paper we present a combined MILP/CP approach for solving the multistage batch scheduling problem by Pinto and Grossmann [6]. This problem deals with batch plants consisting of parallel equipment in each stage, and in which a set of orders corresponding to different products must be delivered as close as possible to the due dates. In this paper, the chosen objective is the minimization of assignment cost of the orders to the equipment. The rigorous MILP optimization for this problem has proved to be intractable. Pinto and Grossmann [6] proposed a two stage procedure in which the assignment of orders is performed in the first stage by minimizing the in-process time. In the second stage the tardiness is minimized for that assignment. *correspondingauthor, email: [email protected]
878 M6ndez et al. [4] developed an alternate MILP model in which the assignment and sequencing variables are treated separately thereby reducing the number of binary variables, but at the expense of introducing undesirable big-M constraints that exhibit poor LP relaxations. Based on recent work by Jain and Grossmann [3], we propose a combined MILP/CP model for solving this multistage scheduling problem. A decomposition strategy is used in which the assignment of orders to equipment is performed with a relaxed MILP master model, while the sequencing is performed with a CP model. Solutions of the CP problem are used to generate cuts that are incorporated to the MILP master problem. This decomposition strategy has been implemented in OPL in which the MILP master problem is solved through CPLEX and the CP through ILOG-Solver and Scheduler. Results are shown for the MILP model and the proposed combined MILP/CP model. As will be shown, very significant computational savings can be achieved with the proposed method without compromising optimality.
2. PROBLEM DEFINITION We consider a multistage batch plant where all the orders need to be processed at each stage in one of the alternative equipment. Each product has a due date that must be met. Also, the processing times on each equipment vary depending on the order (product) being processed. We further assume that each equipment has a cost associated to the processing of each order that depends on the duration and use of raw materials and utilities. The objective is to minimize processing costs for the orders while meeting the due dates. Here, we divide the problem into an optimal assignment problem and a sequencing feasibility problem. It is natural to relax an MILP formulation by deleting the sequencing constraints which often have poor relaxations due to their big-M structure. Using CP for sequencing is often more efficient due to the fact that it provides special constructs for handling and managing sequences although it does not necessarily fit well for optimization purposes. The main difference with the work of Jain and Grossmann [3] is that in the problem of this paper we have multiple stages and therefore we cannot divide the problem by solving a separate sequencing problem for each parallel equipment. Furthermore, the generation of cuts is non-trivial since it is much more difficult to identify which of the assignments is the actual cause of a delay in the schedule.
2.1. Assignment Problem The assignment problem (MILP) can be written as follows: min
~
Cjm.T, jm.Yjm-k-C~.zm
(1)
jEJ,mEM
subject to Vj c J, s C S
E yJ m = 1, mEMs Yjm ~ Zm, Yjm = 0,
(2)
Vj E J,m C M
(3)
Vj, m E B
Yj,ml + Yj,m2 <-- 1,
VjEJ,
(4) ml,m2Ct3
(5)
879 +cuts Yjm E {0,
1}
Zjm E R +
The objective function (1) minimizes the cost for assignments added by a one-time cost for each equipment assigned. The assignment variables Yjm are defined in Eq. (2) which states that each job j has to be assigned to exactly one equipment m in each stage s. In Eq. (3), the variable Zm is forced to take the value one if any job j is assigned to it. Constraint (4) prohibits assignments defined in the set/~ and the topology constraint (5) ensures that a job does not follow disconnected paths given in the set/3. This sub-problem results in a least cost assignment that may not be feasible since the sequencing requirements have not been included explicitly in this model.
2.2. Sequencing Problem Having solved the assignment problem, the sequencing (CP) problem consists of finding a feasible schedule for a given assignment. In this part of the problem there is one complication. Constraint programming does not provide any information if a feasible solution cannot be found. In the single-stage case by [3], where each equipment was scheduled separately, not finding a valid sequence indicated that the assignment was infeasible. This was then used as a basis for deriving integer cuts for the MILP. In contrast, here we need to sequence all orders, which in turn requires that the due dates be relaxed in order to guarantee obtaining feasible solutions in the CP. The relaxation is performed by defining a slack for the due dates and minimizing the number of orders that do not meet their due dates. The problem formulation is as follows. min ~ d j
(6)
jEJ
subject to stage(j, last(t)).end _< Tj~ + sj, Sj ~> O r
dj = 1,
Vj E J
Vj E J
(7)
(8)
stage(j*, t*) requires equipment(m*)
(9)
stage(j*, t*).duration = "Cj*m*+ Tf, m,
(10)
stage(j,t) precedes stage(j, next(t))
Vj E J,t E T
(11)
The objective function (6) minimizes the number of orders that violate their due dates. A zero objective value indicates that all assignments are feasible and the problem is thus solved. The first constraint (7) relaxes the problem by allowing a slack sj on the due date TjP. Constraint (8) defines the relation between the lateness flag dj and the corresponding slack-variable. Constraints (9) and (10) build the link between an equipment, stage and order using the Yjm values from the assignment problem and the knowledge of which equipment belong to which stage. These fixed triplets are marked with a star (*). The parameter "r,j,m, is the processing time and TjS.m, is the setup time. The last constraint (11) defines the precedence between the stages. The proposed hybrid procedure combining CP and MILP is shown in Fig. 1.
880
Problem not solvable L Infeasible (Solve assignment problemL STOP r ~ MILP: (1- 5) + cuts I)"
I
(~Analyzesolution~ ~ addnewcuts
Fix assignments of orders
l
OPtim al s~ g luti~nSf~und "T ]~ N~slacks O ~. (~SP~lve seCP: quen(6cin11~rob lem]
Fig. 1. The hybrid procedure
Fig. 2. A schedule with late orders
2.3. Generating cuts If the relaxed sequencing problem results in positive slack variables (late orders), cuts need to be added in order to avoid these infeasible assignments. The cuts eliminate existing assignments and have the same general form as in [3].
E ajm'Yjm <_ E a j m - 1 jEJ
Vm E M
(12)
jEJ
The coefficients aim have 0-1 values depending on the current assignment. A number of different strategies have been tried but due to the lack of space we will just present one of them that eliminates the bottlenecking assignments by tracing a zero-wait path for each late job, starting from the last stage and advancing towards the first one. The method would produce for the case in Fig. 2 the following two cuts: YA3 + YA2 -+-YD2 nt- YC2 nt- YE2 -k- YB2 <_ 5 YD3 + YD2 + YC2 + YE2 + YB2 <_ 4
881 Table 1 Results of examples Strategy
5o/3s/6e 780
Objective
6o/2s/4e 1593
MILP*
19554/15.32
Hybrid**
41/1.70/0.40 7/0.09/0.17
9o/2s/5e 2243
10o/3s/6e 3815
8o/2s/6e 1987
10091/7.12 186799/461.26 10348/45.18 2730654/6056.34 8/0.19/1.45
5/0.12/4.62
18/1.50/2.03
*(Number of nodes/CPU-s) **(Iterations/CPU-s MILP/CPU-s CP)
3. NUMERICAL EXAMPLE In the following, a number of problems are solved to show the possible benefits of this strategy. In this comparison, we list the number of orders, stages and equipment and solve the problems using the proposed hybrid strategy and a simplified MILP-formulation from [4]. The results are presented in Table 1. Five different-size problems were solved and the number of orders, stages and equipment are listed in the header. The objective function values are given in the first row. For MILP, we report the number of nodes and CPU time, and for the hybrid strategy, the number of iterations is given followed by the CPU-s spent by the MILP assignment problems and the CP sequencing problems. As can be seen from Table 1 the computational savings are significant, especially when the problem size increase. The proposed hybrid strategy works well for all examples. It should be noted, however, that for large problems the sequencing (CP) may become inefficient due to the need of minimization of (6). Also, the robustness of the cuts depends on the fact that the sequencing part should minimize the completion times, by not forming any unnecessary slacks between both orders and stages. This, however, can easily be accomplished with a shortest path calculation after solving each CP problem. 4. CONCLUSIONS A hybrid method for solving multistage scheduling problem has been proposed. The major challenge in this method lies in generating cuts for assignments that do not exclude feasible solutions, but are strong enough to reduce the search space. The results have shown that order magnitude reductions can be achieved in computation. 5. ACKNOWLEDGMENTS The authors would like to acknowledge financial support from the Center for Advanced Process Decision-making at Carnegie Mellon University, the Finnish Cultural Foundation as well as Abo Akademi University. REFERENCES 1. Carlier J. and Pinson E. (1989), An Algorithm for Solving the Job-Shop Problem Management Science, 35 (2), 164-176 2. Hentenryck P.V. (1989), Constraint Satisfaction in Logic Programming MIT Press, Cam-
bridge, MA.
882 3. Jain V. and Grossmann I. E. (2000), Algorithms for Hybrid MILP/CLP Models for a Class of Optimization Problems Paper submitted for publication. 4. Mdndez C. A., Henning G. P. and Cerd~i J. (2000), A Continuous-Time Approach to ShortTerm Scheduling of Resource-Constrained Multistage Batch Facilities Proceedings, European Symposium on Computer Aided Process Engineering - 10, 1045-1050. 5. Pekny J.F. and Reklaitis G.V. (1998), Towards the Convergence of Theory and Practice: A Technology Guide for Scheduling/Planning Methodology AIChE Symposium Series, 94 (320), 91-111. 6. Pinto J. M. and Grossmann I.E. (1995), A Continuous Time Mixed Integer Linear Programming Model for Short Term Scheduling of Multistage Batch Plants Ind. Eng. Chem. Res., 34 (9), 3037-3051. 7. Pinto J. and Grossmann I.E. (1998), Assignment and Sequencing Models for the Scheduling of Chemical Processes Annals of Operations Research, 81,433-466. 8. Raman R. and Grossmann I.E. (1994), Modelling and Computational Techniques for Logic Based Integer Programming Computers chem. Engng, 18, 563-578. 9. Shah N. (1998), Single- and Multisite Planning and Scheduling: Current Status and Future Challenges AIChE Symposium Series, 94 (320), 75-90.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
883
Development of Interactive Facilities in a Knowledge-Based Scheduling Framework G. P. Henning INTEC (Universidad Nac. del Litoral - CONICET), Giiemes 3450 - 3000 Santa Fe - Argentina E-mail: ghenning @intec.unl.edu.ar This contribution presents an interactive scheduling framework aimed at supporting the construction and evolutionary modification of schedules by means of mouse-and-click actions over Gantt diagrams. The proposed approach pursues the engagement of the user in the process of maintaining a schedule in a dynamic environment, in a non-disruptive fashion, by providing him/her with interactive facilities for schedule modification. The proposed framework relies on an explicit object-oriented representation of the schedule and supports, up to now, three categories of user-driven revision actions: operation-based, production orderbased and resource-based schedule modification actions. Before executing any action proposed by the scheduler possible conflicts are checked. A rich underlying representation of the domain layer keeps track of different kinds of soft and hard constraints and prevents users from possible mistakes. Indeed, any manipulation is done under the supervision of consistency enforcing methods that send warning messages in case soft-constraints are violated and forbid modifications that infringe hard constraints. Consequences of accepted revision actions are propagated and immediately depicted. During this constraint propagation phase an opportunistic approach is adopted to improve the schedule's quality. 1. INTRODUCTION Complex production scheduling problems require a 'solution' to be more than the mere implementation of an algorithm. Schedule building in many real environments is an extended, iterated process that may involve conflicts between competing decision makers. Moreover, the scheduling problem is not a static one. Quite the opposite, it might be considered as one that requires continuous revision and reaction. Indeed, scheduling is an ongoing process where evolving and changing circumstances continually force reconsideration and revision of previously conceived plans. Due to these reasons, the scheduling function may be envisioned as an evolutionary one that operates in the following way. An initial schedule is built, problematic or unsatisfactory aspects of the pre-established schedule are identified, requirements are relaxed or strengthened, schedule modifications are made, and so on. Changes are introduced by means of mouse-and-click actions over a Gantt diagram and their consequences are immediately depicted on the diagram. Thus, the current schedule provides the context for identifying and negotiating changes in a non-disruptive fashion. This "process view" of the scheduling problem has also been stressed by [1 ] and [2]. Research activities related to reactive scheduling and scheduling revision techniques has gained importance within the field of Artificial Intelligence and knowledge-based methods only in recent years [3-5]. In relation to manufacturing scheduling, several alternative
884 methodologies have been proposed [5]. Nevertheless, the number of knowledge-based approaches employed to address scheduling problems pertaining to the chemical engineering domain is still very low as compared to other fields. This paper is organized as follows. First, the interactive scheduling view being addressed is presented. Then, the three different categories of revision actions developed up to now are introduced and the implementation of actions representative of the last category are discussed. Finally, conclusions based on the use of the facilities in industrial environments are presented.
2. "PROCESS VIEW" OF THE SCHEDULING P R O B L E M Scheduling is not an isolated, stand-alone function. Quite the opposite, it involves multiple decision-makers that belong to different departments of the organization and generally pursue distinct competitive goals. Even more, some of their goals might have not been considered during the initial schedule generation. For those decision-makers, a schedule is a context for identifying conflicts or weak points of a plan, for negotiating and suggesting changes as well as performing "what-if" type of analyses. Figure 1 shows a state transition diagram (STD) that represents the conceptual life-cycle of a production plan during the solution of the predictive scheduling problem under such circumstances.
States A Current accepted schedule B Scheduleto be modified C Analyzed schedule D Modified schedule E Final accepted schedule
Transitions tl modificationdecision t2 analyze and propose changes t3 selectand make modification t4 analyze and reject change t5 analyze and accept change t6 analyze and accept as final schedule
Fig. 1. STD depicting a conceptual representation of an schedule's interactive improvement process As seen in Figure 1, state A represents the initial schedule or any schedule under the interactive repair process. When the decision to modified it is taken (state B), it is first required to analyze the current production plan and to propose modifications (state C). When a change is introduced (transition t3) a modified schedule is generated (state D). After a new analysis process, such a schedule can be (i) accepted as the final production plan (state E), (ii) rejected, returning to the previously proposed schedule (transition t4 to state B), or (iii) accepted as a new intermediate schedule to be further improved (transition t5 to state A). In industrial environments, no matter how well defined a predictive schedule is, reactive facilities are needed to cope with unexpected events/disturbances on the shop floor as well as changes in production orders. The occurrence of unforeseen events such as the arrival, cancellation or modification of production orders, the break down of certain equipment items, the late arrival of raw materials, personnel illness, etc., all conspire to make a previously generated schedule invalid. As depicted in Figure 2, an interactive repair process can be performed in these situations. However, there is a difference between this case and the evolutionary improvement of a predictive schedule. It is due to the fact that it is necessary to represent the current active or partially executed schedule and to evaluate the consequences of unexpected events.
885
tl
J
t2 t6
~4 ts
States Transitions A Current active schedule tl unforeseenevent triggering a repair activity B Scheduleto be repaired t2analyze & propose repair changes C Analyzed schedule t3 selectand make modification D Modified schedule t4 analyze and reject change E Intermediateschedule t5 analyze and accept change t6 new modification decision t7 stop iterative repair process
Fig.2. STD depicting a conceptual representation of an schedule's interactive repair process after an unforeseen event occurs. 3. I N T E R A C T I V E R E V I S I O N ACTIONS This section presents different classes of interactive revision facilities that have been added to an existing scheduling framework which is under continuous expansion and improvement [2]. The implementation of these revision actions relies on an object-oriented (OO) model [6]. The representation of the underlying domain is not discussed here due to lack of space. However, the interested reader can fin its details in [2]. One of the key differences between this proposal and the one adopted by other authors is the explicit representation of a production plan. In case the interactive scheduling environment is used in a reactive fashion, the representation of the production plan distinguishes between already executed operations, activities currently under execution and others to be performed in the future. In this contribution, the methodology for generating the current working schedule is not presented. The schedule might have been developed by any technique, but the underlying assumption is that it will be represented by means of the OO model reported in [2], on top of which the interactive facilities are being built. Revision actions are executed by means of mouse-and-click actions, either selected from a tool-bar menu or from the menu that pops-up when clicking on the icons displayed on the graphical interface. Thus, the scheduler can modify a schedule by shifting or reordering production runs, splitting a long production run acting as a bottleneck (only for continuous and semi-continuous operations), adding new work shifts to overloaded units, and so on. In all cases, before actually executing the user requested action, consistency enforcing methods monitor the fulfillment of constraints defined in the system's knowledge base. Indeed, the system performs a look-ahead analysis and generates warnings in case soft constraints are not satisfied and prohibits proposed modifications that would violate hard constraints. The rationale behind this proposal is that human schedulers have accumulated enough experience to conduct the schedule revision process in a non-disruptive manner, provided they are given appropriate guidance tools to cope with the complexity of interacting constraints. Moreover, the underlying idea is to take advantage of an active graphical interface through which the user directly manipulates the OO knowledge base. For example, working under the schedule evolutionary improvement mode, the scheduler analyzes the Gantt chart and finds out that in a given processing stage, having several units operating in parallel, there is an unbalanced work load. Therefore, he/she decides to move certain production runs from the overloaded units to equipment having a smaller load. Before executing each proposed movement the system will automatically check whether or not the new assigned unit is an acceptable one, if the new predecessor and successor production runs are feasible neighbours,
886 etc. Even though other authors [1] consider that humans may adopt myopic "fire-fighting" tactics, industrial schedulers think they are the ones having ultimate responsibility for all decisions. So, they claim for "hands-on" facilities to interactively introduce modifications that might improve the current schedule's quality or make feasible an infeasible production plan. Three categories of user-driven revision actions have been implemented up to now: operation-based (move, delete, swap, merge, etc.), production order-based (insert, delete, modify) and resource-based (add/delete working shift, modify equipment availability period, etc.). Due to lack of space only examples of this last category will be provided.
3.1. Operation-based revision actions There are two big categories of operation-based actions: The ones that apply to production runs and the actions that are specific of changeover and setup operations. More details about the implementation of these particular revision actions can be found in [7]. Production Run-Based Revision Actions. Two main categories are again distinguished: simple actions, applicable to single production runs and composite actions appropriate for production run pairs. Move, Eliminate, Modify-Time-Bounds and Divide pertain to the first group and Swap and Merge to the second one. While Move and Eliminate are actions that may be executed on any type of production run, Divide cannot be applied to batch operations. Similarly, while Swap is relevant to any type of production run, Merge can only be applied in the case of two continuous production runs. Changeover and SetUp-Based Revision Actions. Modify-Time-Bounds and Eliminate are the only operations that have been implemented up to now pertaining this type of operations. 3.2. Production order-based revision actions The implemented revision actions allow the scheduler to work both at the level of whole ProductionOrders or at the level of StageProductionOrders. In the first case the actions refer to: (i) the deletion of an existing order (Eliminate), that results in the elimination of the associated stage orders, (ii) the definition of a new order (Create), resulting in the automatic creation of the corresponding stage orders, and (iii) the modification of the order, by changing the required amount product and/or it due-date. At the level of a StageProductionOrder the scheduler is given tools to insert it in the schedule of a unit, creating its associated
ProductionRun. 3.3. Resource-based revision actions Only renewable-resources have been addressed in the current implementation of the framework. Specifically, the modification of unit's availability periods (Create, Delete and Modify), working shifts (Add and Delete) and operating parameters (ModifyProcessingTime and ModifyProcessingSpeed) have been implemented. Future work will incorporate the consideration of other renewable resources, such as manpower, and non-renewable resources such as raw-materials and supplies. Figure 3 depicts partial views of the interactions diagrams [6] associated to the implementation of revision actions that operate on a unit's availability period. For example, Figure 3.a shows one of the diagrams that corresponds to the Delete action. As seen, first it is verified whether the period to be eliminated has associated scheduled operations. In case this is true, the system informs the user and asks him/her to confirm the deletion of the period and its associated operations. Only if the scheduler agrees, the action is actually executed. In turn, Figures 3.b and 3.c show partial views of the Modify action implementation. While Figure 3.b. depicts a situation that corresponds to a wrong specification
887 of the end points of the interval, Figure 3.c shows a case associated to a situation where conflicts with other availability periods of the same unit are found. The system detects this situation, informs the overlapping interval to the user and solicits him/her to redefine the end points of the unit availability period. All these cases are examples of the system's look-ahead capabilities that perform a series of checking activities before executing any action. The implementation of such facilities is made possible due to the explicit representation of the domain in which the approach relies. 4. CONCLUSIONS This interactive scheduling approach pursues the engagement of the user in the process of maintaining a schedule in a dynamic environment by providing him/her tools for schedule modification. While it recognizes the important role that human expertise may play, incorporates facilities for keeping track of constraints. Indeed, any manipulation is done under the supervision of consistency enforcing methods that send warning messages in case softconstraints are violated and forbid modifications that infringe hard constraints. The consequences of accepted revision actions are propagated and immediately depicted. During this constraint propagation phase an opportunistic approach is adopted to improve the schedule's quality. The proposed facilities have been implemented in the KAPPA-PC environment [8] and incorporated into industrial support systems being used on a daily basis. Revision actions have been very well accepted by industrial schedulers who prefer this approach to automatic repair facilities. Indeed, they claim for more active graphical interfaces and the implementation of more sophisticated revision actions involving non-renewable resources. One of the features that schedulers like most is the scenario management system. It allows them to simultaneously handle several prospective schedules and play "what-if" type of analysis by simply switching between distinct graphic interfaces.
Acknowledgments. The author acknowledges financial support from FONCYT under Grant 14-00356, and from "Universidad Nacional del Litoral" under CAI+Ds 048 and 121.
REFERENCES 1. Smith, S., OPIS: A Methodology and Architecture for Reactive Scheduling. In Zweben, M., Fox M. (eds), Intelligent Scheduling. Morgan Kaufmann, San Francisco (1994) 29-66. 2. Henning, G.P., Cerd~i, J., Knowledge-based predictive and reactive scheduling in industrial environments: Computers and Chemical Engineering, 24, (2000) 2315-2338. 3. Sauer, J., Bruns, R., Knowledge-Based Scheduling in Industry and Medicine, IEEE Expert, 12 (1997) 24-31. 4. Kerr, R., Szelke E. (eds.), Artificial Intelligence in Reactive Scheduling, Chapman & Hall, London, 1994. 5. Zweben, M., Fox M. (eds), Intelligent Scheduling. Morgan Kaufmann, 1994. 6. Quatrani, T., Visual Modeling with Rational Rose and UML Addison-Wesley, 1998. 7. Henning, G., Knowledge-Based Interactive Scheduling of Multiproduct Batch Plants, Lecture Notes in Artificial Intelligence 1952, Monard M.C. and Sichman J.S. (Eds), Springer-Verlag, 2000. 8. Intellicorp Inc. KAPPA-PC User's Guide. Version 2.2, 1994.
888
Fig. 3. Interaction diagrams showing how actions applicable to availability periods are implemented. (a) One of the implementations of the Delete operation (b - c) Modify associated implementations.
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V.All rightsreserved.
889
Computer aided system for short term scheduling of batch processes based on hybrid simulation G. H6treux a, J.Perret a, H. Pingaud b a Laboratoire de G6nie Chimique (CNRS - UMR 5503), INPT-ENSIGC 18, Chemin de la Loge, F-31078 Toulouse Cedex 04, France b Ecole des Mines d'Albi Carmaux / Centre de G6nie Industriel Campus jarlard - route de Teillet, 81013 Albi CT Cedex 09, France Short term scheduling of batch processes is a complex problem on account of its hybrid nature. It could be tackled either with optimisation methods or with simulation approach. The aim of this contribution is to present the structure of a decisional system which combines in a single environment a hybrid simulator coupled with various decisions-making mechanisms (local or global decisions established either with heuristics, optimisation or user control). 1. INTRODUCTION In the recent years, the competitive economic context confers relevance in the study of batch processes (low volume production of high value added products). Batch plants are characterised by a necessary flexibility in utilising the various resources (human and material) and a reactive behaviour in presence of frequently changing customer requirements. If planning and scheduling of discrete manufacturing operations are well-established (1), the scheduling of batch processes is less well-developed (2). Indeed, as neither their discrete nor their continuous aspects can be under-estimated, batch processes are described by dynamic hybrid systems (3), and so, the classical methods cannot be applied as is. Although a solution is generally achieved rapidly by simulation, unresolved conflicts can appear in some cases, and a more global analysis of the problem is then necessary. The aim of this contribution is to present the structure of a decisional environment based on a hybrid simulator coupled with various decision-making mechanisms (heuristics, optimisation or direct interaction with operators).
2. PRINCIPLES OF THE HYBRID SIMULATOR Hybrid problematic deals with systems where continuous phenomena and discrete events occur explicitly and simultaneously. Separated theory and methodology have been developed in each domain and efficient solutions are provided for homogeneous problems. Consequently, complex hierarchical organisations have to be developed, inducing
890 interactions between discrete scheduling algorithms and continuous control algorithms. In the proposed modelling, the discrete part (recipe phases sequence as feed, reaction, etc.) is represented by Petri nets. The continuous part is described by means of differential and algebraic equations that simulate the evolution of continuous state variables relative to each operation (reaction ~ mass and thermodynamic balances, PID control, etc.). As they are defined in both parts, events provide the link between the discrete supervision level and the continuous control level. So, an event firing means the transition crossing and the switch from a continuous phase to the next one. The models for recipes are separated from the resource models. When a unit is engaged to deal with a recipe task, the interaction between the two Petri net levels is made by way of the dynamic transitions merging mechanism (4). In order to organise the plant information, Object Petri nets are used and a net hierarchical architecture is defined. Tokens, places and transitions are instances of different classes. A computer implementation has been achieved with an objected oriented language (5). Moreover, this hybrid simulator is implemented in a software package and it is used in an important French agri-food company as a plant retrofitting tool (6). 3. SIMULATION AND SCHEDULING Nevertheless, an efficient exploitation of this hybrid simulator for scheduling still requires a few works. The global model made up with the set of Petri nets (units, recipes, etc) describes a superstructure of the various patterns that can be reached by the system. A single simulation represents one pattern Ex: A 6-days simulation, with 30 units, 150 jobs, l0 recipes owning 11 among the whole set of possibilities. The tasks each o n e ~ 85 . . . . . d . . . . Pentium 266 MHz with 64 Mo of RAM. completion of a simulation necessitates much decision-making (resource allocation, lot sizing, operations starting date, etc.). Some decision variables have flexibility and are currently set using simple heuristics finalised by the industrial experience. A Gantt chart is the main output. In particular, it provides a possible schedule of operations, defines batch sizes and processing times and settles the choice and the allocation of each resource. Nevertheless, there is a major drawback in the use of simulation in order to solve scheduling problems: if some constraints are explicitly taken into account during computation (level in tanks, mass balances, etc.), others are only validated subsequently at the end of the simulation (maximum time lags, etc.). Indeed, these heuristics which are fast to compute, use information about what happened before the current simulation time, and try to make predictions on the immediate future. But a decision at time t can lead to the violation of a constraint at time t+At. Since a past decision can not be modified, the violated constraint is only indicated on the Gantt chart. So, in some cases, this mechanism can lead to poor decisions when applied to constraints requiring a whole time horizon view or complete process knowledge is necessary (for example, respect of maximum time lags or determination of an efficient recipe sequencing strategy when cleaning operations have to be carried out). In this context, a basic solution is that the user runs several simulations in which some choices are fixed at each iteration. But, the complex nature of interactions makes this iterative procedure exhaustive; furthermore, Fig. 1. Decision-making structure
891 there is no guarantee of convergence. The proposed approach defines a two level decision module including the hybrid simulator : 9A "detailed" level (scheduling oriented) which mainly deals with the resource allocation and temporal aspects (starting dates, cleaning operations, etc.). To do this, the hybrid simulator is used. 9A "aggregate" level (planning oriented) which has a more global view of the problem over time. Its objective is : - on the one hand, to evaluate lot sizing, - o n the other hand, to generate additional constraints (post-simulation decisions) within the model in order to lead the system toward a proper solution. To do this, a complete reference graph of the flow-sheet is built. Processing times and resource allocation established by the simulator are taken into account in this reference graph and a modification procedure is run and compared to the initial simulated solution. According to the type of modification carried out, the decision module directly provides the final solution or interacts with the simulator in an automatic iterative fashion until a satisfying solution is found. So, the main topic of the decision-making module is the research of a feasible plan among whole possibilities. It is based on a disjunctive graph and a combinatory exploration that leans on a Branch & Bound method. 4. DECISION-MAKING MODULE 4.1. Principles
disjunctive graph has to be built based on recipes defined with the recipe editor software. This gives a monolithic representation of the entire problem and each task appears explicitly. For instance, if production orders require the manufacturing of 2 batches, only one flow appears on the recipe editor representation whereas the disjunctive graph, which models the path of each batch, is made up of two different paths (cf Figure 2.).
4.2. Disjunctive graph modelling
The graph G(T,U) used to describe the flow-sheet is an extension of classical disjunctive arc based graphs used for job shop scheduling (7) (8). It translates the sequencing decisions of tasks on shared resources. So, it is made up of: 9A set T of vertices representing the set of operations. Each vertex corresponds to a task To., i.e. the jth operation of job i, associated with an amount of material B~ (batch size) and a fixed duration Po. Let call to., the effective starting date of task To.. The dummy nodes s and t represent respectively the origin and the end of the schedule. They start and close the manufacturing of batches. 9A set U of arcs representing temporal relationships between operations. U can be decomposed into 2 subsets : Uc (conjunctive arcs with Uc=UpC)UNCJUL cJUs), and Up (disjunctive arcs).
892 o:-
Conjunctive
ares
Uc"
9 Subset Up: each arc uu, u e U e is defined between two subsequent tasks Tg and Tkt. They represent the recipe relative to each job. An arc uij,kt is associated with a weight set equal to the processing time Pu of the task connected to its origin. An arc between s and each task without predecessor is weighted by 0. An arc between each task T o. without successor and t is weighted bypo.. 9 Subset Us : these arcs model delay constraints : - If a maximum time lag Au, kt>O exists between the end time of task T o. and the starting time of the next task Tkt, a process arc ukt,Oe U s weighted by -(po.+A,j, kt) is defined. - I f Au,kt = 0, this arc induces a no-wait constraint making task Tkt start exactly at the completion time of task To.. 9 Subset UL : these arcs model calendar constraints [r,j, do.] : - If a task T o. has an earliest starting date ro., then an arc weighted by r o. is defined between s and TO.. - If a task T o. has a latest completion time do., then an arc weighted with-(do.-pg) is defined between T 0. and s. 9 Subset Us : these arcs model the fact that an operation needs several resources and so, it is made up of several sub-tasks. The sub-tasks TUare carried out simultaneously, so they have the same starting date and the same duration. They are linked with arcs always weighted with O. 9"
Disjunctive
arcs Up"
Each disjunctive arc can be represented by two opposite conjunctive arcs. These arcs uo.,kt and ukt, o are created between two tasks that possibly share a resource. Each arc uu,kt is weighted by Po. 4.3 S e a r c h o f a f e a s i b l e s o l u t i o n In our procedure, the idea is to use the schedule provided by the simulator as an initial basis of exploration. Indeed, it yields the processing time, the batch sizes and a sequence of tasks. A complete graph of the problem is built where all disjunctive arcs are explicitly shown. Then, the resource allocation and the direction of disjunctive arcs are fixed, based on the previous schedule. The calculation of earliest and latest starting dates determines if this graph really has a solution, in which case, available margins are evaluated. Then, all that remains is to move within the margin the starting date of each task which has raised an alarm during simulation.
5. E X A M P L E The built of the disjunctive graph has to be built by following a well-defined procedure. This is illustrated on an example. Two kinds of product with a specific recipe are produced in the flow-sheet (cf Figure 3.). This last is made up with 1 reactor, 2 buffer tanks, 2 conveyance systems and 1 packaging line. We suppose that each resource performs only one kind of operation apart from the reactor and the conditioning line that are shared between the two recipes. E a c h recipe is b r o k e n into five c o n s e c u t i v e steps : S,t: Material raw drain (T,I) // T r a n s p o r t (Tia) // F e e d (T,3) S,2: R e a c t i o n (Ti4) Si3: Drain (Ti5) // T r a n s p o r t
(T,6) // Feed (TiT)
S,4: Idle (T,8) Sis: Drain
(T,9) // Packaging
(T,10)
T h e processing times are the f o l l o w i n g : Pll = PiE = PI3 = Pal = PE2 = P23 = 15 m i n ; PI5 = PI6 = PI7 = P25 = P26 = P27 = 15 m i n ; PI9 = PlI0 = P29 = P210 = 10 min. F i g . 3. R e c i p e s o f e a c h p r o d u c t
893 Reactions in batches 1 and 2 last 2 and 1 hours. The idle time lasts from 0 to 1 hour in the first tank (pls=0, A18=60), and from 15 minutes to 4 hours in the second one (p28=15, A28=240). Moreover, we suppose that the 2 batches must be delivered between 23.00 and midnight. In these conditions, the Gantt chart provided by the hybrid simulator alone is shown in Figure 4. One can notice that an impossibility has been generated regarding the use of the reactor. Indeed, an alarm is raised (ALA_TPS) because the maximal delay constraint is violated (75 minutes instead of 60 minutes). The drain of the reactor for the first batch can not be achieved in the same time that the feed of this same resource for the second batch. Fig. 4 : Gantt chart generated by the simulation software As all constraints are not satisfied, the decision-making module is run. The corresponding disjunctive graph is built : - The vertices representing a packaging and a drain of raw materials are respectively connected to the end and the origin dummy node by arcs belonging to the subset Up - Arcs UN modelling delay constraints are associated with the arcs Up. - Drain, transport and feed operations are simultaneous; the associated vertices are connected themselves by simultaneity arcs belonging to the subset Us. - Arcs UL modelling calendar constraints connect the origin dummy node to the vertices representing packagers. For each one, two arcs - (d19 " p19) convey the instructions S~tt[,20,.~ S 113 r 1"r S~-~[255' SIps,~2+70'~270~ given by the production ,z, o , ; ~ ! Su!270 270j.:~"~~,] o ' 0 slices. The disjunctive arcs -P , -P . ~J/ Uo connect vertices sharing the same resources. The corresponding disjunctive graph is shown in Figure 5. " /11"[300, 300] -p23 ", p.... Process arcs Up, UN, Us, UL [30, 30 correspond to thin arcs whereas resource arcs \ e 130.301 1~__ ~241~,~v, p28"~" I correspond to bolded arcs. x~~ 't $23i105, 120]$241240,~0f The scheduling generated by simulation alone - ( d 2 9 - P29) correspond to the Fig. 5. Solution graph conjunctive graph where arcs 1 and 3 have been chosen. Earliest and starting dates are computed by using the Bellman-Ford algorithm. It shows that in this case, no solution exists with this sequence of tasks. So, other sequences have to be tried until the decision-making module find a satisfying solution (Branch & Bound algorithm). In our example, a feasible solution is provided when arcs 2 and 4 are chosen. Earliest and latest starting times are displayed near each node between brackets. It appears that to generate a feasible schedule, the reaction of the second batch has to be first achieved. The Figure 6 shows the Gantt chart of the feasible solution. s
"
\
' :
894
Fig. 6. Gantt chart of the solution 6. CONCLUSION The hybrid simulator described in this paper is implemented in a software package and it is already used in an important French agri-food company as a plant-retrofitting tool. In order to meet the needs of batch process scheduling, a global decision module has to be added. The work presented in this contribution provides a first response against a type of implausibility often encountered in the schedule established by only simulation. This module will be soon integrated into the software. Nevertheless, many areas for further investigations can be identified: 9Concerning the continuous aspect, it is envisaged to link the simulator to a general DAE integration module. This feature would allow the undertaking of more detailed and generic models of processes. 9Concerning the scheduling aspect, the second phase of the proposed method has to be developed in order to improve the final solution. In particular, it seems interesting to take into account set-up operations (such as cleaning operations) more precisely in order to reduce lost time in the decision-making module. The batches splitting, is another feature under development. Indeed, in the case where the decision module can not find a solution, some batches may be splitted into smaller batches and then, a new simulation can be computed. REFERENCES
1. 2. 3. 4.
5.
6. 7.
8.
M. Pinedo, Scheduling ." Theory, algorithms and systems, Prentice Hall, Englewood Cliffs, 1995. E. Kondili, C.C. Pantelides, R.W.H. Sargent, A general algorithm for short-term scheduling of batch operations : MILPformulation, Comp. & Chem. Eng., 17, 2, pp211-227, 1993. A Pages et al. , An hybrid process model based on Petri nets applied to short term scheduling of batch~semi continuous plants, ADEDOPS'95, London, April 1995. R. Champagnat, Supervision des syst~mes discontinus : d6finition d'un modMe hybride et pilotage en temps r6el, Th~se de doctorat de l'Universit6 Paul Sabatier, LAAS, Toulouse, 1998. B. Palomino, Conception d'un simulateur hybride pour l'aide ~ la d6cision en ordonnancement d'ateliers batch de l'agroalimentaire, th~se de doctorat de l'Institut National Polytechnique, LGC, Toulouse, 2000. B. Palomino, H. Pingaud, X. Joulia, ProMixt An environment for simulation and scheduling of batch processes, ECCE'2, Montpellier, 5-7 October 1999. G.H~treux, B.Palomino, H.Pingaud, (2000), Decision making tool for short scheduling in batch processes, The fourth Korea-France workshop on simulation, optimization and control in process systems engineering, Chejudo Island, Korea, February 8-9, 2000. P. Esquirol, P. Lopez, L'ordonnancement, Economica, Paris, 1999.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
895
A u t o m a t i c A c c i d e n t Scenario Generation for Petrochemical Processes Dongwoon Kim a, Yongsun Lee a, I1 Moon at, Youngsoon Lee b, and Donghyun Yoon c aDepartment of Chemical Engineering, Yonsei University, Seoul 120-749, Korea bDepartment of Safety Engineering, Seoul National University of Technology, Seoul 139-743, Korea CKorea Occupational Safety and Health Agency, Inchon 403-120, Korea This paper presents a methodology for the automatic generation of accident scenarios in chemical process industries. Accident scenarios as a basis of hazard analysis are considered as a part of the design criteria. Generating accident scenarios is a time and labor consuming procedure. To reduce the effort and resources devoted to this, we developed a systematic method of generating accident scenarios and implemented software called CAGAS (Computer Aided Generation of Accident Scenarios). This system is composed of a scenario generation model and a scenario verification model. In the scenario generation model, many accident scenarios are generated by accident scenario factors including process information and accident progress factors. The scenario verification model verifies the generated accident scenarios based on an expert knowledgebase and an accident history database. This system is useful in the detection and the identification of dangerous states in complex chemical plants during the design stage. This is also valuable in providing guidelines to prevent accidents and to improve process safety. 1. INTRODUCTION The safety of chemical processes needs to be verified from the first step of the design. Among many hazard analysis methods, testing possible accident scenarios is widely used to find hazardous location and the sequence of accidents. This study proposes a systematic method of generating accident scenarios to evaluate various chemical processes. Various approaches of accident scenario generation have been proposed [2-5]. However, a consistent application of standardized procedures for considering accident sequences has not been achieved although the proposed procedures seek to identify the credible accident scenarios. In this study the accident sequence is expressed as accident progress factors by analyzing the actual accident history and the results of safety evaluation methods such as HAZOP, FTA and SMV. Here, generated accident scenarios describe the expected situation with the accident sequence factors for the given process. This gives guidelines on the basis of safety aspects during the design stage. tCorresponding address: Tel) +82 2 361 2761, Fax) +82 2 312 6401, E-mail) [email protected]
896 2. ACCIDENT SCENARIOS IN HAZARD ANALYSIS The most important step in the hazard analysis procedure is the selection of study nodes because the required time for conducting hazard analysis increases rapidly as the number of study nodes increases. In conventional hazard analysis, Preliminary Hazard Analysis (PHA) is used to decide the study nodes and to determine the significant hazards associated with process. This analysis need a team of analysts for the complex chemical process and highly depends on heuristics.
Figure 1. Hazard Analysis Method using the Accident Scenarios Once accident scenarios are generated by the proposed methodology at the stage of PHA for conducting a hazard analysis, the hazardous location and status in the process can be easily identified. It helps reducing the time and cost for conducting hazard analysis. 3. C O M P U T E R AIDED GENERATION OF ACCIDENT SCENARIOS In general, the generation of reliable accident scenarios strongly depends on an expert's experience. But experts tend to consider a scenario as the frequency of accidents and ignore the consequence of an accident. This approach based on an expert's experience is a time and labor consuming procedure. To reduce the effort and resources devoted to this and to consider the frequency and consequences of an accident, we developed a systematic method of generating accident scenarios and implemented software called CAGAS (Computer Aided Generation of Accident Scenarios). 3.1. Structure of CAGAS
CAGAS consists of main modules and three sub modules. The main module includes process data input-output interface and visualization of the generated accident scenarios in various graphs and documents. Three sub-modules are 1) PIM (Process Information Module) 2) PFM (Progress Factor Module) 3) SVM (Scenario Verification Module). PIM describes characteristics of process units such as process equipment, process condition, and site status. The PFM is composed of two linked databases: APD (Accident Progress Database) and Progress Check Database (PCD). The APD contains more than 3,000 accident sequences. Each record contains the cause of the accident, formula and sequence code information. The PCD provides detailed response information about the probability factor of each scenario
897 especially for unit, equipment and site status. It is determined by expert heuristics based on operation experience and accident histories. SVM verify the result of generation to avoid unexpected results. It determines whether the generated accident scenarios have a reality and supplies a priority of accident scenario.
3.2. Methodology for the Generation of Accident Scenarios Accident scenarios are combinations of hazard locations and accident sequences. Hazard locations are determined by process risk. In this system process risk (Ranking Values) is defined by the likelihood (UFL: Unit Factor for Likelihood) and consequence (UFC: Unit Factor for Consequence) for each accident scenario factors as shown in figure 2. These are comparative values of accident frequency and consequence for each accident scenario factors. It ranges from 0 to 1 for frequency and from 1 to 10 for consequence and weighting factors are introduced for material factors and operating condition factors. For operating temperature, weighting factors are decided by using the following condition.
IF Operating Temperature > Flash Point IF Operating Temperature >_ Boiling Point wf=l. 1 Else wf=l Else wf:l
(1)
Since this condition gives information to determine outcome event of accident scenarios, weighting effect follows consequences of the final accident event. Table 1. Accident Scenario Factors Accident Scenario Factors RankingValues Unit Factors UFL, UFC Process Factors PFL, PFC Equipment Factors EFL, EFC Material Factors MFC Status Factors SFL, SFC Accident Progress Factors RFL, RFC Ignition Source Factors IFL, IFC Process Condition Factors
Examples LDPE, HRDS, CDU Storage, Reaction, Distillation, Separation Reactor, Pump, Pipe Hydrogen, Benzene Startup, Normal Operation Leak, Rupture, Crack/Hole Spark, Hot Spot Temperature, Pressure, Capacity
To predict accident sequences for given input data, we compare input data to APD and decide the accident sequences depend on PCD. The potential risk of each accident scenario is also derived from the component factors to produce a numerical risk ranking and comparative analysis of those risks. The priority of generated accident scenarios is defined by ranking value.
Ranking Value = ~ Consequence, x I-Ii Likelihoodi
(2)
Once an accident scenario has been developed, for each step of the accident sequence, the factors affecting the sequence and the variable conditions could be verified by related actual accident data. The system analyzed the causes of accidents, represented the effect of failure combinations, and found the most probable dangerous scenarios in complex processes.
898
Figure 2. Flowchart of Accident Scenario Generation 4. ACCIDENT SCENARIOS FOR HYDROCRACKING PROCESS Using the CAGAS, accident scenario generation is carried out for a typical hydrocracking process, which is shown in figure 3. This process is comprised of several common process units such as heat exchanger, reactor, pump, pipe and valve. The hydrocracking reaction generates heat that increases temperature and causes the reaction rate to accelerate. Since this process is operated under high pressure and high temperature, there exist lots of potential hazards. An accident history at selected hydrocracking units in three countries is illustrated in Table 2. These accidents caused losses of property, human life, and environmental quality. Major causes of these outcomes are hydrogen blistering, erosion, overpressure, crack and partial overheating. Using the CAGAS system, various accident scenarios are visualized at different process conditions. This system generated almost three hundred accident scenarios and their risk ranks. And the system describes the accident progress sequences as well as determines root causes and contributing factors. These are summarized in figure 4 and table 3. Table 2. List of Accidents at Hydrocracking Units Unit Process
Section
Causes
Events
Loss
Location
Year
Hydrocracking Unit
Heat Exchanger
IgnitionSource
Fire
4 injuries
Korea
1997.07
Hydrocracking Unit
RecycleOil Pipe
HydrogenBlistering
Explosion
$77,000,000 Korea
1999.05
Hydrocracking Unit
Pipe(Elbow)
Erosion
Fire, Explosion
-
Korea
1997.03
Hydrocracking Unit
Separator
Overpressure
Fire, Explosion
$73,000,000
UK
1987.03
Hydrocracking Unit
Reactor
Crack
Fire
$15,000,000
USA
1980.12
Hydrocracking Unit
Reactor
Partial Overheating
Explosion,Fire
$25,000,000 USA
1970.12
899
Figure 3. Process Flow Diagram for Hydrocracker High Pressure System Reprinted from: US EPA, EPA Chemical Accident Investigation Report, 1998. The results for sequence of accident scenario in these units are as follows.
Highly Ranked Accident Sequence 1 Intemal corrosion and erosion at the piping system in a reactor causes a material weakening and rupture of a pipe. It leads to fire and explosion in the present of ignitior source.
Highly Ranked Accident Sequence 2 Relief valve trouble occurs at the separator section caused by overpressure. It leads to release of material and fire develops under high pressure and high temperature.
Highly Ranked Accident Sequence 3 Hydrocarbon is released and a fire subsequently develops in the reactor effluent pipe du~ to excessively high temperature caused by reactor temperature excursion. The highly ranked accident scenarios involve the release and auto ignition of a mixture of flammable hydrocarbons and hydrogen under high temperature and pressure as known T.A. refinery plant.
Figure 4. Major Causes of Accident for Hydrocracking Process
900 Table 3. Examples of Accident Scenario Progress No. Accident Scenario Progress 1 2 3 4 5 6 7 8 9 10
InternalCorrosion --) Weakening--)Rupture Rapid Evaporation --) Overpressure--) Rupture Changeof Condition --) TemperatureExcursion--) Rupture WrongConstruction/Design --) Rupture Erosion--) Weakening--) Rupture Poor Flow --) Changeof Condition (TemperatureExcursion) --) Rupture VentBlockage--) Overpressure--)Rupture InadequatePurification --) Runaway/SideReaction --)InternalExplosion/Fire--) Rupture WrongManual --) Mal-Operation/PoorMaintenance--) Overpressure--) Rupture InstrumentFailure --) IncorrectCondition --) Runaway/SideReaction--)InternalExplosion/Fire--)Rupture
5. C O N C L U S I O N A new method for the generation of accident scenarios was proposed for chemical process industries, and the computer aided generation of accident scenarios (CAGAS) system was developed in this study. As a result of applying CAGAS to the hydrocracking process, hazardous locations and dangerous states are found, and the system generated almost three hundred accident scenarios and their risk ranks. The highly ranked accident scenarios are fire and explosion, which are caused by material weakening from the internal corrosion and erosion in the piping system of the reactor. The current limit of this system is that it is applicable only to a domain specific process, but it will be expanded to general chemical processes by building generic libraries. If hazard analysis such as HAZOP covers most possible hazardous states, which are identified by this system during the design stage, major accidents in chemical plants could be prevented. This study proposed an approach to improve the safety of chemical plants by generating accident scenarios systematically. REFERENCES 1. F. I. Khan and S. A. Abbasi, Inherently safer design based on rapid risk analysis, Journal of Loss Prevention in the Process Industries, 11, (1998). 2. I. Moon, D. Ko, S. T. Probst and G. J. Powers, A Symbolic Model Verifier for Safe Chemical Process Control Systems, J. of Chem. of Japan, Vol. 30, No. 1 (1997). 3. S. H. Lee, J. K. Kim and I1 Moon, Safety Analysis of Boiler Process Operating Procedures using SMV, Journal of the Korean Institute of Chemical Engineering, Vol.37, No. 5 (1999). 4. H. R.Greenberg and J. J. Cramer, Risk Assessment and Risk Management for the Chemical Process Industry, New York: Van Nostrand Reinhold, 1991. 5. L.C. Cadwallader, H. Djerassi, I. Lampin and J. Rouillard, A Comparison of U.S. and European Methods for Accident Scenario Identification, Selection, and Quantification, IEEE Thirteenth Symposium (1990). 6. US EPA, EPA Chemical Accident Investigation Report, 1998. 7. NFPA, NFPA Code 325M, Fire Hazard Properties of Flammable Liquids, Gases, and Volatile Solids, National Fire Protection Association, 1991.
European Symposium on Computer Aided Process Engineering - 11 R. Gain and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
901
The integration of production plan and operating schedule in a pharmaceutical pilot plant L. Mockus, J.M. Vinson, K. Luo Pharmacia Corp., 4901 Searle Parkway, Skokie, IL 60077 In reality, planning and scheduling activities are loosely integrated. In the case of a complex supply chain, it might be too difficult to maintain both long-term and short-term information within one large model. However, in the case of a single facility and relative short time horizon this task becomes feasible. This is the case in a pharmaceutical pilot plant, where a two- or three-year planning horizon is the most one would expect due to frequent changes in the products and their demands. In earlier work, we decomposed the pilot plant planning and scheduling problem into long-term planning of resources and shortterm scheduling of operations [ 1]. The long-term plan is mainly concerned with allocation of human resource, such as operators, engineers, chemists, and analysts to projects that have uncertain timelines. The short-term schedule fixes the human resources and considers the scheduling of activities in the pilot plant itself. The algorithm resolves uncertainty in future projects by over-allocating resources so that, on average, resource constraints are met. In this paper, we continue this exploration. In particular, we explore techniques for combining the long- and short-term into one plan without creating undue burden on the solver and the planning group. 1. INTRODUCTION We present a methodology that combines a production plan and a daily operation schedule in a single model. Normally, it is unnecessary and computationally infeasible to represent detailed, everyday operational activities in the production plan. In a pilot facility, detailed activities for running a production step become finalized only a few weeks, or even a few days before the actual run. Our approach is to use general information for future campaigns and add more detail to campaigns closer to the start of the scheduling horizon. How detailed should an operation schedule be? The computational aspect is just part of the problem. The level of detail carries information such as duration and resources needed. For scheduling purposes, manpower is our major concem; clearly, the schedule should not exceed the resource limitations. Thus, we decided to model only operations with different manpower requirements, while combining those with similar manpower requirements into super-operations. For example, two charges followed by a heat operation might be combined into one operation, called "reaction preparation," if they all require the same resources. Another aspect of the proposed approach is to keep the production plan and operating schedule within one model. Currently, we create the production plan and operation schedule separately; with the operation schedule following production plan. The creation of these plans is manual and requires quite a lot of human intervention (phone calls, meetings, etc.) in
902 order to develop an acceptable schedule. Unfortunately, the two schedules can easily become unsynchronized over time, requiring continual oversight. Integration of the production plan and the operation schedule permits closer delay monitoring and faster resource conflict resolution. We automate this process by integrating the production plan and operation schedule in one model and by scheduling the resulting model with a commercial solver developed by Advanced Process Combinatorics (APC). A natural extension of this methodology gives us an operation rescheduling mechanism where operations that are already completed become frozen for the scheduler. As a result, we only have to schedule remaining operations, and thus the impact on the near-term schedule is minimal. The advantage of implementing various level of detail is two-fold. It allows keeping the model small. Second, and probably the most important benefit is that maintenance of the model is simplified in the case of daily changes. 2. BACKGROUND
Batch scheduling literature is mostly concerned with the optimal allocation of resources to tasks and optimal scheduling of tasks to meet a given demand schedule. The classic examples are from Kondili and others [2, 3] and Sahinidis and Grossmann [4]. The recent work by Honkomp and others [5] considers uncertainty prevailing in the pharmaceutical industry, but this uncertainty is just in processing time. They assume that demands are known a priori for the duration of scheduling horizon. These papers mostly reflect the manufacturing environment. In the R&D environment, this assumption might not hold primarily because the demands are dictated by clinical, toxicological, and formulation studies and are quite uncertain. In addition, production runs are unique to each campaign, so the timing and sequencing of tasks will vary from campaign to campaign. There are works addressing the problem of reacting to processing uncertainties in the chemical industry. Cott and Macchietto studied the performance of on-line scheduling with several time shift algorithms [6]. In the algorithms, the deviations between the target schedule and the actual schedule are detected at short intervals and the schedule is shifted in order to minimize the effects of processing time variations. Kanakamedala and others presented an algorithm that uses a least impact heuristic and provides significant improvement over the earliest finishing unit heuristic by emphasizing the importance of maintaining the original schedule [7]. Kim and Kim propose a mechanism for reactive scheduling where they use simulation- in conjunction with job dispatching rules - to dynamically generate a schedule for discrete manufacturing systems [8]. Knowledge based approaches involve human expertise in planning and scheduling activities (including reactive scheduling) [9]. Instead of using math program to decide the optimal schedule the human expertise is utilized by providing a suite of tools (including 'what-if analysis). Little attention is given to such issues as combining the production plan and operation schedule in a single model. In the work of Zhu and Majozi [10] the integration of planning and scheduling is being considered for the special case of multipurpose batch plants where equipment is dedicated for each process. In our pilot plant, equipment is shared across processes. As mentioned above, this work provides a methodology for tighter integration of the production plan and operation schedule. The uncertainty in the pilot plant environment is handled by continuously decreasing the level of detail for the future batches, thus focusing on the current production schedule while making allowance for the future production. One way
903 to think about it would be imposing a high penalty for the deviations from the near-term plan and continuously decreasing penalties for deviations further into the future. Penalties dynamically change as we move forward in time. In addition, the decreasing level of detail for future batches serves the purpose of a cushion to accommodate the uncertainty of future demand dates and quantities. If we plan to the lowest level of detail and something happens (equipment breakdown, rework, etc.) then much of the work (human or computational) that went into the original schedule will have to be redone. On the other hand, if a future batch is created as a single task with the duration and resources corresponding to the sum of the unit operations, then it is cushioned from uncertainties. Eventually this batch will be ready to run, the level of detail will be increased, and the reserved time slot will provide more flexibility during rescheduling. These batches can be further abstracted into a single step task, representing all the batches. The steps, too, can be rolled-up into a campaign that will act as a placeholder for the resources. 3. PROBLEM STATEMENT
In its simplest form, the planning and scheduling problem can be stated as the minimization of the sum of tardiness for all campaigns (demands) subject to constraints on allocation, sequencing, resources, and production. Allocation constraints allow only one task to be processed on a given piece of equipment at any time. Sequencing constraints give the specific order of operations for a set of tasks. In general, the recipe for the process dictates these constraints, but sometimes we link tasks artificially to ensure administrative control. For example, a task can be started only after equipment maintenance task is finished. Resource constraints limit the sum of resources used at any time by their availability. Production constraints are dictated by production process established in pilot plant. These constraints generally require zero wait between unit operations belonging to the same batch. This is required, for example, when the stability is unknown for the reaction product, as is frequently the case in R&D processes. Good Manufacturing Practices (GMP) require that equipment has to be cleaned before processing another material (equipment changeover). Cleaning and the equipment setup associated with it may take up to week in our pilot plant. Another production constraint is that no task can be started or finished at the beginning of each shift, although it may continue processing. 4. IMPLEMENTATION We have used the commercial scheduling tool VirTecs developed by Advanced Process Combinatorics (APC) for the solution of this problem. The advantage of this tool compared to other similar tools (MIMI from Chesapeake) is that a mathematical model for such a problem was implemented, whereas in MIMI one must create the model from scratch. There is a computational speed advantage as well: the scheduling engine developed by APC is tailored for pharmaceutical scheduling problems and is much faster than CPLEX (general purpose solver used by MIMI). There is a number of software to solve advanced planning and scheduling problems. Many of them are tailored to manufacturing job shop applications, rather than batch chemical manufacturing. VirTecs defines ready-to-use modeling primitives of the state-task network framework (state, equipment, resource, and task). It is quite natural to designate each unit operation as a separate task that requires equipment and operator resources. Allocation, resource, and
904 equipment changeover constraints are already defined in the model provided by VirTecs. The zero wait constraint between unit operations within the same batch is achieved by forcing the states between unit operations to have no capacity. The sequencing constraint is easily realized as a material balance constraint that is already provided by VirTecs. For example, the reaction task produces one unit of reaction product that is consumed by separation task. The only constraint that is not so trivial is the "dead" time constraint, where tasks cannot begin or finish at the shift change. The solution is to create a special resource that is always available except at the shift change. Every task requires this resource for the first and last minute of processing. There is yet another minor complication in our design. The exact sequence and timing of unit operations is taken into account only for batches that are currently running or are to be run in a few weeks. Batches that are far in future are not split into unit operations. This is done to prevent unnecessary complexity, but the main reason is that due to demand uncertainty the number of batches required to produced desired amount of product may be unknown. There is also the possibility that a clinical study may be canceled, and the entire campaign, or part of it, becomes unnecessary. On the other hand, the number and timing of studies might change, forcing a change in the demands and production targets. Another modeling issue arises with the differentiation between batches for which sequencing and timing of unit operations is known and future batches for which sequencing and timing of unit operations is unknown. In the first case, unit operations are considered as separate tasks. In the second case, the full batch is a single task. Although it is quite natural to consider all batches as single tasks, it is more convenient to split batches in close proximity into their corresponding unit operations. This comes into play when scheduling resources over a short horizon. Batches may be able to overlap at specific operations, such as starting the reaction of batch 2 while batch 1 product is in the filtration and drying operations. Similarly, one may be able to overlap batches of other steps when the specific operation resources are known. 5. CASE STUDY We compare several different schedules to test the proposed methodology. All the schedules are derived from the actual two-year production plan. Fifteen campaigns for seven different products were taken into consideration. The first schedule contains full unit operation detail for the entire two-year span. Scenarios 2, 3 and 4 are built off the base schedule, adding extra processing time to every operation. Scenarios 5-9 test the effect of our philosophy of only adding details to the near-term campaigns, while leaving information out of future campaigns. The remaining two scenarios test the effect of adding demand (new batches) to existing campaigns that are scheduled to run at the beginning of the planning horizon. The schedule for each scenario was optimized to remove resource constraints and meet its deadlines as closely as possible. Table 1 shows the results for each of the scenarios, listing tardiness as the prime indicator of schedule quality. While the base case schedule is one of the best in terms of tardiness, the effort of adding full details to every batch of every campaign is quite cumbersome. In fact, all of that effort is lost when comparing scenarios three and six. The delays of 5% in scenario three are not uncommon, while the effort to construct a schedule with only six months worth of detail is much lower (scenario six). In fact, it is rather difficult to determine the details for campaigns that are to run more than six
905 months into the future. In an R&D environment, the operations required to make a chemical can change quite frequently. This is another source of the time variability that we see when planning campaigns. Table 1 Computational results Scenario Brief Description 1 Full details in all campaigns 2 Full, 1% added to time of all campaigns 3 Full, 5% added to time of all campaigns 4 Full, 10% added to time of all campaigns 5 No detail in any campaign 6 First six months have details 7 First nine months 8 Year and a half of detail 9 Full details, increase demand in one campaign 10 Full details, increase demand in two campaigns
Tardiness (hours) 46 0 2207 5296 21399 2788 t 973 3 268 770
Increase by 1% in the duration of each activity (scenario 2) leads to the slight decrease of the total tardiness because the solver provides only "good" sub-optimal solution. The same happens for scenario 6. However, such a decrease may be treated as negligible when compared to other scenarios. The effect of slightly increased demands is quite modest, as seen in the last two scenarios. We have modeled increased demands as the addition of one batch to campaigns at the start of horizon. Adding batches at the start of horizon should maximize the total impact of the changes. The explanation of the modest effect might be the different equipment units used for those campaigns. Although the common resource shared between campaigns are operators, the optimizer manages to shift campaigns just by small amount. However, the disturbances caused by increase in demands may exceed the benefit provided by the rigor provided by detailed schedule. 6. CONCLUSIONS We have presented a methodology that combines a production plan and a daily operation schedule in a single model. This work is demonstrated on a pharmaceutical pilot plant example. The approach of combining production plan and operating schedule is tailored to the uncertainty prevailing in the pilot plant environment. The example shows that a modest increase in the processing times or demands disturbs base plan, suggesting that adding every detail throughout the planning horizon provides no benefit as far as total tardiness is concerned. In other words, the benefit of having an exact schedule disappears in the face of uncertainty. An important future research direction might be simulation of the production plan. Monte Carlo simulation is a valuable tool to estimate the robustness of the plan, as suggested in recent work by Subramanian and other for testing pipeline management philosophies [ 11 ]. In addition, we have only presented a snapshot of the schedule. Evolution of the plan over the course of several months might give a much better feeling for the uselessness of setting up campaigns with full details far into the future. Stochastic simulation of the plan will help
906 determine the quality of the schedule under known uncertainty. By introducing probability distributions for processing times, equipment breakdowns, and demand sizes one could determine how often rescheduling might be necessary, based on the level of uncertainty, and establish the desired level of plan detail. For example, the simulation might show that it is best to have a detailed schedule for three months under one set of uncertainty conditions, and six months under another. REFERENCES
1. L. Mockus, J. Vinson and R.B. Houston, Planning and Scheduling in a Pharmaceutical Research and Development. Computer Aided Process Engineering, 8, pp. 1057-1062, 2000. 2. E. Kondili, C.C. Pantelides and R.W.H. Sargent, A general algorithm for short-term scheduling of batch operations-I. MILP formulation. Computers Chem. Engng., 17, pp. 211227, 1993. 3. E. Kondili, C.C. Pantelides and R.W.H. Sargent, A general algorithm for short-term scheduling of batch operations-II. Computational issues. Computers Chem. Engng., 17, pp. 229-244, 1993. 4. N.V. Sahinidis and I.E. Grossmann, Reformulation of multiperiod MILP models for planning and scheduling of chemical processes. Computers Chem. Engng., 15, pp. 255-272, 1991. 5. S.J. Honkomp, L. Mockus and G.V. Reklaitis, A framework for schedule evaluation with processing uncertainty. Computers Chem. Engng., 23, pp. 595-609, 1999. 6. B.J. Cott and S. Macchietto, Minimizing the effects of batch process variability using online schedule modification. Computers Chem. Engng., 13, pp. 105-113, 1989. 7. K.B. Kanakamedala, G.V. Reklaitis and V. Venkatasubramanian, Reactive schedule modification in multipurpose batch chemical plants. Industrial and Engineering Chemistry Research, 33, pp. 77-90, 1994. 8. M.H. Kim and Y.D. Kim, Simulation-based real-time scheduling in a flexible manufacturing system. Journal of Manufacturing Systems, 13, pp. 85-93, 1994. 9. R. Jacobs and W. Jansweijer, A knowledge-based system for reactor selection. Computers Chem. Engng., 24, pp. 1781-1801, 2000. 10. X.X. Zhu and T. Majozi, A novel continuous time MILP formulation for multipurpose batch plants. Submitted to Industrial and Engineering Chemistry Research. 11. D. Subramanian, J.F. Pekny and G.V. Reklaitis, A simulation-optimization framework for addressing combinatorial and stochastic aspects of an R&D pipeline management problem. Computers Chem. Engng., 24, pp. 1005-1011, 2000.
European Symposiumon ComputerAided ProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rights reserved.
907
Mixed-Integer Multiperiod Model for the Planning of Oilfield Production A. Ortfz G6mez, V. Rico-Ramfrez* and R. Vdzquez-Rom~in Instituto Tecnol6gico de Celaya, Departamento de Ingenierfa Qufmica, Av. Tecnol6gico y Garcia Cubas S/N, Celaya, Gto., CP 38010, MEXICO We present a multiperiod optimization model for oil production planning in the wells of an oil reservoir. The problem considers a fixed topology and is concerned with the decisions involving the oil production of the wells in each period of time. We assume logarithmic behavior for the well flowing pressure (with respect to time) while calculating the oil production (Home, 1998) and consider time periods of uniform length. A numerical example has been solved through the GAMS modeling system (Brooke et al., 1998) to highlight the scope of the proposed representation.
I. I N T R O D U C T I O N Multiperiod optimization in the chemical industry has recently received considerable attention. Multiperiod planning considers a fixed topology and is concerned with the decisions involving the startup/shutdown of the operation of the process in each period of time. Oilfield operation is a multiperiod problem because the cost and the demands of oil production vary from period to period due to market or seasonal changes. An oilfield infrastructure consists of production platforms and a number of reservoirs including onshore and offshore facilities. In oilfield multiperiod models, design decisions involve the capacities of the production platforms as well as decisions regarding which production platforms and wells to install over the operating horizon (van den Heever and Grossmann, 2000). Planning decisions involve the oil production profiles in each time period. In the past, decisions regarding platforms capacities, drilling schedules and production profiles had often been made separately under certain assumptions to ease the computational burden. Simultaneous models emerged with the works of Bohannon (1970) and Sullivan (1982). Bohannon (1970) proposed simultaneous MILP models for the oilfield design and production planning whereas Sullivan (1982) developed a simultaneous MILP model for gasfield design and production planning. Also, Iyer et al.(1998) proposed a multiperiod MILP model for the planning and scheduling of investment and operation of offshore facilities. Oilfield facilities are often in operation over several decades. So, although changes in the reservoir pressure with respect to time are not significant in the short run, such changes cannot be ignored for a simulation involving future planning and investment decisions. It is known that the reservoir behavior represents a nonlinear constraint but, in all the works described above, the
* Author to whom all correspondence should be addressed
908 reservoir behavior as a function of the cumulative oil produced has been approximated by linear constraints. Recently, van den Heever and Grossmann (2000) proposed a simultaneous approach for the oilfield planning which deals with nonlinearities directly and can be solved in a reasonable computer time. In such a model it is assumed that the operating conditions are constant across the planning horizon. So, the productivity index, p, is assumed to be constant for a given period of time. The productivity index depends on the conductivity of the well and allows the calculation of the oil flow rate as a function of the pressure drop between the reservoir and the well bore: q, =p.(pr _pW) (1) where qt is the oil flow rate in period t, pr is the reservoir pressure and pW is the pressure of the well. Well analysis (Home, 1998) reveals, however, that the well flowing pressure presents a time dependent nonlinear behavior and, as a consequence, the assumption of constant operating conditions may not apply. Fig. 1 illustrates the oil extraction from a reservoir. As given by Equation (1), the driving force for the oil production from a well is the pressure difference between the reservoir and the well bore. Also notice that when the well has just been drilled (or when it has been shut in for a significant period of time), we can assume that the pressure of the well bore is the same as that of the reservoir. So, at the beginning of the operation, when the well is open to flow, oil can be extracted because of the pressure difference between the well bore and the well head. As the operation time increases, the well bore pressure decreases and that also causes an oil flow from the reservoir to the well. However, the oil flow rate from the reservoir to the well depends also on the geological properties of the well surroundings, such as permeability, thickness, porosity, etc., which determine the well production capacity. Hence, because of the resistance to the flow between the reservoir and the well bore, oil production causes the well bore pressure to decrease with time. A simple expression has been often used (Home, 1998) to represent such a behavior:
p f = pin --Cl qt [In(t)+ c2l
(2)
where cl y c2 are constants experimentally determined which result from combinations of the geological properties characterizing the well,fi in is the pressure of the well bore at the beginning of the operation (reservoir pressure) and t4 is the (final) pressure of the well bore after an operation time t. On the other hand, if the well is shut in, the well bore pressure will increase because of the effect of oil flow from the reservoir to the well. Fig. 2 shows the behavior of the well bore pressure when the oil is flowing and when the well is shut in.
Fig. 1
Extraction of Oil From a Reservoir
909 In this paper we are concerned with the short term planning of oil production in the wells of an oil reservoir. Planning decisions consist on determining the oil flow rates and the operation/ shut in times for each well of the reservoir in each period. Such decisions are based on practical considerations, which avoid the well bore pressure to decrease beyond a minimum allowable value. Also, one should remember that the oil production rate has to satisfy the oil demand for each period of time. Since we are focusing on short term decisions, we consider that the pressure of the reservoir is a constant over the time horizon. On the other hand we assume logarithmic behavior of the well bore pressure for the calculation of the oil production as given by Equation (2), and it is assumed that the values of the geological properties of the well are known.
2. PROBLEM STATEMENT This work considers the short term planning of the oil production in the wells of a reservoir over a Horizon H divided into N P time periods of length T. Hence, given the oil production demands for each period of time, planning decisions involve finding the oil flow rates and operation/shut in times of the wells. The main constraints of the problem involve avoiding the well bore pressure to decrease beyond a minimum allowable value and satisfying the oil production demands. Assumptions made in this paper include: 1) Multiple wells in a reservoir produce independently from each other. 2) Nonlinear behavior for the well bore pressure as function of the oil flow rate and time (Equation (2)). 3) The objective function is calculated in terms of cost coefficients which change for each period of time due to seasonal changes.
2.1 A Simplified Model We have developed several models of varying complexity for solving the problem described here. So, MINLP models assuming acyclic and cyclic operation for the wells which use convex envelopes for dealing with the nonlinearities are now under investigation. In this section we present a simplified model which considers time periods of uniform length. Also, in this model we assume that a well is either flowing at a constant rate or shut in across the complete time period. Such a simplification significantly reduces the problem complexity. The following sets, indices, variables, parameters and equations are defined. Sets and indices: P= set of time periods, [1...NP] W= set of wells
pUp . . . . . . . . . .
plow ~ . . . . hL
r"
t~
Fig. 2
t
t~
Time Dependent Behavior of Well Bore Pressure
t
910
i,j= indices corresponding to well and time period, iE W andjE P Continuous Variables:
qii= oil flow rate in well i and period j piniij= well bore pressure of well i at the beginning of period j //~/= well bore pressure of well i at the end of period j
Dij, Iu= pressure differential in the well bore when the well is producing and when the well is shut in, correspondingly Binary and Boolean Variables: Y~/=TRUE if well i is producing in period j (Boolean)
yij= binary associated to Yij. 1 if well i is producing in period j Wijl = 1 if the well is shut in and the well bore pressure does not go beyond the maximum allowable value
wi/2= 1 if the well is shut in and the well bore pressure reaches the maximum allowable value Parameters: a,5,T= cost coefficients
pUp,plOW=maximum and minimum allowable pressure of a well q~i - parameter for calculating the pressure increase of the well when it has been shut in T-time period length M=suitable upper limit for the residual of the equations involving pressure drop calculation
Objective:
Minimize
Z Z ruqif +Z Z 8uyuT +Z Z aoO- YO}1" i
j
i
j
i
(3)
j
Constraints: qoT > d j
Vj e P
(4)
i
Pd = Poi. - Do
v
Puy = p Ui,, + I o
v
PiiI =p~p
Vi~ W, j e P
P~7 + Iij < P:P J P~'~+ Iii > P7 Dq =qu{clOn(T)+c:~ Vie W, j e P Iij =q;{c,[ln(T)+c2]}(l- yu) Vie W, j e P q~..x {q [ln(T) + c2]}= (p~" - p:OW) max
qij < qo
Vi ~ W, j ~ P
Vie W, j ~ P
(5)
(6)
911
q,, < q~r YO + qlOWO_ YO) Vi E W, j ~ P q,i >_qlow V i e W , j e P
(7)
tn Po = P,~-~
(8)
Vi ~ W, j ~ P
The objective function of the model consists on minimizing the production cost (Equation (3)). It is assumed that there is a variable cost associated to the oil production rate (7U). Also, we assume that there is a cost associated to each well which changes depending on whether the well is open to flow (80) or shut in (o~0). The minimization is subject to the satisfaction of the demand on each period of time (Equation (4)). Equation (5) represents the behavior of the well flowing pressure. Notice that if the well is open to flow then the well flowing pressure decreases, but if the well is shut in then the pressure increases. Equation (6) establishes the upper limit for the oil flow rate in a time period so that the pressure does not decrease beyond a minimum allowable value. Equation (7) suggests that, because of operational constraints, even when a well has been shut in there is a minimum oil flow rate equal to q~OW(although in this work we use ql~ Equation (8) corresponds to the linking constraints from a time period to the next one. Finally, the disjunction in Equation (5) can be reformulated by using the Big-M representation and writing the relations among the binary variables: f
ptf -ptiJln.4-Oij > - M ( 1 - y,j) Pi~ - P,~ + Do < M O - Yo ) pti; _p:OW_Oi j > _ M O _ yu)
m
2[
>
Pij -- PiJm-- tj - - M (1- Wijl ) p,: _ p ; -I0 <MO_wo,) p; + l j - p r p <MO-w,j,) w,jz + wo2 = 1- y,j
Ptf _ pU~ ~ _M O _ wq2 ) , P,~ - P,P < MO-w,j2) Pij -bltj _pUp ~_MO_wij2)
(9)
3. RESULTS An illustrative example is presented to highlight the scope of the proposed representation. The problem has been solved by using OSL, a commercial solver available through the GAMS modeling system (Brooke et al., 1998). The system consists of a reservoir containing 8 wells, a configuration which is commonly known as the "spider" problem. The planning horizon has been divided in 6 time periods. The geological properties (well flowing pressure behavior) differ for each well of the reservoir. Furthermore, the cost coefficients vary not only for each well but also for each period of time. Seeking simplicity, here we only show the well flowing pressure profile for well 8 (Fig. 3) and the oil production (thousands of barrels per day) for each well in each period of time (Table 1). The solution time is 20.1 seconds in a Pentium III, 550 MHz. The objective function is $35.1 million dollar. 4. DISCUSSION We have developed a multiperiod optimization model for oil production planning in the wells of an oil reservoir. The problem is concerned with the decisions involving the oil production of the wells in each period of time. We consider logarithmic behavior of the well flowing pressure; however, by assuming that a well is either open to flow or shut in over a complete time period (of fixed length), the resulting model is a MILP problem. Currently, MINLP models which remove such an assumption and also consider the nonlinear behavior of the reservoir pressure are under investigation.
912
Oil Productionof Well8 T5 (thousandsof barrels~day) 8.34 53.33
t
t
~ /
34.21 I
Fig. 3
WellFlowingPressure(psia)for Well8
oo9/ oo
~ 9 7
1"~1573911~/~
v
Period
I ""15763~9 Period
Oil Production and Well flowing Pressure Profile of Well 8
Table 1. Oil Production of the wells of a reservoir (thousands of barrels/day) Period 1 2 3 4 5 6
Demand 233 150 100 100 150 200
Welll Well2 ..... 25.66 ..... 21.10 30 .......... 74.83 ..... 6.85
Well3 46.66
Well4 64.16
29.17 46.66 46.66
46.66
Well5 Well6 Well7 .......... 38.5 53.72 75.17 ..... 40.83 . . . . . . . . . . 75.17 24.79 40.83
Well8 58.34
53.33 34.21
5. A C K N O W L E D G M E N T S A. Ortfz-G6mez and V. Rico-Ramfrez would like to thank the financial support provided by the Mexican National Council for Science and Technology (CONACYT). REFERENCES Bohannon J., A Linear Programming Model for Optimum Development of Multi-Reservoir Pipeline Systems, J. Pet. Technol.22, 1429 (1970). Brooke A., D. Kendrick, A. Meeraus and R. Raman, GAMS- A User's Guide, GAMS Development Corporation, Washington, DC (1998). Home, R. N., Modern Well Test Analysis, Second Edition, Petroway Inc, Palo Alto, CA (1998). Iyer R. R., I. E. Grossmann, S. Vasantharajan and A. S. Cullick, Optimal Planning and Scheduling of Offshore Oil Field Infrastructure Investment and Operations, Ind. Eng. Chem. Res. 37, 1380 (1998). Sullivan J. A., Computer Model for Planning the Development of an Offshore Oil Field, J. Pet. Technol.34, 1555 (1982). van den Heever S. A. and I. E. Grossmann, An iterative Aggregation/Disaggregation Approach for the Solution of a Mixed-Integer Nonlinear Oilfield Infrastructure Planning Model, Ind. Eng. Chem.Res.39, 1955 (2000).
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
913
Decision-making framework for the scheduling of cleaning/maintenance tasks in continuous parallel lines with time-decreasing performance Sebasti/m Eloy Sequeira, Mois6s Graells and Luis Puigjaner. Chemical Engineering Department, Universitat Polit6nica de Catalunya ETSEIB, Avinguda Diagonal 647. Pavell6 G, 2~ 08028-Barcelona, Spain
Keywords: Scheduling, Maintenance, Optimisation, Cape. The efficiency of equipment units decreases depending on the time they have been operating due to fouling, formation of subproducts, decay of conversion etc. Therefore, equipment maintenance and/or cleaning is periodically required to restore the original conditions that allow higher productivity rates. This poses a trade-off between the cost of equipment shutdown and the resulting productivity improvement. The problem of determining the maintenance frequency and the timing of the specific tasks in each equipment unit is studied a hierarchical way (planing and scheduling). A general approach is first introduced and next the implementation is presented. The solution approach proposed is based on a practical decision-making framework allowing both, the user driven what if analysis and the optimisation carried out by plugged modules. Finally, an example is shown and the results are discussed. 1. INTRODUCTION The problem addressed in this paper is quite a common one in industrial practice. The efficiency of equipment units decreases depending on the time they have been operating or the amount of different materials they have processed. Therefore, action is required to reestablish the initial performance level. The decision-making problem is determining when such actions are to be carried out, which leads to the trade-off between the cost of the action itself (resources required, production break, etc.) and the benefit it generates (productivity increase, reduction of operational costs, etc.). In the case of continuous lines, this situation is exemplified by catalysed reactors, evaporators, etc. The problem in catalysed reactors is catalyst progressive deactivation. In the case of continuous evaporation units, efficiency decreases due to different factors like fouling, deposits of insoluble materials, etc. Furthermore, the case of parallel lines introduces additional constraints such as keeping total processing rate and buffer tanks levels within certain limits. Several approaches have been proposed to cope with these problems using mathematical programming, which allows determining the optimal values for operation variables such as starting time and duration of cleaning operations. However, mathematical programming is usually a rigid approach strongly dependent on the way the models are formulated. This leads to simplified models for the physical behaviour of the equipment, the way they operate (cyclic schedules are forced, etc.) and the objective function.
914 The approach introduced in this work is the development of a general computer aided decision-making tool for the plant engineer. This tool has been conceived as a basic framework for managing different models and optimisation techniques as well as a basic user interface for addressing these kind of problems. Such a tool is designed for practical routine work including not only the optimisation procedures but also the "what if" analysis based on simulation as well as the upgrading of plant status information for subsequent rescheduling of cleaning tasks. The scheduling strategy adopted is based on the determination of the starting time and duration of the next cleaning/maintenance task (or tasks in the general case) for each processing line given the initial conditions for each one. This allows the use of the tool for scheduling/rescheduling purposes following a rolling horizon strategy. 2. GENERAL FRAMEWORK Consider an operation scheme as illustrated in Figure 1. The equipment units can be filters, catalytic reactors with decreasing activity, or any units with performance affected by time.
Figure 1: General system structure Each equipment unit will have an associated model, which represents numerically its performance as a function of time. There is also a global performance measure, which results from the individual contributions. The holdup tanks try to compensate the perturbations to and from the parallel lines, and can not be necessarily present. The problem statement is as follows: Given: 9 The material to be processed for a period of time under consideration 9 The equipment models, parameters and initial status 9 The individuals and global performances as time function Determine: 9 The frequency of cleaning 9 The material to be processed by each equipment 9 The time assignment for maintenance of each equipment" The solution of such a problem is usually presented under the form of mathematical programming [1 ]. The mathematical models include several variables and very often integer or binary variables. This kind of approach, for one hand, leads to complex models that are expensive in terms of development and calculation time (or the models are oversimplified for
915 that reason). On the other hand, the models are formulated for specific mathematical tools, which are hardly used or even understood by the common plant engineer. This work proposes to solve the problem using a hierarchical approach, dividing the problem and considering a more detailed view of its nature. Once the subproblems are identified, more practical and familiar tools can be used to address the decision-making problem. Hence, two different subproblems are identified. Determining how much material is to be processed in each unit before maintenance and which is the maintenance frequency to be adopted is a planning issue. In contrast, the timing of the specific maintenance tasks is related to scheduling. Such a problem fragmentation allows a practical and simplified problem solution. 3. IMPLEMENTATION Figure 2 shows the proposed approach. In first place the individual model is developed. It contains the equipment physicochemical behaviour and acts as a black box, returning some variable values when changes on parameters and/or decision variables occur. Secondly, the overall performance model, which evaluates the objective function using the information generated by the individual model applied in every equipment calculation. The solver is plugged then in the system for getting the best overall performance value. The same scheme is considered in both subproblems, scheduling and planning. But the input conditions for scheduling have been already optimised.
Figure 2: Proposed approach. 4. EXAMPLE The example considered consists of five parallel lines in the evaporation section of a sugar plant [2]. During the operation, the heat exchange capacity of the evaporators decreases with time due to fouling and solids deposition. Since the evaporators are fed with constant mass flow, the outlet concentration of each evaporator decreases. Therefore, system performance representing overall plant goals is assumed to be given by final concentration of the mixture in the product holdup tank. Following the guidelines of the previous sections, the different system components were developed. In first place, the evaporator model gives the output concentration according to its
916 parameters and feed conditions, the cleaning frequency and the time horizon. On second place, the objective function takes the information generated by each evaporator and calculates the overall plant performance.
Figure 3: User interface for the planing subproblem The components were written in Visual Basic for Application (VBA). Data are entered to an Excel spreadsheet [3]. The object-oriented features of VBA simplify the model development, while Excel allows entering data directly to a graph. This provides to the user an easy way for performing "what if?." analysis and interacting with tables and graphs. Additionally, a solver can be plugged to the system for solving the planing subproblem [4]. Figure 3 shows a screenshot of the spreadsheet and the colour convention used to distinguish parameters, and model inputs and outputs. Model parameters are heat exchange data (area, steam condition, etc.) and the main inputs are: 9 System feed condition (rate and composition) and fraction fed to each evaporator. 9 Time horizon and cleaning frequency for each evaporator. 9 Cleaning time required for each evaporator. The corresponding model outputs are: 9 Final outlet rate and composition for each evaporator. 9 Final rate and composition of the mixture in the product holdup tank.
917 A solution procedure may be run to obtain an improved problem output. A tailored heuristic can be used or standard solver may be plugged in the system to get an optimal solution [2] as well. In any case, the outputs are the values of the decision variables: time horizon and cleaning frequency and flow rate for each evaporator. Once the planning solution is established, the scheduling problem must be solved. Taking the time horizon, the cleaning time and the cleaning frequency for each evaporator, the time allocation for the cleaning operations is decided. The performance criterion adopted was the minimisation of the overlapping between different cleaning tasks. The reason encouraging the adoption of this criterion is double. On one hand, cleaning needs manpower and manpower overlapping is not desired. On the other hand, changes in product flow should be reduced, and a simultaneous cleaning will produce them. This problem has multiple degenerated solutions, which a solver would not distinguish. To avoid a solution depending upon the starting point, a simple heuristic (the stairway heuristic) was used to determine a most convenient solution. This heuristic assigns sequentially the cleaning tasks to the evaporators along the time horizon by starting a task as soon as the previous is finished. If overlapping can not be avoided, it is distributed in uniform way between all the maintenance tasks. The user is the one deciding which solution wants among all the degenerated ones by giving the sequence establishing how cleaning tasks must be dispatched. In addition the user may also enter the initial cleaning time if a delay must be introduced in the schedule. Figure 4 shows the user interface for the scheduling subproblem. Once the data is entered, the timing is performed automatically by the heuristic and displayed on a Gantt Chart.
Figure 4: User interface for the scheduling problem The user can also change the start and end time for the cleaning tasks, thus modifying the proposed solutions according to his own experience and information. Finally, a chart showing the resource consumption as manpower or storage needs was included as well for supporting this simulation capability. Figure 5 illustrates the level of the product holdup tank along the time horizon.
918
5. CONCLUSIONS This work introduces a practical way to solve the planning and scheduling problems of maintenance tasks in continuous parallel lines. This work shows how problems of such nature can be solved in a hierarchical way. Once the subproblems are defined, the solution procedure is simplified and tools more familiar to the engineer can be used to provide a decision-making framework for the routine work of the plant manager. An example of the application of such approach was presented, showing that this way to solve the problem offers a more practical solution to the engineers. The decision-making framework allows not only obtaining optimum solutions, but also the possibility to perform "what if?." analysis via its simulation capability. In addition, the graphical representation of information allows a fast comprehension and analysis of a problem that is formulated in a way closer to plant engineers and their every day needs. Furthermore, the practical significance of the solution procedure proposed is demonstrated by the application developed for a sugar plant. The enthusiasm the plant engineers showed for a tool they understand at the first sight shows that the tool is giving answer to a most practical question: when this evaporator should be cleaned? Finally, the modular approach adopted is easy for maintenance and for adaptability to new problems, to changes on models and objective functions, thus showing the great generality of the approach. Perfll de Navel
2 E+06
:~ 5 E+05 0E+00
-5 E+05 -1 E+06
~
j
i
~
___
1 I0 50 i '____i--i.......... _ ~ 80'____~f _ ......... -
.
,--,,
i_t .
.
.
.
.
.
.
.
Tmmpo
Figure 5: User interface for the scheduling problem Acknowledgements
Sebastian Eloy Sequeira wishes to acknowledge to the Spanish "Ministerio de Educaci6n, Cultura y Deporte" for his financial support (F.P.I. grant). References
1. V. Jain and I. Grossmann, "Cyclic scheduling of continuous parallel process units with decaying performance". AIChE Journal 44, 1623-1636, 1998. 2. S. E. Sequeira, H. Heluane, M. Colombo, M. Hernandez, M. Graells and L. Puigjaner "Scheduling of continuous parallel evaporators with decreasing heat transfer coefficient". L. A. AIChE meeting proceedings, November 2000. 3. E. M. Rosen, "A Perspective: the use of the Spreadsheet for Chemical Engineering Computations", Ind. Chem. Res., 39, 1612-1613, 2000. 4. S. E. Sequeira, M. L. Graells and L. Puigjaner, Synergic use of the available cape tools: an application to real time optimisation. L.A. AIChE meeting proceedings, November 2000.
European Symposiumon ComputerAidedProcessEngineering- 11 R. Ganiand S.B.Jorgensen(Editors) 9 2001 ElsevierScienceB.V.All rightsreserved.
919
A Conceptual Optimisation Approach for the Multiperiod Planning of Utility Networks Alexandros M. Strouvalisa and Antonis C. Kokossis b* a Department of Process Integration, UMIST, P.O. Box 88, Manchester M60 1QD, UK. b Department of Chemical and Process Engineering, School of Engineering in the Environment, University of Surrey, Guildford, GU2 7XH, UK. The paper presents a paradigm of addressing large-scale optimisation problems using customised tools. Tailor-made techniques limited to specific applications are employed to effectively invest resources for solution searches. Integration of special logic is implemented at the modelling level with emphasis on the multiperiod planning of steam turbine networks. Logic extracted from conceptual preprocessing is employed to reduce combinatorial load. Logical constraints screen redundant degrees of freedom and customise the solution space to the specific application. The methodology decreases considerably the computational cost while offering a high degree of physical insights. 1. INTRODUCTION Previous efforts explain the importance of taking advantage of knowledge at the modelling stage. Floudas and Grossmann (1995) reviewed the methods for reducing combinatorial complexity of discrete problems employing logic-based models. Raman and Grossmann (1992) reported improvements in the solution of MINLP problems with a combined use of logic and heuristics. They illustrated their ideas in process synthesis problems (1993), employed inference logic for the branching of the decision variables, and studied the use of logical disjunctions as mixed-integer constraints (1994). Turkay and Grossmann (1996) extended the application of logic disjunctions to MINLP's using logic-based versions of OA and GBD algorithms. Hooker et al. (1994) applied logic cuts (as inequalities or by symbolic inference) to decrease the number of nodes visited in MILP process networks models. Integration of information with modelling is a validated contribution for improving computational efficiencies. Applications though usually comprise of algorithmic developments that lack insights and understanding. The use of logic supported primarily by conceptual analysis is alternatively propounded. The Hardware Composites (Mavromatis and Kokossis (1998), Strouvalis et al. (1998)), a graphical tool developed for the optimisation of steam turbine networks, is employed to support analysis of operations. The optimal solution space is visualised by the Hardware Composites. The graphs consist of linear segments (m and g) representing the steam turbine expansion paths. The most efficient expansion paths are first fully loaded before less efficient are used. Boiler operation is represented by zones (dashed lines). Each set of Corresponding author.
920 demands is directly traced on graphs (power output versus passout demand) as well as the optimum loading of turbines and boilers. 2. P R O B L E M DESCRIPTION
The problem considered is the multiperiod planning of utility systems with steam turbines and boilers. Given are demand patterns in power and heat, efficiencies of units and operational costs (fixed, incremental, changeover). Objective is satisfaction of demands with minimum cost. Units change modes over time to accommodate demands while allowing for efficient operations. The optimal planning is determined in terms of on/off status and modes of units (steam consumption/generation, power output). The horizon is divided into periods of equal length. The period is adequate to model start-ups or shut-downs for each turbine and boiler. Demands are considered constant over each period. The utility cost consists of unit (i) operational costs (fixed and incremental) and (//) transitional costs (start-up, shut-down costs). The optimal planning balances two trends: "coarse" profiles where each period is operated by the most efficient combination of units (stand - alone optimal modes) irrespective of preceding and succeeding periods. They usually involve several switch on and offs to accommodate the dissimilar status of units between periods. "smooth" profiles include less transitions but more operational costly modes. Changeovers are limited by employing more steady over time configurations of units at the expense of not necessarily using the most efficient unit combinations per period. 3. PROPOSED F R A M E W O R K
The proposed methodology features a blend of engineering knowledge, algorithmic methods and conceptual representations. Screening is accomplished through qualitative and quantitative assessment of operations. Options associated with promising trade-offs are embedded in operational superstructures while infeasible/inferior modes are made redundant. The operational analysis is performed according to the hierarchy:
A. Solution Space Analysis The full solution space of the Hardware Composites is reduced by selectively removing units from the graphical representations. Selection of units is justified by trade-offs between: (i) operational/transitional costs and (ii) fixed/incremental efficiencies of units. The analysis determines a customised selection of alternatives in view of the specific variation of demands.
B. Logic Extraction Demands are projected on the alternative solution spaces of Hardware Composites and their reduced versions. Periods requiring operation of specific units due to hardware limits are detected. By tracing demands over spaces, possible operating units per period and changeovers between periods are revealed. The identification of candidate configurations and transitions over time defines the connectivities of sequential modes. In that manner degrees of freedom are decreased and the operational superstructures are set up.
921
C. Integration of Logic in the Model Logic as extracted and included in superstructures is transferred to models by fixing / aggregating variables and imposing logical constraints. The model formulation is thus customised to the characteristics of the particular application in a user-defined way. The philosophy of the proposed approach is demonstrated qualitatively by Fig. 1. "Coarse" profile (i) corresponds to unit operations proposed by Hardware Composites. Each period employs the units of the domain where the set of demands locates. Depending on distribution of demands on the Hardware Composites profile (i) reflects the transitions over time. Qualified reduced Hardware Composites give rise to alternative unit combinations defined, for example, by profiles (ii), (iii), (iv) which present "smoother" planning options. Any of the four profiles or a combination of them may stand for the optimal planning.
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.
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,\
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.
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Fig. 1. Alternative solution spaces generate alternative planning profiles. 4. ILLUSTRATIVE EXAMPLE The network of Fig. 2(a) features 3 backpressure passout turbines (T1, T2, T3) and 3 boilers (Bl, B2, B3). Operational horizon includes 30 periods each associated with a set of demands in power and heat. Power is given in power units [PU], heat in passout units [SU] and costs in cost units [CU]. The Hardware Composites of the network are shown in Fig. 2(b). They include all units and projected demands (P1 to P30) on the full solution space. Each domain involves operation of specific units. For a transition Pt to Pt+l the units in operation and possible start-ups and shut-downs are identified. Trade-off analysis justifies the alternative reduced space which excludes turbine 3. The sole operation of Tl and T2 introduces less transitions and competitive planning profiles. The reduced Hardware Composites are presented in Fig. 2(c). The spaces under consideration generate the planning profiles for turbines and boilers shown in Fig. 3(a) and (b) respectively. They encompass only feasible modes interconnected according to permissible changeovers.
922
Fig. 2. (a) Utility network, (b) Hardware Composites, and (c) Reduced Hardware Composites for illustrative example. Transition from P 1 to P2 is presented for illustration. According to solution spaces of Fig. 2(b) and (c), P1 requires units T1 and T2 while P2 only T1. These modes are embedded in the operational superstructure of Fig. 3(a) in addition to the stand-by mode of T2 in P2 (T2 is again operational in P3). Similarly, P1 and P2 are served by boilers B1 and BE or B3 (Fig. 2(b), (c)). Alternatively P2 is operated by B1 (Fig. 2(c)). Boiler modes and their allowed interconnections are shown in Fig. 3(b). Following the same procedure the entire operational superstructures are shaped.
4.1. Computational Results Two approaches are examined: (a) full solution search with elementary implemented knowledge and (b) reduced solution search based on operational superstructures. The first experiment navigates the entire space; search is restrained only by basic feasibility constraints imposing a minimum number of operational units per period. The second experiment performs search on degrees of freedom embedded in superstructures. The sizes of the MILP models for
923 each case are presented in Table 1. The increased number of equations in (b) represents the expense of transferring logic to the model. Fixing and aggregation of variables reduces continuous and binary variables. ~
LI3
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PERIODS
(d)
PERIODS
Fig. 3. Operational superstructures for (a) turbines, (b) boilers and optimal profiles for (c) turbines and (d) boilers. Table 1: Sizes of models for illustrative example.
Full search (a) Reduced search (b)
Continuous Variables 756 679
Binary Variables 180 44
Constraints 1,382 1,789
Non-zero Elements 4,037 5,128
Optimisation is performed with application of OSL solver of GAMS. After the limit of 1,000 CPU (sec) the Branch-and-Bound solver did not find the optimum solution. Alternatively, the solution of (b) was more efficient. Table 2 presents the computational results and Fig. 3(c) and (d) the proposed operational planning profiles. Table 2: Computational results for illustrative example. Iterations
Nodes
CPU(sec) 200 MHz PC
Full search (a)
52,629
7,787
Reduced search (b)
3,349
1,111
1,000 (interrupted) 104
Objective Value [CU/Oper. Horizon] (Relaxed Objective) 529,999.3 (381,397.9) 480,861.5 (454,842.6)
Under full solution search considerable CPU resources were spent for obtaining suboptimal operations while reduced search revealed the solution in orders of magnitude less effort. Combinatorial complexity, relaxation gaps and flat objective function profiles are major reasons
924 why full search was inefficient. The solver was trapped in suboptimal operations and had to visit many nodes to fathom the binary tree. 5. CONCLUSIONS A hybrid scheme is introduced for extracting and implementing conceptual knowledge to the solution search. The analysis identifies possible unit transitions (start-up and / or shut-down) between successive periods eliminating redundant enumerations. Operational superstructures are built involving scenarios associated with promising trade-offs. Variables are fixed and aggregated and logical constraints are imposed to adapt the search to the reduced space. The merits of the approach are not only computational. Deep insights offer overall perspective of operations. The user can study the importance of units to operations, the impact and range of trade-offs in view of alternative configurations and most important of specific demands over solution spaces. Instead of an automated algorithmic screening technology, engineering understanding is employed to drive the selection of degrees of freedom and customise the model to the particular problem. REFERENCES 1. Floudas, C.A. and Grossmann, I.E. (1995) Algorithmic approaches to process synthesis: logic and global optimisation, AIChE Symposium Series 304, Fourth International Conference on Foundations of Computer-Aided Process Design, 91, 198. 2. Hooker, J.N., Yan, H., Grossmann, I.E. and Raman, R. (1994) Logic cuts for processing networks with fixed charges, Computers and Operations Research, 21 (3), 265. 3. Mavromatis, S.P., and Kokossis, A.C. (1998). Hardware Composites: a new conceptual tool for the analysis and optimisation of steam turbine networks in chemical process industries Parts I & II. Chemical Engineering Science, 53(7), 1405. 4. Raman, R. and Grossmann, I.E. (1992) Integration of logic and heuristic knowledge in M1NLP optimisation for process synthesis, Computers and Chemical Engineering, 16(3), 155. 5. Raman, R. and Grossmann, I.E. (1993) Symbolic Integration of logic in Mixed-Integer Linear Programming Techniques for process synthesis, Computers and Chemical Engineering, 17(9), 909. 6. Raman, R. and Grossmann, I.E. (1994) Modelling and computational techniques for logic based integer programming, Computers and Chemical Engineering, 18(7), 563. 7. Strouvalis, A.M., Mavromatis S.P., and Kokossis, A.C. (1998). Conceptual optimisation of utility networks using hardware and comprehensive hardware composites. Computers and Chemical Engineering, 22, S 175. 8. Turkay, M. and Grossmann, I.E. (1996) Logic-based MINLP algorithms for the optimal synthesis of process networks, Computers and Chemical Engineering, 20(8), 959.
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rights reserved.
925
An integrated environment for batch process development- from recipe to manufacture Venkat Venkatasubramanian, Jinsong Zhao, Shankar Viswanathan, Chunhua Zhao and Fangping Mu School of Chemical Engineering, Purdue University, West Lafayette, IN 47907, USA Peter Harper, Boris Russell Computer Aided Process Engineering Center, Technical University of Denmark, Lyngby, DK Batch process development involves the process of converting a chemical synthesis into an optimum, safe, robust, and economical process for manufacturing the chemical of desired quality at the ultimate desired scale. In this paper we describe a strategy for developing a set of integrated decision support tools to facilitate this activity. The primary motivators for doing this is to perform fast, and high quality process development since there is significant economic leverage that can be gained by speeding up this activity while not compromising on quality. The application of the strategy to an industrial case study is also presented. 1. I N T R O D U C T I O N In the specialty chemicals, pharmaceuticals, agrochemicals, and advanced materials sectors the process of successfully bringing a product discovery, which usually takes the form of a new molecule or formulation, to the marketplace is complex, expensive, time-consuming, as well as information, knowledge, and computation intensive. The information flows between the various stages of this process are dynamic and voluminous, the level of uncertainty in the underlying technical and commercial data high, the technical specifications and information base which must be developed to support production very extensive, and the decisions typically highly combinatorial. Thus, it usually takes an enormous amount of time, effort and money to develop a new product. In these businesses the first to market captures the advantage of establishing a market position which is difficult for later entrants to erode. Given the limited patent life of the discovery, the shorter the time from discovery to manufacture the longer the time during which revenue generation can be protected. This is a major consideration: in the case of a successful pharmaceutical, a day gained or lost in the commercial life of a product typically represents $1 million in net revenue. Thus, the really compelling drivers, for the development process, are speed to market and getting it right the first time. As Gary Pisano[ 1] notes "Although superior process development can reduce manufacturing cost, in itself an important dimension, the power of process development more often lies in how it helps companies achieve accelerated times to market". These are therefore key motivators for developing integrated computer decision support Corresponding author. Tel: 1-765-494-4083.Email: [email protected]
926 tools for batch process development activities because they can offer significant advantages. By increasing speed of development these tools would help focus key scientists on innovation, save resources for doing more value added activities, allow teams to work on more projects simultaneously, permit companies to wait longer if necessary to initiate a project. Integrated tools will help facilitate communication between team members in a multi-disciplinary team, integrate their knowledge by serving as bridge between different paradigms. Use of these tools would minimize "eye-lash" learning and facilitates faster integration of new engineers/chemists into teams. Integrated development tools would foster standardization of information by sharing data across the company and help standardize practices across the company from R&D through manufacturing. The aim of this work is to propose a systematic framework to provide fully automated decision support for batch process development activities from recipe to manufacture and develop a prototype integrated workbench based on this framework and demonstrate its use on an industrial case study. 2. STRATEGY The strategy that we have adopted towards developing integrated process development tools consists of the following steps: 1. Systematically identifying different stages of a process development pipeline. 2. Developing an information flow and functional interaction model between different aspects of the pipeline. 3. Identifying/developing pipeline management tools and methodologies for different stages of the pipeline. 4. Developing techniques that would allow integration of these tools based on the information and flow interaction model. The batch process development pipeline can be viewed as consisting of the following aspects - process alternatives synthesis, process alternatives evaluation, process operations development, and process operations assessment. Each of these in turn entails several sub-functions. Process synthesis involves sub-functions such as route selection, flowsheet selection, and solvent selection. Process evaluation consists of sub-functions such as preliminary scale-up, material and energy balance calculations, costing, resource planning, capacity planning, and sensitivity analysis. Process operations development entails development of batch records. Process operations assessment includes sub-functions such as process hazards analysis, vent sizing, and emission calculations. There is a lot of interaction between these activities in terms of data sharing and information transfer. Current automated systems that provide support for these activities routinely run into formulation, calculation, and analysis bottlenecks because of inadequate support for the flow and transfer of information across these activities. Also there is a predominantly sequential view of the process development activities. This is however restrictive because traditional downstream activities can provide insights that can be used to modify decisions that were made in traditional upstream activities. For e.g. results from process hazards analysis (downstream activity) can be used to modify decisions made during flowsheet or solvent selection (upstream activity). It is therefore necessary to support bi-directional flow of information to provide a fully integrated set of decision tools. The information flow and functional interaction model shown in Figure 1 captures the informational and functional interactions between the process development activities identified.
927
Figure 1 Information Flow and Function Interaction Model Table 1 Pipeline Management Tool ProSyn ProPred ProCamd DATA ReacDef TMS PAS Simul and DynSim BRIC ITOPS BatchHAZOPexpert
Tools Purpose Generation of process alternatives Pure component property prediction Design and selection of solvents and process fluids Collection of measured data for parameter estimation Definition of reaction kinetics information Thermodynamic model selection Process Analysis System Steady state and Dynamic simulation systems Simulation of batch records Intelligent Tool for Operating procedure synthesis Hazard and operability analysis
We propose a solution strategy based on total integration of processing systems that uses the two software systems - ICAS[2] and PHASuite[3,4,5,6,7]. ICAS or Integrated Computer Aided System addresses the process synthesis and process evaluation aspects of batch process development, while PHASuite addresses the operations development and assessment aspects. The different tools within these two systems and the process development aspects they address are shown in Table 1.
928 These tools share information and data to provide contiguous support during the process development lifecycle as per the function and information flow interaction model shown in Figure 2. This paradigm requires less data collection effort during the various activities and thereby ensures data consistency across the process development lifecycle. This paradigm therefore provides a better alternative to batch process development resulting in enhanced productivity. For e.g. the function and information flow interaction model dictates that iTOPS and BatchHAZOPexpert need to share both information and function as they address operations development and operations assessment aspects of process development. The detailed function and information flow across these tools is shown in Figure 2.
Figure 2 Function and Information Flow between iTOPS and BatchHAZOPexpert The critical aspects of the interaction between the two tools that need to be observed is that the overlapping information about materials, equipment, and chemistry is pulled in from a database. This ensures consistency of information and avoids redundant input of information. Also the operating procedures, another piece of input, necessary for doing safety analysis inside BatchHAZOPexpert is generated within iTOPS by invoking its functionality. Similarly safety critical information about different operations that needs to be incorporated inside the operating instructions is generated by invoking BatchHAZOPexpert's functionality and by feeding back the information to iTOPS. This illustrates how iTOPS and BatchHAZOPexpert share information and function to perform operating procedure synthesis and process hazards analysis more efficiently and effectively. This paradigm is being expanded to include the entire range of process development tools and is being tested on an industrial case study. Some of the results for this case study are highlighted in the next section.
929
3. CASE STUDY RI
R2
Toluene
Solvent_ 1
MeOH
Cat
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I2
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Solvent 2 ~ ~ . . . . Reaction 3 2E +H 20 ->D+2 MeOH
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Product
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<5 Purelnt _ 2
Figure 3 Process Description for Case Study The strategy described in Section 2 has been tested using the process shown in Figure 3 consisting of 6 major processing steps (3 reactions, and 3 distillations). This process was adapted from Allgor et. al. [8]. The following tasks were performed for the different reaction and distillation steps - properties for different compounds were obtained within ProPred, reactions were defined using ReacDef, streams and units were defined in ICAS, preliminary material and energy balances were performed in Simul using the information input in ICAS, the information was then transferred to iTOPS to generate the operating procedures. The completion times were then modified through dynamic simulation in BRIC. The modified information was then transferred from iTOPS to BatchHAZOPexpert(BHE) and qualitative hazards analysis was performed in BHE. The results of the analysis were then fedback to iTOPS to incorporate the safety information in the final operating procedures for the first reaction which are shown in Table 2. 4. CONCLUSIONS
This paper has described a strategy for integrating batch process development tools. It has identified the different stages of a batch process development pipeline and developed a function and information flow model that is used to integrate a set of pipeline management tools. This integrated system has been tested on an industrial case study. The application of this strategy on the industrial case study demonstrates the advantages it offers. This method minimizes rework by avoiding multiple problem set-ups thereby allowing faster process development. The quality of the development activity is also enhanced through sharing of the function across the different tools.
930
Table 2 Operating Procedures for First Reaction of Case Study No. Batch Operation Instructions Potential Hazards 1 Charge 500L R2 to Reactor-1 2 Purge Reactor-1 for 0.25 hour 3 Charge 500L Toluene to Reactor-1 for 0.5 hour Purge Reactor-1 for 0.25 hour Heat Reactor-1 to 70~ for 0.5 hour High temperature can lead to vaporization of volatile materials Load Catalyst Slurry (5 mol/L) 1 L 1. Low agitation leads to poor mixing 2. Hightemperature leads to potential fire hazard 7 Purge Reactor-1 for 0.25 hour 8 Charge R2 250L to Reactor-1 High charge flow rate leads to static charge 9 Load Catalyst Slurry (5 mol/L) 1 L to 1. Low agitation leads to poor mixing Reactor-1 2. Hightemperature leads to potential fire hazard 10 11 12 13
Hold for 2 hours Heat Reactor-1 to 90~ for 2 hours
Short time leads to incomplete reaction High temperature can lead to decomposition of A and R2, high gas generation rate of H and swelling reactor content Take a sample of the product from Operatorexposure to hot and hazardous materials Reactorl for purity acceptance testing during sampling Cool to 25~ for 0.25 hour
REFERENCES 1. G. Pisano, The Development Factory: Unlocking the Potential of Process Innovation, Harvard Business School Pr., 1996 2. R. Gani, G. Hytoft, C. Jaksland and A.K. Jensen, Integrated Computer Aided System for Integrated Design of Chemical Processes. Computers & Chem Eng. 21, pp. 1135-1146, 1997 3. R. Srinivasan and V. Venkatasubramanian, Automating HAZOP analysis of Batch Chemical Plants: Part II. Algorithms and Application. Computers & Chem Eng, 22, pp.1357-1370, 1998 4. S. Viswanathan, C. Johnsson, R. Srinivasan, V. Venkatasubramanian and K.E. Arzen, Automating Operating Procedure Synthesis for Batch Processes - Part I. Knowledge Representation and Planning Framework. Computers & Chem Eng, 22, 1673-1685, 1998. 5. V. Venkatasubramanian, J. Zhao and S. Viswanathan, Intelligent systems for HAZOP analysis of complex process plants. Computers & Chem. Eng, 24, pp. 2291-2302, 2000 6. S. Viswanathan, J. Zhao and V. Venkatasubramanian, Integrating operating procedure synthesis and hazards analysis automation tools for batch processes. Computers & Chem. Eng., Supplement, $747-750, 1999 7. J. Zhao, S. Viswanathan, C. Zhao, F. Mu, and V. Venkatasubramanian, Computer-integrated tools for batch process development. Computers & Chem. Eng, 24, pp.1529-1533, 2000 8. R.J. Allgor, M. D. Barrera, P.I. Barton and L.B. Evans, Optimal Batch process Development, Computers & Chem Eng, 20, 885-896, 1996
European Symposiumon ComputerAidedProcess Engineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rights reserved.
931
Knowledge-based management of change in chemical process industry Jinsong Zhao, Shankar Viswanathan, Chunhua Zhao, Fangping Mu and Venkat Venkatasubramanian* Laboratory for Intelligent Process Systems, School of Chemical Engineering Purdue University, West Lafayette, IN 47907, USA The only constant in chemical process industry (CPI) seems to be change. Although the management of change (MOC) concept has been a common practice in CPI, incidents caused by inadequate MOC on existing systems and processes continue to occur. With the complexity of the processes in CPI increasing, a minor improper change has the potential to lead to serious disruptions. Currently, MOC is mainly performed by human teams. To overcome the disadvantages of the human team-based MOC, an automated knowledge-based MOC expert (MOCexpert) system is presented in this paper. Potential hazards resulting from the changes to the processes can be captured and reported by MOCexpert. A case study is given to illustrate the features of this system. 1.
INTRODUCTION
The only constant in the chemical process industry (CPI) seems to be change. Changes are made in order to improve quality, productivity, safety or reduce cost. There are few processes that operate for long without undergoing some changes. Incidents, however, caused by inadequate management of change (MOC) on existing systems and processes continue to occur, such as the catastrophic explosion at the Flixborough Works of U.K. Nypro Ltd in 1974 which killed 28 people [ 1]. With the complexity of the processes in CPI increasing, a minor improper change may lead to disruptions in process operations. Therefore, for promoting employee safety and protecting the environment, MOC has been the most important element of any process safety management (PSM) system. The concept of MOC was first introduced by the nuclear power industry in the early1960s. The first specific mention of MOC in the CPI literature was the American Petroleum Institute's "Management of Process Hazards" in 1990 [2]. During the recent ten years period, MOC has received increased attention as a common practice in the CPI. The importance of MOC is also underlined by U.S. Occupational Safety and Health Association (OSHA) PSM standard 29 CRT 1910 and U.S. Environment Protection Agency (EPA) Risk Management Program 40 CFR Part 68. These standards regulate that all changes except replacement-in-kind are subject to the MOC procedure. Currently, MOC in the CPI is mainly performed by human teams. Little effort has been seen in the research and development of computer-aided tools for MOC. Human team-based MOC has the following disadvantages: (1) It is hard for the team to thoroughly find out all potential adverse consequences of every change to the processes; (2) Consistent MOC is difficult to be Corresponding author. Tel: 1-765-494-4083. Email: [email protected]
932 achieved since the quality of MOC heavily depends on the experience and knowledge of the team; (3) Human errors may lead to severe mistakes in MOC. To overcome the disadvantages of the human team-based MOC, an automated knowledge-based MOC expert (MOCexpert) system is developed and presented in this paper. Potential hazards resulting from the changes to the processes can be captured and reported by this system.
2. MANAGEMENT OF CHANGE (MOC) To understand MOC, the relevant definitions must be given first: 9 Change: any addition or modification to a process, equipment, procedures, raw materials or a replacement item that is not a replacement of kind. MOC: The discipline of identifying the components of a continuously evolving system for the purposes of controlling changes to the components and maintaining integrity and traceability throughout the system life cycle [3].
I
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Fig. 2 Framework of MOCexpert
An effective MOC system requires significant commitments from managements and employees. To effectively implement MOC, systematic management program should be formalized. Fig.1 shows a stepwise MOC system flow diagram for implementation to ensure regulatory compliance and safe operation of the plant. In the diagram, there are six steps: 1. Identify need for change. Questions such as "Is the change necessary?" are often asked in this step. 2. Determine approval steps. The people who need to review the changes are determined in this step. People with expertise in relevant engineering fields are organized to form a MOC team. 3. Conduct required hazards review. This step is the most time-consuming step. To identify all potential problems caused by the changes, process hazards analysis techniques, such as check-list review, what-if analysis, failure mode and effects analysis, event tree and fault tree analysis, and hazard and operability analysis (HAZOP), should be used here. Among these methods HAZOP is the most widely used in CPI.
933 4.
Make recommendations to control the identified hazards. For each identified hazard, control actions should be recommended based on the current protection facilities. If the existing protection facilities are not sufficient, new protection methods should be suggested. 5. Obtain authorization to proceed. Based on the changes and the recommendations, appropriate levels of authorization should be obtained. Authorization is not only endorsement, but also a systematic review of the MOC program. Questions such as "Have all identified hazards and associated tasks required before implementation of the changes been addressed and documented?" should be asked by the authorizations in this step. 6. Implement. Before implementation, it is also critical to have a final review of the MOC program. There are several types of change that can take place in CPI [4]: 9 change in technology 9 change in temporary facilities including temporary piping, connections or hoses
9 change in procedures including operating procedures, safe working practices, risk standards, administrative procedures, maintenance/inspection procedures 9 change in site equipment such as adding new buildings 9 change in personnel 9 change in organization function and structure 9 change in policy including government laws and company internal policies In this paper, only issues involved in change in technology are considered by MOCexpert. Change in technology may occur as a result of changes in raw materials, catalysts, solvents, product specifications, by-products, design inventories, instrumentation and control systems, equipment, operating procedures, operating conditions such as temperature, pressure and flowrate and so on. In the next section, MOCexpert is presented to address issues involved in the third and fourth steps, i.e. hazards review and recommendation of controls, for the changes in technology.
3. K N O W L E D G E - B A S E D MOC
According to the steps in the above MOC system flow diagram, it can be seen that MOC is a time and effort consuming procedure that requires technical expertise, manpower and capital. The quality of MOC heavily depends upon the experience of the MOC team, which makes it hard to keep the consistency of MOC. Therefore, there exists substantial motivation to develop computer-based approaches for MOC. Authors have successfully developed and implemented knowledge-based computer tools for automating process hazards analysis [5-10]. These tools include iTOPS (an intelligent Tool for Operating Procedure Synthesis), BHE (an automated HAZOP expert system for Batch processes), HE (an automated HAZOP expert system for continuous processes). Given a process recipe, iTOPS can generate the operating procedures based on which BHE can perform HAZOP analysis. In BHE, there are about fifty built-in operation models that are often encountered in batch chemical processes. Reasoning rules were constructed to capture the adverse consequences
934 and abnormal causes of the hazards. Similarly, given pipe and instrument diagram (P&ID) of a process, HE can automatically identify all potential hazards in the process. Based on these tools, a knowledge-based MOC expert (MOCexpert) system is developed. Figure 2 shows the framework of this system. MOCexpert can work with process or product design tools by which alternative processes or products can be designed. Once a better process or product is generated by the design tools and the management decides to replace the existing one with the better one, the relevant changes such as the changes in the raw materials, operating conditions, P&ID and operating procedures are input to MOCexpert. If the process is a batch process, changes can be made through the recipe input of iTOPS. If the process is a continuous chemical process, the changes can be directly made in the P&ID. Before the changes are introduced, potential hazards are identified for the original process and saved in the hazard knowledge base (HKB). Changes are then made through the input such as process materials, process equipment, process recipe and P&ID. MOCexpert records the changes. After the changes are made, HAZOP analysis is performed for the changed process. The captured hazards are also saved in the HKB. Based on the information in the HKB, MOCexpert can identify the added hazards and eliminated hazards caused by each change, and a MOC Report reflecting the impact of the changes can be generated. 4. CASE STUDY A simple case study, which is extracted from a waste water treatment batch process, is used to illustrate the features of MOCexpert. The process includes two main s t e p s - extraction and distillation as shown in the process sequence diagram (Fig.3). 300L waste water containing product P is first charged to extractor K1001, then 300L solvent toluene is charged to the same vessel. Extraction is performed in K1001 F,g. 3ProcessSequenceDlagram over 30 minutes. Samples are taken to determine when to end the extraction. Once the extraction is done, the organic phase containing toluene and the product P is transferred to Reactor K 1002. The aqueous phase is drained to sewer. In Reactor K 1002, the content is heated to reflux, and toluene and the product is separated by distillation. The solvent toluene as the condensate is recycled and stored in a toluene tank. The capacity of K 1001 and K 1002 is 1000L. Use of toluene, however, causes a problem. According to relevant environmental regulations, the presence of a small amount of toluene lost in the waste water after the extraction is not acceptable. Therefore, an environmentally benign substitute of toluene has to be found out. Butyl acetate, sec-butyl-3-methyl-butanoate and isobutyl-isoprentanoate were reported in literature [ 11,12] as the substitutes. In this case study, butyl acetate is used to replace toluene. To accommodate the change of the solvent, the following changes in the operating procedure are made: 1. In the second operation of the first main step, 500L butyl acetate, instead of 300L toluene, is charged to extractor K 1001.
935
.
In the second main step, the butyl acetate tank, instead of the toluene tank, is used to receive the condensate - butyl acetate.
The hazards of the original process are listed in Table 1. Recommendations for controlling the hazards are also given in the table. Table 2 shows the MOC report after the changes are introduced to the process recipe. Since butyl acetate is an environmentally benign compound, the high concentration hazard "operator exposure to highly hazardous material toluene during sampling" was eliminated. The fire hazard was also eliminated because the flash point of butyl acetate is 22~ which is much higher than that of toluene (4.4~ After the 500L butyl acetate is charged to the extractor, the total volume is added up to 800L which is close to the maximum operating level limit of the extractor. That will lead to the new hazards (High level hazard and high agitation hazard in Table 2). The recommendations to control the added hazards are: "calculate the required vessel capacity" and "control the agitation speed". Table 1 HAZOP results Step Deviations 1 High temperature
before changes are made Consequences 1. Vaporization of materials 2. Potential fire hazard during charging due to presence of flammable toluene above flash point.
Recommendations 1.Calculate vent size 2. Use POGO stick in charge operation
Short time
Poor extraction effect
Control operation time
High concentration Low temperature High temperature
Operator exposure to highly hazardous material toluene during sampling Poor distillation effect
Personal protection equipment should be used Install temperature controller
1. Viscous bottom due to over distillation 2. Operator exposure to hot material during sampling 3. Potential foaming Potential foaming Viscous bottom due to over distillation Operator exposure to highly hazardous material toluene during sampling
1.Install temperature controller 2.Personal protection equipment should be used Control agitation speed Control operation time Personal protection equipment should be used
Low agitation Low level High concentration
Table 2 MOC report Changes Hazards eliminated 1.Replace 1. High temperature hazard: toluene with Potential fire hazard during charging butyl acetate due to presence of flammable 2.Change the toluene above flash point. amount of 2.High concentration hazard: butyl acetate Operator exposure to highly to 500L hazardous material Toluene during sampling
Hazards added 1.High level hazard: potential overflow (at K1001 & K1002) 2.High agitation hazard: potential overflow (at K1001 & K1002)
Recommendations 1. Calculate the required vessel capacity 2. Control agitation speed
936 5. CONCLUSIONS This paper presents a knowledge-based system - MOCexpert for automating MOC in CPI. MOC is the most important element in any PSM system. But, accidents caused by improper MOC continues to happen due to human errors in the MOC procedure. To reduce the human errors, the time and effort involved in MOC, and make MOC more consistent, we developed this system to bridge the gap. MOCexpert is not meant to replace the MOC team, but to speed up the MOC process by providing consistent hazard review and relevant recommendations to control the hazards. The future work will be focused on addressing issues involved in other kinds of changes and integrating MOCexpert with process/product design tools. REFERENCES
1. F. P. Lees, Loss prevention in the process industries: hazard identification, assessment and control (Second Edition). Volume 3, Butterworth Heinemann, London, (1996) 2. H. H. West, M. S. Mannan, R. Danna, E. M. Stafford, Make plants safer with a proper management of change program. Chem. Eng. Prog., 94(6): 25-36, (1998) 3. British Standards Institute, Configuration Management. British Standard 6488, BSI London, (1988) 4. W. Schlechter, Managing your process hazards as a means of conforming to O S H A requirements. International Journal of Process Vessel and Piping, 66(1-3), 403-415, (1996) 5. R. Srinivasan and V. Venkatasubramanian, Automating HAZOP analysis of batch chemical plants: part I&II. the knowledge representation framework. Computers & Chem. Eng., 22 (9):1345-1370, (1998) 6. V. Venkatasubramanian and R. Vaidhyanathan, A knowledge-based framework for automating HAZOP analysis. AIChE Journal, 40:496-505, (1994) 7. V. Venkatasubramanian, J. Zhao and S. Viswanathan, Intelligent systems for HAZOP analysis of complex process plants. Computers & Chem. Eng., 24(9-10): 2291-2302, (2000) 8. S. Viswanathan, C. Johnsson, R. Srinivasan, V. Venkatasubramanian and A.K. Erik, Automating operating procedure synthesis for batch processes: part I&II. knowledge representation and planning framework. Computers & Chem. Eng., 22(11): 1673-1698, (1998) 9. S. Viswanathan, J. Zhao and V. Venkatasubramanian, Integrating operating procedure synthesis and hazards analysis automation tools for batch processes. Computers & Chem. Eng., Supplement, $747-750, (1999) 10. J. Zhao, S. Viswanathan, C. Zhao, F. Mu and V. Venkatasubramanian, Computer-integrated tools for batch process development. Computers & Chem. Eng., 24(2-7): 1529-1533, (2000) 11. A.E. Hodel, Butyl acetate replaces toluene to remove phenol from water. Chemical Processing, 56(3): 53, (1993) 12. P.M. Harper, R. Gani, P. Kolar, and T. Ishikawa, Computer aided molecular design with combined molecular modeling and group contribution. Fluid Phase Equilibria, 158-160: 337347, (1999)
European Symposiumon ComputerAided Process Engineering - 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScience B.V. All rights reserved.
937
A Novel Continuous Time MILP Formulation for Multipurpose Batch PlantsIntegrated Planning, Design and Scheduling X.X. Zhu + and T. Majozi Department of Process Integration, University of Manchester Institute of Science and Technology, P.O. Box 88, Manchester M60 1QD, United Kingdom This paper presents a methodology for the integration of planning, design and scheduling in multipurpose batch plants. In dealing with this problem, the method presented in this paper exploits the mathematical structure of the overall plant model. It is discovered that the overall model exhibits a block angular structure that is decomposed by raw material allocation. If raw materials can be allocated optimally to individual plants, solving individual models for each plant can produce same results as solving an overall model for the site. This discovery leads to a decomposition strategy which consists of two levels. In the first level, only planning decisions are made, and the objective function is the maximisation of the overall profit. The results from solving the planning model give optimal raw material allocation to different plants. In the second level, the raw material targets from the first (planning) level are incorporated into the scheduling submodels for each plant, which are solved independently without compromising global optimality. The objective function for each scheduling submodel is the maximisation of product throughput. The scheduling level uses the concept of the state sequence network (Majozi and Zhul). Solving scheduling submodels for individual plants rather than the overall site model leads to problems with much smaller number of binary variables, hence shorter CPU times. If conflicts arise, i.e. the planning targets are too optimistic to be realised at the scheduling level, the planning model is revisited with more realistic targets. This eventually becomes an iterative procedure which terminates once the planning and scheduling solutions converge within a specified tolerance. In this way, the planning model acts as coordination for scheduling models for individual plants. An industrial case study with three chemical processes is presented to demonstrate the effectiveness of this approach. Keywords: Batch processes, Mathematical programming, Integrated planning and scheduling 1.
INTRODUCTION
Conflicts usually exist between planning and scheduling targets. Most chemical sites with distinct planning and scheduling departments experience this problem on a daily basis. This occurs due to the inconsistency between planning and scheduling in terms of time horizons and the focus of decision making. Whilst scheduling focuses on short-term operational issues, planning is aimed at long-term economic issues and tends to overlook operational aspects and +Corresponding author. Email:[email protected]
938 disturbances occurring on a daily basis. However, these aspects affect the selection and allocation of raw materials to individual plants. Whenever these disturbances occur, the predetermined schedules may not be valid anymore and the production throughput may be affected. This causes the incoherence between planning and scheduling. Therefore, there is a need to develop methods that can reconcile planning and scheduling targets to guarantee the coherence between them. Most of the work done on batch processes has focused on developing better models for scheduling alone without involving planning. A significant amount of work has also been dedicated to developing planning models for chemical processes, with more emphasis on multiperiod models (Sahinidis, Grossmann, Fornari & Chathrathi2; Paules & Floudas3; Papalexandri & Pistikopoulos4; Karimi and McDonaldS; Iyer & Grossmann6). However, not much attention has been paid on integrating planning and scheduling decisions. Further, multiplant models have received limited attention whereas they are frequently encountered in practice. In addressing the problem of integrated planning and scheduling, Subrahmanyam e t al. 7 and Bassett e t al. 8 attempted to tackle the problem of applying distributed computing to batch plant design and scheduling. Their approach is based on the decomposition of the problem into design and scheduling levels, yielding a Design Super Problem (DSP) and Scheduling Sub Problem (SSP), respectively. The DSP problem entails aggregation of plant life span time horizon into nonuniform time periods. The boundaries of the time periods define aggregate production deadlines. The SSP problem addresses scheduling within short-term time horizons, i.e. time periods from the DSP, to cater for the day to day plant operations. Uniform time discretization is applied within each short-term time horizon. The scenario concept is adopted to yield an MILP formulation for DSP problem. The scenario is a collection of predicted demand levels associated with their probabilities. Once the schedules are optimised within a given time period, they are then spliced/linked together to yield an overall solution to the original problem. It was acknowledged, however, that the complexity behind solving problems of this nature has prevented the application of this methodology to large-scale problems. The problem formulation results in a colossal model, which cannot be solved easily by the existing computational techniques, even for very simple problems. The fact that this methodology also includes uniform time discretization renders it suboptimal due to the restriction imposed on time distribution over the time horizon. This paper proposes a novel procedure for the integration of planning and scheduling in multiplant operations. These plants do not necessarily have to be confined within the same site. Also, this procedure can handle both multiproduct and multipurpose operations. Scheduling is based on the SSN representation and the continuous time formulation proposed by Majozi and Zhu ~. 2.
P R O B L E M S T A T E M E N T AND DISCUSSIONS
An (i) (ii) (iii) (iv)
integrated planning and scheduling problem can be stated as follows. Given: the cost of raw materials and product selling price, effluent and waste disposal costs, stoichiometric relations between raw materials and products, the production recipe for each product, including mean processing times in each unit operation, the available units and their capacities,
(v)
939
(vi) the maximum storage capacity for each material, and (iv) the time horizon of interest, determine: (i) the selection and allocation of raw materials to be used, (ii) products to be produced and their quantities in order to maximise profit, (iii) the optimal schedule for tasks within the time horizon of interest, (iv) the amount of material processed at any particular point in time within the scheduling time horizon, and (v) the amount delivered to customers over the scheduling time horizon. An overall model involving planning and scheduling may lead to large size problems, which might be difficult to handle mathematically and computationally. To overcome this problem, the solution procedure presented in this paper involves the decomposition of the overall problem into subproblems. This is achieved by exploiting the block angular structure of the overall model by applying the idea of structured programming Williams 9. A model with the block angular structure with n plants is shown in Figure 1. The b-column consisting of constants forms the right-hand side. The A-blocks represent common rows relating to common constraints. In multiplant models, these are concerned with the allocation of resources including raw materials, utilities and labour. The B-blocks represent the equations corresponding to each individual process. The first level of the proposed decomposition is aimed at determining the most optimal allocation of raw materials to different plants. Therefore, the model for this level only consists of the A-blocks. At this level only planning decisions are determined. However, the capacity limits of the plants are considered. The second level is aimed at determining the optimal schedule for each plant in order to meet the targets set at the first level. This level consists of the B-blocks, and each B-block forms an individual submodel.
Generalised
b Ao t
A1
A2
t
rn o d e l
I A~
B1 Ba
bo
[~
~_ I
I
_.
, I
b2
I ~ - constraints common Cinstraints
, !
~ _ fo
I
Bn
bn
Figure 1. Generalized block angular structure model
in ividual process
940
Cost data z(s)
Stoichiometnc data
1
1
Time horizon (I-I)
I
-'97 l~lanning ]~-
I ~z(s'),rs.~,,s,s'~S ~z(s"),r~r Scheduling for processl r, >_~.m,, (s, p)
Scheduling for process2 r,, >_~.,~(s,p)
p
P
P
l
P
No
(S (n)),rs~...... S, S (n) E S
Scheduling for processN r, >__~m.(s,p) Zmp(S,,,p )
r
~mp(.r p) [
~
S
p [Zmp(s(n),p) P
z(s)-~mp(s,
IYes Stop Figure 2. Procedure for integration of planning and scheduling
3.
STRATEGY FOR INTEGRATION OF PLANNING AND SCHEDULING
The connection between the above planning and scheduling models is achieved by incorporating the following constraint in each of the scheduling models. r,. >_Z mu (s, p), V s = f e e d
(1)
p
where mu(S, p) is the amount of raw material s used at time point p. The right hand side of constraint (1) represents the amount of raw material s required at the scheduling level. The left hand side (rs) represents the amount of raw material s predicted at the planning level. Figure 2 shows the overall procedure for integrated planning and scheduling. The planning model requires stoichiometfic data, cost data, capacity constraint data and the time horizon of interest. The capacity constraint data that provide the upper bound on production, i.e. zmax(s), is obtained by performing scheduling without raw material limits, over the time horizon of interest. If conflicts between planning and scheduling arise, i.e. the planning targets are too optimistic to be realised at the scheduling level, the planning model is revisited with more realistic targets. This eventually becomes an iterative procedure which terminates once the planning and scheduling targets are within specified tolerance (~').
941
4.
CASE STUDY
To illustrate the performance of this approach, a case study was conducted on a site consisting of three batch chemical processes. The processing time variations for processes 1, 2 and 3 were 20 %, 30 % and 25 %, respectively. Processes 2 and 3 shared raw materials, with the exception of one raw material, which resulted in different products from each of these processes. The cost data for the planning model, as well as the scheduling data for the scheduling model were provided. 5.
RESULTS
All the results were obtained using GAMS/OSL solver in a 333 MHz AMD K6-2 processor. The planning and scheduling models were based on one-year (8004 hours) and 12-hour time horizons, respectively. According to the results, processes 1, 2 and 3 were supposed to produce 6000, 22233.33 and 5000 tons of products per annum (8004 hours), respectively, in order to meet the overall optimal profit of s million. Over a 12-hour time horizon, these targets are equivalent to 8.996, 33.333 and 7.496 tons, respectively. It was noted that the shared raw material allocation was biased more towards process 2 than process 3. This was ascribed to the economic data supplied to the planning model. Had an arbitrary raw material allocation been used, suboptimal results would be obtained. For example, 50:50 distribution of all common raw materials between processes 2 and 3 gave an overall profit of s million compared with the optimal profit of s million. A 70:30 allocation that was biased more towards process 3 than process 2 gave an overall profit of s million. Solving the problem with decomposition using the procedure of Figure 2 required CPU time of about 0.93 seconds, whilst solving the overall model (consisting of the planning model and the three scheduling models) required 4.57 seconds. This was mainly due to the smaller number of binary variables associated with individual models. Whilst the overall model required 75 binary variables, the individual models for processes 1, 2 and 3 required 35, 20 and 20 binary variables, respectively. Moreover, the overall model required 1185 total variables (binary and continuous) and 1721 constraints, whereas the decomposed model required an average of 424 total variables and 609 constraints. All the results converged within a tolerance of 10-6. 6.
CONCLUSIONS
A new procedure for linking planning and scheduling decisions has been presented. This procedure exploits the block angular structure of the overall multiplant model, and using structured programming develops a generic decomposition approach. The solution procedure consists of two levels. In the first level, the planning model is formulated and solved for the optimal allocation of raw materials to individual processes. In the second level, the raw material targets obtained from the planning model are incorporated into the scheduling models for individual processes. These models are then solved independently. Therefore, the planning model serves as coordination between scheduling models of individual processes. The SSN representation (Majozi and Zhu 1) is used for the formulation of the scheduling models. If the targets set at the planning level are too optimistic to be realised at the scheduling level, the planning model is revisited with more realistic targets. This iteration only terminates when the planning and scheduling targets reach a specified tolerance. Application of this decomposition procedure to an industrial case study comprising of three processes showed a significant
942 reduction in problem size which is concomitant with shorter CPU times compared to the overall model. The overall model requires 75 binary variables, 1185 total variables (binary and continuous) and 1721 constraints with 4.57 CPU seconds. The size of the decomposed models is reduced to an average of 25 binary variables, 424 total variables (binary and continuous) and 609 constraints, with an average CPU time of 0.93 seconds. This reduction in problem size allows the application of this decomposition procedure to large-scale industrial problems. Significance of optimal allocation of raw materials manifests the necessity of using a planning model to guide the scheduling models to maximise the overall profit. REFERENCES
1. 2. 3.
4.
5. 6. 7. 8. 9.
T. Majozi and X.X. Zhu, A novel continuous time MILP formulation for multipurpose batch plants. I - Short-term scheduling, sumitted to Ind. Eng. Chem. Res. 2000. N.V. Sahinidis, I.E. Grossmann, R.E. Fornari and M. Chathrathi, Optimization model for long range planning in the chemical industry, Comp. Chem. Eng., 13(9): 1049-1063. 1989. G.E. Paules and C.A. Floudas, Stochastic programming in process synthesis: a two stage model with MINLP recourse for multiperiod heat-integrated distillation sequences, Comp. Chem. Eng., 16(3): 189 - 210, 1992 K.P. Papalexandri and E.N. Pistikopoulos, A multiperiod MINLP model for the synthesis of flexible heat and mass exchange networks, Comp. Chem. Eng., 18(11/12): 1125 - 1139, 1994. I.A. Karimi and C.M. McDonald, Planning and scheduling of parallel semicontinuous processes. 1. Production Planning, Ind. Eng. Chem. Res., 36:2691-2700, 1997. R.R. Iyer and I.E. Grossmann, A bilevel decomposition algorithm for long-range planning of process networks, Ind. Eng. Chem. Res., 37(2): 4 7 4 - 481, 1998. S. Subrahmanyan, K.G. Kudva, M.H. Bassett and J.F. Pekny, Application of distributed computing to batch plant design and scheduling, AIChE, 42 (6): 1648 - 1661, 1996. H.M. Bassett, J.F. Pekny and G.V. Reklaitis, Decomposition techniques for the solution of large-scale scheduling problems, AIChE, 42 (12): 3 3 7 3 - 3387, 1996. H.P. Williams, Model Building in Mathematical Programming. Third Edition-Revised, 4564, John Wiley and Sons, 1997.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
943
Retrofitting of Mass Exchange Networks with Temperature Effects H.E. Alfadala a, A.K. Sunol ~and M.M. E1-Halwagib aDepartment of Chemical Engineering, University of South Florida, Tampa, FL 33620 bDepartment of Chemical Engineering, Auburn University, Aubum, AL 36849 A systematic procedure for the retrofitting of MENs with temperature effects is developed. Initially, alternative structural configurations of interest are examined. Both series and parallel structures are addressed. Two primary retrofitting strategies are employed: those restricted by no-capital constraints and those involving capital expenditure. The no-capital alternatives include enhancing performance of current system as well as solvent substitution. The capitalbased alternatives include the addition of new equipment. A new type of temperature-based mass-pinch retrofitting analysis is developed to maximize the utilization of existing capital while reconciling added capital with operating cost. The main concept in this novel pinch diagram is identifying maximum performance as dictated not only by thermodynamic limitations but also by physical size limitations. Different process alternatives are considered and screened to attain the optimum design. 1. INTRODUCTION The area of synthesizing MSA-based separation networks has witnessed active research over the past decade. A key contribution in this field is the introduction of the concept of mass-exchange networks (MENs) [ 1]. The design task of MEN is to synthesize a network for selectively transferring certain species from a set of rich streams to a set of lean streams or mass separating agents (MSAs). To optimize the cost of mass separating agents, the masspinch analysis can be utilized [1, 2]. Systematic techniques can be used to screen the external MSAs and to trade off capital versus operating costs [2-5]. Several important categories of MEN synthesis problem have been recently addressed. These include MENs with: a single transferable compound [1,3], multiple species [6], regeneration of MSAs [7], simultaneous mass and heat transfer [8], and chemically-reactive separations [9, 10]. In spite of the usefulness of these contributions, they have a common limitation. They are all grass-root design procedures in which it is assumed that all mass-exchange units are new. Another important class of problems entails a combination of already existing units with the possibility of adding new units if warranted. This is referred to as the retrofitting problem and is the focus of this paper. Grass-root design procedures are not applicable to retrofitting applications because they do not address the utilization of existing units. In retrofitting applications, maximum utilization of existing capital should be sought. Furthermore, the trade-off between operating cost, existing capital, and new capital investments should be systematically reconciled.
944 2. APPROACH AND MODELING
The initial effort concentrated on development of a super structure that included plausible alternatives. The subsequent model development involved rigorous simulation using ASPEN PLUS simulator to identify simplified unit model parameters and bounds. The simplified unit models were subsequently used in formulation of the problem as a Mixed Integer Non-Linear Mathematical programming, which was solved using LINGO. The model exploits the generalized formulations discussed in Grossmann's work [13]. The key attributes of our implementation of this generalized formulation are mass exchange rates which are bounded by the flooding rates; Henry's constants which is temperature dependent and are treated as discrete operating conditions; and the retrofit realities such as zero capital cost and fixed unit sizes that are coordinated through logical constraints. 3. SOLUTION METHODOLOGY To illustrate the proposed procedure for basic cases, we employ the following simplifying assumptions: 1. A maximum of one additional mass exchanger is allowed. This assumption is driven by, practical reasoning (from layout consideration, size limitations, cost inhibition, process control, etc.). 2. Each column operates isothermally. For the mass-exchange units, three main configurations are examined: series structures with the new column preceding and following the existing column, as well as parallel structure. Since retrofitting design entails the utilization of the existing column, the grass-root mass-pinch analysis is not applicable anymore. The grass-root pinch diagram is aimed at maximizing the use of process lean streams with the objective of minimizing operating cost. The retrofit mass pinch diagram will have to maximize the use of the process unit(s) with the objective of minimizing capital cost. Then, capital cost should be traded off with operating cost. Now, let us develop a new mass-pinch diagram to tackle the retrofit problem, starting with the first case. Series structure with the new column located after the existing column: In addition to the component material balances, the column performance model (e.g., Kremser equation, H = H T U x NTU, empirical model, etc.) can be stated as follows: yOUt = f(G, yS, L, x in, m, column sizing criteria, operating conditions) (1) With the existing column of a given size, we now have "equipment-based limitation" in addition to the conventional equilibrium-based limitations. Therefore, we must first identify maximum performance target for the existing column with thermodynamic and size limitations. This is achieved by passing maximum allowable flowrate of MSA referred to as L max (e.g., at flooding velocity for the column, maximum capacity of existing pump) through the column. When the flowrate of the MSA through the existing column is maximum, its operating cost is also maximum and mass-exchanged within it, is also maximum. Now, for the given supply composition of the rich stream and maximum flowrate of the MSA, the column performance model can be used to calculate the outlet compositions for the rich and lean streams (yOUt, *and Xl ~ *, respectively), i.e., yOUt, * = f(G, ks, Lmax, xin, m, column sizing criteria, operating conditions) (2)
945 In this case, the load exchanged within the existing column is at its maximum, thereby requiring minimum load to be exchanged within the new column (and, consequently, minimum capital cost). These aspects lead to the new retrofit mass-exchange pinch diagram shown in Fig. 1. It captures thermodynamic as well as equipment (physical size) limitations. Mass Exchangec
....' ~ .....7
Retrof~ing M .... Exchange Pinch ~
##%/ . / y
Lean / i / Stream/ .~ //... [
y'.,."
i i i [
......."~....
[ 1MaximumIntegrated /"a~' ~xo.~nged (byExistingColumn)
.................... f Minim~Load /to be Exchanged , wi~m~w Column y, y--
~
Xlm
X l out,•
xI
Fig. 1. Retrofit Mass-Exchange Pinch Diagram (when existing column precedes new column)
This pinch diagram corresponds to maximum operating cost for the existing column and minimum fixed cost for the new column. In order to trade off the fixed versus the operating costs, mass-exchanged load must be shifted from the existing to the new column. This load shifting is accomplished by using flowrates of the MSA less than the maximum allowable. Therefore, the outlet composition for the rich stream leaving the existing column is varied to the right of yOUt, *, thereby yielding an intermediate composition for the rich stream leaving the existing column and entering the new column to be yintex. Let us define the trade-off composition difference to be: COex = yintex _ yOUt, * (3) This variable reflects the extent of load shifting from the existing to the new column. For a given co, the retrofit mass-exchange pinch diagram is shown in Fig. 2.
Mass
Exchanged
R,~o~.g Mass-Exchange Pinch
k ~
i
......""?,......... . ~'""q ........ .
] "
/
Lean / i /" Stream/. 7
~
l
i lintegratedMass ! /Exchange(by i .~.ExistingC~ ..........!...............!.............
Rich / i S;ream/ i / ~ y y'.'," y%,,,,,~ X Im
X I~
TL~ Exchanged I withinNew ~C~ L Y Y X1
Fig. 2. Cost Trade-off via Load Shifting from Existing to New Column Similar pinch diagrams can be developed for the cases when the new column is located before the existing column and for the case of parallel structure [ 11]. Combined Heat and Mass Exchange Networks Temperature effects can be used to affect the system performance by modifying rich-lean equilibrium relation. Therefore, mass and heat exchange aspects should be addressed in
946 tandem. In order to address the combined heat and mass exchange problem, a decomposition approach is proposed. The foregoing procedure for retrofitting MENs will continue to form the core of the synthesis procedure. Nonetheless, the temperatures of the lean streams entering the mass exchangers will be varied iteratively. For each stream, we define the admissible range of temperature for each lean stream (e.g., between supply temperature and freezing temperature or decomposition temperature). This interval will be referred to as: [T/j TUj]. The admissible temperature range is discretized into nik temperature. This discretization significantly simplifies the problem as it decomposes the MEN synthesis problem from the heat exchange network (HEN) synthesis problem. For each set of iterative lean-stream temperatures, the previously-mentioned MEN retrofitting procedure is employed. Since the supply and iterative cooling/heating temperatures for each lean stream are now known, the synthesis of a HEN can be undertaken through conventional HEN synthesis techniques [12]. The pinch diagram for the combined heat and mass exchange networks is constructed for each temperature in the interval [T~ TUj] as shown in Fig. 3.
Mass ExchangedR.,.oj.,.g (s .... Z', --f .... ? ,,'..,,~,,--~<-,~. s=&/~ / i / ~x,m,~. ,~,,eh~ ~'Y I / ', /Integrated / Z ', /MassExchange " ~ //! ! / (byExisting R ~ - ~ - ------iI.........!i!~Column)I~c~dum N!_wl yt
, i
yut. *
!
',
y,
y
Fig. 3. The Retrofit Temperature-Dependent Mass-Exchange Pinch diagram Mass Exchanged
/ :,~ //i i
Pinch[
k
1~ /
~,t,*
i
]
[
[[ within New
~ut
!
ys
~a/i w[
y
|(by Existing
[Column)
~i
i
e;,,i
i/r176
y
4 r''~ )
Fig. 4. Retrofit Heat~Mass-Exchange Pinch Diagram with Load Shifting The composition difference between the intermediate composition and the maximum inlet composition is an optimization variable that can be used to trade off fixed versus operating costs so as to minimize the total almualized cost of the network. This is an iterative process which will be done at each T* k within the temperature interval. Likewise, the trade-off composition difference co for this case will be:
947 09 = yint _ y (4) This variable reflects the extent of load shifting from the existing to the new column at each temperature. For a given co, the retrofit heat/mass-exchange pinch diagram can be represented as shown in Fig. 4. The total annualized cost is calculated by adding up the costs of the retrofitted MEN and the HEN. The iterations are continued to cover the admissible temperature ranges for the lean streams and the minimum total annualized cost of all iterations is identified. 4. CASE STUDY: REMOVAL OF CARBON DIOXIDE FROM AN AMMONIA PLANT The detailed data for the case study are given in reference [11]. Figure 5 illustrates the basic features of the problem. At present, K2CO3 is used to remove CO2 down to 0.002%. The objective is to reach a target composition of 0.001% via a combination of solvent substitution, equipment addition, and temperature variation. The admissible temperature (~ range for K2CO 3 is [210 240]. I
i
K2CO3 in LI? ~@ T=239~ Rich Feed G = 3.86 lbmole/s[ / = 18.75 mol%'~
~MDEAin L2?
RetrofittedMEN & HEN
K2CO3 out ~ x1~
Outlet Rich y' = 0.001 mol%
x2~
MDEA out
Fig. 5. Removal o f Carbon Dioxide in an Ammonia Plant By optimizing heat effects, the optimum cooling temperature was identified to be 228~ which allows the target to be achieved within the existing column (Fig. 6). The TAC of the system is $ 30.0 MM/yr from which the cooling cost (both fixed and operating) is $ 16.2 MM/yr and the annualized operating cost for the lean stream is $13.8 MM / yr. K2CO3 from Stripper @ 239~
~
Outlet Rich ..&./ = 0.001 mol%
IT = 228~
L = 100 lbmole/s & x s = O.0 m o l t o "
72 m;!%k~t~
E~Colamn Stripper
[
Rich Feed~ G = 3.86 lbmole/s / = 18.75 tool%
Fig. 6. Optimum solution with cooling only K2CO3 Solutionin
Outlet Rich
L = 100 l b m o l e / s ~Xinne w = O. 0 m o l % N e w C o l u m n = 15 ft
X int = 5.5 x ] O "4 m o l %
I
E x i s t i n g C o l u m n = 76 ft
xOUtexisting = 0.75 mo1% ~
K2CO3 to Regeneration
Ty~
I ~/ ' = I
0.001 mol% 0.015 tool%
Rich FeedT G = 3.86 lbmole/s Y"exi.~,i,g= 18. 75 mol%
Fig. 7. Optimum solution with Addition o f a New Mass Exchanger Next, we by allowing for an addition of a new mass exchanger, the optimal solution is found to involve a new mass exchanger that uses hot potassium carbonate and operates at an inlet lean-stream temperature of 239~ and has a TAC of $14.1 MM/yr (Fig. 7).
948 5. CONCLUSION A systematic methodology for retrofitting mass exchange networks with temperature effects is presented. This methodology examines the various process configurations for the rich and lean streams including series and parallel structures. A new mass-pinch representation is developed to address the retrofitting problem. In addition to thermodynamic limitations, it embeds sizing limitations of the existing units. Maximum achievable performance is identified by, driving the existing-unit performance to the limit. Next, capital cost of the new units is traded off versus the operating costs so as to reach minimum total annualized cost. NOMENCLATURE G = rich stream flowrate i = index for temperature intervals (pollutant-free basis), lbmole/s j = index for lean streams superscripts: H = Height of column, ft H T U = Height of transfer units * = equilibrium L = lean stream flowrate, lbmole/sin = inlet value m = slope of equilibrium line int = intermediate value N T U = number of transfer units k = iteration index T = interval temperature, K l = lower bound t = solution temperature, K max = maximum practically feasible out = outlet value x = composition of pollutant in an MSA (pollutant-free basis) s = supply value y = composition of pollutant t = target value in rich phase (pollutant-free basis) u = upper bound subscripts: ex = existing column
Greek letters:
co = trade-off composition difference
REFERENCES 1. M.M. E1-Halwagi and V. Manousiouthakis, AIChE J., 35 (8) (1989) 1233. 2. M.M. E1-Halwagi, Pollution prevention through process integration through process integration: systematic design tools, San Diego, Academic Press, 1997. 3. M.M. E1-Halwagi and V. Manousiouthakis, Chem. Eng. Sci., 45 (9) (1990) 2813. 4. K.P. Papalexandri, E.N. Pistikopoulos and C.A. Floudas, Trans IChemE 72(A)(1994) 279. 5. N. Hallale and D.M. Fraser, Trans I Chem E 78 Part A (Mar. 2000) 202. 6. M.M. E1-Halwagi and V. Manousiouthakis, AIChE Annual Meeting (1989). 7. M.M. E1-Halwagi and V. Monousiouthakis, AIChE J., 36 (8) (1990) 1209. 8. B.K. Srinivas and M.M. E1-Halwagi, AIChE J., 40 (3) (1994) 463. 9. M.M. E1-Halwagi and B.K. Srinivas, Chem. Eng. Sci., 47 (8) (1992) 2113. 10. B.K. Srinivas and M.M. E1-Halwagi, Chem. Eng. Sci., 49 (13) (1994) 2059. 11. H.E. Alfadala, A.K. Sunol and M.M. E1-Halwagi, J. Clean Prod. Proc., (in press, 2000). 12. B. Linnhoff, Trans. Inst. Chem. Eng. Chem. Eng. Res. Des., 71, Part A5, (1993) 503. 13. L.T. Biegler, I.E. Grossmann and A.W. Westerberg, Systematic methods of chemical process design, New Jersey, Prentice Hall, 1997.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
949
Implementation of multiobjective optimisation for multipurpose batch plant planning Catherine Azzaro-Pantel, Andr6 Davin, Luc Pibouleau, Pascal Floquet and Serge Domenech Laboratoire de G6nie Chimique - UMR CNRS/INP/UPS 5503 E N S I A C E T - 18, C h e m i n de la L o g e - 31078 T o u l o u s e C e d e x 04 - France
Fax : (33) 5 62 25 23 18 e-mail : [email protected] As until now, optimisation of multipurpose batch plants was restricted at only a single objective function, generally an economic one. Today some factors such as time, quality, safety and environmental purposes have to be taken into account. It therefore becomes necessary to implement multiobjective optimisation procedures. This is possible with genetic algorithms (GA) who are especially suited in dealing with the multiobjective case due to numerous local solutions, which allows numerous criteria to be simultaneously computed. The retained approach to the multiobjective genetic algorithm is to select "Pareto-optimum" solutions by adaptation versus several independent criteria on a part of the whole population. A comparison with a monobjective GA that only uses profit to produce optimal solutions shows that significant advances can be observed from a multiobjective approach. 1. INTRODUCTION Up and tmtil now, optimisation of multipurpose batch plants approach was restricted to a single criterion, typically an economic valuation of the process. However in actual real word problems, factors such as campaign completion time, quality, safety and environmental restrictions are fast becoming key points in designing and operating any industrial plant. It therefore becomes necessary to implement multiobjective optimisation procedures to deal with these problems. Due to a high combinatorial nature, classical mathematical methods appear to be inefficient to tackle this kind of problem. Consequently, a stochastic method, a GA procedure (Goldberg, 1989 - Cartwright and Long, 1993), whose efficiency for solving similar problems has been proved (Bernal-Haro et al., 1998) is used in this study. Furthermore, GAs are especially suited in dealing with the multiobjective case due to the generation of numerous solutions, which allows numerous criteria to be simultaneously evaluated. Consequently, the aim of this study is to suggest several approaches for solving multiobjective problems for multipurpose batch plant design and operation by using genetic algorithms together with a discrete event simulator for computing the problem objectives. The approach consists in the use of a multiobjective genetic algorithm, which extracts the "Paretooptimum" solutions from a population. Let us recall that a given solution is said to be Paretooptimum if it is strictly better than the other solutions with regard to at least one objective. The proposed method is then compared with a monobjective genetic algorithm that minimises only profit to produce optimal solutions.
950 2. BATCH PLANT SIMULATION BY A DISCRETE-EVENT SIMULATOR A special configuration of a batch facility, the most complex one, is the multipurpose plant. It consists in general purpose equipment items used to manufacture a variety of products following different routes through the plant. The process flow in a multipurpose plant is not only complex, but also altered by unpredictable situations such as operator or ingredient unavailability, storage facility limitations, flow bottlenecks and equipment failures. Let us note that the development of general-multipurpose simulation systems for modelling batch plants is a quite hard task (Baudet et al. 1998). Consequently, existing simulators are usually dedicated to the description of specific situations and cannot be extended to different production cases. A classical approach to solve multipurpose plants planning problems is to use a MINLP (Mixed Integer NonLinear Programming) formulation (Mauderli and Rippin 1979, Papageourgiou and Pantelides 1996, Schilling and Pantelides 1997), where the problem is generally decomposed into two levels : (a) organisation of production campaigns, i.e., determination of production (product nature and quantities) and completion time according to given planning objectives; (b) production scheduling, i.e. allocating tasks to resources and fixing task starting dates. The main limitations of MINLP approaches are linked to a combinatorial explosion when real world problems are under consideration. Globally speaking, a batch plant may be viewed as a physical system changing its state in response to individual events occurring at discrete times. To model its behaviour, a discreteevent approach was used. The implementation of a discrete-event simulator is an efficient solution to job-shop scheduling problems. In this context, two software tools, Melissa ~(Peyrol et al. 1993) and Ad-Hoc 2 (Baudet et al. 1998), have been first developed using FORTRAN language, and recently a complete rewriting of the two packages in C++ was carried out to suit the tool to various industrial fields (B4rard et al. 1999, Brrard 2000) The decomposition of AD'HOC into four different layers makes easier the reusability of the simulator. In fact, this basic simulator implementation shows that all the contexts where discrete-event modelling can be applied, present a common structure. But, if the first layer (simulation core) is context independent, the dependencies in the following ones (event layer, job-shop layer and supervisor) are stronger and stronger. Thus, only the two first layers are part of the toolkit, the last two ones are specific of each application. This toolkit was modelled using OMT (Object Modelling Technique) by Navarre et al. (1998). For solving batch plant planning problems a general framework for modelling and optimising production conditions has been developed and includes: (a) the Discrete-Event Simulation (DES) model for describing the global dynamic behaviour of the production system and determining scheduling feasibility. The model includes the main features of production conditions. (b) the confrontation with industrial data will allow to fit simulation parameters introduced in the DES model. (c) this global simulation tool can easily be coupled with a stochastic optimisation procedure to efficiently solve the highly combinatorial problem of production planning applied to a specific production framework.
MicroElectronique Logiciel Industriel de Simulation et de Suivi d'Ateliers 2 Ateliers Discontinus-Heuristiques et Ordonnancement gt Court-terme
951 3. GENETIC ALGORITHMS Genetic algorithms (GA) are stochastic optimization methods based on the biological principles of natural selection (Beasly et al., 1993; Goldberg, 1989; Holland, 1975; Michalewicz, 1994). The basic feature of GA is to place parameters of the problem to be optimized within what is referred to as chromosome (or individual) which consists in genes. Each parameter is mapped to a gene in the chromosome. A genetic algorithm extracts a population of chromosomes and generates new populations using a variety of genetic operators including crossover and mutation. Members involved in genetic operators are chosen from the population by using a fitness function (selection procedure). A large variety of operators and selection methods can be used. Their common underlying feature is the use of randomness for selection, manipulation and generation of chromosomes. The fitness of a population member is a measure of how good or useful the particular solution encoded by the chromosome is. From given values of parameters encoded by the chromosome, the fitness is generally computed as a function of the objective function under consideration, either as an explicit function or as the solution of a LP or a NLP problem. A GA terminates when a user specified criterion is reached. Typically, this is some expected fitness, a number of iterations (known as generations), or some threshold on the diversity of the current population. There are several advantages for the use of GAs : (a) They are less trapped in local optima than classical mathematical programming methods, so they can reach in many cases the global solution; (b) Multiple solutions are available at the end of the search. For these reasons, a GA has been retained for the multiobjective optimization of multipurpose batch plants simulated by AD'HOC. 4. GLOBAL PRODUCTION PLANNING OPTIMISATION The planning production optimisation problem may be solved through different strategies corresponding to various plant scenarios, i.e., cyclic production, finite horizon production, "just-in-time" production. In each case, a set of production management parameters and a performance criterion computed by the DES simulator have to be defined. Let us recall that a GA computes a set of individuals (the population) and a set of biologically inspired operators that can generate new individuals from parents. According to the evolutionary theory, only the most suited elements of a population can survive and generate offspring, thus transmitting their biological heredity to new generations. The heredity, enclosed in the chromosomes of the individuals, is subjected to mutation and to crossover mechanisms. The offspring generated by the genetic manipulation is evaluated, via a fitness value, measuring the quality of the solution represented by the chromosome. In the following, the problem formulation corresponding to a finite horizon production is presented. This strategy deals with a production organised in only one campaign for a given horizon time H. The objective is to maximise product quantities produced by the workshop during the campaign.
952
4.1 Production management parameters and encoding procedure The production management parameters are as follows: - Q1 : quantity of a given final product manufactured in each campaign; - n~ : number of batches for each final product FPi of size si; - Pi : priority order for a manufactured product ; - hi, h2 : heuristic rules to select equipment items and batch to load in case of conflicts. A solution is encoded in a chromosome by using of an integer-coded gene string. After decoding, this chromosome represents the production management parameters of a potential solution. The encoding procedure is illustrated as follows : Production level
Batch Numbers
Recipe Priorities
Heuristics
I QxI nl [ n21 n31pllp2 [p3 Ip41psIh~ Ih21
4.2 Evaluation function The performance criterion is the profit achieved by the production penalised by an outer penalty term if the campaign duration (D) exceeds the given horizon. Then, the evaluation function (or fitness) is expressed as 9 Fitness = 2 Oi "B, + p . ( D - T)~ where Bi is the sale price of the final product j and D.is a penalty PFi
coefficient. 5. M U L T I O B J E C T I V E G E N E T I C A L G O R I T H M In the Pareto approach, the optimal solutions may be undominated, i.e. solution where an improvement in any way is not possible without degradation of another target. The undominated solutions are : Xl, X2, X3 E ~ such that x2 is under domination of x3, if f(x2) is partially lower than f(x3), i.e. f(x2) [3 f(x3) Xl is undominated if there is no x egt where x dominates xl. Several practices are reported in the literature to select these Pareto-optima solutions 9 N Relative weight. The general criterion is ~ coifi(x), where coi is a given value lying in the i=1
range (0, 1), with sum equals 1.
I"
1/r
Demand level vector. The criterion is [
f ( x ) - y ~ ] , where r generally equals 2 and yi i=1 (the demand level vector) is a parameter related to targets and quite difficult to specify. Min-Max formulation. This approach minimises differences between optimum solutions for each criterion and request solutions Min [F(x) = Max (Zj(x)), where Zj(x)-
j0-_ jS .
All these approaches have some inconveniences and their uses depend on the problem under consideration. With AGs, there are several ways to select undominated sets 9
953 (a) Vector Evaluated Genetic Algorithm (VEGA). The method is the classical multiobjectve GA. The population is divided upon each criterion and selection is applied on the partial population. After this phase, the population is mixed before a new iteration. An improvement is proposed with classification between dominate and undominate solutions, by giving an advantage to the undominate ones. (b) Non Dominated Sorting Genetic Algorithm (NDSGA). In this technique, the selection procedure is referred to a new visit. Before each step of selection, the population is arranged by a non dominancy criterion. The reproduction operates in proportion of calculated variable fitness, and the repartition function is:
Sh(d,j):l-( d,]
)2 i f d,j ~O'share and Sh(d~)=Oelsewhere
O'share
with d,j distance between 2 individuals i and j and [-]share minimum distance of two individuals of the same type. 6. APPLICATION The Pareto reservation strategies VEGA as well as NDSGA have been implemented to solve multiobjective problems. These two kinds of algorithms have been compared on problem 2 already treated by B6rard (2000) with a classical AG. In the problem, 3 products have to be manufactured in a plant involving 10 equipment units. The horizon time of the campaign is one week and no predeterminated events or maintenance breaks can occur. In addition to the profit maximum, several objectives can be also considered: processing time (fig. 1) and improvement of material working time (fig. 2)
Comparison Mono( u )/multiobjective(,) approaches on the variation of the campaign duration (left fig. 1, right fig. 2) The AG parameters are the same as used by B6rard (2000) for a monobjective (profit) optimisation problem: Mutation rate : 20%; Crossover rate : 60%; Population Size : 70 individuals and Number of generations : 250 Best individual found by the two multiobjectve GAs is the same that Berard indicates in all cases, but the comparison of the 10 best individuals shows significant differences between the mono/multiobjective approaches.
954
Comparison Maximizing utilisation rate / Chosing non dominate solutions 7. CONCLUSION For the problem presented here, the best solution related to a profit criterion is also the best one for the other criteria. Multiobjective GA has found these solutions over an empty set of better solutions related to a second criterion, not reached by the monobjective GA. By implementation of a multiobjective GA, the decision maker is assured to have at its disposal a good solution. By additional information provided by a GANTT diagram, he can verify how good solutions have been settled. REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
Baudet P., Azzaro-Pantel C., Domenech S. and Pibouleau L., Can. J. of Chem. Eng., 76(4), 300, 1998 Beasly D., Bull DR, Univ. Comput, 15(2), 58, 1993 Brrard F., PhD Dissertation, INP Toulouse, 2000 Brrard F., Azzaro-Pantel C., Pibouleau L. Domenech S., Navarre D. and Pantel M., Comp. & Chem. Eng., 23S, 565, 1999 Bernal-Haro L., Azzaro-Pantel C., Domenech S. and Pibouleau L., Comp. & Chem. Eng., 22S, $777, 1998 Cartwright H. M. and Long R. A., Ind. Eng. Chem. Res., 32, 2706, 1993 Goldberg D. E., "Genetic Algorithms in Search, Optimization and Machine Learning", Addison Wesley, Reading MA, 1989 Holland J. H., "Adaptation in Natural and Artificial Systems", MIT Press, Cambridge MA, 1975 Mauderli A. and Rippin D. W. T., Comp. & Chem. Eng., 3, 199, 1979 Michalewicz Z., "Genetic Algorithm + Data Structures = Evolution Programs", Springer, New York, 1994 Navarre D., Pantel M., Brrard F., Pantel C. and Pibouleau L., 17th IASTED Int. Conf. MIC'98, Grindelwald, Switzerland Papageourgiou L. and Pantelides C., Ind. Eng. Chem. Res., 35,488, 1996 Peyrol E., Floquet P., Pibouleau L. and Domenech S., Comp. & Chem. Eng., 17S, $39, 1993 Schilling G. and Pantelides C., Comp. & Chem. Eng., 21 S, S 1191, 1997
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
Concurrent Process Engineering & The Implications for
955
CAPE 1
Rene Banares-Alcantara a, Eric S Fraga b and Tony Perris c aUniversitat Rovira i Virgili, Tarragona ([email protected]) bUniversity College London ([email protected]) CConsultant ([email protected]) This paper explores the potential for introducing the concepts of concurrent engineering into process development and design and examines the implications of such an approach for CAPE tools and environments. Two specific examples (tools for conceptual process development and a supporting information environment) are examined and key issues and challenges are identified which must be addressed by the CAPE community if effective progress is to be made. 1. INTRODUCTION: THE NATURE OF THE PROBLEM The nature of any design process, especially where innovation is involved, is one of trial and error, since much of the information will be either uncertain or even completely unknown at the outset. Consequently, there will be a cyclic process of revision: new ideas will need to be incorporated, errors & omissions will need to be corrected. This is both necessary and productive - some at least of these revisions constitute improvements to the design, either to improve profitability or to ensure legislative compliance and the most effective time to make such innovations is as early as possible in the development process, high on the influence curve.
With the growing importance of issues such as flexibility, safety, environment, etc, the scope for growth rather than reduction of the revision cycle is substantial and action is required to reverse this trend and to make the underlying processes for revision management markedly more efficient by, for example, improved information management and sharing. The following are some of the critical issues: 9 The range of aspects which must be examined and "traded off" is large and is still growing: Steady-State Flowsheet Calculations
Heat Transfer
Separations
1 A c k n o w l e d g e m e n t s : this paper is based on the work of the Concurrent Process Engineering and Whole Process Synthesis & Integration working groups of CAPE.NET, an EU-supported Thematic Network. Detailed background, references, etc, may be found on the CAPE.NET website: http://CAPENET.chemeng.ucl.ac.uk and the work is now being taken forward by the EUREKAsupported CAPE-21 Project (website: http://CAPE-21.ucl.org.uk).
956 Batch & Cyclic Operations Control & Instrumentation Campaign Scheduling & Planning Process & Utilities Flowsheets Equipment Datasheets Uncertainty & Design Margins Project Evaluation & Cost Optimisation Value Engineering
Environmental Effluents & Byproducts Safety & Relief Reaction Engineering Reliability & Availability Preliminary Layouts Operating Procedures Commissioning Studies
Resilience Fluid Flow Flexibility Operability Piping Materials Utilities QA
9 Process engineering is always on the critical path and there is always severe pressure to reduce timescales, to deliver earlier and better information to other design disciplines and to reduce the number of changes (these are, of course, mutually conflicting requirements!) 9 The current approach to process development and design is largely sequential - specific issues are addressed one at a time and, as new ideas & concepts are developed and as problems are exposed, revisions are made and the sequence repeated 9 The current generation of design tools both reflect and reinforce this approach, resulting in such problems as revision loops which build up in both process and plant design as uncertainties & unknowns are resolved and inevitably lead to avalanches of late changes 9 Schedule pressures build up and eventually the revision cycles are cut short. Inevitably, errors remain, waiting for the worst possible moment to manifest themselves 9 The inevitable result of such an approach is problems with quality (ie. residual errors), cost and schedule over-runs and, potentially at least, a serious impact on the success of the business. QA & audit trails are becoming a serious problem 2. CONCURRENT ENGINEERING: A BETTER WAY? The key to a reduction in the revision cycle is to achieve a "good balanced design" (ie. one which addresses all the critical issues of manufacturing excellence, as they apply to the case in hand) as early as possible. The design then develops by a process of "broadside" evolution/refinement of all the aspects, rather than by a process of major changes as each new aspect is addressed. The objective must be to improve our ability to get it right first time, whilst supporting & encouraging innovation, and to improve our processes for early identification and implementation of necessary revisions, whilst reducing the volume of nonessential revisions. In order to achieve this * All operational issues must be tackled side-by-side, ensuring that the inevitable sequence of revisions is a process of refinement and optimisation rather than of wholesale change. . A broad range of multi-disciplinary skills must brought to bear at an early stage so that latent problems are unearthed promptly, whilst the design is still sufficiently "fluid" that modifying the design is simple and cost-effective. 9 Problems must be "designed out", not suppressed by the later addition of costly extras. Major recycling of the design can thereby be avoided and the design process becomes significantly more effective and efficient. Concurrent Engineering concepts are now well established in several sectors (eg. vehicles and aerospace design) and are seen as one of the keys to rapid protoyping or fast-track projects. In much of what has been done to date, the basic idea is to bring forward issues which typically arise in activities such as fabrication and assembly into the early design process as constraints - typically, these are "hard", based on geometry or materials/machining
957 limitations, and they are often implemented via sophisticated CAE (Computer Aided Engineering) systems. Plant engineering (ie. piping, vessels, electrical, steelwork, etc) has significant synergy with other AEC (Architecture Engineering & Construction) sectors and quite a lot is being done, once again mainly centred around CAE systems (for example, the vital function of space allocation/clash detection), taking advantage of the experience that has been accumulated and the systems which have been developed. Conceptual process design, however, is quite different from plant design in that it deals with physical flows of materials and energy and the underlying logic of the process, control & safety systems and not with physical equipment. The issues to be addressed, especially those now receiving greater emphasis, such as safety and environmental legislation, are often "softer" and more diverse (typically judgemental, rather than, for example, geometric) and are thus more difficult to codify and deal with. The present generation of CAE systems does not address such problems and we need a different set of tools & environments. Companies have been experimenting with multi-functional task forces and so-called "tiger teams": in essence, these imitate the "perfect engineer", working very closely together and sharing information, knowledge and experience. Typically, this approach has been implemented by ensuring physical proximity of key personnel (often by sharing an office) in order to improve communications and sharing of information. With recent and expected improvements in information management systems, however, the opportunity is arising to achieve this information sharing electronically: teamworking and remote working (including alliances, etc) are thereby facilitated, avoiding the costly and time-consuming relocations of key personnel. 3. IMPLICATIONS FOR CAPE Our objective is to achieve a development & design process wherein the various aspects are addressed side-byside and the design evolves in a "broadside" manner, rather than by major change. The adoption of such a concurrent approach will have major implications for 9 CAPE tools - multi-aspect tools will be required and the earlier they are applied within the development and design process (ie. the higher up the influence curve) the greater the impact will be. Not only must we try to "parallelise" existing process engineering activities but we must also try to bring some of what have always been regarded as downstream issues "up front" and try to convert them into constraints which can be applied at the very earliest stages of process development, thus eliminating potential problems before they arise, rather than fixing them later. 9 Our business processes - the ways in which we organise ourselves to use these tools. Many businesses are analysing and reorganising themselves in order, for example, to
958 facilitate more flexible working practices, team working, alliances, partnerships, remote working, ... but it has been remarked that "it is no use changing your business processes i f you can't share and manage the information". The intrinsic business tasks (in this case, the various aspects of process and plant design) and the technological capabilities required to execute them in an optimal manner are inextricably bound together and must evolve alongside one another - either, on its own, will have only a limited impact. "People issues" (ie. such things as cultural, organisational and human factors) are often the hardest to solve but a detailed discussion lies beyond scope of this paper. 9 O u r needs to share information - without comprehensive information sharing between people, CAPE & non-CAPE tools, departments, organisations, ..., such an approach cannot work. Indeed it can be argued that, for the development & design process for a complex artifact, such as a chemical process/plant, designers must live in an "information environment" - like the air we breathe, information must be available where & when needed and in the form required, so that it is not necessary to go looking for it and then to pre-process it in any way! The remainder of this paper presents two examples of how CAPE will need to adapt to the requirements of concurrent process engineering: 1. a multi-aspect tool for conceptual process development 2. management & sharing of information - a supporting infrastructure 4. TOOLS FOR CONCEPTUAL PROCESS DEVELOPMENT The design process is a complex task made up of many steps and requires input from a variety of sources. At the core of this process (in the case of process development & design) is the initial development of the process flowsheet (taken to include the definition of the main plant items and major control loops but to exclude significant equipment detail - this flowsheet is then the foundation upon which the detailed design is subsequently built using major process simulation and equipment design tools). CAPE tools for process design and development have been the focus of significant research effort for the past 30 or more years. This research has generated useful tools and techniques: for instance, the pinch method for heat exchanger network synthesis has become an indispensable part of the designer's "toolkit". Although in the past the primary emphasis has been on energy savings, more recently the problem of design has changed and has grown to include many different aspects, in response to increasing competitive pressures in the global market and to the tighter regulatory processes. For example: 9 Changing business pressures are leading to more demanding requirements in such areas as flexibility & responsiveness, multiproduct plants, speed to market, innovation and so on. Although some of these issues are currently being tackled individually, a comprehensive methodology has not been defined. 9 The growing importance of issues such as safety, waste management, controllability, and so on, means that these issues must be addressed at the earliest stages - ie. concurrently with the traditional issues of raw materials & energy consumption. 9 The information available at this early stage will be seriously incomplete and much of what is available will be uncertain. A complete & consistent mathematical description of the problem is therefore not necessarily achievable and so methods which rely on such a description (such as MINLP) may not be suitable at this stage.
959 9 Partial solutions will be valuable in guiding the developers towards a greater understanding of the problem and towards areas where some basic research (for example, into novel types of unit process) may pay handsome rewards. A longer-term objective may be a tool for process invention, rather than analysis and development? 9 The engineer is a crucial part of the process of design. The engineer brings flexibility, experience, judgement and intuition to the design process and is ultimately responsible for the decisions made. Furthermore, design is a team based activity and a synergistic relationship between the engineers and the computer tools is required for the development of truly novel and effective process designs. This has significant implications in ease of use and in the use of information visualization and analysis tools for gaining insight. 9 QA and SHE requirements are such that the rationale of the design - its history and the reasons for decisions, and so o n - must be captured. Knowledge-based decision support/capture will be required. 9 Site wide issues (utility systems, waste management, operation, layout, supply chains, and so on) may have an inordinate impact on the process design but are often not considered until the basic process structure has been fixed, leading to major recycling of the design. It is our premise that such a concurrent approach must underpin effective tools for conceptual process development. Automated design is unlikely to be feasible and is arguably not desirable, except perhaps in small and very-well-defined areas. Many of today's key issues require the engineer's direct input based on insight and previous experience. The way forward is by encouraging and supporting interaction between and amongst the engineers and the computer tools used in design, and in process engineering as a whole. 5. MANAGEMENT & SHARING OF PROCESS DESIGN INFORMATION Over the last few decades, there has been a variety of attempts to manage process engineering data, ranging from DPLS (Chiyoda), PEDB (ICI), PRODABAS (Prosys Technology), PROCEDE (Leeds University) and now ZYQAD (Aspen Technology) and AXSYS (AEA Hyprotech) and, of course, there are a variety of such systems aimed at managing data "downstream" of the P&ID. These initiatives/systems have typically focussed on interfacing CAPE tools, providing graphical interfaces to simulators, and so on. None has yet been widely adopted into routine practice, despite the very obvious benefits which could be derived from comprehensive information management and sharing within process development and design. In the authors' opinion, there are a number of issues/arguments which may help to explain why attempts to date have not been widely adopted, for example: 9 We must manage information, not just data (ie. data plus drawings, reports .... and the models, knowledge, decisions .... which underpin the design). Careful thought must also be given to recording its quality or credibility (for example, is an item of information accurate, uncertain or even "guesstimated"?). 9 We must capture the Design Rationale (how and why decisions were made): The evolution of a design involves a wide range of trade-offs, judgements and decisions: the "rationale of the process design". Without this rationale, effective re-use of designs is potentially both difficult & dangerous and the audit trail is broken. 9 We must manage and share Knowledge (a mixture of experience, databases, vendor information, earlier designs, rationale and subsequent performance and/or problems)
960 9 This information comes from a wide variety of diverse sources, not just CAPE tools, such as graphics systems, word processors, spreadsheets, ... and from the users themselves, in the form of selections, decisions, etc. 9 It is widely shared by people, teams, CAPE and non-CAPE tools, departments, organisations, ... and these interactions may be both local and remote 9 The information must be managed, not just exchanged between computer systems 9 Models & Model Information: Process models are a particular form of knowledge regarding the process and its behaviour. Many different models may be prepared during process development and design, covering a wide variety of scopes (for example, an equipment item or a whole flowsheet) and at different levels of detail. The "deep knowledge" which these models contain is a significant corporate asset and its re-use and exploitation is an important opportunity for further increases in, for example, production efficiencies, flexibility, etc. Model development has its own information and rationale too! 9 Project management issues (such as revision, release and change management, QA, audit trails) are often ignored 9 Standards: We need appropriate and effective standards which 9 support and do not obstruct evolution in both tools and working practices 9 cover the whole problem (or can be combined with other standards to do so) 9 lead to efficient and effective solutions 9 can evolve: do not anchor us in the past and lock us into old solutions/systems. Note that the information must be archived for at least as long as the plant exists and this is typically many times longer than the lifetime of the computer systems within which the information was generated. Note also the growing requirement for webcompatibility. In other words, these previous attempts have only partially addressed the actual job of the end-user and have failed to support "the process of process development & design" and the interactions between the ways in which designers do their job and the tools and infrastructures available to them. This last aspect is especially important - we are, after all, trying to provide new capabilities in order that development & design can be done in new and better ways. Future progress will depend on addressing these broader issues and perhaps the most important issues of all will arise in the multidisciplinarity and "people issues" which will be required to fully understand the end-users' requirements and then to deliver, via a series of useful intermediate steps, a solution which will be robust and adaptable to evolution in both the tools and the business processes (ie. the ways in which they are used) and will thus be able to stand the test of time and deliver the benefits which are so urgently required. 6. CONCLUSIONS The introduction of the principles of concurrent engineering into process development and design is not simple and the implications for CAPE developers are substantial. Many of the challenges are not directly concerned with calculations but with "people issues" or with business processes and with the ways in which end-users will need to interact with their design information, with its uncertainty and with the tools with which they manipulate it. Unless these "non-CAPE" issues are successfully addressed, the CAPE community will be increasingly unable to provide the capabilities and the level of support which their end-users require in order to implement new and more effective approaches to development and design.
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rights reserved.
961
A Unified Framework for the Flexibility Analysis and Design of Non-Linear Systems via Parametric Programming V. Bansal a*, J.D. Perkins a, and E.N. Pistikopoulos at aCentre for Process Systems Engineering, Imperial College, London SW7 2BY, U.K. This paper outlines a new framework based on parametric programming that unifies the solution of the various flexibility analysis and design optimization problems that arise for linear, convex and non-convex systems with deterministic or stochastic uncertainties and provides new information on the dependence of a system's flexibility on the values of the design variables. 1. I N T R O D U C T I O N All chemical plants are subject to uncertainties and variations during their design and operation. Given this fact, it is clearly important for an engineer to be able to quantify the ability of a system to be operated feasibly in the presence of uncertainties (i.e. conduct flexibility analysis) and to have systematic methods for designing systems which are both economically optimal and flexible. The past two decades have seen considerable advances in the development of such methods, both for deterministic cases, where the uncertain parameters are described through sets of lower and upper bounds on their values, and stochastic cases, where the uncertain parameters are described through probability distributions (for a full review see Bansal, 2000 [ 1]). Bansal et al. (2000) recently proposed a new approach for the flexibility analysis and design of linear systems, based on parametric programming [2]. One of the key advantages of this approach is that it provides explicit information about the dependence of a system's flexiblity on the values of the design variables. The purpose of this paper is to propose a new framework that generalises this parametric programming approach for the flexibility analysis and design of general, non-linear systems and which, for the first time, allows a unified solution approach to be used for the various flexibility analysis and design optimization problems that arise for different types of process model and uncertainty model. 2. A N E W F R A M E W O R K FOR FLEXIBILITY ANALYSIS AND DESIGN
A process system at steady-state can be described by the following mathematical model of equality and inequality constraints: hm (x, z, O, d, y)
=
gt (x, z, O, d, Y) <
O, m 6 M, O, 1CL,
(1) (2)
*Current address: Process Systems Enterprise Ltd., 107a Hammersmith Bridge Rd., London W6 9DA, U.K. tCorresponding author. Tel: +44 20 7594 6620. Fax: +44 20 7594 6606. E-mail: [email protected].
962 where x is the vector of state variables; z is the vector of control variables that can be manipulated during plant operation depending on the uncertain parameters 0; d is the vector of continuous design variables; and y is the vector of integer design variables. The new framework for flexibility analysis and design of systems is illustrated schematically in Fig. 1. For both linear and non-linear systems, the common starting point is to solve the feasibility function problem [3] shown in Fig. 1 as a multi-parametric program using specialised algorithms. For example, for a convex, non-linear model, the feasibility function problem is solved as a convex, multi-parametric non-linear program (convex mp-NLP) [4], while for a non-convex model, it is solved as a non-convex mp-NLP [5]. In all cases, the outcome of solving the multi-parametric program is a set of linear expressions for the feasibility function ~ in terms of 0, d and y, which are exact for linear systems and globally accurate within a user-specified tolerance for non-linear systems, and an associated set of regions described by linear inequalities in 0, d and y in which these solutions are optimal. For systems with deterministic parameters, the critical values of the uncertain parameters can be identified through vertex properties for linear and convex models and through the solution of further mp-LPs for non-convex models, as shown in Fig. 1. It is then possible to express the flexibility test measure Z, and the flexibility index F as explicit linear functions of the design variables. This reduces their evaluation to simple function evaluations for a given design and enables a designer to know a priori the regions in the design space for which feasible operation can be guaranteed. The critical parameter information and the expressions for Z and F can be used to formulate design optimization problems which do not require the iterative strategies used in current approaches [6]. Instead, the optimal design for a fixed degree of flexibility is determined through the solution of the single (mixed-integer) non-linear program shown in Fig. 1, while the algebraic form of the trade-off curve of cost against flexibility index can be generated explicitly by solving a single-parameter (mixed-integer) non-linear program. For systems with stochastic parameters described by any kind of continuous probability distribution, the procedures for evaluating the stochastic flexibility and the expected stochastic flexibility metrics are identical for both linear and non-linear models once the parametric feasibility function expressions have been generated. The use of these expressions is especially significant for non-linear systems because they remove all non-linearity from the intermediate optimization sub-problems, something that would not be possible using non-parametric approaches. Furthermore, by considering the sub-problems as multi-parametric linear programs, the number of problems that needs to solved compared to existing approaches (e.g. [7]) is drastically reduced since it only increases linearly with the number of uncertain parameters as opposed to exponentially, and parametric information is obtained that allows the metrics to be evaluated for any structure and design through a series of function evaluations.
3. NON-LINEAR PROCESS EXAMPLES (SEE [1] FOR MORE DETAILS) 3.1. Convex Problem Fig. 2 shows a system [8] where three plants each convert a raw material A into an intermediate B. This is mixed with a limited amount of fresh B before passing through plant 4 which produces the final product C. It is assumed that the supplies of raw materials A and B, denoted by 01 and 02, respectively, and the demand for product C, 03, are uncertain, while there are
0"Q
FEASIBILITY FUNCTION Solve
~ = rain u s.t. h (x,z,0,d,y) = 0 g (x,z,0,d,y)~< u . .
L O 0Q
.
.
FLEXIBILITY INDEX In each of the K regions:
.
.
.
.
.
LL;n
- solve the mpLP: max gtk(0,d,y) s.t. 0 in CR k 0 . .
-
ear z k (d,y), k = 1..... K2, and associated CR k
1'
OPTIMAL DESIGN WITH FIXED DEGREE OF FLEXIBILITY
,7 >
- obtain 0c'k(8k). from signs of gradients of W" w.r.t. 0 solve ~.k[0c'k(Sk),d,y]=0 L
Compare 5k (d,y) to retain lower bounds . . . .
.
Linear F k (d,y), k = 1..... KF, and associated CR k
OPTIMAL DESIGN WITH OPTIMAL DEGREE OF FLEXIBILITY Solve the single parametric program:
min
E wtC (x~,z~,0t ,d,y)
Cost (F t ) = min E wt C [xt,zi,0 t (F t ) ,d,y]
s.t.
h (x~,z~,0~ ,d,y) = 0, for all 1,
s.t.
STOCHASTIC FLEXIBILITY See Steps 2 and 3 of Algorithm 4.2 in Bansal (2000) - solve mpLPs to obtain parameter bounds as functions of d, y and lower-dimensional parameters - calculate quadrature points
L
Evaluate SF for any d and y through ,function evaluations
EXPECTED STOCHASTIC FLEXIBILITY - use parametric mformatlon from stochastic flexibility sub-problems - ESF(d) = Zs SF (d,yS). P (ys)
Z k ~ 0, for all k. "
l " !
h (x',z~,0i ( ~ ) , d , y ) = 0, for all i, g (x t,z' ,0 t (1zt ),d,y) < 0, for all i,
g (xt,za, 01 ,d,y) < 0, for all i,
~..t. 0~
- solve the mpLP: ~k = min ~5 i s.t. 0 i n C R k a n d 0 i n 0 +_ 5A0+'-
Solve the single opmmzation problem:
~it~
~7
Non-Convex
Linear and Convex
Non-Convex
Compare ~k(0c'k, d, y) to retain upper bounds
O
1..... K, and associated CR k
In each of the K regions:
i.d. critical values, 0 c'k, from signs of gradients of ~k w.r.t. 0 .
Linear ~k(0,d,y), k
FLEXIBILITY TEST Linear and Convex
.
as an: - mpLP (linear h and g) - convex mpNLP (linear h, convex g) - non-convex mpNLP (non-convex h ~ d g)
.
.
.
F k >/ F t, for all k. . . . ~
.......
"
Iterative Strategy Not Required
Algebraic Form of Cost vs. Flexibility Trade-Off Curve
L
Evaluate ESF for any d .... through function evaluations
964
-
1
Sp~les
f4~[
2
F6- Fs-! Fl~ 4 - SP~Bcies-
3
F7
1 Fl'_ JSpecies
Figure 2. Convex Model Example.
Figure 3. Flexibility Index Solutions.
three design variables, d = [dl,d2,d3] r, which correspond to the processing capacities of plants 1, 2 and 3. Solving the feasibility function problem as a convex mp-NLP with a tolerance of ~ -- 0.1 leads to seven feasibility function expressions. These are shown below (without their respective regions of optimality): ~1 (0, d)
=
-0.162401 - 0.333702 + 0.370803 - 0.0586dl - 0.0744d2 - 0.4931,
~t2 (0, d)
--
-0.248702 + 0.276403 - 0.1729dl - 0.1850d2 - 0.1170d3 - 0.2572,
~3 (0,d)
=
-0.221701 - 0 . 3 6 8 7 0 2 + 0 . 4 0 9 6 0 3 - 1.0537,
~q/4 (0, d)
--
--0.172901 -- 0.356802 + 0.396403 -- 0.0241dl - 0.0497d2 - 1.1949,
~t5 (0,d)
--
-0.262802 +0.291903 - 0 . 1 6 9 4 d l - 0 . 1 5 9 7 d 2 - 0 . 1 1 6 2 d 3 - 0 . 7 2 9 3 ,
~q/6 (0, d)
--
-0.131001 - 0.329602 + 0.366203 - 0.0789dl - 0.0943d2 - 0.8535,
~t7 (0, d)
--
-0.197701 - 0.352802 + 0.391903 - 0.0055dl - 0.0521d2 - 0.7372.
In all cases the critical directions for the uncertain parameters are towards the lower bounds for 01 and 02, and towards the upper bound for 03. One expression for the flexibility test measure is then obtained which is independent of the values of the design variables, and is valid over the whole range of d: %(d)
-
V3(0c'3,d) = 2.2963.
Since ~ > 0, this indicates that there are no plant capacities within the given ranges for which the system can be operated feasibly over the whole ranges of the uncertain supplies and demand, even with the manipulation of the control variables during operation. Four expressions are obtained for the flexibility index, corresponding to ~ l = 0, ~3 = 0, ~4 = 0 and ~7 __ 0. These are all independent of the processing capacity of plant 3, as seen in Fig. 3, which graphically illustrates the parametric expressions in the design space.
965 2 kW/K
1.2 kW/K ~< 0 ~< 1.6 kW/K
TI
563
K
323 K
Figure 4. Non-Convex Model Example.
Figure 5. ~ vs. O.
3.2. Non-Convex Problem Fig. 4 shows a heat exchanger network [9] where the heat capacity flowrate, 0, of one of the hot streams is uncertain. Solving the feasibility function problem for this example as a nonconvex mp-NLP with a tolerance of e = 0.05 leads to a number of linear expressions for ~(0). The resulting feasibility function values lead to a predicted flexibility test measure, Z = 0.7549 at the non-vertex point, 0 - 1.3831. Note that since ~ > 0 in the whole range of 0 considered, the flexibility index is zero. For this particular example, the analytical solution for ~ can be derived. Fig. 5 plots the predicted feasibility function from the application of the parametric programming framework against 0, and compares it with the actual solution. It can be seen that the parametric expressions do indeed over-estimate the real-solution to within e - 0.05 for the whole range of 0. The flexibility test measure is also over-estimated within this tolerance since the analytical value is % -- 0.7077 occurring at 0 - 1.3982. 4. E X T E N S I O N TO M U L T I - P U R P O S E P R O C E S S E S A multi-purpose system such as an in-line blending system, used to make a range of product recipes, can be described by the following model [ 10]:
hm(x,z, yz,O,d,y)
:
0,
m C M,
(3)
g l ( x , z , Yz,0,d,Y)
<
0,
lCL.
(4)
The only difference between (3) and (4) and the equalities and inequalities of the systems considered thus far in this paper, (1) and (2), is the presence of the variables Yz. These binary variables correspond to discrete modes of operation which can be altered depending on the values of the uncertain parameters 0. If flexibility analysis of such a system is to be conducted, the starting feasibility function problem in the framework of Fig. 1 corresponds to a multiparametric, mixed-integer linear program (mp-MILP), for which an algorithm such as that of [ 11 ] can be used to obtain linear parametric expressions for the feasibility functions. Once these expressions have been obtained the various flexibility analysis and design problems can tack-
966 led using the framework in Fig. 1 in exactly the same manner as that described in Section 2. Examples of this kind can be found in [12].
5. CONCLUDING REMARKS A new framework has been described for flexibility analysis and design. This provides a unified solution approach for different types of process model (linear, convex, non-convex, discrete controls) and different types of uncertainty (deterministic, stochastic) and allows explicit information to be obtained on the dependence of the flexibility of a general, non-linear system on the values of the uncertain parameters and the design variables.
REFERENCES 1. V. Bansal, Analysis, Design and Control Optimization of Process Systems under Uncertainty, PhD Thesis, University of London (2000). 2. V. Bansal, J.D. Perkins and E.N. Pistikopoulos, Flexibility Analysis and Design of Linear Systems by Parametric Programming, AIChE. J. 46 (2000) 335. 3. K.P. Halemane and I.E. Grossmann, Optimal Process Design under Uncertainty, AIChE. J. 29 (1983) 425. 4. V. Dua and E.N. Pistikopoulos, Algorithms for the Solution of Multiparametric Mixed Integer Nonlinear Optimization Problems, Ind. Eng. Chem. Res. 38 (1999) 3976. 5. V. Dua, K.P. Papalexandri and E.N. Pistikopoulos, A Parametric Mixed-Integer Global Optimization Framework for the Solution of Process Engineering Problems under Uncertainty, Comput. Chem. Eng. 23 (1999) S 19. 6. I.E. Grossmann and M. Morari, Operability, Resiliency and Flexibility - Process Design Objectives for a Changing World, In: Proc. 2nd Int'l Conf. on FOCAPD (A.W. Westerberg and H.H. Chien, eds), CACHE (1983) 931. 7. D.A. Straub and I.E. Grossmann, Integrated Stochastic Metric of Flexibility for Systems with Discrete State and Continuous Parameter Uncertainties, Comput. Chem. Eng. 14 (1990) 967. 8. G.R. Kocis and I.E. Grossmann, Relaxation Strategy for the Structural Optimization of Process Flow Sheets, Ind. Eng. Chem. Res. 26 (1987) 1869. 9. I.E. Grossmann and C.A. Floudas, Active Constraint Strategy for Flexibility Analysis in Chemical Processes, Comput. Chem. Eng. 11 (1987) 675. 10. E.N. Pistikopoulos, T.A. Mazzuchi, K.D. Maranas and T.V. Thomaidis, Simultaneous Assessment of Flexibility, Reliability & Availability for In-Line Blending Systems: A Unified Framework for Analysis and Retrofit Design, In: PSE '91 Proceedings, Vol. I. 11. V. Dua and E.N. Pistikopoulos, An Algorithm for the Solution of Multiparametric Mixed Integer Linear Programming Problems, Annals Oper. Res. In Press (2000). 12. A. Salzano, Flexibility Analysis of Mixed-Integer Linear Systems via Parametric Programming, Internal Report, Imperial College, University of London (2000).
European Symposiumon ComputerAidedProcess Engineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rights reserved.
967
Open software architecture for numerical solvers : design, implementation and validation Jean-Pierre Belaud a, Karim Alloula a, Jean-Marc Le Lann a and Xavier Joulia a Laboratoire de G6nie Chimique (LGC, UMR CNRS 5503), I N P T - ENSIGCT, 18 Chemin de la loge, F-31078 Toulouse cedex 4, France; [email protected]
a
Open software architectures are the way forward for the next generation of CAPE tools. The CAPE-OPEN (CO) standard achieves true plug and play of industry business components in enterprise software. This paper deals with the development of numerical solvers for application within the scope of an open architecture framework. We will first discuss the CO standard specification of numerical solvers. Then, we will give an overview on Numerical Services Provider (NSP) software. Finally, some process applications using the services from NSP will be considered. 1. I N T R O D U C T I O N Traditional simulation environments are closed monolithic systems; and the resulting bottlenecks in interoperability, reuse and innovation have led to the CAPE-OPEN and GLOBAL-CAPE-OPEN projects*. These projects represent a collaboration between the chemical and oil industries, academics, and software suppliers; with a view to defining a standard for component-based approach to process simulation. The resulting standard, CAPEOPEN [ 1] [2], is now widely disseminated. This standard distinguishes two kinds of software components: Process Modelling Components (PMC) and Process Modelling Environments (PME), the latter making use of the services provided by the PMC. Typically the PME are environments that support the construction of a process model and that allow the end-user to perform a variety of different tasks, such as process simulation or optimisation [3]. The distinction between these two components is not readily apparent; and furthermore, it is worth noting that in the near it will be possible to assemble any number of PMC to deal with a specific task. The current version 0.9.3 of the standard defines several PMC including Thermodynamic and Physical Properties, Unit Operations and Numerical Solvers. From an analysis point of view these are represented by the packages Thrm, Unit and Numr. Each package encloses a set of interfaces. Clearly as there is a wide range of materials, unit operations and solvers currently used by the process industries, these generic packages can be further subdivided into * CAPE-OPEN and Global CAPE-OPEN are funded by the European Community under the Industrial and Materials Technologies Programme (Brite-EuRam III), under contracts BRPR CT96-0293 and BPR-CT98-9005. In addition, Global CAPE-OPEN follows the Intelligent Manufacturing Systems initiative promoting collaboration between six intemational regions.
968 more specific package structures. In addition to these business packages, the standard introduces two other additional packages: Base, which describes the elementary types (such as CapeString, CapeDate, CapeURL, ...) and Common, which defines the CO common interfaces such as Identification, Parameter and Error Handling. This set of specifications incorporates the CO architecture which is based on the distributed component (heterogeneous) system and the object-oriented paradigm. The involved technologies are the UML notation [4], the CORBA [5] and (D)COM [6] middleware, the CO Work Process and object-oriented languages.
2. CO NUMERICAL SOLVERS: Analysis, design and specification 2.1 Analysis The Numr package is subdivided into five packages: 9 The Solver package focuses on the solution algorithms that are necessary for carrying out steady-state and dynamic simulation of lumped systems. In particular, this includes algorithms for the solution of large and sparse systems of linear algebraic equations (LAE), non linear algebraic equations (NLE) and mixed differential and algebraic equations (DAE). 9 The Eso package contains the Equations Set Object (ESO) concept which is an abstraction representing a square or rectangular set of equations. These equations define the physical behaviour of the process. An ESO is a purely continuous mathematical description: the equations remain the same for all the possible values of the variables. 9 The Model package introduces the Model object to embody the general mathematical description of a physical system. The fundamental building block employed for this purpose is a set of ESO. However, many physical systems also involve discontinuities, and this fact must be reflected in their mathematical description. Accordingly, a Model may additionally encompass one or more State Transition Networks (STN) [7]. These are formal descriptions of discontinuous phenomena. 9 The Utility package contains the public parameter concept which allows some customisation of each Solver component. 9 The Smst package characterises the flowsheet solvers that analyse the process flowsheet in order to determine a suitable calculation sequence. This specification is only dedicated to sequential modular simulation systems. These well-established operations are partitioning, ordering, tearing and sequencing [8]. From these five packages, the utility package being basic, four components are set up. The following package diagram details the various dependencies between them (the grey packages). The black arrows within the picture display the relations that are in the CO scope. The standard defines the services proposed by the Solver and Smst components. Currently the way one builds an Eso or a Model component, or access them, is not standardised by CO. This task is set to the flowsheeting tool suppliers. However the publication of services to the Solver component is CO standardised. So, software suppliers, industrials and academics can provide CO compliant Solver or Smst components, or use those CO compliant PMC. In the latter case, they may have to adapt some interfaces of their legacy codes.
969
2.2 Design The Solver package which is responsible for driving the resolution of the problem using all the information from the Model and the Eso contains the five following interfaces: 9 ICapeNumericSolverManager acts as a factory and creates any kind of solver for a specific ESO from a specific type, either linear, non linear or differential. 9 ICapeNumericSolver is the base interface of the solver hierarchy and so defines general facilities for identifying the various algorithmic parameters that are recognised by a numerical solver, for altering their values if necessary. 9 ICapeNumericLASolver defines facilities which are specific to solvers of LAE systems. No specific methods have been defined for this kind of solver. It is assumed that the Solve method gets the A matrix and the b vector of the A. x = b system using the already defined methods. 9 ICapeNumericNLASolver defines facilities which are specific to solvers of NLAE systems. It defines methods which allow to obtain and set convergence tolerance and the number of iterations. 9 ICapeNumericDAESolver defines facilities which are specific to solvers of DAE systems. It defines methods which allow to obtain and set relative and absolute tolerance. The next diagram represents the interface diagram of the Solver package.
970
2.3 Specification The packages described in 2.1 are translated in module for the CORBA system. Hence the specification of CO Numerical Solvers is enclosed in the Numr module within the CAPEOPEN library version 0.9.3. Interface Diagramof the SolverPackage
[~
|
] <> iCapeNumericSolverManagerI CreateSolver0
<> ICapeNumericSolver manages
/1
CapeSolverType
LA NLA
GetParameterList0setParameter0 ~->/
1 GetSolution0S~Interface0||| "'n> Destroy() / /~
+lowerBound/I
/~upperBound
~/ CapePublic=,Paramet .y) er(.omI +name~1~CapeStriBe.ng(.e)om
owns
~176
I§ description
+currentValue~ ~ +defautValue CapeVariant (fromBase)
DAE
<> ICapeNumericLASolver
<> ICapeNumericNLASolver SetCvgTolerance0 GetCvgTolerance0 SetMaxlterations0 GetMaxlterations0 DoNIteration0
<> ICapeNumericDAESolver SetRelTolerance0 GetRelTolerance0 SetAbsTolerance0 GetAbsTolerance0 AdvanceToNextEvent(
)
3. NUMERICAL SERVICES PROVIDER" Implementation Our Solver component compliant with CO version 0.9.3 is called NSP, Numerical Services Provider. According to the component scheme introduced in 2.1, this software realises the Solver package depending on the Model and Eso packages. Following the CO architecture for CORBA system, NSP acts as a numerical server through the Object Request Broker, employs the Identification Common Interface and follows the CO Error Handling strategy. It stands for the second stage (the business model) within the three-stage architecture. Obviously, this stage is a separate process that can be used on a single machine, or across the network within an internet-based enterprise business system. The NSP application combines Java and C/C++ codes as well as Fortran 77 legacy codes thanks to the wrapping technique. It supplies the LAE, NLAE and DAE objects, jointly to the configuration parameter objects. 9 The linear algebraic solver allows to solve the system A. x = b and is the result of the wrapping of the UMFPACK solver [9]. It is really efficient and offers a large choice of configuration.
971 9 The non linear algebraic solver deals with the system F(x)= 0. It relies on the Newton~F ~ k~ Raphson algorithm and profits from the linear solver for solving -~(ox j=-F(xk). 9 The differential algebraic equation solver manages the system
F t,x,-~ - 0 . It wraps
the integrator DISCo [10]. Its strategy is based on the Gear method with variable order and step. About the NSP design there are classes which implement the CO interfaces: SolverManager, Solver, LASolver, NLASolver, DAESolver, Identification, CapeSolverType and CapePublicParameter. These objects are distributed thanks to CORBA technical classes which manage the communication and put in place a delegation implementation through the tie mechanism. In order to decouple these classes fixed by the CO specification from the semantic implementation classes, the bridge design pattern is applied. Then we have in parallel the CO classification of Solver class (this "abstraction" side defines the CO higher-level methods) and our own classification of SolverImpl class (this "implementation" side provides only primitive methods). The SolverImpl hierarchy keeps the same base classes structure of Solver hierarchy in order to accelerate the development but a more detailed classification of solver could be set up. This bridging allows us to set up our own design. It decouples the objects implementing the CO interfaces and our semantic objects and improves extensibility, independently ensuring the extensions of the two hierarchies. Keeping the same approach, the model side and the solving side are fully disconnected thanks to the SystemMap concept. The resulting SystemMap objects are in charge of the appropriateness and adaptation of models into the solver formats. They manage the storage of matrices and the variables (unknowns x and independent variable t). In consequence the SolverImpl object communicates with the Model object only through a SystemMap object. The main advantage is that the SystemMap objects factorise the code guaranteeing the link with the Model and Eso components. The SolverImpl object uses an ErrorsManagement object in order to generate a CO error from an error specific to a solving algorithm. It can also produce a log file, aiming at administrating NSP. The NSP application acts as a real framework. It is a set of co-operating classes that make up a reusable design for disseminating any solver algorithm through the CO standard. It dictates the overall structure taking advantage of the CO architecture. Especially through the bridge and the map concepts, the resolution method developer (from legacy codes or not) inserts his design within the SolverImpl hierarchy and needs only to interact with the NSP objects. The CO standard life cycling has no direct effect on his new and former objects. Moreover the comprehension of the CO Numerical Solvers specification would be not required. The developer can concentrate on the specifics of his business. 4. APPLICATION USING THE NSP SOFTWARE COMPONENT: Validation
A PME application which is compliant with the CO Numerical Solvers uses the NSP in order to solve its process mathematical model. It relies on the CO architecture for CORBA
972 system. In fact it corresponds to the first stage and has to supply the Model and the Eso components to the NSP. This basic PME application illustrates the NSP component through three test cases. 9 A linear system has been generated and solved. For all those values the accuracy criterion has been satisfied. 9 An isothermal flash model has been calculated using the NSP services in order to solve the resulting non linear system. 9 The Raleigh distillation has been performed using the NSP services in order to solve its differential and algebraic equation system. 5. CONCLUSIONS The current CO Numerical Solvers specification is introduced. The first complete implementation of this standard leads to the NSP software component which can provide numerical services to any CO compliant software. Its realisation validates not only the CAPE business interfaces but also the overall architecture for a CO process simulation. GLOSSARY UML: Unified Modelling Language DCOM: Distributed Component Object Model NLE: Non Linear Equations ESO: Equations Set Object PMC: Process Modelling Components
NSP: Numerical Services Provider LAE: Linear Algebraic Equations DAE: Differential Algebraic Equations STN: State transition Networks PME: Process Modelling Environments
REFERENCES
1. CAPE-OPEN standard: www.global-cape-open.org 2. B. L. Braunschweig, C. C. Pantelides, H. I. Britt and S. Sama, Open software architectures for process modelling: current status and futures perspectives, FOCAPD, 1999. 3. C.C. Pantelides and H. I. Britt, Multipurpose Process Modeling Environments. In L.T. Biegler and M.F. Doherty (Eds.), Proc. Conf. on Foundations of Computer-Aided Process Design '94. CACHE Publications, Austin, Texas. pp. 128-141, 1995. 4. J. Rumbaugh, I. Jacobsen and G. Booch, Unified Modeling Language Reference Manual, Addison Wesley, 1997. 5. Object Management Group's CORBA/IIOP: www.omg.org 6. Microsoft's COM: www.microsoft.com 7. M. Avraam, N. Shah and C.C. Pantelides, Modelling and Optimisation of General Hybrid Systems in the Continuous Time Domain, Comput. Chem. Engng., 22S, $221-$228, 1998. 8. A.W Westerberg, , H.P. Hutchinson, R.L. Motard and P. Winter, Process Flowsheeting. Cambridge University Press, Cambridge, U.K, 1979. 9. I.S. Duff, R.G. Grimes and J.G. Lewis, User's guide for the Harwell-Boeing sparse matrix collection, Release I, Technical report, Rutherford Appleton Laboratory, 1992. 10. A. Sargousse, Noyau num6rique orient6-objet d6di6 /t la simulation des syst6mes dynamiques hybrides, PhD thesis, INPT, 1999.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
973
Multiplicity and stability of CSTR-Separator-Recycle Systems Costin S. Bildea, Alexandre C. Dimian and Piet D. Iedema University of Amsterdam, Nieuwe Achtergracht 166, 1018 WV Amsterdam, The Netherlands The nonlinear behaviour of several CSTR-Separator-Recycle systems is studied by rigorous application of the singularity theory. The plant Damkohler number (Da) is introduced as a parameter of the dimensionless balance equations. A feasible operating point exist iff Da > Da er, where the critical value Da cr corresponds to a singular point of the model equations. For one-reactant nth-order isothermal reaction, the feasible operating point is unique and stable. In other cases (second-order isothermal reaction involving two reactants; first-order adiabatic reaction), multiple steady states exist, the range of achievable conversion being limited by the instability of the low-conversion state. 1. INTRODUCTION The nonlinear behaviour of stand-alone chemical reactors, including state multiplicity, isolated solutions, and sustained oscillations, have been demonstrated by a large number of articles 1~ In all cases, the source of nonlinearity was the dependence of reaction rate on temperature, coupled with some form of energy feedback. Recently 5, we have shown that coupling chemical reactors and separation units through material recycles is an additional source of nonlinearity. This work extends previous results by including stability assessment and considering non-isothermal reactor operation. 2. DYNAMIC MODELLING For the chemical reactor, a dynamic model can be derived based on unsteady mass and energy balance. The model contains a few nonlinear differential equations, being amenable to analytic or numerical investigation. In the case of an isothermal CSTR, the mass balance for the k th chemical species has the following dimensionless form:
Z(fj'Zk,j)--fout'Zk--Da" HZ,
dZk = dt j~inlet
streams
(1)
i~reactants
where t, f a n d z are dimensionless time, flow rates and concentration, respectively. For reactors in recycle systems (for example, Figure 1), it is convenient to use the plant Damkohler number 5 which includes the reaction constant (k), reactor holdup (V), feed flow rate (F0) and concentration of the key component (CA,0): Da = kc~,~ V / F o .This definition is different from the classical one, which uses the flow rate and composition at reactor inlet (F1 and CA,l) as reference values. Dynamic modelling of the separation units is more difficult. Even a simplified dynamic distillation model might contain about one hundred differential equations. For such model, analysis and generalization of the results is not easy. For this reason, we consider a separation section where all separation units are lumped. We assume that the separation section is under
974 local control, which is achieved by manipulating internal flow rates or heat duties. This way, the composition of the outlet streams is kept constant. Then, changing the flow rate or composition of the inlet streams is reflected by a gradual change of the flow rate of outlet streams. When the recycle streams are considered and complete reactant recovery is assumed, a simple model describing the dynamic behaviour consists of a first-order differential equation: ~'kZ~
Tdf~
=
fin Zin,k -- Lut,k Zout,k
(2)
where v, f and z are dimensionless time constant, flow rate and concentration, respectively. The index k refers to the k th component, Zout,k is fixed, and fn, Zin,k are determined by the reactor performance.
3. ISOTHERMAL CSTR- S E P A R A T O R - RECYCLE 3.1. One reactant, nth-order reaction This section considers a n th-order reaction A --> B, taking place in an isothermal CSTRReactor-Separator system (Figure 1). The model has two steady state solutions: (ZA,2,f3)~ =(ZA,3,oO) and (zA,2,f3): = , x / ~ , z A , 3 . , ~ - ~ _ 1 The first solution is unfeasible, corresponding to infinite reactant accumulation. The
Da
(4) second solution is feasible (positive recycle flow rate) when z~,3 9 > 1 Figure 3 presents the conversion of the two solutions vs. Damkohler number. The stability of a steady state changes when one real eigenvalue or a pair of complex-conjugate eigenvalues crosses the imaginary axis, corresponding to one of the following conditions, respectively:
det(Js.s,2)=n(Za,3""~Sa-1)=O
(5)
~ffs " Z A,3
(6)
trace(Js.s, 2)= 0, or, equivalently, r, = -
where Js.s. is the Jacobian matrix evaluated at steady state Eqs. 5 and 6 show that the operating point satisfying Eq. 4 is stable (solid line in Figure 2). A, ZA,3
05
l_)i
ZA,3
X
o -05
Z B,4 = 1
-1 1
v_
Figure 1. One-reactant, nth-order reaction
2 D a
3
4
5
( z A3) n
Figure 2. Stability of steady solutions in CSTR-SeparatorRecycle system
9'/5
A fee_~._._~,~ ~-~1
A recycle , ZA3) ,
B recycle
{CS 1
I I
1
A recycle (f3,ZA,3) _
tcs
/
~fe~d~..,,>
_
2
(&oo..)
=
A , ZA,2, ZB,2
SP=fv,~,s
e~, ~ f ~ ~ f5, ZB,5)
Figure 3. Two-reactants, second-order reaction in CSTR-Separator-Recycle system
3.2. Two-reactants, second-order reaction This section analyses the second order reaction A + B ~ P , taking place in an isothermal CSTR-Separator-Recycle system. When the reactants are completely recycled, feasible operation is possible only if the ratio of reactants in the feed matches exactly the reaction stoichiometry. For this reason, only one reactant feed may be on flow control (fA,0=l), while the feed flow rate of the second reactant (fB,0) must be used to control its inventory. Two possible control structures 6 are presented in Figure 3: one recycle stream or the reactor effluent, respectively, on flow control. The dynamic model includes reactor and separation equations, as well as the relation for the feed flow rate of second component imposed by the control structure: CS 1:fB,0 = fRec,a -- f5
(7)
CS 2:fa,0 = fz --0 + f3 + fs - D a ' z a , 2 " z,,2)
(8)
In both cases, two steady state solutions are possible. They have complex analytical expressions, not reproduced here. The conversion of the key component, Xa, is presented in Figure 4 for the case of high purity separation (ZA,3 = ZB,5 = 1 ). For given values of the fixed flow r a t e VRec,B orj~) and separation performance (ZA,3 and ZB,5), two feasible steady states exist when the plant Damkohler number exceeds the critical value corresponding to the turning point of the D a - X A diagram. 08-
cs l]
CS 21 0.8
06-
0.6
04-
0.4
i
02- ~.=~o~,\,,,. " .... 0
[,
0
'
5
-"--:-~-..=" U..',~;...... ;..:..,;~, . . . . . . ; ....................
10 D I 5
i
0.2
29
2
0
o
,
,
5
10
D
,
,
15
20
25
Figure 4. Multiple steady states of two-reactants, second-order reaction in isothermal
CSTR-Separator-Recycle system.
The critical value D a cr represents a limit point 7 of the balance equations. Then, the following feasibility conditions (existence of steady states) can be derived:
976
CS 1" Da > Da ~r = 4
Dacr fRec,B ZA,3 9ZB,5(fRo~,B__ 1)' and CS 2: Da > -
4 ~ ZA,3"ZB,
( f2 ) 2 5
(9)
f2-1
We emphasize that a recycle system designed near Da cr (an optimisation procedure is likely to suggest this!) can suffer from serious operability problems. If the reaction kinetics is over-estimated, or the feed flow rate deviates from the nominal design value, the operating point falls at the left of the turning point in the D a - XA map, in the region where no steady state exists. In this case, infinite reactant accumulation occurs, and the plant has to be shut down. This situation was observed by dynamic simulation using a more detailed model 6. In Figure 4, the lower steady state is unstable and has an unusual behaviour: larger reactor gives lower conversion. The instability can be proven based on steady state considerations only, showing that the analogue of CSTR's slope condition is not fulfilled. Note that the lowconversion state is closed-loop unstable. Moreover, it is independent on the dynamic separation model 9Because this instability cannot be removed by control, operation is possible when the following requirements, necessary but not sufficient, are met: CS 1" X A >
1
and CS 2: X A >
ZB,5 9fR~.B + 1 '
2
(10)
2 + ZA,3"(f2 --1)
A dynamic model is needed to prove the stability of the upper solution branch, which is not guaranteed by Eq. 10. More precisely, differences in the dynamics of reactants' recycle might lead to oscillatory behaviour, because of the violation of a dynamic stability condition. Because analytical computation of the eigenvalues of the dynamic model is difficult, we had to recourse to numerical methods. For the simple separation model presented here and a wide range of parameter values, time-dependent solutions converged to the upper steady state. Numerical computation revealed negative eigenvalues of the linearised model. Direct methods for computation of Hopf bifurcation points failed to find a solution. Similar results were obtained for other dynamic separation models (for example, series of first-order elements, or time constants dependent on the feed flow rate). Although all these results indicate that the upper steady state is stable, we do not exclude, for other dynamic models of the separation section, the possibility of oscillatory behaviour on the high-conversion branch of Figure 4. The lower limit of the conversion achievable at a stable operating point decreases as the flow rates increase (Eq. 10) and Da cr increases (Eq. 9). There is, however, a lower limit of the reactor size that can be used in a recycle system, given by D a . ZA,3 "ZB,5 > 4. 4. A D I A B A T I C C S T R - S E P A R A T O R - RECYCLE
Occurrence of multiple or unstable steady states of chemical reactors can be explained by the dependence of reaction rate on temperature, coupled with some sort of heat feedback. These undesired phenomena are avoided in practice by a suitable control system, for example manipulating coolant flow rate to keep constant reactor temperature. However, many industrial reactors are operated adiabatically. In this case, although the inlet temperature is controlled, state multiplicity and instability are possible. This section considers a first-order reaction, taking place in an adiabatic CSTR (Figure 1). The steady state dimensionless model of a stand-alone reactor is2: -X+Da*.O-X).ex
p I+7.B,.x)=O
(11)
977 where the conversion X is the state variable. Activation energy (y), Damkohler number (Da*) and adiabatic temperature rise (B*) are dimensionless model parameters. Eq. 11 admits three solutions for yB* > 4(1 +B*). Because usually B* has small values, an approximate criteria for state unicity is yB* < 4. When the reactor is coupled with separation throut~h recycle, the (reactor) Damkohler number (Da*) and (reactor) adiabatic temperature rise (B) depend on the state variable X. For this reason, they have to be replaced by dimensionless parameters containing only independent variables. This can be achieved using the flow rate and concentration at plant inlet as reference values in dimensionless parameters. Hence, we introduce the plant Damkohler number (Da) and plant adiabatic temperature rise (B). Then, the following equations can be derived:
Da* = Da
X . ZA,3 . I_XI(I_zA,3) B*= B zA'3 ' l _ y . (I_ZA,3)
(12)
The steady-state model of the adiabatic CSTR-Separator-Recycle system is obtained by combining Eqs. 11 and 12. If the feed and recycle streams have the same concentration (ZA,3 = 1 ), the model reduces to:
f (X, Da, B, 7") = - X + Da . X . (1 - X). exp
B X
In addition to the trivial solution X = 0, Eq. 13 admits one or three solutions. An analytical expression is not possible, but the defining condition for a limit point singularity v, g = Og/OX = 0, has two solutions (X, Da), indicating an S-shaped X vs. Da diagram and at most three steady states. The cusp singularity 7, where the two limit points disappear, is located in the unfeasible parameter range:
(X, Da, B ) = ( - 2
y+2 y+4 y+4 / (14) ZA,aY-- 2?' --4' ZA.aYexP0' + 2)' 4 A part of the S-shaped X vs. Da diagram corresponds to negative conversion, which has no physical meaning. Hence, the number of feasible steady states changes when one of the limit points enters the region of positive conversion. This is a boundary limit point 7. Its defining condition, g = Og/OX = X = 0, has the solution:
(X, Da, B)= (O,1/z Aa,3,1/7")
(15)
Compared with the stand alone adiabatic CSTR (TB* <4), the range of adiabatic temperature rise for which a unique steady state is obtained is four times smaller (TB < 1). We also note that the point defined by Eq. 15 satisfies the defining condition for an isola singularity7,g =Og/OX =Og/OB = 0 , when the adiabatic temperature rise is considered as bifurcation parameter. First diagram of Figure 5 presents conversion vs. plant Damkohler number, for different values of the adiabatic temperature rise. When fly > 1, zero, two or one steady state exist for
Da < Da r Da cr < Da < 1/ZA,3, and Da > 1/Zh3, respectively. When fly < 1, a unique steady state exists for Da > 1/ZA,3 . Second diagram of Figure 5 shows the conversion vs. adiabatic temperature rise, for different values of the plant Damkohler number. When Da > 1/zA,3 , one
978 r=25 I
1
~
08
"
0.6 X 0.4
X
iiI
,
///o.,;",0
// ',,,,
02 "'" ":':'-: "[:'" ." ", 0
0.5
Da
B=0 04=11), 1
,
1.5
-0.1
-0.05
o
',,, o5
"4
0
0.05 B
".-
.....
"
0 1
0.15
. . . .
0.2
0.25
Figure 5. Bifurcation diagrams of adiabatic CSTR-Separator-Recycle system
steady state exists for any B value. When D a < 1/ZA.3 , there is a minimum value B cr of the adiabatic temperature rise for which two steady states exist. The critical values D a cr and B ~ can be obtained by finding the limit point singularity at fixed B or Da, respectively. When multiple steady states exist, the low-conversion one is unstable. If the simple dynamic model presented above describes the separation section, the high-conversion steady state is stable. When low conversion is desired (for example, due to selectivity requirements) stable operation requires that the adiabatic temperature rise does not exceed the corresponding critical value. The conservative unicity condition zB < 1 ensures also stability. Similar with the case discussed in Section 3.2., operation near the limit points is not recommended. 4. CONCLUSIONS Interaction between Reaction and Separation through material recycles generates nonlinear phenomena. In contrast with stand-alone reactors, a zero-conversion, infinite-recycle steady state always exists. This state is stable if the reactor volume is below a critical value, for a given feed flow rate and reaction kinetics. New steady states appear when the reactor volume exceeds the critical value. Two mechanisms are presented: a) occurrence of one stable steady state at a transcritical bifurcation, when the infinite-recycle state loses stability; b) occurrence of two steady states at a fold bifurcation. In this case, only one of the new states is stable. The designer has to avoid the unstable solution, which puts a lower limit on the achievable conversion. We argue that designs close to the critical points are dangerous, as disturbances or design parameter uncertainty may shift the operating point in the region where no feasible state exists. REFERENCES 1. Adoimatis, R.A. and Cinar, A., 1988, Chem. Eng. Sci, 43,887-898. 2. Balakotaiah, V. and Luss, D., 1983, Chem. Eng. Sci, 38, 1709-1721. 3. Subramanian, S. and Balakotaiah, V., 1996, Chem. Eng. Sci., 51, 401-421. 4. Uppal, A., Ray, W.H. and Poore, A.B., 1976, Chem. Eng. Sci., 31,205-214. 5. Bildea, C.S., Dimian, A.D. and Iedema, P.D., 2000, Comp. Chem. Eng., 2-7, 209-215. 6. Luyben, M.L and Luyben, W.L., 1997, Essentials of Process Control, McGraw-Hill, New York. 7. Golubitsky, M. and Schaeffer, D. 1985, Singularities and groups in bifurcation theory, Springer-Verlag, New York.
European Symposiumon ComputerAidedProcessEngineering- 11 R. Ganiand S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rightsreserved.
979
A New Multiparametric Mixed-Integer Quadratic Programming Algorithm V. Dua, N. A. Bozinis and E. N. Pistikopoulos* Department of Chemical Engineering, Centre for Process Systems Engineering, Imperial College, London SW7 2BY, U.K.
A number of important engineering problems, such as mixed logical dynamical systems, which simultaneously involve process dynamics and logical constraints can be reformulated as multi-parametric mixed integer quadratic programs (mp-MIQP) by treating the control variables as the optimization variables and the state variables as parameters - the quadratic terms appear only in the objective function (typically associated with minimization of least square errors). This paper presents an algorithm for the solution of mp-MIQPs. The solution of mp-MIQPs is given by an enclosure of the nonlinear profiles of the control variables as a function of the state variables. The optimal solution is then obtained, on-line, by evaluating the profiles for a given value of the state variables and then choosing the minimum of the values corresponding to different profiles. 1. I N T R O D U C T I O N Process engineering problems usually involve uncertain parameters. These uncertain parameters arise, for example, due to variations in demand and supply for a planning problem [1]. Such problems can be addressed by using the fundamentals of parametric programming. The key advantage of using parametric programming to address process engineering problems under uncertainty is that a complete map of optimal solutions is obtained as a function of parameters, without exhaustively enumerating the entire space of these varying parameters [2-8]. On-line control problems can also be reformulated as parametric programming problems by treating control variables as optimization variables and state-variables as parameters so as to obtain the control variables as a function of state variables [9]. The on-line control problem therefore reduces to a function evaluation problem since for all the given states of the plant, the optimal control actions are available as a function of the state variables. For the case when control problems also involve logical constraints and/or discrete choices, such as the startup and shutdown of certain equipments under certain operating conditions, binary variables are introduced to formulate the logical constraints [ 11-13]. Such systems, which simultaneously involve logic, dynamics and operating constraints are known as mixed logical dynamical (MLD) systems and can be mathematically formulated as mixed-integer programs [14]. Similar to the concepts of formulating the on-line control problems as multi-parametric *Corresponding author. E-maih [email protected], Tel.:+44 (0) 20 7594 6620, Fax: +44 (0) 20 7594 6606. The authors gratefullyacknowledge the financialsupport from DETRTETSU.
980 programs [9], MLD systems with a linear objective function can also be formulated as multiparametric mixed-integer linear programs [ 15] and solved by using the algorithms described in [3,6,8]. In this work, we address the case when the objective function in the MLD problems involves quadratic terms (such as least square errors), resulting in a multi-parametric mixed integer quadratic program (mp-MIQP). Note that mp-MIQPs are a special case of the multiparametric mixed-integer nonlinear programs (mp-MINLPs) for which we presented algorithms in our earlier work [7]. Here we present a specialized algorithm for mp-MIQPs which involves the solution of multi-parametric quadratic programs (mp-QPs) instead of general and more difficult multi-parametric nonlinear programs (mp-NLPs) (involved in the solution of mp-MINLPs). While the solution of mp-NLPs requires linear approximations, the solution of mp-QPs is given by exact quadratic profiles. The rest of the paper is organized as follows. The mathematical foundations and the algorithm for the solution of mp-MIQPs is proposed in Section 2; an illustrative example is presented in Section 3, while some concluding remarks are given in Section 4. 2. M U L T I P A R A M E T R I C M I X E D - I N T E G E R QUADRATIC P R O G R A M M I N G
2.1. Mathematical Formulation Consider an mp-MIQP problem of the following form [ 16]" z(O) -- mincTx + lxTQx
+ dTy,
s.t. Ax + Ey <_b + FO,
x,y
(1)
where x C X C_ 9~n is a vector of continuous variables, y C {0, 1 }l is vector of 0-1 binary variables, 0 E 19 C_ 9~s is a vector of parameters, Q is an n • n constant, symmetric and positive definite matrix, A, E and F are m • n, m • l and m • s constant matrices respectively, b, c and d are constant vectors of dimension m, n and I respectively and X and 19 are compact polyhedral convex sets of dimensions n and s respectively. The procedure proposed in this section is based upon decomposing (1) into a multi-parametric quadratic program (mp-QP) and a mixed-integer nonlinear program (MINLP). The solution of the mp-QP, which is obtained by fixing the vector of binary variables, provides a parametric upper bound, whereas the solution of the MINLP, which is obtained by treating 0 as the vector of free variables provides a new integer vector. The parametric solutions corresponding to two different integer solutions are then retained to keep as tight upper bounds as possible. The steps of the procedure are described in detail in the following sections.
2.2. Multiparametric QP Subproblem An initial feasible y is obtained by solving (1) by treating 0 as the vector of free variables. Let the solution be given by y - y. Fix y -- 37in (1) to obtain an mp-QP problem of the following form:
s
lxTQx,
s.t. Ax<__(b-Ey)+FO.
(2)
x
The solution of (2) is given by a set of linear parametric profiles, 2(0) i, and the corresponding critical regions, CR i, where i - 1 , . . . , I [10]. Note that the optimal objective function, ~,(0), is continuous, convex and piecewise quadratic. This solution represents a parametric upper bound to the solution of (1). Let ~,(0) in CRi be denoted by s i. Note that CR i also includes the regions where 37is infeasible and in these regions ~,(0)i -- ~o [8].
981 2.3. M I N L P Subproblem For each critical region, CR i, obtained from the solution of the mp-QP subproblem in (2), an MINLP subproblem is formulated as follows [7,8]: z = min cTx + lxr Qx + dTy x,y,O
s.t.
A x - FO + Ey <_b cTx--[- lxT Q x - z(O) ik .-+-dry <_O, k - 1,... ,K i
y, y~k_ ~ yi.k < l j i k l _ l j E J ik
j E L lk
J
~
k--1 ~
(3)
K i, ~ . 9. ~
where 0 E CR i is treated as the vector of free variables bounded by the set of inequalities which define CRi; jik _ (j[yf = 1) and L ik -- (jly~k - 0), and Ijik I is the cardinality of jik and K i is the number of integer solutions that have already been analysed in CR i. Note that K i also corresponds to the number of iterations that have taken place in CR i. It may also be noted that the set of inequalities, cTx + l x T Q x - ~(0) ik + dTy <_ 0, known as parametric cuts, excludes integer solutions with higher values than the current upper bound, ~(0) ik, where ~(0) ik denotes ~(0) i for a given k; the set of inequalities, J CLJ ' k y~k_ J~ ikYjik < -- [jik I-- 1, corresponds to integer cuts prohibiting previous integer solutions from appearing again. The integer solution, y = 371, obtained by solving (3), and the corresponding CRs are returned back to the mp-QP subproblem, (2), to obtain another set of parametric profiles. Note that (3), in general, is a non-convex MINLP, where the nonconvexity is due to ~(0) ik, which is present in the second set of constraints. This problem can be solved to global optimality, for example, by using the algorithms described in [ 17]. Since solving this problem by using global optimization can be computationally intensive, following tests are suggested to alleviate this problem. Solve (3) without the, nonconvex, parametric cuts; let the solution set be given by z*,x*,y* and 0". If z* < z(O*) ik for each k, then y* is the next integer solution. Otherwise, for each k, evaluate z(O) ik at the vertices of CRi; let this set be given by 2kv, where v -- 1,2,..., V is the index for the vertices of CR i. Also let 2v be the maximum of 2kv at a given vertex v. Also let u(0) be affine in 0 such that u(O v) > U, for v = 1,2,..., V, where 0 v is the value of 0 at the vertex v. Note that u(0) represents a linear overestimator of ~(0) ik in CR i and can be obtained by solving a linear program. If z* _ u(0*), then no better integer solution exists in CR i and hence search in CR i is fathomed. Otherwise, (3) should be solved by using global optimization. Another simple approach is to linearize ~(0) ik at various points in 0 and obtain appropriate upper bounds and hence avoid the issue of global optimization - this will however require solving MINLPs for all the linear upper bounds. If there is no feasible solution to the MINLP subproblem, (3), in a CR, that region is excluded from further consideration and the current upper bounds in that region represent the final solution. The final solution of the MINLP subproblem is given by a set of integer solutions and the CRs in which they were identified.
2.4. Envelope of Parametric Solutions The set of parametric solutions corresponding to the new integer solution, y -- 371,are included into the set of parametric current upper bounds, given by the parametric solutions corresponding to 37, in the corresponding CRs. Note that a comparison of the parametric (quadratic) solutions corresponding to two different integer solution is not performed. If this comparison were made
982 it would result in the lower of the two profiles as an upper bound. Instead of the comparison, all the parametric profiles corresponding to all the integer solutions obtained in a CR are retained and the best solution is obtained at a particular vector point 0, on-line, by taking the minimum of the values obtained by function evaluation of parametric profiles for a given 0. Also note that a rigorous procedure for comparing nonlinear profiles, similar to the one for linear profiles [3], can be developed and will result, in general, in nonlinear and nonconvex CRs, further complicating the solution of (3). Note that for a CR i, some of the profiles from the set ~(0) ik can be eliminated without affecting the final parametric solution, min{~,(0) ik}. These profiles arise due to the subdivision of a given k critical region, for a fixed integer solution, to smaller critical regions, for another fixed integer solution, and the resulting accumulation of all the parametric profiles. Identification of such redundant profiles involves recursively solving MINLPs of the form similar to the one given in
(3). 2.5. Enclosure of the Solution of Multiparametric M I Q P Based upon the above discussion, the steps of the proposed solution procedure can be summarized as follows:
i. (Initialization) Define an initial region of 0, CR, and an initial integer solution 37. ii. (mp-QP Subproblem) For each region with a new integer solution, 37:
a. Solve mp-QP subproblem (2) to obtain a set of parametric upper bounds s corresponding critical regions CR. b. If an infeasibility is found in some region CR, go to iii.
and
iii. (Master Subproblem) For each region CR, formulate and solve the MINLP master problem in (3) by treating 0 as the vector of free variables bounded in the region CR, introducing integer cuts and introducing parametric cuts. Return to ii with new integer solutions and corresponding CRs.
iv. (Convergence) The algorithm terminates in a region where the solution of the MINLP subproblem is infeasible. The solution is given by an envelope of the current upper bounds s integer solutions, )7, and the corresponding CRs. v. (Redundancy Check) For all the CRs remove the redundant parametric solutions. 3. N U M E R I C A L EXAMPLE Consider an mp-MIQP problem of the form given in (1) with
C=
b=
[0.02] i00196000631 E030] 0.03 ' Q= 0.417425 3.582575 0.413225 0.467075 1.090200 2.909800 1
0.0063 0.0199 , d =
,A=
1 0 -1 0 -0.0609 0 -0.0064 0 0 1 0 -1 0 0
,E=
-0.31
'
-1 0 -1 0 0 -0.5 0 -0.7 -0.6 0 -0.5 0 1 1
,F=
3.16515 3.7546 -3.16515 -3.7546 0.17355 -0.2717 0.06585 0.4714 1.81960 -3.2841 - 1.81960 3.2841 0 0
983 Table 1 Parametric expressions: X(O) i - DiO + d i, CR i" EiO < e i D1 =
El=
D2 =
E2 =
[ 0.0000 0.0000
=
=
eI
3.1651 -1.0020
3.7546 -1.1886
d2 = [ -4.5825 t -0.0567
1.000~
0 1.0000 1.001~ -1.1862
e2
0.0000 00000
d3
[ 0.0000 00000
0 1.1862 1.8048 -1.0000 -1.0000
=
e3 =
3.1651 - 1375 ] 1886
0 -1.3465 1 3465 -1.0000
/)5=[
E5 =
3.1651 1.81960
- 1.3894 1.3465
1.0000 1.0000 -1.0000 -1.1862
=
=
-1.0020
E4 =
-0.5965 -1.3186
0 1.1862 1.0000 - 1.0000 -1.1862
- 1.0000 1.0000 -1.0000 -1.3894 1.3465
E3 =
dl
1.0000 1.0000 - 1.3465 1.3465 -1.0000
- 1.3465 -1.00130
D3
0.0000 0.0000
=
e4 =
3.7546] - 3 2841
d5
- 1.0000 - 1.0000
e5 =
=
]
1.0000 1.2593 0.6984 1.2101 -0.9434
1.0000 1.0000 0.6984 - 1 2593 [ -0.5965 -1.3186
I ~ 1.2593 1.6535 - 0 8194 -0.6984
[_4.5825 -0.0567 1.0000 0.8336 -0.6984 - 1.2593 -4.5825 ] 1.6902 1.01300 ] -1.1185 - 0 8336
D6 =
[ -0.5848 1.8196 0 1 3894 -1.0000 1 0000
E6 =
D7
[ 0.0000 0.0000
=
1.0556] -3.2841
d6
1.0000 1.0000 0 - 1.8048
e6
0.0000 0 0000
=
=
0 1.8048 - 1.0000 1.1862 - 1.8048
[ - 0 5848 = [. 1 8196
1.0556 -3.2841
=
1.3894 1.0000
1 0000 - 1.8048
e8
D9
[ 3.1651 = [ 1 8196
3.7546 - 3 2841
d9
- 1.00013 - 1.0000 1.0000
e9
- 1.3894 - 1.3465 -0.5848 1.8196
1 0556 -3.2841
dl 0
1 3894 1.0000
1.0000 1.8048
el0=
D8
E 8
E9 =
Dl o =
E 1~ = -
1.0000 1.1185 0 -1.6535
d7 = [ -0.5965 - 1 3186 ]
- 1 0000 - 1.0000 0 1.0000 1.0000
E7 =
[ -1.5636 1.6902 ]
e7 =
0 1.3238 0 0.9434 0.8740
d8 = [ - 1.3708 L 109020 ]
0] [ o] =
=
=
=
0 8194 - 1.3238
-3.5825 -2.9090
]
- 1.2957 - 1.2101 - 0 0853 ] -2.9090
1.2957 - 0 8740
and 0 ___01 < 1,0 < 02 _ 1. The solution of this problem by using the algorithm described in Section 2.5 is given in Tables 1 and 2. Note that the mp-QPs were solved by using POP [18] and the MINLPs were solved by using DICOPT++ [19]. This example required the solution of 3 mp-QPs and 25 MINLPs. 4. CONCLUDING REMARKS On-line control and optimization problems can be reformulated as multi-parametric programs. The solution of such problems is given by explicit expressions for the control variables as a function of the state variables. For the case when on-line control problems also involve logical constraints, integer variables are introduced. Such problems are known as MLD problems and are reformulated as mp-MIQPs. In this paper a decomposition algorithm for obtaining an enclosure of the solution of mp-MIQP was proposed and illustrated with an example. REFERENCES 1. 2. 3.
E.N. Pistikopoulos and V. Dua, Proc. 3rd Int. Conf. on FOCAPO, J. E Pekny and G. E. Blau, Eds., (1998) 164. J. Acevedo and E. N. Pistikopoulos, Ind. Eng. Chem. Res. 35 (1996) 147. J. Acevedo and E. N. Pistikopoulos, Ind. Eng. Chem. Res. 36 (1997) 717.
984 Table 2 Envelope of the parametric solution CR 1 CR 2 CR 3 CR 4 CR 5 Region y 'i [0,1] [1,0] [0,1] [0,1] [1,0] [0,1] I' [1,0] [0,1] X(0) 1 X(0)2 X(0)2 X(0)2 X(0)4 x(0) ~ x(0)-' x(0) .1 x(0) 1 C R lo CR 6 CR 7 CR 8 CR 9 Region [. [0,1] [0,1] [0,11 [0,11 Y
x(O)
,,
[o,1] I x(0) [1,o]6
x(0) ~
x(0) 7
x(0) s
~(0) 9
x(0) 1~
,,
4. A. Pertsinidis, I. E. Grossmann and G. J. McRae, Comput. Chem. Engng. (Suppl.) 22 (1998) 205. 5. K.P. Papalexandri and T. I. Dimkou, Ind. Eng. Chem. Res. 37 (1998) 1866. 6. J. Acevedo and E. N. Pistikopoulos, Oper. Res. Lett. 24 (1999) 139. 7. V. Dua and E. N. Pistikopoulos, Ind. Eng. Chem. Res. 38 (1999) 3976. 8. V. Dua and E. N. Pistikopoulos, An algorithm for the solution of multiparametric mixed integer linear programming problems, Annals of Operations Research, in press (2000). 9. E.N. Pistikopoulos, V. Dua, N. A. Bozinis, A. Bemporad and M. Morari, Comput. Chem. Engng. 24 (2000) 183. 10. V. Dua, N. A. Bozinis, E. N. Pistikopoulos, A. Bemporad and M. Morari, An algorithm for the solution of multi-parametric quadratic programs, Tech. Rep. D00.7, Centre for Process Systems Engineering, Imperial College, London, U.K. (1999). 11. R. Raman and I. E. Grossmann, Comput. Chem. Engng. 15 (1991) 73. 12. M. Ttirkay and I. E. Grossmann, Comput. Chem. Engng. 20 (1996) 959. 13. A. Vecchietti and I. E. Grossmann, Comput. Chem. Engng. 24 (2000) 2143. 14. A. Bemporad and M. Morari, Automatica 35 (1999) 407. 15. A. Bemporad, F. Borrelli and M. Morari, Proc. American Control Conf. (2000). 16. V. Dua, N. A. Bozinis and E. N. Pistikopoulos, Multi-parametric mixed-integer quadratic programming: an enclosure of the solution, Tech. Rep., Centre for Process Systems Engineering, Imperial College, London, U.K. (2000). 17. C. S. Adjiman, I. P. Androulakis and C. A. Floudas, AIChE J. 46 (2000) 1769. 18. N. A. Bozinis, V. Dua and E. N. Pistikopoulos, A MATLAB (@ The Math Works, Inc.) implementation of multi-parametric quadratic programming algorithm, Centre for Process Systems Engineering, Imperial College (1999). 19. J. Viswanathan and I. E. Grossmann, Comput. Chem. Engng. 14 (1990) 769.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
985
A Process Integration Design Method For Water Conservation and Wastewater Reduction in Industry R. F. Dunn a and H. Wenzel b aNylon Technology, Solutia, 3000 Old Chemstrand Road, Cantonment, FL, bDepartment of Manufacturing Engineering, Technical University of Denmark, Lyngby, Denmark This paper addresses an operational technique for applying mass integration design in industry with special focus on water conservation and wastewater reduction. The presented techniques derive from merging US and Danish experience with industrial applications of process integration as a result of a recent established co-operation between the US company Solutia Inc. and the Technical University of Denmark. In brief, the technique comprises a two stage graphical approach. In the first stage, the water pinch diagram is used to identify key design targets such as the minimum amount of fresh water required by the studied system, the amount of water recycle and reuse that is achievable and the water quality concentration bottleneck. Practical key insights provided by the water pinch diagram are discussed. In the second stage, source-sink mapping diagrams are used to identify the water recycle and reuse network, and any alternative networks, that achieve the identified targets. The approach has been used for continuous processes in the US and batch processes in Denmark independently and is found to be the most operational and cost-beneficial for both applications. 1. I N T R O D U C T I O N Wastewater reduction and water conservation are becoming increasingly more important issues in process industries. More stringent environmental regulations, concerns over longterm health effects on humans and nature, and the future availability of "clean" water resources are important factors driving efforts toward improvements in water conservation and wastewater reduction in manufacturing processes. As these issues continue to receive intense government scrutiny and heightened concern from community-organized environmental groups, the ability of industry to address these issues may soon impact their right-to-operate within these communities and their sustainability of future operations. Several recent research efforts have focused on the development of process design methodologies and tools for water recycling that are generic and can be utilized across a wide variety of industries. The design methodologies and approaches cover a variety of techniques ranging from the graphical based "water pinch" analysis [1-2], the source-sink graphical methodology [3] to mathematical optimization based approaches [4-5]. This paper
986 presents a new combined methodology to assist the industrial engineer in designing and redesigning the production system to minimize water consumption and waste water generation. This methodology draws from several of the innovative design tools that have been created by different researchers and links these tools together to provide a cohesive design methodology. The design approach is based on a two-stage procedure. In the first stage, the water pinch diagram is used to identify key design targets such as the minimum amount of fresh water required by the studied system, the amount of water recycle and reuse that is achievable and the water quality concentration bottleneck. In the second stage, source-sink mapping-diagrams are used to identify the water recycle and reuse network, in addition to any alternative networks, that achieve the identified targets.
2. EXAMPLE: IDENTIFYING THE WATER RECYCLE NETWORK FOR STREAMS CONTAINING A SINGLE CONTAMINANT In order to illustrate the methodology, an example will be used throughout the document. This example is based on the data provided by Wang and Smith and is summarized in Table 1 below. The example involves four unit operations (process numbers 1, 2, 3 and 4) that use water at the flowrates provided. Water quality requirements for each unit (inlet concentration) are provided and wastewater discharge quality (outlet concentration) are also provided. Within each unit operation the mass of the contaminant in the water stream is increased. Simple examples of these types of operations would be water for rinsing, water used in scrubbers, cooling towers with blowdowns, etc. This case study involves analysis of water streams containing a single component; however, the methodology presented in this paper can easily be extended to multi-component situations. The design objective is to identify mixing and direct recycle opportunities to minimize wastewater discharge and freshwater usage. The solution should not result in changes to the water flowrates or compositions exiting the units, nor should it violate the inlet water quality requirements. Table 1 Case Study Data Provided by Wang and Smith (1994) Process Mass Load of Inlet Concentration Outlet Number Contaminant Requirement Concentration (kg/hr) (ppm) (ppm) 1 2 0 100 2 5 50 100 3 30 50 800 4 4 400 800
Water Flow xl 03 (kg/hr)
20 100 40 10
2.1. Identifying the Design Targets The identification of water reuse and recycle process designs (networks) using a graphical tool called the water pinch diagram was introduced by Wang and Smith in 1994. The water pinch diagram is indeed a simple and useful tool for identifying several key design targets. In general, the water pinch diagram consists of a graphical plot of the composition change (from the inlet composition to the outlet composition) of the water within a unit operation
987 versus the cumulative mass transferred within the unit operation. The water pinch diagram for this case study has been reported in literature [ 1] and Figure 1 is included to highlight the key aspects of this diagram. A line starting at the intercept between the y-axis and the composite curve is drawn that touches the lowest point on the limiting composite curve as indicated in Figure 1. This line is referred to as the water supply line, the point of contact (100 ppm composition) is referred to as the water quality pinch point, and several key targets can be obtained from this graphical representation.
Key insights provided by the water pinch diagram The water pinch diagram provides a number of key insights to the engineer including the overall design targets: 1. The inverse slope of this line represents the theoretical minimum amount of freshwater required to satisfy all unit operation requirements (inlet and outlet compositions and mass transferred within unit operations. 2. Subtraction of the inverse slope value from the total cumulative water required for all unit operations equals the amount of water that can be directly recycled/reused within the process units. 3. The pinch point represents a thermodynamic water quality bottleneck within the system of process units. This represents the composition location where fresh water will always be added within a network- besides to processes having fresh water requirements. This composition target also represents the point within a network where water purification technology (sometimes referred to as interception or regeneration technology) should be employed to improve the water quality. Thus, for the case study provided, the minimum amount of freshwater required is 90xl03kg/s, the total potential for direct recycle/reuse of wastewater to process units is 80xl03kg/s and the thermodynamic water quality bottleneck within the system of process units is 100 ppm. This composition represents the point where freshwater- besides to process 1 requiring fresh water inlet- will be added within the water recycle. After these targets have been established through the use of a water pinch graph, the task is to design the appropriate network of process units that uses the minimum amount of freshwater required and allows the recycle/reuse of the maximum amount of wastewater. This is accomplished in stage 2 of the graphical design procedure.
Fig. 1. Water Pinch Diagram (Wang and Smith, 1994 [ 1])
988
2.2. Identifying the Water Recycle Network Wang and Smith [ 1] have presented a methodology for identifying the network from the water pinch diagram, but the approach becomes very complex as the number of water sink streams and water source streams increases. Instead, a methodology using graphical sourcesink diagrams (also referred to as mapping diagrams) to identify the water reuse/recycle network has been found much easier and time efficient to apply in systems comprising both continuous and batch processes, and this method is recommended and will be employed here. The source-sink stream mapping diagrams are graphical tools that were developed to assist the design engineer in identifying where segregation/recycle and segregation/interception/recycle opportunities exist within a process [3, 6-7]. A "source" stream refers to any stream that contains some amount of water and exits any unit operation within a production system. A "sink" stream refers to any stream that contains some amount of water and enters any unit operation within the system. Streams, containing water, which enter or exit unit operations in a complex system, are commonly found at a number of locations within the system. Source streams are identified typically from the plant process flow diagram (PFD). In some cases, data availability may be limited; therefore, streams with larger flow rates, greater waste loads, and/or higher heat duties are generally given more emphasis during the data collection phase of the design strategy. An example mass mapping diagram is provided as Figure 2. Multiple mapping diagrams may need to be developed for the same source and sink streams and examples of several additional mapping diagrams are available in literature [8].
Water Load (kg/s)
0
$4 D
Wlc 0
Source Streams (W)
D Sink Streams (S)
S5 []
Mixing&Recycle W l a ~ ~
wib
W2
Direct Recycle
o W3
Interce~ 0
W4
$1
Composition of Key Species (weight %)
Fig. 2. Mass Mapping Diagram for Water Conservation and Reuse Design Figure 2 highlights direct recycle, mixing and direct recycle and interception (regeneration) opportunities as indicated on the figure. In the case study example provided there are four sinks (entering streams) and four sources (exiting streams) that have been identified. The source-sink mapping diagram is generated using the source and sink streams previously indicated in Table 1. These streams identified are plotted as single points on a source-sink
989 mass mapping diagram as indicated on Figure 3. The steps for using the mapping diagram
have been previously provided in literature [8].
Water Load
0
(kg/s x 1o -3)
Source Streams
[] Sink Streams
loo [] | 90 80 70 60 50 4O
[]
30 20
@
[]
1o
o
I 1oo
|
I
I
I
I
I
[
200
300
400
500
600
700
I'800
Composition of Key Species (ppm)
Fig. 3. Mass Mapping Diagram for the Case Study 2.3 Water Recycle Network Solutions Using the Mapping Diagram: Five independent networks can be identified for the case study provided. recycle networks have been provided in Figure 4 below.
All five water
Networ___~k1:
T~I~
Network 5:
Network 3:
T ftJli i
T
i
~"~
Fig. 4. A simultaneous comparison of all water reuse/recycle networks identified
990 All of these networks achieve the minimum freshwater usage target of 90x103kg/s and the total wastewater recycle/reuse target of 80xl03kg/s. Several factors such as network economics, safety, distance between unit operations, etc. should be considered prior to selecting the network desired for implementation. 3. EXPERIENCE WITH PRACTICAL APPLICATIONS The described methodology combines, to the experience of the authors, the best of existing methods to identify water reuse targets and to design water reuse systems. It has been applied to a wide range of production systems ranging from continuous processes in large chemical and petro-chemical industries to batch processes in small industrial laundries and textile dyehouses [9]. Experience and results from these applications will be presented at the conference. 4. CONCLUSIONS This paper provided an operational technique for applying mass integration design in industry with special focus on water conservation and wastewater reduction. The technique comprises a two stage graphical approach. In the first stage, the water pinch diagram is used to identify key design targets such as the minimum amount of fresh water required by the studied system, the amount of water recycle and reuse that is achievable and the water quality concentration bottleneck. In the second stage, source-sink mapping diagrams are used to identify the water recycle and reuse network, and any alternative networks, that achieve the identified targets. An example has been included to illustrate the proposed procedure. REFERENCES
1. Y.P. Wang and R. Smith, Chem. Eng. Sci. 49 (1994) 981. 2. N. Hallale, Energy and the Environment: AIChE Topical Conference Proceedings, (2000) 242. 3. M.M. E1-Halwagi, Pollution Prevention Through Process Integration, San Diego: Academic Press, 1997. 4. A. Alva-Argaez', A. Villianatos and A. Kokossis, Comp. and Chem. Engr., 23 (1999) 1439. 5. S.E. Keckler, and D.T. Allen, J. Ind. Ecology 2 (1999) 79. 6. A.M. Dobson, and R.F. Dunn, AIChE 1997 Spring Meeting, Houston (1997). 7. G. Parthasarathy, R.F. Dunn, and M.M. E1-Halwagi, AIChE 2000 Spring Meeting, Atlanta (2000). 8. R.F. Dunn and G. E. Bush, J. of Cleaner Prod., 9 (2000), 1. 9. R.F. Dunn and B. K. Srinivas, in Emerging Separation and Separative Reaction Technologies for Process Waste Reduction: Adsorption and Membrane Systems, AIChE, New York, 1999, 277.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
991
Retrofit of Mass Exchange Networks D. M. Fraser, N. Harding, and C. Matthews Department of Chemical Engineering, University of Cape Town Private Bag, Rondebosch, Western Cape, 7701 South Africa
In this paper we examine the retrofit of an industrial mass exchange network for the removal of hydrogen sulphide from a set of gaseous streams, in order to reduce the sulphur emission from the plant. The system used by Sasol for hydrogen sulphide absorption from the gas streams is one in which the gas-liquid contact is effected in three venturi scrubbers with a combined cross and co-current flow arrangement. Previous mass exchange network design was only for counter-current systems, so we needed to develop new methodology for solving this system. Once the new method had been developed it was applied to the system, in order to evaluate the impact of three different strategies on the overall sulphur emission from the plant: segregation of the gas streams, increase of contact time in the scrubbers, and increase of the gas flow rate through the scrubbers. The paper concludes by evaluating the relative merits of these alternative strategies. 1. INTRODUCTION Mass exchanger network synthesis (MENS) has developed over the past twelve years, following the pioneering work of E1-Hawagi and Manouthiousakis (1989). Mass exchange networks (MENs) are systems in which particular components (often pollutants) are removed from rich process streams using lean process streams or external mass separating agents (MSAs) in units such as absorbers, strippers, ion exchange columns, adsorbers and mixersettlers. E1-Halwagi and Manouthiousakis introduced methods for targeting the minimum amount of MSA required, and hence the minimum operating cost for such systems. More recently Hallale and Fraser (1998a-b, 2000a-d) introduced methods for also targeting capital costs for MENs, which allow one to trade off operating and capital costs before design. These techniques were developed for grassroots designs, but have also been applied to retrofit of existing systems, either to simply reduce operating costs (Fraser and Hallale 2000a), or to reduce pollution (Fraser and Hallale 2000b). All the systems studied to date involved counter-current flow of the rich and lean streams in the contacting units, whether tray columns, packed columns, or a series of mixer-settlers. In this study we examined the retrofit of an industrial MEN system which involved cross-flow of the rich and lean streams in three successive devices in which the internal flows were cocurrent. It was therefore necessary to develop methodology for handling this sort of system.
992 2. P R O B L E M STATEMENT
In the system under consideration, shown in Figure 1, gas streams from seven different sources were combined and then split into four scrubbing units in which the hydrogen sulphide was removed before the gas stream was sent to incineration and then disposed of up a stack. A stream (Stream 8) also containing hydrogen sulphide joined the combined exit stream from the four units (Stream 9) before the incinerator. This stream contained phenol and could not be treated in the four units. The hydrogen sulphide absorbed in the four units is converted to sulphur (note that Figure 1 only shows the gas streams).
~r
,,
Figure 1. Sulphur recovery system. Each of the four scrubbing units consisted of three venturi scrubbers, with the gas flowing through them in series, and the liquid in parallel, as shown in Figure 2. The overall flow of gas and liquid though each unit is thus clearly cross flow, while within each scrubber the gas and liquid flow together in a co-current manner (see Figure 2 insert). Rich Stream, G .......
Lean Stream, xLn
~"1
Figure 2. Arrangement of the three Venturi scrubbers in each unit. Nominal flows and hydrogen sulphide compositions for the eight feed streams are given in Table 1, together with the outlet composition of the scrubbing units (Stream 9) and the overall stream to the incinerator (Stream 10).
993
Stream Flow (% of stream 10) H2S
(%)
1 24.48 1.11
Table 1. Stream flows and compositions. 2 3 4 5 6 7 25.78 17.85 11.38 8.53 1.71 1.04 1.32
1.52
1.76
2.05
10.74
5.16
8 5.43
9 94.57
10 100.0
2.37
0.62
0.72
The objective of this study was to reduce the sulphur emission up the stack. Plant personnel were already investigating increasing the contact time in the scrubbers, as well as increasing the gas flow through the scrubbers which was limited by the capacity of the exit cyclones. We decided to quantify the extent of reduction in sulphur emissions for increased gas flow, and also compare what could be achieved by segregation of the streams, instead of the current arrangement where they are all mixed together before going to the scrubbing units. The only data available for this study were the flows and compositions of all the feed streams to the process and the outlet composition from the scrubbing units, the maximum flow to each of the units (18.33% of stream 10), and a relationship between unit throughput and absorption in it. Unfortunately, there was no reliable equilibrium data for this system. 3. APPROACH TAKEN The approach we took to solving this problem was first of all to develop a model for the combined cross and co-current flows in each of the four scrubbing units. This model was then used to estimate a pseudo-equilibrium for the system, by fitting it to the plant's current operation (assuming the equilibrium to be linear). In doing this an efficiency was assumed for each of the venturi scrubbers, with a view to possibly estimating the effect of increased contact time. Where the flow to a unit exceeded the maximum specified, the balance was assumed to bypass it. When the results obtained like this were compared to the relationship between throughput and absorption, a discrepancy was found which was accounted for by adjusting the relationship to fit the plant data. 3.1. Analysis of combined cross and co-current flow system In a co-current flow scrubber, standard analysis shows that the operating line has a slope of -L/G (Treybal, 1981). The system under consideration has three co-current flow scrubbers, with the gas flowing in series between them, and fresh liquid entering each scrubber (so the liquid is in cross flow). The liquid flows to each of the scrubbers had been optimised, and it had been found that the best arrangement was for the flow to the first scrubber to be double the flow to the second and third scrubbers. The slope of the operating line for the first scrubber is therefore double that of the other two scrubbers. This is shown in Figure 3(a), where it may be seen that the operating lines for the second and third scrubbers start at the unknown exit gas composition, y, of the preceding scrubber (Treybal, 1981).
994 10A
10=~ A
Stage I
--'J
Stage 2
--"
Stage 3
8
g ~
4
- - -)~ - - Equilibrium
2
Line
0 0
5
x I%1
10
15
(a)
A
6 ~~............~ 4 .~~ . . . . . . . . . . . . _ 2
.""
~
01('" 0
o
. - "
""'"~
,
,
5
10
x I%1
15
(b)
Figure 3. y-x diagram showing construction of operating lines: (a) 100% efficiency; (b) 80% efficiency. Knowing the inlet and outlet compositions from the unit as a whole, and the liquid and gas flows to each scrubber, one can postulate a linear equilibrium line, and then solve for the slope of the equilibrium line which intersects each of the three operating lines, as shown in Figure 3(a). It is however more likely, given the short contact time in each scrubber, that they do not operate at equilibrium. Figure 3(b) shows the solution for this system if it is assumed that each stage operates at 80% efficiency, where it has been assumed that this means the final outlet composition from each stage may be obtained as 80% of the change which would have occurred if it had been at equilibrium. The results obtained were checked to see to what extent they were affected by the approach to equilibrium assumed, and it was found that they were independent of it. This is because the fitted equilibrium line simply shifts to accommodate the different efficiencies, as may be seen by comparing Figure 3(a) with Figure 3(b), in which the only difference is that the slope of the equilibrium line in (b) is 80% of the slope in (a). 3.2.
Mass transfer composite curve
A helpful way of comparing different options was developed by analogy with the mass transfer composite curve introduced by Hallale and Fraser (1998a) for counter-current flow systems. A typical mass transfer composite curve for cross flow is shown in Figure 4, for a system with three units, with each unit having a feed of different composition, and each one removing 50% of the transferred species. This figure plots rich stream composition against mass transferred, and is derived by vector addition of the individual mass transfer lines. The way this composite curve is used is follows. The streams entering the four units are segregated and then different combinations are put together, so that each scrubber unit has a different feed. This aims to maximise mass transfer driving force, while maintaining a minimum flow close to the maximum flow though each of the four units. Any combined flow greater than the maximum is bypassed around the unit. Figure 5 compares the mass transfer composite curve for the original configuration with that for the best option found (Option 7). This shows clearly the improvement for Option 7 compared to the original configuration. This method requires some programming to produce the mass transfer composite curves, but it does provide a clear way of seeing which option will transfer the most mass.
995
2 "
A
- - ~
0
m
m
m
m
m
m
l
1
2
3
4
5
6
7
Original - - Option
7
Mass Transferred
10 - - ~
A6 =< v
9
9149
- -Stream
1
- - -e- - - S t r e a m
2
- - ~lr - -Stream
3
9
>'4 "A I
0
.
9
"=
.
.
I
|
9
.
9
5
i
10
.
.
|
Total
|
15
Mass Transferred
Figure 4. Mass transfer composite curve for cross flow system. Figure 5. Mass transfer composite curve for Option 7. 4. R E S U L T S The methods described above were used to determine the improvement in sulphur removal for different retrofit possibilities. It was found that segregation of the streams before scrubbing (rather than mixing them all together) gave a definite increase in sulphur recovery. The best option (Option 7) was to send Stream 1 on its own to Unit 1, Streams 2 and 7 together to Unit 2, Streams 3 and 6 to Unit 3, and finally Streams 4 and 5 to Unit 4. This gave a 10.5 % increase in sulphur recovery. Using this option, the liquid split to the three stages in each unit was varied. This indicated that the present 2:1:1 split could be improved, and that, in fact, for the best segregation option, a split of 1:1:1 gave a further 2.4% increase in sulphur recovery. Another possibility that was examined was the effect of including stream 8 (the one containing phenol) in the scrubbing system (presuming the phenol had been removed). This indicated that a maximum sulphur recovery increase of 8.8% was possible. This is less than the best option without including this stream, so there does not appear to be an incentive to remove the phenol so that stream 8 can be sent to the scrubbing units. The final possibility considered was a debottlenecking of the cyclones to allow a larger gas flow through the scrubbing units. It was found that a 15% increase in throughput (for the best
996 segregation option) would result in an additional 11,6% increase in sulphur recovery. There is therefore a good incentive to increase the cyclone capacity. 5. CONCLUSIONS Two diagrams are presented for the modelling and analysis of the combined cross and cocurrent flow venturi scrubber unit examined in this study: one is the y-x diagram for a combined cross and co-current flow system, and the other is the mass transfer composite curve for such a system. These diagrams enabled us to examine a range of retrofit options to improve the sulphur recovery in these units. It was found that both segregation of the feed streams and changing the liquid feed ratios to the three stages, both of which could be implemented without major costs, would give improvements. There seems to be no incentive to include the phenol containing stream in the sulphur recovery units. Debottlenecking of the cyclones, which could be quite costly, would give a major boost in the ability of the units to recover sulphur. ACKNOWLEDGMENTS The authors wish to acknowledge the collaboration and support of Sasol Technology (Sastech), particularly Mr Daan le Roux, in the work of this project. REFERENCES
1.
E1-Halwagi, M. M.; Manousiouthakis, V. (1989) Synthesis of Mass Exchange Networks,
AIChE J., 35(8), 1233-1244. 2. 3. 4. 5. 6.
7.
8. 9. 10.
Fraser, D. M.; Hallale, N. (2000a) Retrofit of Mass Exchange Networks Using Pinch Technology, AIChE Journal 46(10), 2112-2117. Fraser, D. M.; Hallale, N. (2000b) Determination of Effluent Reduction and Capital Cost Targets Through Pinch Technology, Environmental Science and Technology, in press. Hallale, N.; Fraser, D. M. (1998a) Capital cost targets for mass exchange networks. A special case: Water minimisation, Chem. Eng. Sc., 53(2), 293-313. Hallale, N.; Fraser, D. M. (1998b) Synthesis of a cost-optimum gas-treating process using pinch analysis, Advances in Environmental Research, 2(2), 167-178. Hallale, N.; Fraser, D. M. (2000a) Capital and Total Cost Targets for Mass Exchange Networks, Part 1: Simple Capital Cost Models, Computers and Chemical Engineering, 23(11-12), 1661-1679. Hallale, N.; Fraser, D. M. (2000b) Capital and Total Cost Targets for Mass Exchange Networks, Part 2: Detailed Capital Cost Models, Computers and Chemical Engineering, 23(11-12), 1681-1699. Hallale, N.; Fraser, D. M. (2000c) Supertargeting for Mass Exchange Networks: Part I Targeting and Design Techniques, Trans IChemE (Part A), 78(Mar), 202-207. Hallale, N.; Fraser, D. M. (2000d) Supertargeting for Mass Exchange Networks: Part I I Applications, Trans 1ChemE (Part A), 78(Mar), 208-216. Treybal, R. E. (1981) Mass Transfer Operations, 3rd ed., McGraw-Hill, Singapore.
EuropeanSymposiumon ComputerAidedProcessEngineering- 11 R. Ganiand S.B.Jorgensen(Editors) 9 2001 ElsevierScienceB.V.All rightsreserved.
997
The Interactions of Design, Control and Operability in Reactive Distillation Systems Michael C. Georgiadis a , Myrian Schenk b, Rafiqul Gani c and Efstratios N. Pistikopoulos b* aCentre for Research and Technology - Hellas, Chemical Process Engineering Research Institute, P.O. Box 361, Thermi 57001, Thessaloniki, Greece. bCentre for Process Systems Engineering, Department of Chemical Engineering, Imperial College, Prince Consort Road, London SW7 2BY, U.K. CCAPEC, Department of Chemical Engineering, Technical University of Denmark, DK-2800, Denmark. The aim of this work is to systematically study the interactions of process design, process control and operability in reactive distillation systems. Based on a rigorous and high fidelity dynamic model that has been validated against experimental data. The study is performed in a system involving the production of ethyl acetate from the esterification of acetic acid with ethanol. The problem is posed as a dynamic optimization problem under uncertainty and solved using control vector parameterization techniques. Two state-of-the-art optimization strategies are then employed. First, the design and control tasks are considered sequentially and, for the second, design and control are optimized simultaneously and the potential synergistic benefits of such an approach are investigated. In both cases, multi-loop proportional-integral (PI) controllers are used. Optimization results of the simultaneous approach indicate promising process designs and illustrate how controllability is a strong function of the process design and how careful design can result in significantly improved controllability without necessarily having to pay an economic penalty. 1. INTRODUCTION Reactive distillation systems involving reaction and separation in a single unit have the potential to greatly reduce capital and operating costs. However they are not extensively used in industry perhaps because it is perceived that the operation of a reactive distillation system will always be more difficult and will pose higher requirements on the quality of the design and control system than the conventional reactor and train of distillation columns. This can be mainly attributed to the presence of reactions coupled with the separation, which will have a significant influence on the robust operation under variations. In the light of this, the problem of designing and operating reactive distillation systems is one that would benefit from a detailed study of the interactions of process design, process control and operability. However, most of the studies that have appeared in the literature have solely concentrated on steady-state design/synthesis aspects, linear controllability analysis or on dynamic modelling and control of fixed designs [1-3]. Furthermore, in most approaches relatively simple models were used: tray hydraulics and pressure dynamic were normally Corresponding author. E-mail:[email protected]
998 ignored [3]. On the other hand, recently the interactions of design and control issues have been systematically studied in complex distillation systems using dynamic optimisation-based approaches. The economic and operability benefits that could have been obtained through an integrated approach have been clearly demonstrated [4-5]. In this work, a reactive distillation example problem is described by a high-fidelity dynamic model in which the operability characteristics can be captured over time. The interactions of design and control are studied through the use of two strategies and the potential synergistic benefits of the integrated simultaneous approach are investigated. 2. P R O B L E M DESCRIPTION AND DYNAMIC M O D E L
The production of ethyl acetate from the esterification of acetic acid with ethanol and water is considered (see Figure 1). The saturated liquid mixture is fed at a rate of 4885 mol/h in order to produce a top product with at least 0.52% ethyl acetate composition and a bottom product of no more than 0.26% ethyl acetate. Reaction takes place in all 13 trays of the column. The objective is then to design the column and the control scheme at minimum total cost, able to maintain feasible operation over a finite time horizon of interest (24 hours); subject to (i) high-frequency sinusoidal disturbances in the acetic acid inlet composition; (ii) "slow-moving" disturbance in the cooling water inlet temperature; (iii) product quality specifications; (iv) flooding, entrainment and minimum column diameter requirements; (v) thermodynamic feasibility constraints for the heat exchangers and (vi) operating pressure limits for the column. [ " S low-tnoving" disturbance
High-frequency disturbance Zac.= a cO-496-~O.05sin(0~t) to
[
c o = 2 x / 3 6 0 0 r a d s- t
Ethy lAcetate
<____ Acetic
Acid/Ethan
Xd>=
6(5~
ol/VCater
4 8 8 5 naol/h 13
Xb<= 0.26~ Fig. 1. Reactive distillation example The basis of the detailed model has been presented in our previous work [6]. The model includes details that are normally neglected, such as: 9 Detailed flooding and entrainment calculations for each tray and 'subsequent' calculation of'critical' points in the column and the minimum allowable column diameter. 9 Equation for the pressure drop for each tray 9 Liquid hydraulics and liquid level on each tray and in the auxiliary units by using modified Francis weir formulae.
999 The liquid-vapour equilibrium has been represented accurately using non-ideal models [67]. The model and its steady-state analogue have been implemented within gPROMS [8]. 3. SEQUENTIAL DESIGN AND CONTROL The steps of the sequential approach used for optimal design and control of the reactive distillation system are outlined in Figure 2. The step-by-step results obtained using this strategy are briefly reported in the following sections. I
Fix U n c e r t a i n t i e e a n d ] di~u~bancesat nominal| value$ .J
5 Optimize steady-state ]-L pr..... design ,.r
contraints
vlol~,~d
__f
Test for feasibility in of uncertainties ]
L t~'esence
t:~oces s
design minimis~d violations
OK
~ Constraints violated
design and control system
Fig. 2. Outline of Sequential Strategy
3.1. Nominal and Flexible Steady-State Designs First, a steady-state optimization using gOPT/gPROMS [8] was carried out with the uncertain cooling water inlet temperature and feed composition of acetic acid fixed at their nominal values. The nominal column design obtained is not feasible for the whole range of uncertain cooling water temperatures and so a multi-period optimization approach [9] was used to obtain an economically optimal and flexible design. Three degrees of freedom (reflux ratio, steam flow rate and cooling water flow rate) can, in principle, be adjusted to offset the effects of the uncertainty. The following cases were considered (i) all three degrees of freedom allowed to vary ("best-case" design) and (ii) no degrees of freedom allowed to vary ("worstcase" design). The different optimal designs and resulting annual costs for the nominal and the two cases considered are shown in Table 1. Note that D refers to column diameter and S refers to the surface area of the heat exchange coil in the reboiler, Reb, or the condenser, Cond. It can be seen that for the "best-case" design with all three degrees of freedom working together, a "back-off" of only 1.13% extra is required of the nominal economic optimum of $4.40 million and that this is almost entirely due to increased operating cost. The "worst-case" flexible design, where the degrees of freedom are treated as design variables, requires an increased "back-off' of 9.5% from the nominal economic optimum. This is due to an increase in capital
1000 costs, with considerable over-design in the condenser area, and to a substantially increase in operating costs.
3.2. Dynamically Operable Design Both, the "best-case" and the "worst-case" flexible designs were dynamically tested in the presence of the sinusoidal feed composition disturbance and the "slow-moving" profile for the cooling water inlet temperature which ranges between suitable lower and upper bounds. As expected there were a large number of constraints violations for both designs, and so they both require a control scheme in order for feasibility to be maintained. The control structure considered here has been proven to be stable and possible the best [10]. The control loops are (R - X d ~ (Fsteam - X b ) and (Fwate r - Pc ) where R is the reflux ratio, Fsteam is the steam flowrate and F~aterthe cooling water flowrate. Finally, x j and X bare the distillate and bottom compositions and Pc the pressure in the condenser. The dynamics equations of the PI controllers are properly incorporated into the model. No set of controller's tuning parameters (gains, reset times, set-points and biases) could be found for either design that would enable all the system feasibility constraints to be satisfied over the entire time horizon. This was conformed by solving a dynamic optimisation problem for each design where the controller's tuning parameters were selected to minimize the sum of the constraint violations. For the "best-case" design, the solution of this problem gave large constraint violations in a number of constraints, making it difficult to modify the design to make it more controllable. For the "worst-case" design the solution of the dynamic optimisation problem showed that the main operability bottleneck was due to violations in the minimum column diameter constraints (related to flooding). There was also one small violation in a thermodynamic feasibility constraint in the reboiler. Then, only the minimum column diameter modified accordingly and a new steady-state flexible design was obtained (see Table 1). It is interesting to note that the modified design naturally has a larger column diameter, the heat exchanger areas are the same and the total costs of the two designs are very similar. This illustrates how controllability is a strong function of the process design and how careful design can result in significantly controllability without necessarily having to pay a high economic penalty. Table 1" Comparison of different Designs Design Variable
Nominal
Case 1
Case 2
D (m)
6.09 280
6.09 286
6.12 325
Modified Case 2 6.3 325
SReb (m2)
417
458
498
498
0.45 3.95 4.40
0.46 3.99 4.45
0 47 4.35 4.82
0.48 4.35 4.83
SCond (m 2) Capital Cost ($ m/yr) Operating Cost ($ m/yr) Total Cost ($ m/yr)
.....
3.3. Optimal Controller Tuning Optimal tuning of the controller's gains, reset times, set-points and biases was carried out for two different cases. In the first case, the expected total annualized cost of the system over the whole time horizon was minimized. In the second case, a normalized integral square error (ISE) function was minimized in order to keep the controlled variables' values as close as to their set-points as possible. These problems correspond to large-scale dynamic optimization
1001 problems consisting of approximately 2000 variables (120 differentiable states). The system where the ISE is minimized leads to an ISE of 38 and a total cost of $4.99 million. On the other hand, the system where the total cost is minimized gives an ISE of 135 and a total cost of $4.85 million. This clearly demonstrates the potential trade-off between the quality of control and the economics of the process. A pareto curve of ISE v s . cost could be drawn up by solving the economic problem with an extra inequality constraint to bound the value of ISE. However, given the very high price one has to pay in order to minimize the ISE ($140,000 per year in utility cost), and that all the process feasibility constraints are enforced in both cases, it is logical to use the economically tuned control scheme. Furthermore, the difference in the two ISEs over 24 hours operation is not significant and this does not translate to much noticeable tighter control. Thus is better to choose the controller's tuning parameters based on the minimum cost results. Here, the total cost of the system, $4.85 million, is very close to the cost of the original steady-state flexible design, $4.83 million. By paying a slight economic penalty of $20,000 per year it is possible to obtain a fully operational design. 4. SIMULTANEOUS DESIGN AND CONTROL The sequential strategy outlined above has illustrated that interactions do exist between process design and process control. However, a more systematic approach for exploiting these interactions is to also include the process design variables as optimization variables whilst optimizing the controller tuning. The potential economic benefits of such a simultaneous approach were therefore investigated. The steps of the general mathematical framework have been presented by the work of Mohideen et. al [9] and recently in a more general approach by Bansal et. al [5]. For the example in this work where the control and column's structure is fixed the approach reduces to a dynamic optimization problem minimizing the total expected cost subject to the differential algebraic process model; the PI-control scheme equations; the disturbances profiles; and the inequality feasibility constraints. The optimization variables are the design variables (column diameter and heat exchanger areas) and the gains, reset time, set points and biases of the controllers. The problem is again solved as a large scale dynamic optimization problem with 12 optimization variables and a number of path and end-point inequality constraints describing feasible operation of the process. The optimal design, controller tuning parameters and associated costs are shown in Table 2 and compared with the results obtained using the sequential strategy with economically tuned controllers. The simultaneous strategy has the same capital costs and lower operating costs leading to a 5% total annual cost savings ($220,000 per year). It is interesting to note that the simultaneous approach is able to give a fully operational system with an annual cost in between the costs of the "best-case" and "worst case" while the sequential approach gives a system which is more expensive than the "worst-case" optimal flexible design. This clearly demonstrates how a simultaneous approach can exploit the interactions between design and control to give process designs that are cheaper and more easily controlled than those found by even state-of the-art sequential approaches. It is also interesting to observe the different control settings adopted by the two approaches. As can be seen from Table 2 and the corresponding dynamic profiles (not shown here due to space limitations) the simultaneous approach gives tighter bottom product composition that is closer to the constraint boundary; also tighter top product control and almost identical pressure control. The economic impact of this control action is the reduction in the operating costs.
1002 5. CONCLUDING REMARKS
The design and operation of reactive distillation systems was tackled through systematic approaches to design and control. The use of rigorous and high fidelity dynamic models aids in identifying potential operability bottlenecks. An economically attractive and fully operable reactive distillation and control scheme was obtained which exploits the interactions between design and control. The economic benefits that results from using such an approach are likely to increase when structural issues (for example, number of reactive and separation trays, feed tray location, control structure) are taken into consideration. The current developments in mixedinteger optimization techniques [5],[11] provide enough hope for further investigation in this area. Table 2: Comparison of sequential and simultaneous strategies Quantity
Sequential
Simultaneous
D (m)
6.3
6.37
325
315
498
425
747 1 0.54 1959 12000 0.1 -9500 0.603 1.023 0.48 4.37 4.85
181 0.59 0.527 4573 7294 0.22 - 10.400 1.32 1.023 0.48 4.17 4.63
SReb
SCond
(m2) (m 2 )
Gain, PI 1 Reset Time, PI1 Set-Point, PI1 Gain, PI2 Reset Time, PI2 Set-Point, PI2 Gain, PI3 Reset Time, PI3 Set-Point, PI3
Capital Cost Operating Cost Total Cost
ACKNOWLEDGEMENTS
Financial support from EU (HYSEP project- JOE3-CT97-0085) is gratefully acknowledge. REFERENCES
1. A1-Arfaj, M., and W.L. Luyben, lnd. Eng. Chem. Res, 39 (2000) 3298. 2. Luyben, W.L., Ind. Eng. Chem. Res, 39 (2000) 2935. 3. Sneesby, M.G., M.O. Tade, R. Datta and T.N, Smith, Ind. Eng. Chem. Res, 36 (1997) 1870. 4. Bansal, V., R. Ross, J.D. Perkins and E.N. Pistikopoulos, Journal of Process Control, 10 (2000) 219. 5. Bansal, V., J.D. Perkins, E.N. Pistikopoulos, R. Ross and J.M.C. van Schijndel, Computers and Chemical Engineering, 24 (2000) 261. 6. Schenk, M., D. Bogle, R. Gani and E.N. Pistikopoulos, Chem. Eng. Res. Des., 77 (1999) 519. 7. Pilavachi, P.A., M. Schenk, E. Perez-Cisneros and R. Gani, Ind. Eng. Chem. Res, 36 (1997) 3188. 8. gPROMS Advanced User Guide, Process Systems Enterprise Ltd, London (2000). 9. Mohideen, M.J., J.D. Perkins and E.N. Pistikopoulos, AIChE J., 42 (1996) 2251. 10. Schenk, M., Ph.D. Thesis, University of London, (1999). 11. Bansal, V., Ph.D. Thesis, University of London, (2000).
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rightsreserved.
1003
General strategy for decision support in integrated process synthesis, design and control Alexandra Grancharova Department of Engineering Cybernetics, Norwegian University of Science and Technology N-7491 Trondheim, Norway E-mail: [email protected] In this paper, a general strategy for decision support in integrated process synthesis, design and control is presented. The strategy is based on the fuzzy sets theory and is general in sense that it can be applied to various types of industrial processes. 1. D E S C R I P T I O N OF T H E G E N E R A L S T R A T E G Y F O R D E C I S I O N S U P P O R T IN I N T E G R A T E D P R O C E S S SYNTHESIS, DESIGN AND C O N T R O L The general strategy for decision support in integrated process synthesis, design and control is applied to solve the following problem: Given information on the feed stream and the desired products specifications, determine the best combination of process flowsheet, process operating conditions and process control system, so that a number of criteria are optimized, respectively order all the combinations from best to worst. The developed strategy solves this problem through comparison between different alternatives by means of fuzzy preference relations. The strategy is represented by 4 stages described in the following paragraphs. 1.1. Stage I" Determination of the ordered set of the satisfactory alternatives of process
flowsheets. This stage includes the following main steps:
1). Determine the set F = {F~,F2, ... Fn} of all alternatives Fi of process flowsheets that can be used to solve the formulated problem. This can be achieved by applying wellknown methods for synthesizing process flowsheets, for example the method described in [ 1] can be used for separation process synthesis. 2). Rank all alternatives of process flowsheets. It should be noted that at this stage of the strategy, no process design is performed yet and in order to rank the alternatives only qualitative criteria are used. They are: -
criterion !pq.(Fi) for product q u a l i t y - it is suggested for this criterion to take the linguistic values "low", "medium" and "high";
-
criterion Iflowsheetcost(El ) reflecting the costs needed to realize the respective flowsheet - it represents only an approximate evaluation of the costs and it is suggested to take the same linguistic values as above criterion;
1004 criterion lecoZ"(F i) reflecting the impact on the environment of the respective flowsheet - it also takes the linguistic values proposed above depending on whether there are hazardous substances in the flowsheet and how dangerous they are; - criterion laine ( ~ ) reflecting the time needed for process start-up. The alternatives are ordered from best to worst by introducing four types of fuzzy preference relations: -
-
preference relation
QI,.,. = { ( F ~ , F j ) . F~ ,Fj ~ F, lzz,,. ( F z,Fj)} which shows how
"good" the alternative 17. is compared to alternative Fj with respect to product quality criterion; -
-
preference relation Q,,,~h...... = {(F z,Fj )" F z,Fj ~ F, lZz,o.,h...... ( F i , F j ) } - w.r.t, costs; preference relation Qz,,o,. = {(F~,Fj) : F i,Fj ~ F, lz I..... ( F z,Fj)} - w.r.t, impact on the environment; preference relation Q,,,., = {(F t , F j ) : F i , F j ~ F,/z,,~., (F~,Fj)} - w.r.t, process startup time
and /.tz,+ ( ~ ,Fj ), ~zzj,~' ...... ( F i ,Fj ), /zZ.,o,.( F i ,Fj ) and /zz~., ( F i ,Fj ) are the respective fuzzy membership functions. It is proposed for these functions to be determined in the following way (here, only the expression for lzz,,.(Fz,Fj) is given, since the others are similar):
O,
if (Ip.q.(Fi)="low" /~ Ipq.(Fj)="medium") v (Ipq. (Fi) ="medium" /~ Ip.q. (Fj) ="high") It
tt
tt
0.5, /f (/p.q.(F~) = low A Ip.q.(Fj)= low ) v 9
It i,,, ( Fi, Fj ) =
It
(I p q. ( F~ ) =" medium" ^ I p.q. ( Fj ) =" m e d i u m ") v I t
9
tt
It
9
( 1)
It
(Ipq ( F i ) - h,gh ^ Ip.q.(Fj)= high ) 1,
if (Ip.q.(Fi)="medium" /x Ip.q.(f j):"low") v (Ip.q.(Fi) ="high" /x Ip.q (Fj)="medium")
Then, a well-known algorithm [2] is applied to determine the fuzzy subset of nondominated alternatives with membership function/z "a ( F z) and the alternatives are ordered from best to worst in decreasing order of kz "a ( F i ). 3). Determine the ordered set of the satisfactory alternatives for process flowsheets by taking only those alternatives Fi whose it "a ( F i ) is greater than a specified value. This set is a sub-set of the set of all alternatives, however it will be expressed in the same way in order to avoid numerous notations. All further steps of the decision strategy will take into account only these satisfactory alternatives.
1005 1.2. Stage II: Determination of the ordered set of satisfactory alternatives of process design. This stage includes the following main steps: 1). For each satisfactory altemative Fi of process flowsheet determined in stage I, determine the respective optimal design Di=D(Fd (the optimal operating conditions, the optimal sizing parameters). For this purpose the following quantitative criteria have to be optimized: - criterion Ip.q. (D i) for product quality;
-
criterion lflowsheetcost ( D i )
-
criterion Iecol"(D i ) reflecting the impact on the environment;
for costs;
- criterion Itime (D i ) for process start-up time. These criteria are named process design criteria and generally they can have different expressions depending on the task to be solved. For example, the product quality criterion Ip.q.(Di) can include both product purity and product rate. The cost criterion
Iflowsheetcost(Di ) represents the sum of the process equipment costs and the energy costs. The ecological criterion Iecol"(D i ) can have the following expression:
Nu Iecot (Di) = E wj . H fl~ 9
j=l
(D i ). Hco. c j(Di ) "' Hflow,j
"Hconc.,J
,
(2)
where Hflow,j(D i ) and Hco,,c.,j(Di) are respectively the flowrate and the concentration of the j-th hazardous substance produced by the flowsheet Fi with associated optimal design Di, r
H~ow, j and Hco,c.,j are the allowed flowrate and concentration of this substance, wj is a weighting coefficient reflecting the degree of dangerousness of the j-th substance and it takes values between 0 and 1, NH is the number of the hazardous substances. Then, the problem of optimal design determination represents a multiple-criteria optimization problem (MCOP): max Ip q ( D i ) D,
""
min Di l flowsheetcos t ( D i ) min leco, (D i ) Di
(3)
min laine ( D i ) Di
The MCOP (3) can be solved by applying standard techniques like finding Pareto-optimal solutions, scalarizing the problem through introducing utility function or function of losses, etc. In this way, a set D F = {(F1,D1),(F2,D2) .... ( F . , D , ) } of alternative couples of process flowsheet and respective optimal process design is obtained. 2). Rank the alternatives (Fi,Dd. This is done in a similar way to that described in stage I of the strategy. 3). Determine the ordered set of the satisfactory alternatives of the couples of process flowsheet Fi and process design Di by taking only those alternatives (Fi, D~ whose pn.a. (F~ ,D i ) is greater than a specified value. The next step of the decision strategy will take into account only these satisfactory alternatives.
1006 1.3. Stage III: Determination of the ordered set of satisfactory alternatives of process control system. This stage includes the following main steps: 1). For each satisfactory alternative (Fi,DO of process flowsheet and process optimal design determined in stage II, determine the set
C(~.D,) ={CI(F'i,Di),C2(F'i,Di), ... Cu(Fi,Di) } of all alternatives Cj(Fi,D~ of control systems. These control system alternatives result from the fact that for each process flowsheet and process design, different control structures (selection of input-output variables) and different types of controllers (PID, IMC, fuzzy, neural, model predictive controllers) can be applied. In order to generate the control system alternatives, the methodologies given in [3,4] can be used. 2). For each satisfactory alternative (Fi,D~) of process flowsheet and process design, rank all alternatives of process control systems Cj(Fi,Dd. In order to rank the alternatives the following two types of quantitative criteria can be used: A). Control quality criteria: - criterion Isse(C j ) for steady state error; -
criterion Iue (Cj) for integral-squared error;
-
criterion lame( C j ) for settling time;
-
criterion lovershoot(Cj ) for overshoot.
13). Cost of control equipment: criterion Icontr.eq"cost(Cj ). All these criteria are named process control criteria. Ranking of control system alternatives can be done by the decision support system presented in [5]. 3). For each satisfactory alternative (Fi,Dd determine the ordered set of the satisfactory alternatives of process control systems by taking only those alternatives Cj(Fi,Dd whose ,1.1n'd'[cj(.Fi,Di)] is greater than a specified value. 1.4. Stage IV: Determination of the best combination of process flowsheet, process design and process control. As result of the stages I, II and III, a set of different satisfactory combinations of process flowsheet, process design and process control is obtained. Each combination is represented as FDCk = [Fi, Di, Cj(Fi,DO], where Fi is a satisfactory alternative of process flowsheet, Di is its respective optimal process design and Cj(Fi,Dd is a satisfactory alternative for control system corresponding to Fi and Di. Thus the set of combinations is denoted as FDC = {FDC~,FDC2, ... FDC N }. The combinations FDCk are ordered from best to worst by applying a similar method to that described in stage I of the strategy (by introducing fuzzy preference relations). The combinations are compared with regard to simultaneous optimization of the two groups of criteria: process design criteria (given in stage II) and process control criteria (given in stage III). The result of the strategy is that all satisfactory combinations of process flowsheet, process design and process control are obtained and they are ordered from best to worst.
1007 2. EXAMPLE. As an example, the problem of selecting the best alternative of process flowsheet for production of Triple Super Phosphate (TSP) is considered. It is shown in [6] that there are 3 alternative flowsheets for production of TSP: A l t e r n a t i v e 1: Den process for production of granular TSP (Fig. 1); A l t e r n a t i v e 2: Ex-den direct granulation of TSP. The flowsheet of this alternative is similar to that of alternative 1 except that the stage of storage curing of TSP is excluded and therefore it is not shown here; A l t e r n a t i v e 3: Slurry-type process for production of granular TSP (Fig.2). These three alternatives are compared by applying the approach given in section 1.1. The linguistic values of the optimality criteria for each alternative are given in Table 1. Results of the comparison are given in Table 2, where two possible cases are considered. In case 1, the first three criteria are accepted to be equally important to each other and much more important than the last criterion. In this case, A l t e r n a t i v e 3 is the best one. In case 2, the costs are considered as the most important criterion and for this case A l t e r n a t i v e 1 is the best one. Table 1. Linguistic values of the optimality criteria for the different alternatives.
~
iteria Productquality (11) Alternatives ~ "low" Alternative 1 Alternative 2 Alternative 3
"medium" "high
Costs (12)
Impact on Start-up time environment (13) (14)
"low"
"high"
"big"
"medium"
"medium"
"medium"
"high"
"low"
"small"
Table 2. Order of alternatives from best to worst for different cases. Weighting coefficients for Order of alternatives from Cases optimality criteria best to worst wt = 0.3125 9 w2=0.3125 Alternative 3 Case 1 w3=0.3125; w4=0.0625 Alternative 1, 2 w1=0.2 9 w2=0.6 Alternative 1 Case 2 w3=0.1 w4:0.1 Alternative 2, 3 9
REFERENCES 1. C. Jaksland, R. Gani and K. Lien, Separation process design and synthesis based on thermodynamic insights, Chem. Eng. Sci., 50 (1995) 511. 2. S. Orlovski, Decision making with a fuzzy preference relation, Fuzzy Sets and Systems, 1 (1978) 155. 3. C. Ng and G. Stephanopoulos, Synthesis of control structures for chemical plants, IFAC Symposium on Large Scale Systems: Theory and Applications, Rio Patras, Greece, 15-17 July (1998) 16. 4. J. B. Jorgensen and S. Bay Jorgensen, Plantwide control structure selection, Process Systems Engineering'2000, Colorado, USA, 16-21 July (2000). 5. A. Grancharova and J. Zaprianov, Decision support system for rating control alternatives, 16th IMACS World Congress'2000, Lausanne, Switzerland, 21-25 August (2000). 6. I. Karshev, I. Gruncharov, P. Bozadjiev and F. Tudjarova, Production of phosphate fertilizers, Intelteh-3, Sofia (1992).
1008 Phosphate Rock(PR)
highlyconcentrated H~P04(~---93%) H20
[grinding
I ~ H~P04(~68%)
I
[ fl. . . . ter
do,a~o I
[
I I
..... '
I
~slurry I don I freshTSP ~ .. I storagecuring
of TSP I non~granularTSP
[ H20
neutralizingsubstance (hmestone)
mixing I ]neutralizedTSP
~ ,~ recycle I granulation I (fra...... lmm)
"~
TSP(dust)
I wetgranules ~(16-18%moisture)
fuel "~1
~,io~
vI
I
~ dus:ry~t~
~~
drygranules
f~....... 5mm
~
~176 (fraction
1-5mm)
I
gases~ '~ H20
[ro~mm~v::Wg~u=,In~~ o~.... dg.... I I=~=: .... (to atmosphere); etreatmentplant)
Fig. 1. Alternative I: Flowsheet of the den process for production of granular TSP (from [6]). Phosphate Rock(PR) H~P04 (38-39%P:O0 [
'~ grinding
I
In.... terl I dosage ] PR ~ decompositionof PR m 2-3 cascade reactors (T=90-100~
[
steam
exhaustedgases (to atmosphere) ~ A
,ls u;,
TSP(dust)
mixmgand I granulation lwet granules naturalgas air ~ ,,q
i
drying
[ gases
1
H:O
T'
fume scrubberfor removingfluorine gases [ dUSts~vlng r l
(tSUwas~St~ tr. . . . plant)
!
I
drygranules
product (fraction1-5ram)
Fig.2. Alternative 3: Flowsheet of the slurry-type process for production of granular TSP (from [6]).
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jmgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1009
Static and dynamic optimisation using a Model Testbed J.P. Henriksen and B.M. Russel CAPEC, Department of Chemical Engineering, Technical University of Denmark, DK-2800 Lyngby This paper presents a new computer aided tool for setting up and solving CAPE related static and dynamic optimisation problems. The Model Testbed (MOT) offers an integrated environment for setting up and solving a very large range of CAPE problems, including complex optimisation problems in the different stages of the process lifecycle. The usefulness of MoT is illustrated through a case study, which highlights the design and optimisation of a continuous stirred tank reactor as steps in the process model lifecycle. It is demonstrated how this integrated environment is used for both static and dynamic optimisation, and how interfacing of solvers and seamless information flow can lead to more efficient solution of process design problems. 1 INTRODUCTION When considering the models during the lifecycle of a process, the requirements, goals, solution techniques and manipulative needs of the process model, changes with the age of the process. The process design phase of the lifeSimulation'~ cycle could thus require a solution of the Engine process model in steady state mode, dynamic mode and/or optimisation mode, depending on the active stage in the lifecycle. A modelling environment that allows such flexible and ef! ficient use of the mathematical model throughout the process lifecycle deserves further investigation.This paper will present the Models idea, the structure and functionality of such an ~ntegrated tools library environment and illustrate how this tool can assist designers at different stages of the process lifecycle in terms of model development I f .~ WI--J "-4 -o,o.~176 and use of optimisation techniques early in the / t ~.ratbi~ I I ""-r "~ design phase. The Model Testbed (MOT) is an IM "" -'/ - l I ~Thermodynamic/ integrated environment to build, analyse, manipulate, solve and visualise mathematical models. The aim of MoT is to provide an Figure 1" Model Testbed in relation to the simulaintuitive, flexible and efficient way of tion engine and the common integrated libraries integrating different aspects of the modelling needs. One such example is the transition from process model development (the initial
1010 parameter estimation to determine model parameters) to the dynamic optimisation solution (in order to determine optimal operational parameters), via an intermediate solution in the steady state mode. In order to provide the necessary flexibility and robustness for the solution of the model equations, MoT is linked with a range of solvers, a properties database and a library of property models. Through an application example, it will be demonstrated how one model can be manipulated and solved at different stages of the process lifecycle without changing the core model equations, and how the results from one stage are utilised later in another stage during the various phases of process (design) lifecycle. A workflow model of the design process will illustrate the different stages and roles involved in the design process. 2 M O D E L TESTBED (MOT) The main feature of MoT is the text based mathematical equations, representing the model. The model equations can [ either be imported from a model generation tool or written directly by the user. The typical solution of the model equations, in any stage of the process lifecycle, involve the following steps: Translation, Analysis, Solution and Integration. The Translation step dissects the textbased equations using a Reverse Polish Model equations
]
OriginalEquatl....
I
I
Pre.... forvahdlty
I
Relationstups L~
[
VariableExpansion
II
"' EquationExpansion
"]
Expanded Equalaon Set Defined
LHSClassification RHSClasslficaUon ModelReadyToSolve
I
Figure 3" Translation algorithm Notation (RPN) algorithm and classifies 4l" I equations and variables using a rulesSolution based classification system. The Translacomplete tion algorithm, illustrated in Figure 2, / I Variable [., .. I starts by scanning the text based equaupdate V tions for validity to ensure mathematical consistency. After the equations have ..... '- !- ..... been validated, each equation and variSolver library Ordering Equation able is expanded to match the current size translation of the problem, usually dependant on the , L ............... number of compounds present in the system. The expansion step ensures that only Figure 2: Solution mode strategy in MoT the core model equations needs to be specified and the model is thus not fixed or limited to a specific condition. Should the condition change, all one needs to do is the repeat the translation step once more. Once the equations have been expanded, giving the actual equations to be solved, the variable classification Administrator
t
f
1011 step is invoked. The classification step determines the base variable layer for each variable and sets per default all right hand side variables as parameters. The user must afterwards classify the variable to suit the model needs. There are essentially four layers in the classification of a variable: 1. Left-Hand-Side / Right-Hand-Side variables 2. Variable type (Explicit, Parameter, Unknown, Dependant, Dependant Prime) 3. Optimisation Layer (Manipulated variable, Constraint Value) 4. Data Type (Real, Integer, Binary) The Analysis step consists of a number of tools and algorithms, including: Generation/analysis of incidence matrix, check for singularity of matrix, equation trace-back, equation-by-equation solution mode, decomposition, partitioning and ordering of the model equations. Furthermore a degree of freedom analysis is automatically performed to ensure that the problem is not ill-posed before going to the solution step. The general strategy in the Solution step in MoT (see Figure 3) can be divided into three parts: 1) Administration of the solution proLocal Optimizer administrator cedure. This part drives the solution Check lists U Set initial procedure. 2) Solution of the text-based equations: This part solves the translated equa~.-J SOP ~l~ tions. 3) Solver perturbation and iteration: Update design vadabis list This part is linked to the solver liTime Function Evaluation brary and could, for instance, use a Local dlfnamic administrator Newton-based technique to deteri Ch.k I Upd~e*in I / Design I Mod,I. I variablelist | Create local I mine the unknown variables of a J Jand update if I ~ I process model. i
The administrator of the solution procedure can be divided into two parts: A global administrator and one or more local administrator(s). Each local administrator instantiates one or more solvers from the solver library. The possible local administrators are: Algebraic administrator Dynamic administrator 9 Optimisation administrator 9 9
I
j SolvelA~g I
Until J Takea step
J via local J
I administratorJ J ,
J
T Store
constraint values in list
UpdatedYdT T
Figure 4: Structure of local administrators instantiated for a dynamic optimisation problem
The global administrator combines all the required local administrators needed to solve a given problem and handles the overall solution sequence. The local administrators are only connected to the model via the global administrator. What makes MoT flexible and powerful is that it from the model classification automatically detects, which of the above local administrators can be applied to a given model, and that the user can choose any of these, e.g. a dynamic optimisation problem with time-variant variables can also be solved in a steady state simulation mode by just changing one solver argument.
1012 An example of the interfaces used and links instantiated by the global administrator, the solution of a dynamic optimisation problem, is illustrated in Figure 4. In this example a local dynamic administrator and a local optimisation administrator is instantiated by the global administrator. For dynamic optimisation problems, manipulated variables and constraints are in principle divided into two categories: Point (time in-variant) and Path (time variant). Path variables and constraints are handled using intervals. The total integration horizon is divided into a number of intervals, where the controlled variables (path) can be manipulated as piecewise constants, piecewise linear or continuous piecewise linear. The controlled variables can either be fixed or specified as optimisation variables. These features provide the flexibility needed in order to introduce transients and disturbances as well as optimisation problems during the determination of an Optimal Control strategy. 3 APPLICATION EXAMPLE
3.1
Problem Description
The usefulness of MoT can be explored when looking at the modelling needs at different stages of the process lifecycle. Using MoT, the core model can be built and gradually expanded during the process lifecycle for use by different experts in order to generate and pass the results from one stage to another. The scenario selected, to illustrate how a Model Testbed can assist at several stages of the process lifecycle involves a reactor retrofit design problem, which has its roots in an industrial process. The reaction involves a complex reaction model with 3 reactants, 1 product and 10 intermediates/by-products. The model involves 4 equilibrium reactions and 7 kinetic reactions. The temperature dependency is modelled using Arrhenius expressions. One main concern is to keep by-product formation low as the cost of disposal is known to be high. The scenario at different stages of the process (design) lifecycle is modelled using a C3formalism [ 1] and is described in terms of "roles", activities, information and tools (see Figure 5). The roles involved in the scenario are: Business Unit Management, Process Systems Engineering, Laboratory, Control Expert and Plant Manager. Process Systems Engineering has the role to provide continuity of the model and to collect information and distribute to the other roles at the different stages of the lifecycle. MoT is used to build, manipulate and solve the model throughout the process lifecycle. As shown in figure 5, the different stages of the process development lifecycle, requires different integrated solvers and solution strategies for the corresponding model at each stage.
3.2
Process Objective Definition and Model Development
The objective of the problem to be solved in this scenario is to determine the optimal size and operating conditions in the retrofit of an existing CSTR reactor. The target presented by the Business Unit Management is to increase production rate by 20%. The reactor has not previously been modelled so a kinetic model is needed. The kinetic model is generated in cooperation between the laboratory and the process systems engineering (PSE) department. The experiments are designed and performed, and from the experimental data, kinetic model parameters are estimated. In this stage, an Arrhenius type model is used to model the temperature dependency of the reaction rates. (1) k = A * exp(-E,~ / RT) rn = k , H W t m
(2)
1013 Laboratory
Control Expert
Process Systems Engineering
Business Unit Management
Plant Manager
Definition of Process Objective and allocation of resources
Meassured Data
Controlabihty issues Control variables Meassured vanables
Target values Revenue functions Economical constraints ...
,,"
,
"
;,," ,"
II L I
Ion Optimal and .~."1 "~ L I Flex,hie design I ...."~'-F
Business Decision on Continuation with Detailed Desi~in
[ ~'~
~
Set-points for I / /r I control vars ]...... 1. .._ I . I Values for .... m
~
........
Inital choice of control structure
i , /" Oblective function 1,.,~" /'- ~ -
va.a~,es
'
Dynamic Analysis
~176
Cost functions Operatbihty constraints
I~1 I
I
.I
"
I. I
/
Dynamic | / Analysis
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,L
T
!
"
Dynamic
"x~
-
-. 9 ~-- .I I. . . . I o~t,~,...... ,ex,~,e I...[ ....... ~ IOptImIsat,on-~., ~Opt,misat,on lanad'~:;i;:a~ -- L I P,Control ~ LI P, Contro,
Evaluation of "~
1 9-4
~8 Evaluationof ~
.=
I
/
) Eva es,0n ua'~1761
Figure 5" Workflow model of Reactor Retrofit illustrating the solution methods used at each stage of the model development. The model and parameter values are brought from one stage to the next without information loss within the Model Testbed framework
1014 The CSTR reactor can be modelled in steady state and dynamic by the following set of equations: D = Feed(i) + Recycle(i) - Output(i) + Weight * r(i) (3) Where D=0 for steady state and D= dn/ for dynamic. dt
The separation system is treated as a black box in this study, and from the plant it is estimated that around 98% of the reactants and recyclable intermediates can be recovered and recycled. This is modelled by: Recycle(i) = RecVector(i) * Output(i) (4)
3.3
Steady State Simulation and Optimisation
Steady state simulations, made with plant operating parameters, are performed to validate and compare the quality of the kinetic model to the plant data. In this case the plant data and laboratory data does not correlate fully, so optimisation results based on this model should be treated with caution. Steady state simulations are useful for both design and analysis of the reactor. The MoT model can be exported to a flowsheet simulator format [3], and thus enable a full-scale plant analysis. The exported model is used by e.g. the control expert in order to make a first judgement on future controllability issues arising from the analysis of the simulation results [4]. This information is passed back to the process engineer who can then concentrate the modelling effort in order to accommodate these operational issues. Steady state analysis is also important for reactor optimisation. The process engineer specifies the needed cost functions and requests information on operational constraints, business constraints, legislatory constraints, target values and possibly revenue functions. The information is supplied by business unit management, the plant manager and the control expert. In this case information for the following optimisation problem is available: min Cost = ReactorCost + SeparationCost + DisposalCost s.L
ProductRate = 25 kmole/h Temperature E [530K, 555K] ReactorDiameter E [1 m, 3 m] Height/Diameter E [1,31
(5)
FeedComp
Z Xi=I i=1
ReactorCost = 25.04 * Diameter~~ SeparationCost = $400 / t reactants DisposaICost = $512 / t light by-product DisposaICost = $ 6 1 0 / t heavy tars
~176 ( $ / h )
The steady state optimisation variables are: Reactor height, reactor diameter, temperature, feed flow rate, feed composition The static optimisation is performed in MoT by adding the extra equations and identifying the optimisation variables and constraint equations. The resulting non-linear optimisation problem is solved using a Sequential Quadratic Programming (SQP) solver from the solver library. Other solvers can also be interfaced and used.
1015
3.4
Dynamic Simulation, Optimisation & Control
After the optimal sizing and operating conditions have been found through static optimisation, a dynamic analysis is performed. The objective of the dynamic analysis is to determine a flexible design, which incorporates the dynamic effects in the system [2]. In this study, the analysis falls in three stages: Flexible design, optimal control and implemention of PI control. The solver in MoT is switched to dynamic mode and the dynamic problem is initialised and solved. The problem is extended to incorporate dynamic effects. This stage involves communication between the process engineer, the control expert and the plant manager. It is identified that an extemal disturbance causes the separation system to oscillate. In order to represent that in the model, we, for the reason of simplicity, introduce the following function as a worst case: Recycle(i) - RecVector * Output(i) * (1 - 0.1 * sin(0.1 * time)) (6) For safety reasons the level in the reactor must not exceed 95% of the reactor height. In this case a shutdown procedure is activated. Furthermore, it is decided that it is acceptable for the product rate to vary up to 2% from the target value. We include two extra path constraints: Level (7) < 0.95 Height
24.5 < ProductRate < 25.5
(8)
First, we identify a flexible and optimal design, i.e. the optimal set of design values and set-points which satisfy the constraints at all times assuming a given disturbance. The Flexibility problem is set up and solved as a dynamic optimisation problem over a time horizon of 100 hours. The flexible design represents a 'worst case scenario', which may now be improved by introducing a control system. Following the workflow model (Figure 5), communication between the process engineer, the control expert and the plant manager leads to a selection of measured and controlled variables. Then, secondly, the Optimal Control problem is solved, i.e. the controlled variables are treated as time-variant optimisation variables and modelled as piecewise constant functions. The chosen optimisation variables are: 9 Design variables (time in-variant): Diameter, Height, Feed comp., Temperature 9 Controlled variables (time variant): Feed flow rate, Valve constant Finally, we introduce two PI controllers in the model, pairing the meassured level in the tank with the feed flow rate, and pairing the meassured outflow rate with the valve constant: LeveIERR = L e v e l - LevelSetPt (9) OuO'7owERR = OuU7ow - O u ~ o w S e t P t (10) FeedFlow = Kcf *( 1 + 1/~ *LevelERR ) ValveConst = Kcv * ( 1 + 1/Tv * OuOTowERR )
(11) (12)
The objective function is changed to give as tight control as possible, i.e. minimise the accumulated deviation between actual values and setpoints for level and outflow. As optimisation variables we choose reactor height and the controller settings: Kcf, Kcv, '~f and Xv. Using dynamic optimisation we then determine the optimal controller settings.
3.5
Results
The results from the different steps in the reactor optimisation are shown in Table 1. It is seen from the results that the dynamic optimisation results gives a good starting point for the
1016
flexibility study and control design and controller tuning, thus illustrating the need for a seamless information transfer. For PI control of the outflow it is seen that it for the given disturbance is optimal only to have a contoller on the feed flow.
Variable
Dynamic Optimisation
Flexible Design
Optimal Control
PI Control
Objective 2015.09 2015.51 2014.29 Diameter 3.00 2.897 3.00 3.00 Height 3.00 3.527 3.0398 3.008 Controlled Controlled FeedFlow 110.804969 109.730355 x(1) 0.3254 0.326705 0.325445 0.325445 x(2) 0.416 0.415 0.416 0.416 x(3) 0.257 0.258 0.257 0.257 Temperature 530 530 530 530 ValveConst 32.361614 29.70 Controlled Controlled Volume 21.892297 21.993542 22.01 21.90 Kcf 80.0 Kcv 0 xf, 0.50 Xv inf Table 1: Optimisation results (the cells in italic have not been used as optimisation variables) 4 CONCLUSIONS This paper has described a Model Testbed (MOT), which is a multipurpose tool for static and dynamic simulation and optimisation. The structure of MoT has been outlined demonstrating how interfaces are instantiated in the integrated model testbed tool. The usefulness of a model testbed has been demonstrated by relating the tool to various stages of the process lifecycle. In a reactor design case study the model testbed was used from earliest model development to the determination of flexible and controllable design of the reactor. By solving the model with increasing complexity within the same tool ensures seamless data transfer and less risk of error. It is shown how proper information and data flow in relation to the process design lifecycle leads to a more efficient design and a better approach to the integration of various design issues.
ACKNOWLEDGEMENTS Jens Peter Henriksen would like to thank the Global CAPE-Open project for the financial support.
REFERENCES 1. Killich, S., Luczak, H., Schlick, C., Weissenbach, M., Wiedenmaier, S., Ziegler, J., 1999, "Task modelling for cooperative work", Behavior&Information Technology,18,p.325-338 2. Mohideen, J., Perkins, J.D., Pistikopoulos, E.N., 1997, "Optimal Design of Dynamic Systems under Uncertainty", AIChe Journal, 42, 8, p. 2251-2272 3. Gani, R, Hytoft, G., Jaksland, C., Jensen, A. K., 1997, An integrated computer aided system for integrated design of chemical processes, Comp. & Chem Engng, 21, 1135-1146 4. Personal communication: Friedrich, M., Bayer AG, Leverkusen, Germany, 2000
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1017
A P r o t o t y p e S y s t e m for E c o n o m i c , E n v i r o n m e n t a l and Sustainable Optimization of a Chemical Complex T. A. Hertwig a, A . XUb, A. B. Nagy c, R. W. Pike b, J. R. Hopper a and C. L. Yaws d a Kaiser Aluminum and Chemical Company, Baton Rouge, LA 70809 b Louisiana State University, Baton Rouge, LA 70803 c University of Veszprem, Veszprem, Hungary d Lamar University, Beaumont, TX 77710 A prototype of a chemical complex analysis system has been developed and used to demonstrate optimization of a chemical complex. The system incorporates economic, environmental and sustainable costs, and solves a MINLP for the best configuration of plants. It was applied to expanding production of sulfuric and phosphoric acid capacities and to evaluating heat recovery options at a major chemical company, and the results were compared to the company's case study. The system selected the best site for required new phosphoric and sulfuric acids production capacities and selected, sited, and sized the optional heat-recovery and power-generation facilities. System capability was demonstrated by duplicating and expanding the industrial case study. 1. INTRODUCTION The business focus of chemical companies has moved from a regional to a global basis, and this has redefined how these companies organize and view their activities. As described by H. J. Kohlbrand of Dow Chemical Company (Kohlbrand, 1998), the chemical industry has gone from end-of-pipe treatment to source reduction, recycling and reuse. Pollution prevention was an environmental issue and is now a critical business opportunity. Companies are undergoing difficult institutional transformations, and emphasis on pollution prevention has broadened to include tools such as Total (full) Cost Assessment (accounting) (TCA), Life Cycle Assessment (LCA), sustainable development and eco-efficiency (economic and ecological). At this point in time there is no integrated set of tools, methodologies or programs to perform a consistent and accurate evaluation of new plants and existing processes. Some of these tools are available individually, e.g. TCA and LCA, and some are being developed, e.g. metrics for sustainability. An integrated analysis incorporating TCA, LCA and sustainability is required for proper identification of real, longterm benefits and costs that will result in the best list of prospects to compete for capital investment. Chemical companies and petroleum refiners have applied total cost accounting and found that the cost of environmental compliance was three to five times higher than the original estimates (Constable, et. al., 2000). Total or full cost accounting identifies the real costs
1018 associated with a product or process. It organizes different levels of costs and includes direct, indirect, associated and societal. Direct and indirect costs include those associated with manufacturing. Associated costs include those associated with compliance, fines, penalties and future liabilities. Societal costs are difficult to evaluate since there is no standard, agreed-upon methods to estimate them, and they can include consumer response and employee relations, among others (Kohlbrand, 1998). The Center for Waste Reduction Technology (CWRT) of the American Institute of Chemical Engineers (AIChE) recently completed a detailed report with an Excel spreadsheet on Total Cost Assessment Methodology (Constable, et. al., 2000). The report was the outgrowth of industry representatives working to develop the best methodology for use by the chemical industry. The AIChE/CWRT TCA program uses five types of costs. Type 1 costs are direct costs for the manufacturing site. Type 2 costs are potentially hidden corporate and manufacturing site overhead costs. Type 3 costs are future and contingent liability costs. Type 4 costs are internal intangible costs, and Type 5 costs are external costs that the company does not pay directly including those born by society and from deterioration of the environment by pollution within compliance regulations. This report states that environmental costs made up at least 22% of the nonfeedstock operating costs of the Amoco's Yorktown oil refinery. Also, for one DuPont pesticide, environmental costs were 19% of the total manufacturing costs; and for one Novartis additive these costs were a minimum of 19% of manufacturing costs, excluding raw materials. Also, external costs are the very difficult to quantify, and this report gives some estimates for these costs from a study of environmental cost from pollutant discharge to air from electricity generation. In addition, this TCA methodology was said to have the capability to evaluate the full life cycle and consider environmental and health implications from raw material extraction to end-oflife of the process or product. Sustainable development is the concept that development should meet the needs of the present without sacrificing the ability of the future to meet its needs. An effort is underway to develop these metrics by an industry group through the Center for Waste Reduction Technology of the American Institute of Chemical Engineers, and they have issued two interim reports (Adler, 1999) and held a workshop (Beaver and Beloff, 2000). 2. P R O T O T Y P E SYSTEM FOR OPTIMIZATION OF A C H E M I C A L C O M P L E X Combining economic, environmental and sustainability costs with new methodology for the best configuration of plants is now feasible. The analyses and components exist. This paper describes the prototype system shown in Figure 1 that combines these components into an integrated system for use by plant and design engineers. They have to convert their company's goals and capital into viable projects that are profitable and meet environmental and sustainability requirements and have to perform evaluations for impacts associated with green house gases, finite resources, etc. This program can be used with these projects and evaluations and also can help demonstrate that plants are delivering environmental, social and business benefits that will help ameliorate command and control regulations. The system is being developed in collaboration with engineering groups at Monsanto Enviro Chem, Motiva Enterprises, IMC Agrico and Kaiser Aluminum and Chemicals to ensure it meets the needs of the chemical and petroleum refining industries. The prototype incorporates TCA methodology in a program from the AIChE/CWRT Total Cost Assessment Methodology (Constable, 1999) which provides the criteria for the best economic-
1019
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Figure I Program Structurefor the Chemical ComplexAnalysisSystem
Chemical Complex Analysis System is shown in Figure 1. The system incorporates a flowsheeting component where the simulations of the plants in the complex are entered. Individual processes can be drawn on the flowsheet using a graphics program. The plants are connected in the flowsheet as shown in Figure 2. For each process material and energy balances, rate equations, equilibrium relations and thermodynamic and transport properties are entered through windows and stored in the database to be shared with the other components of the system. Also, the economic model is entered as an equation associated with each process with related information for prices, costs, and sustainablity metrics that are used in the evaluation of the Total Cost Assessment for the complex. The TCA component includes the total profit for the complex that is a function of the economic, environmental and sustainable costs and income from sales of products. Then the information is provided to either GAMS/DICOPT or SYNPHONY for solving the Mixed Integer Nonlinear Carbon Programming (MINLP) problem dioxide ,~ Ammonia for the optimum configuration of Carbon | emissions Carbon dioxide dioxide ,._1 / emissions plants in the complex. Also, the Urea I :1 ~v Urea "q Plant sources of pollutant generation Ammonium are located by the pollution index > A . . . . i,, ,~IA . . . . lum Phosphate Natural Gas I Ammonia Phosphate ._1 >i Plant Steam component of the system using Air Pl~t the EPA pollution index Hydrogen fluodde Sdlcontetraflodde methodology (Cabezas, et. al.,
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All interactions with the system are through the graphical user interface of the system that is written in Visual Basic. As the process flow diagram for the complex is prepared, equations for the process units and
1020 variables for the streams connecting the process units are entered and stored in the database using interactive data forms as shown on the left side in Figure 1. Material and energy balances, rate equations and equilibrium relations for the plants are entered as equality constraints using the format of the GAMS programming language that is similar to Fortran. Process unit capacities, availability of raw materials and demand for product are entered as inequality constraints. Features for developing flowsheets include adding, changing and deleting the equations that describe units and streams and their properties. Usual Windows features include cut, copy, paste, delete, print, zoom, reload, update and grid, among others. A detailed description is provided in a user's manual The system has the TCA component prepare the assessment model for use with determination of the optimum complex configuration. The AIChE/CWRT TCA program (Constable, D. et. al., 2000) is an Excel spreadsheet that has the cost in five types, as describe above. This Excel spreadsheet is an extensive listing of all possible costs. The TCA component combines these five categories of costs into three costs: economic, environmental and sustainable. Types 1 and 2 are included in economic costs, Types 3 and 4 are included in environmental costs, and Type 5 is sustainable costs. Economic costs are estimated by standard methods (Garrett, 1989). Environmental costs are estimated from the data provided by Amoco, DuPont and Novartis in the AIChE/CWRT report. Sustainable costs are estimated by the study of power generation in this report. It is an on-going effort to refine and update better estimates for these costs. As shown in Figure 1, the system will provide an option to select one of two optimization methods. Determining the optimal configuration of plants in a chemical complex is a mixed integer nonlinear programming problem where the equality and inequality constraints include material and energy balances, process unit capacities and others as described above. There are two methodologies to solve this type of optimization problem, GAMS/DCOPT and SYNPHONY. GAMS (General Algebraic Modeling System) was developed at the World Bank for very large economic models, and it can be used to determine the optimal configuration of a chemical complex by solving a M1NLP programming problem using the DICOPT solver (Kocis and Grossmann, 1989). SYNPHONY uses process graph methodology based on the work of Friedler and Fan (Friedler, Varga and Fan, 1995) to solve the MINLP problem. 3. AGRICULTURAL CHEMICAL COMPLEX EXPANSION VALUATION A major agricultural chemical company had performed a case study for expanding production of sulfuric and phosphoric acid along with heat recovery options at two locations that ten miles apart. This multiple site, multiple-plant expansion was used with the prototype system, and the results compared to the case study for validation of the prototype. In this complex, phosphate fertilizers are produced by reacting ammonia and phosphoric acid as illustrated in Figure 2. Phosphoric acid is made by digesting phosphate rock with sulfuric acid. Sulfur, air, and water are used to make sulfuric acid, and in that process, waste heat is recovered as steam to drive turbines for power generation, and to evaporate water from phosphoric acid. With excess ammoniation capacity available, the objective of the case study was to expand phosphoric acid production capacity by 28%. This requires additional sulfuric acid and steam. Since sulfuric acid can be shipped for miles and steam cannot, phosphoric acid evaporators require some steam capacity from an on-site sulfuric acid plant. When
1021 producing the sulfuric acid needed to produce phosphoric acid, the sulfuric plant produces more byproduct steam than is needed to evaporate the phosphoric acid. So, as long as the two-site sulfuric production capacity is adequate, there is some flexibility in how closely the sulfuric vs phosphoric acids production capacities have to match within each site. Also, spare power-generation capacity at a site will encourage the addition of extra heat recovery equipment to old and new plants at that site. Many U. S. fertilizer complexes have justified new power generation equipment. When a MWH sells for less than a bought MWH, the incentive drops Figure 3. Sulfuric Acid Plant Options at One of Two Plant Sites when generated power displaces the Part of Superstructure forSYNPHONY last of the site's purchased power. When utility's "avoided costs" for new construction are high, many fertilizer complexes have justified excess generating capacity to sell power to their local utility. Site power differences could make it profitable to build a sulfuric plant at one site for the steam and ship all the sulfuric acid to the other site to make phosphoric acid. More options were added to challenge the prototype, and the expansion was to be made in two stages where stage two should waste only a minimum of stage one. Stage one should still be a best choice in case stage two is never justified. Each of the two expansion stages will have one phosphoric acid expansion, and the second expansion will be at the "other" site; one sulfuric acid expansion with an option for over-sizing the first to serve as the second; and a second sulfuric acid expansion does not have to be sited away from the first expansion. Also, there are options for adding heat recovery equipment to one old and any new sulfuric plants and for adding one turbo-generator per site per stage. Enough site differences were specified to make the study interesting. The question for the prototype to answer now was what size phosphoric acid, sulfuric, heat recovery, and powergeneration expansions should be built at each site for each stage of expansion. The file input-output version of SYNPHONY was used for the optimal configuration determination. The superstructure for this demonstration had 67 different species (600 psig steam, sulfuric acid, logic switches, etc.) and 75 processing units. In Figure 3, part of the superstructure for multiple sulfuric acid units is shown for one plant site. A sulfuric plant was one unit using 8-10 species. A new turbo-generator took 10 species and 7 units to model. Two of those species were fabricated to properly couple the 7 units to work as one. Computing time for any one case was less than 15 seconds on a Pentium II PC. The results obtained with the system were consistent with the case studies done previously at the actual complexes that were modeled here, and this served to validate that the system was giving consistent and accurate results. The results of using the system gave the
1022 following evaluations. By raising the cost of shipping sulfuric acid between sites, the sites could be forced to be self-sufficient in sulfuric production capacity. This impacted steamand power-generation capacities at each site. Similarly, the cost of extra storage tanks to handle more than a minimum of sulfuric shipping could be made to limit sulfuric shipping and bias the siting of sulfuric production capacity. This happened when the cost of extra tanks overcame the energy efficiencies of specific sites. Production rate for a higheremissions, single absorption sulfuric acid plant was curtailed as expected by voluntarily limiting the two-site SO2 emissions to pre-expansion levels. With this old plant curtailment, the new sulfuric plant was built with corresponding extra capacity. The curtailed, singleabsorption sulfuric plant was converted to double-absorption for expansion stage two when the conversion cost was significantly less than the cost of a new plant and excess capacity was built in expansion stage one. However, few companies would build excess capacity in stage one without a power incentive or strong anticipation of stage two. Extra heat-recovery and power-generation equipment was justified only when longer payback periods were acceptable. Heat-recovery and power-generation equipment was installed or not installed based on installation cost and the value of the power. The value of power varied because incremental power displaced purchase at one site and added to sales at the other site. In conclusion, the prototype of a chemical complex analysis system has been demonstrated on an agricultural chemical complex expansion. 4. CONCLUSIONS A prototype of a chemical complex analysis system has been described and its capability demonstrated by duplicating and expanding an industrial case study. The system selected the best site for required new phosphoric and sulfuric acids production capacities and selected, sited, and sized the optional heat-recovery and power-generation facilities. REFERENCES
1. Adler, S. F. 1999, Sustainability Metrics Interim Report No. 1 and Interim Report No. 2 AIChE/CWRT, 3 Park Avenue, New York, NY. 2. Beaver, E. and B. Beloff, 2000, Sustainability Metrics for Industry Workshop, AIChE/CWRT and BRIDGES to Sustainability, Austin, Texas, May 17-18, 2000. 3. Cabezas, H., J. C. Bare and S. K. Mallick, 1997, Computers Chem Engr, Vol. 21, Supp $305. 4. Constable, D. et al., 2000, Total Cost Assessment Methodology; Internal Managerial Decision Making Tool, AIChE/CWRT, AIChE, 3 Park Avenue, New York, NY, February 10, 2000. 5. Friedler, F., J. B. Varga and L. T. Fan, Chem Eng Science, Vol. 58, No. 11, p. 1755. 6. Garrett, D. E., 1989, Chemical Engineering Economics, Van Nostrand Reinhold, New York, NY. 7. Johnson, J., 1998, Chemical & Engineering News, p. 34, August 17, 1998. 8. Kocis D. and I. Grossmann, 1989, Computers Chem Engr, Vol. 21, No. 7, p. 797-819. 9. Kohlbrand, H. K., 1998, Proceedings of Foundations of Computer Aided Process Operations Conference, Snowbird, Utah, July 5-10, 1998.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1023
Decision Support Tools for Process Design and Selection Soorathep H E A W H O M * and Masahiko HIRAO Department of Chemical System Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
We proposed a new tool that is capable of reducing the complexity of the process synthesis problem and analyzing a trade-off between the environmental impact, economy and robustness of the process. In addition, new efficient process robustness parameters were also proposed. Controllability was indicated by the failure probability which could be calculated with a small number of iterations. The operability was evaluated by the deviation ratio. Applicability of the method was illustrated in a case study. Three types of closed-loop toluene recovery processes, 1) membrane-based, 2) condensation-based, and 3) adsorption-based, were investigated to quantitatively compare the characteristics of each process. By the proposed methodology, it is possible to design an appropriate process with minimal environmental impact and maximal robustness at a desired economic performance. 1. INTRODUCTION The need for an environmentally benign process has been increasing due to the pressure on chemical process industries to improve their environmental performance. Thus, the environmental impact of a process must be considered and minimized at an early stage of process design. The robustness performance is also important for the selection of good designs because of the requirement for a process that is robust under given input variations. Even in a case where there are no input variations, the problem still contains uncertainties. Therefore, the robustness objectives should be considered at the design stage to ensure that a process can properly operate with an acceptable performance under these uncertainties. Conventional methods for indicating robustness are based on the feasibility function, which is a measure of the ability of a process to meet design specifications under uncertainties. However, the calculation of process flexibility parameters involves the solution of a very complex multi-extremal and non-differentiable optimization problem. A synthesis problem has more than one objective. Frequently, this problem is nonlinear and involves discrete variables. Most of the literature on decision support tools for process design focuses only on economic and environmental problems or on economic and robustness problems. Only a few studies have been on the development of methodologies for considering economic, environmental and robustness problems simultaneously. *Corresponding author. Tel.: 813-5841-6876, E-mail address: [email protected]
1024
Fig. 1. Failure probability.
Fig. 2. Performance distribution.
2. E C O N O M I C INDICATOR The economic performance can be represented by a summation of fixed costs and operating costs and the subtraction of product revenues. The fixed cost must be discounted by the life spans of the units. 3. E N V I R O N M E N T A L I N D I C A T O R The Sustainable Process Index (SPI)[1] is designed to deal with various environmental objectives simultaneously. The basic concept of the SPI is to calculate the area required to sustainingly embed a process into an environment. All mass flows that the process either extracts from or emits to the environment must not influence the environment in such a way that brings natural evolution into danger. The total required area consists of the raw material area, energy supply area, infrastructure area, staff area, and product dissipation area.
4. PROCESS ROBUSTNESS INDICATOR Process robustness is based on the feasibility function, which is a measure of the ability of a process to meet design specifications under uncertainties[2]. Several methods involve solving a very complex multi-extremal and concave optimization problem, which in the worst case requires a large number of iterations. In this paper, new efficient parameters for the evaluation of the process controllability and operability are proposed.
4.1. Failure probability The controllability of a process can be evaluated using the failure probability. The calculation of this parameter requires only a small number of iterations, and it provides a picture of feasible and infeasible regions. Failure probability is defined as the probability of failure scenarios. Figure 1 is the probability density surface of two uncertain inputs, which in this case are the feed rate and the concentration of input. The volume under the infeasible region is the failure probability. The region where the process cannot operate without an adjustment of any control variables denotes the infeasible region. A smaller failure probability implies a more controllable process.
1025 Uncertain parameters identification
Attributes identification Attributes classification
MOS
construction
Sub-problem optimization
--Objectives --Goals -Hard constraints -Soft constraints
Pareto set construction Problem ] reformulation
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.
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Fig. 3. Diagram of the synthesis methodology.
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4.2. Deviation ratio
Deviation ratio of parameter x -
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where gj(d,z, O) <~O, Oi* = {Oi* E Oi[Oi* ~ Oi~ and o ~ is a nominal point value, c and p are the standard deviation and an arithmetic mean of the input or output distributions, respectively. The operability can be indicated by the deviation ratio. The basic idea of the deviation ratio is to investigate how cost or environmental impact increases when a fluctuation of input occurs, as shown in Fig.2. The subproblems must be solved for a new set of optimal control variables. The formulation of subproblem is defined as: min zCZ
O(d,z, 0)
s.t.
gj(d,z,
0) ~< 0.
(2)
Here, d is a vector of design variables, z is a vector of control variables, Z is a region of admissible values of control variables, and 0 is the uncertain parameter. 5. SYNTHESIS M E T H O D O L O G Y Figure 3 illustrates a flowchart of the synthesis methodology. The primary attributes of a problem must be identified and classified according to the following quantities: design variables, state variables, control variables and uncertain parameters. A certain set of objectives, goals, hard constraints and soft constraints must be defined. The problem usually involves more than one objective function, such as economic and environmental objectives. Thus, it becomes a simultaneous multi-objective optimization. In this and most other similar cases, it is unlikely that the different objectives can be optimized by the same parameter. Hence, some trade-off between the objectives is required to ensure a satisfactory design. Since SPI is a parameter that makes it possible to deal with various environmental objectives simultaneously as it inherently includes the amount of wastewater, carbon dioxide emission, and all other environmental impacts, the problem can be constructed as a two-objective-functions problem: the cost and the
1026
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Compressor~ Reuse of VOC
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1
C~176 Membraneunit pump VOC recovery
-Decanter ~Wastewater
Fig. 5. The membrane-based closed-loop toluene recovery process. SPI. With these criteria, the Pareto set can be constructed using available multi-objective optimization techniques. In this study, we apply a method called Normal-Boundary Intersection (NBI)[3]. It uses geometrically intuitive parameterization to produce an even spread of points on the Pareto set, giving an accurate picture of the entire surface. The basic calculation method of a failure probability has been described in section 4. Figure 4 shows the flowchart for computing a deviation ratio. The subproblems must be solved to obtain a new set of control variables. By adjusting the control variable set, an aspect of the process performanc such as cost or environmental impact may change. If the performance of the process decreases substantially under uncertainties, the deviation ratio of that parameter becomes high. The muti-criteria optima surface (MOS) is a surface obtained from a plot between objective functions and other criteria. This surface shows how each criterion changes under given circumstances. The sensitivity analysis of each criterion can be retrieved from the created MOS. By considering the Pareto set and MOS, it is possible to design an appropriate process and select operating conditions with minimal environmental impact and maximal robustness at a desired economic performance. If a satisfactory solution cannot be reached in this stage, then the problem must be reformulated. After reformulation, the design problem will be solved again. 6. CASE STUDY The synthesis procedure described above is applied to a case study of the synthesis of a closed-loop volatile organic compound (VOC) recovery process. The closed-loop toluene recovery process is used to dry-off excess toluene in an adhesive tape manufacturing process. The flue gas containing 0.5 to 3 vol% of toluene is fed to the toluene recovery process to recover toluene and the lean gas is used as the inert gas to purge toluene from the manufacturing process. The problem description is to design a process that can treat 2500 kgmol/hr flue gas with 3 vol% toluene concentration. Distributions of flow rate and toluene concentration are assumed to follow a normal distribution with 0 .2 - - 1000 (kgmol/hr) 2 and 0 . 2 _ 1 • 10 -6, respectively. There are several candidate processes for this case study: membrane-based, condensationbased and adsorption-based processes. Figure 5 shows the flowsheet of the membrane-based closed-loop toluene recovery process [4]. To reduce the size of the synthesis search space, we compared the characteristics of all candidate processes. Figures 6 and 7 show the relations of cost and SPI vs toluene feed concentration, respectively. The adsorption-based process is economically attractive for low toluene
1027
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Fig. 8. Pareto trade-off curve between cost and Fig. 9. 2D-plotted MOS between cost and CO2 SPI. emission. concentration, while the membrane-based process becomes more competitive when the toluene concentration increases. However, on the basis of the environmentally benign condition, the membrane-based process is more attractive than the others over the entire range of toluene concentration. After screening, the membrane-based process is selected. The design variables are the membrane area, permeability and selectivity. The control variables are the pressure of both sides of the membrane. Figure 8 shows the Pareto set for the design problem. The SPI decreases when cost increases. The MOS of cost vs CO2 emission, cost vs wastewater, cost vs the failure probability, cost vs the cost deviation ratio and cost vs the SPI deviation ratio are shown in Figs.
21
-
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1028
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Fig. 12. 2D-plotted MOS between cost and cost Fig. 13.2D-plotted MOS between cost and SPI deviation ratio, deviation ratio. 9, 10, 11, 12 and 13, respectively. The most economically attractive solution may not necessarily be environmentally attractive. Hence, a trade-off between economic and environmental objectives is required to ensure a satisfactory design. Furthermore, the failure probability, the cost deviation ratio and the SPI deviation ratio are also considered to ensure that the process is adequately robust. 7. CONCLUSIONS In this paper, a new methodology for environmentally benign process synthesis has been described and demonstrated. The synthesis methodology is capable of designing and selecting the process flowsheet with minimal environmental impact and maximal robustness at a desired economic performance. The new efficient robustness parameters are also presented. The failure probability can be used to identify process controllability. This parameter can be rapidly calculated as it requires only a small number of iterations. The deviation ratio is capable of identifying the operability of the process. REFERENCES
1. Krotscheck C. mid Narodoslawsky M., The Sustainable Process Index. A new dimension in ecological evaluation., Ecological Engineering, 6(1996) 241-258. 2. Biegler L. T., Grossmann I. E. and Westerberg A. W., Systematic methods of chemical process design, Prentice Hall. NJ. 1997. 3. Das I. and Dennis J. E., Normal-Boundary Intersection: An alternate approach for generating Pareto optimal points in multi-criteria optimization problems., SIAM J. Optimization, 8(3):631-657,1998. 4. Sea B. K., Yamaguchi T., Nakao S. and Hirao M., Evaluation of toluene recovery process using membrane separation, 8th APCChE Congress, Seoul, Korea, Aug. 1999
EuropeanSymposiumon ComputerAidedProcessEngineering- 11 R. Ganiand S.B.Jorgensen(Editors) 9 2001 ElsevierScienceB.V.All rightsreserved.
1029
Site-wide Energy Optimization with Steam Pressure Changes Yat-Yeung Kwok, Chi-Wai Hui* Chemical Engineering Department, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong.
A utility plant operating strategy is proposed which requires changing steam pressures to different external conditions such as electricity prices and product demand. Changing steam pressure varies electricity generation ability of steam turbines and affects the production capacity of some production plants. When electricity and fuel prices vary with time and/or production capacity is limited by steam pressure, optimizing steam pressure could be beneficial. A multi-period mixed integer non-linear programming (MINLP) model has been formulated to address the tradeoffs between utility and production by allowing steam pressures and production rates to change. Linear and non-linear models for steam boilers, turbines, etc., are suggested that provide a robust and accurate foundation for the optimization. 1. INTRODUCTION Steam pressure levels at a chemical production site are normally fixed regardless of changes in external and/or internal economic and operating conditions. Changing steam pressures affects the power generation ability of turbine units as well as the processing capacity of production equipment such as heaters and steam-driven compressors. Quite a few studies of steam pressure optimization have been reported in recent years, but they mainly focused on making permanent changes in steam pressures. These approaches can be classified into two main streams: the thermodynamic approach proposed by Zheng and Dhole [1] combined Pinch and Exergy analysis to target exergy losses in a heat exchange system and the potential for shaft-work production. Mathematical programming [2-3], on the other hand, optimizes the operating conditions and configurations of utility plants taking into account both energy and mass balances. Petracci et al. [4] optimized the turbine configuration and de-aerator pressure for an ethylene production plant. In their work, temperature and pressure were not considered as optimization variables. Corvalan S.M [5] reported a study optimizing the utility system of an ethylene plant considered pressure and temperature as continuous variables. The results showed that changing steam conditions could enhance the profit of the overall plant. In this paper, a new utility plant operational strategy is proposed. The strategy requires steam pressure to be optimized according to changes in external economic conditions such as electricity prices, and/or internal operational conditions such as cooling water temperatures.
*Author to whom correspondence should be addressed. Email:[email protected]
1030 These changes happen regularly between day and night operations, resulting in differences in optimum steam pressures. 2. THE SITE MODEL
The chemical production site to be modeled consists of a utility plant and five production plants: an ethylene plant (ETY), a vinyl-chloride Plant (VCM), a poly-propylene plant (PP), a poly-ethylene plant (PE) and a poly-vinyl-chloride plant (PVC), see Figure 1. In the utility plant, boilers B 1 and B2 consume both imported fuel oil and by-product fuel from the ethylene plant. The very-high-pressure steam (VHP, 120atm) used in turbines T1 and T2 comes from the boilers as well as the ethylene plant. Steam at medium (MP, 40atm) and low pressure (LP, 3atm) from T1 and T2 is used inside the utility plant as well as for the production processes. In the production site, an ethylene plant cracks naphtha (NAP) producing ethylene (ETY) and propylene (PPY). Ethylene is the raw material for vinyl-chloride monomer (VCM) production and the production of poly-ethylene (PE). VCM is used for producing final product poly-vinyl-chloride (PVC). Propylene from the ethylene plant is used for polypropylene (PP) production. 3. BOILER MODELS
Assuming that the efficiency of a boiler is independent of its loading, fuel consumption can normally be presented as: Fuel (kg/hr for FO or liter/hr for FG)
= al + a2*
Steam_Flow (kg/hr)
(1)
If the boiler efficiency is a function of steam flow, a non-linear equation is required: Fuel (kg/hr or liter/hr)
= a3 + a4 *
Steam_Flow (kg/hr) + as* Steam_Flow 2 (kg/hr)
(2)
Where al-a5 are parameters that can be regressed from plant data. Although (2) is more descriptive than (1), the difference between (1) and (2) is normally small. In cases where the steam flow varies within a narrow range, (1) is a very good approximation. 4. TURBINE MODELS
To calculate the exact power output of a steam turbine where both steam flows and pressures are varied, a rigorous steam property model is required. However, due to the discontinuity of steam properties, a rigorous steam property model may cause difficult in the optimization. In this work, power output of the turbines are calculated using the following empirical equations: E1 = b l * FliP* PHPb2+ b3* FLPl* PLPb4 E2 = bs* FLP2*PLPb6+ b7* FCOND2 E3 = b8* FLP3 * PEPb9+ bl0* FLP3
(3) (4) (5)
In these equations, bl to bl0 are regression parameters; E, F and P are the corresponding power outputs, steam flows and steam pressures (see Figure 1). These parameters are regressed from ASPEN+ simulation results as PHP varied over the range of 37- 42atm and PEP
1031 2-5atm. The average errors of the three equations were all less than 0.5% and the maximum errors were within 1% in the cases below. 5. P L A N T M O D E L S
The utility consumption rates of the production plants are assumed to be linear functions of the production rates using the following equation: Utility Consumption Rate (kW or kg/hr) = Cl * Raw Material Feed Rate (kg/hr)
(6)
The utility consumption rates are assumed not to be affected by the steam pressures. However, steam pressures vary the performance of some equipment such as reboilers and compressors, which can affect the maximum production capacities of the production plants. In this work, the following constraint was used to represent the change of production limits: Maximum Production
= c2 + c3 *
PHP +
C4 * PLP
(7)
6. CASE STUDIES The example production site shown in Figure 1 requires regular electricity import from public generators. In the following case studies, only the maximum production capacity of PVC plant is affected by changing steam pressures. Assume that the electricity-purchasing contract imposes a minimum electricity import of 1000kW at all times. To encourage offpeak use, low electricity prices are given on night and midnight shifts as well as during weekends (see Table 1). The boiler, turbine and plant models' parameters are given in Tables 2-5.
Case 1 - Fixed Production Targets for PVC, PE a n d PP Assume the followings: Production Targets: PP = 1500 ton/week; PE = 2000 ton/week; PVC = 1000 ton/week Steam Pressure (atm): PVHP=120; 37_
1032 during day shifts. Optimum production rates and electricity import results are shown in Figures 5 and 6 respectively. Figure 5 shows low production rates of PVC and PE during day shifts when electricity is expensive, but high during night and midnight shifts when electricity prices are low. Production of PP has an opposite behavior with high production rate on the day shifts. This may due to the high consumption rate of LP in PP production, which creates opportunity for larger in house electricity generation. Since off-peak electricity prices are lower than the in house electricity generation cost, electricity import is high during evening and midnight (Fig. 6). Allowing the change of steam pressures yields 2% of utility cost saving. This saving is mainly caused by the reduction of electricity cost. Case 2: High Product Demands In case 1, the production capacities of the plants are larger than the respective demand during the planning period. Production exceeding the production targets cannot create additional profit. In the second case, assuming high demand for PE, the PE production target is removed. Increasing PE production can now increase overall profit. Production targets for PVC and PP are also increased and fixed at the following levels:
Production Targets: PP = 2500 ton/week; PVC = 2000 ton/week; PE = no target In this case, varying steam pressure not only reduces utility cost, but also affects the profitability of production. When steam pressures are not allowed to change (Pnv=40atm, PEP = 3atm), the optimum production and electricity import profiles are shown in Figures 7 and 8 respectively. Production of PE is maximized at around 20,000 kg/hr due to the restriction on changing steam pressures. When steam pressures are allowed to change, production increases to 24,000 kg/hr while PHP and PEP are at 43atm and 2atm respectively. To maintain this high level of PE production, electricity import on the night shifts has to be increased, thereby slightly increasing the electricity cost. With steam pressure changes, the overall profits (including overall utility, raw material and product costs) increase 9% mainly due to the increasing of PE production. 7. CONCLUSION This paper has presented a multi-period site model that included both utility and production plants, allowing it to address tradeoffs between the two systems. When external conditions such as electricity prices and product demand are varied, operating conditions such as steam pressures and production rates should be changed to maximize overall profits. The two case studies showed that significant profits can be generated by allowing steam pressure to change. REFERENCES 1. Zheng J, Dhole V. (1995), Targeting for Efficiency Improvement in the Design of Commercial Power Stations, ASME Cogen. Turbo Power '95 conference, Vienna, Austria Brooke, A. 2. Papoulias S.A. and Grossmann I.E. (1983), A Structural Optimization Approach in Process Synthesis- I. Utility systems, Computer Chem. Engng., vol 7, p695-706.
1033 3. Petracci N., Eliceche A.M. and Brignole E. (1991), Utility System Optimal Operation, Computers-Oriented Process Engineering, p387-392. 4. Petracci N., Eliceche A.M. and Brignole E. (1993), Optimal Operation of an Ethylene Plant Utility System, Computers Chem. Engng., vol 17, p147-152. 5. Corvalan S.M. and Eliceche A.M. (1999), Minimization of Natural Gas and Water Consumption in the Operation of Utility Plants, Computer Aided Process Engineering, vol 10, p475-480. 6. Kendrick, D. and Meeraus, A. GAMS - A User's Guide (Release 2.25); The Scientific Press: San Francisco, CA, 1992.
ACKNOWLEDGMENTS The authors would like to acknowledge financial support from the Research Grant Council of Hong Kong under Grant HKUST6036/98P and HKUST6104/99P. ...............Shift .........I[:Weekday
[[Weekend][Noi::0f [I
................
II (Yen/kW)I(YerekW)ll Hours Ii ....._1 .........
It
Day '[I
1t .,Night ....................... ......I1
20 4
711...... 4 .Jl ...... 4
[ Boi]'er II [
B1
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!l ...........ETY II .... vcm
Ii..... 6
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i l .Parameter .... . . . . . . 0.009 .......
"~-
~-Sh~fl I[........ c;: "................ C4.............'~II I!~-......... ................................... .....' C2~::~"< ]i........... ........... !I .... PVC 10000 .............. 2 ......................... 3333 ........11 II 1[......... J[............................ ~i ........................
al (FO) . . . . . . . - ......................... a2 (FO) 0.009 al (FG) a2 (FG) .0079 Table 2:Boiler Parameters ....
...................
.......
....
]!................. PEI: il][_ji__i ill__I[ ._,_6 5 0 - - ] ~ ] [ PP ! ...................... -ll 275 ....... Table 5: Parameters for Calculating ....... Production Limits
13000
II Turbine l['---][Par'~"~eter..... T1
ELECT
IMma~
1_
~,
r
e~
L '~e,
]' uv,v
Figure 1" Example Site
I
.., ..ll 0:2 !1 o1. ....iI~ .... II
Table 4: Parameters for Calculating Utility Consumption
B2
bl 1.3042 ... b2 . . . .0.8235 b3 0.2065 ............................... b4 0.1769 b5 0.2013 T2 b6 0.1832 b7 0.2039 b8 0.8806 T3 b9 -0.0602 bl0 -0.8723 -Tabie 31 T1]rbine Parameters"":
Ii..... EL ...........I!.........Wi~q ............. HP .........II""Le ............il II (kW) ............................................ ][ (kg/lar)][ (kg/hr)1[ (kg(~) II 0 -,! 7 I" ,-01 .......]! i53 ' '01:................... 1'"........... , ....... i!.......... 1[ .... I II 9:53l !131 03,,,, ,ll.........01: .............
I!.........! 2 o
..... !11
Ilmi_d:ightll 3 II '3 Table 1" Electricity prices
Shift ....
I
1034
Figure2: Production Rates (Case I - Steam Pressure Fix)
Figtwe 3" Eleclticity import (Case I - Steam Pnesstme fix) 60000
25000 ~" 2OO00 15000
soooo
PE
. /~ .h'_~ _A AA/X A/V
l
!....
~ 10000
l
l
l ~L
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~- 50oo
o~ V,V
0 1 2 3 4 5 6 7 8 9 10 11 1 2 1 3 1 4 1 5 16 171819 20 21 Shifts
I
II II II II I\1 II II II II III il II II III \1 u
1
V,,,
6
,
"
,',
16
R g u r e 5: Production Rates (Case I - Steam Pressure Change) 25000 35
25
P
..:~w ./~,' .:.
.;,
/
"; 150oo
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g
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loooo
o
.~ lO 5
.:,
--. 20000
~ 30
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V
V
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V
I1.
8 9 10 11 12 13 14 15 16 17 18 19 20 21 Shifts
Shifts
///'"'
F,guree Elmricitylm[mrt ( C a g l - S t m n P r e m r e C h a ~ 69900
Figure 7: Production Rates (Case 2 - Steam Pressure Fix) 25000
-A A '" r\,i,
I-/I/I "/vvv ~
~
"D 20000
':"
,',
,'o
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0
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.
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F i g u r e 11: Electricity I m p o r t (Case 2 - S t e a m P r e s s u r e C h a n g e ) 60000
25000
EL
~o./~ fl I~ f~l 50000
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o
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Figure 10: Production Rates (Case 2 - Steam Pressure Change)
a.
'13'
:7
E= 25
30000
:~ 10000
".:
3s
1
~
- :
!/ V !/ !/
4O
~ :21 ii ~/ II I/ 1
i.
.
Figure 9: Steam Pressures (Case 2 - Steam Pressure Change
Illllll II II II
ol
i .:
'3 '
EL
....
.
Electlkity Import (Caqe 2 - Steam Pre.me Change)
~_, [I I/ fl II :
/
.
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i
Shift Figure 8:
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.
.
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1 2 3 4 5 6 7 8 9 101112131415161718192021 Shifts
,
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116
21
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1035
A combined approach for the overall energy integration and optimal synthesis of low-temperature process systems H.-Q. Li a'*, H.-Z. Chen a , Z.-H. Lia , B.-H. Lib and P.-J. Yaob State Key Laboratory of Biochemical Engineering, Ins. Chem. Met, Chinese Academy of Sciences, RO. Box 353, Beijing 100080, ER China E-mail: [email protected]
a
b School of Chemical Engineering, Dalian University of Technology, RO. Box 158-52, Dalian 116012, ER China E-mail: [email protected] A novel combined methodology is presented for the overall energy integration and optimal synthesis of the low-temperature process system. Combining the benefits of thermodynamic analysis and mathematical optimisation, the proposed approach is divided into two sequential stages: the targeting overall energy integration and the simultaneous optimal synthesis, and it allows the designers to hierarchically address the interactions among all components within a sub-ambient process. The approach has been tested on the cold end retrofit of an existing ethylene plant, resulting into the benefit of around $1.8" 106/yr. 1. INTRODUCTION The need for efficient utilisation and recovery of energy in chemical processes has been established on both economic and environmental grounds. As a key link in chemical process plant, the low-temperature process system is complex, energy and capital intensive. A typical low-temperature process system consists of three components, namely the process (usually distillation sequence), the heat exchange network (HEN) and the refrigeration system. All three components are highly interlinked and interactive so that the task of the overall energy integration and optimal synthesis is very complex. To pursue more benefits, Dhole and Linnhoff combined the pinch and exergy analysis to present a graphical methodology for the over design of sub-ambient processes [1], which addresses those interactions on a global basis and targets the most economic process changes ahead of detailed simulation and design. On the other hand, the mathematical optimisation technique will also carry out the optimal synthesis of low-temperature processes, which involves building and subjecting a general superstructure containing all the process options to optimise both parameters and structural changes in the process [2,3]. Based on the benefits and the shortcomings of simply reviewed approaches above, we combined the benefits of thermodynamic analysis and mathematical optimisation to form a systematic procedure that may efficiently carry out the overall energy integration and optimal synthesis of total sub-ambient process systems. * Author to whom correspondence should be addressed.
1036 2. TARGETING OVERALL ENERGY INTEGRATION---ANALYSIS STAGE The aim at this stage is to bypass the interaction between the HEN and the refrigeration system and to target the overall energy integration for the sub-ambient process. The used graphical tools and strategies are those presented by Dhole and Linnhoff for the overall design of sub-ambient processes [ 1,4-5]. 2.1. Exergy grand composite curve based on the corrected temperatures Based on the thermal data of the base case, the exergy grand composite curve (EGCC) is constructed to show the exergy flows within a low-temperature process. The EGCC enables the designers to quickly locate the inefficiencies of the process. To consider the film heat transfer coefficient values of different streams, the EGCC is defined in the space of corrected temperatures converted from the real ones of streams using the individual temperature contributions [6]. This approach is very useful to screen the promising process changes. 2.2. Splitting exergy grand composite curve After analyzing the EGCC for the sub-ambient process, we obtain a better understanding of the exergy loss distribution among different components and major directions to improve the base case. The thermal data of each distillation column including its associated reboiler and condenser relative to those directions is then taken out of the EGCC of overall sub-ambient process to construct the column exergy grand composite curve (CEGCC), which is a useful tool to quantitatively address the energy-saving potential for possible stand-alone column modifications such as reflux reduction, feed conditioning and scope for side reboiler/condenser. Furthermore, the CEGCC is matched in turn against the rest of EGCC, thus resulting into the splitting exergy grand composite curve (SEGCC) as shown in Fig. 1. Using the SEGCC, we may not only explore opportunities for integration of the split distillation column with the background sub-ambient process but also quickly evaluate the effects of stand-alone column modifications on process integration. When attention is paid on process integration, the heat pumping integration of sub-ambient distillation column with refrigeration system should be preferred since this will bring the dual benefits of reducing operating and capital costs. During screening process improvements by the SEGCC, alternatives will be obtained by rationally adjusting the temperature contributions of process streams.
qc~
Ambient
EGCC
CEGCC Figure 1. Splitting exergy grand composite curve
1037
2.3. Identifying the promising options To quickly evaluate the process changes, we use the shaftwork targeting method proposed by Linnhoff and Dhole [4], which uses the exergy efficiency of the overall refrigeration system to quantify the return of process improvements on shaftwork reduction. Because the heat pumping integration can cause the significant heat-rejection changes, the exergy efficiency of heat pumping integration is formulated to evaluate the effects of heat rejection changes on shaftwork reduction. As a result, the most promising options can be efficiently identified with higher accuracy compared with previous work. The analysis identifies the most promising options for the overall energy integration for low-temperature processes, which and the base case together are then passed on the optimisation stage, where they are implemented into a superstructure for optimisation. Because only the economically viable options are included, the superstructure will be potentially much simpler than a general superstructure containing all the candidate changes. Another benefit of the analysis is that the results determine bounds and initial points for the key parameters like the temperature contributions of streams. With a reduced superstructure, good initial points and feasible bounds, the mathematical optimisation problem can be solved effectively even for large problems. 3. SIMULTANEOUSLY OPTIMAL SYNTHESIS--OPTIMISATION STAGE In the optimisation stage, the energy characteristics of those process units associated with both the most promising process improvements and the base case are first extracted and transformed into equal hot and/or cold streams. Taking these streams, the original HEN is consequently converted into an enlarged HEN (EHEN). Thus, the task is to build a mathematical model for simultaneous optimisation of both the EHEN and the refrigeration system.
3.1. Model representation The starting point of the optimisation model presented in this paper is that proposed by Colmenares and Seider [2]. This consists of three levels as following: Level 1: Following the temperature interval (TI) method [7], the temperature intervals of the hot streams in the EHEN are first assigned on the basis of the corrected temperatures of process streams, together with the cold streams in the EHEN. To reduce the problem-size, the obtained temperature intervals are lumped further. Level 2: Based on the given refrigerant, a superstructure representation for multistage refrigeration system is proposed, which allows for the identification of a number of stages, their operating temperature ranges, the use of economizers, the heat-rejection recovery between intermediate stages. The corresponding parameters are initialized by the result of temperature intervals. To our knowledge, the main features of this superstructure are similar with those presented by Vaidyaraman and Maranas [3]. Level 3: From the above work, a non-linear programming (NLP) formulation is build up
1038 to simultaneously optimise the EHEN and refrigeration system. The selected variables consist of two categories, namely process parameters and structural changes. The former refers to heat transfer duties, temperatures and heat capacity flowrates of refrigerant streams, temperature contributions of streams, etc., and the latter involves number of refrigeration stages, configuration of complex refrigeration cycle, selections of heat pumping process. The complete NLP model includes an economic objective function subject to a set of linear and nonlinear equality and inequality constraints that will be discussed in more detail elsewhere due to the limitation of space here. The objective function follows: M
Min
9
C ro,at
-- ~ [ ~ w a t e r ~z~ T~
+ ~FTo,at ) + Cwork ~ y mAWm + FAe " 1i~Total ~ ~ Com " HEN
(I)
tn=I
Where CTotat is the annual total cost; Cwa,~, and Cwork are the annual operating costs of the and pTotat are the capital costs of the cooling water and the shaftwork respectively; 'FTotat '"HEN ""Cam EHEN and the refrigeration system respectively; ~:~, orator is the annual consumption of cooling Wat e r water; A W m is the annual consumption of shaftwork in the m m refrigeration level; y,, is the index for selecting the mth refrigeration level; F ~ is the annualized factor of capital costs.
3.2. Multiple-population hybrid genetic algorithm Although the problem-size has been reduced, the global optimisation problem produced is still of nonconvexities, high dimensions, noncontinuity and of multi-constraints. In order to effectively solve the problem, the genetic algorithm (GA) is incorporated as a global optimiser [8], and the followings are used to improve the performance of traditional GA: 9 Instead of the complex binary coding, the float coding is used for continuous problem. 9 Metropolis rule is introduced to select the offspring, and this ensures that offspring generation is also satisfied with Boltzmann distribution. 9 In terms of the hill climbing heuristic, the local normal perturbing mutation operator is developed to enhance the performance of local searching. 9 The algorithm is implemented in the way of multiple-populations based on the kaleidoscope topology. So far, a new algorithm named multiple-population hybrid genetic algorithm (MP_HGA) is presented, which is of the higher accuracy and shorter time to converge at the global optimum than the traditional GA. 4. OVERALL FLOW FOR COMBINED METHODOLOGY Fig. 2 illustrates the overall combined methodology in a simplified schematic. A computer-aided prototype system, Advanced Low-temperature Process Energy Integration (ALPEI), has also been developed in the C++ program language.
1039
.....~ D a t ~ preparation ~ n a l result (Rectification and Simulation) (Remaining problem analysis) ~t EGCC ...... ( Process analysis ) ~Solving global optimization) SEGCC ~ MP_HGA @rocess improvement) ( Optimalsynthesis ) $ ShaRworktargeting Optimization model ,, ( Evaluating options ) ( Initializing superstructure ) The most promising options ( ' Thermaldata transfer ~J EHEN ~,(~Temperatureinterval) Figure 2. A simplified schematic for the overall combined methodology 5. CASE S T U D Y m E T H Y L E N E COLD END
To illustrate the utility of the combined method, the cold end retrofit of an ethylene plant is considered. Fig. 3 shows the base case flowsheet, which includes a sequence of distillation columns with a pre-depropaniser. The cold end involves a cascade refrigeration system with a propylene and an ethylene cycle. Detailed data have been published by Yao [6]. In the case study, the task is to reduce the energy requirement of the overall cold end.
Crackinggas Compressor C
................................. ], .................................. 9
......... V---I ~
,176 K
~
r/
ff
1-
~273K
I
~1
~ I"* Product
eq1
eq I
245K
|
9
238K ToDebutaniser ]
ToProp' " yle,nefactionator , ",] qTo u
",~
Figure 3. The cold end of an ethylene plant The most promising process improvements identified in the analysis stage are focused mainly on the demethaniser system, deethaniser system and the ethylene fractionator system. Furthermore, the simultaneous synthesis in the optimisation stage provides the optimal structures and parameters of the HEN and the refrigeration system as shown in Fig. 4 and Fig. 5. According to the result, the final design has been determined and carried out in the retrofit, resulting in a benefit of around $1.8* 106/yr with a short payback time of 8 months.
1040 H1 H2
313 K 278 K
H4
233 K L94K
H5 H6
!36 K
H7 H8 H9
672 kW ~
313K
Ref7
1449kW
307 kW ~'~ 385 kW. RetT,N--J
~297kW
Refl
Ref5 442 kW Ref5 ~ 669 kW
256 K <
Ref6 ~ 106kW
289 K j . Ref6 504kW
<
<
~
278 K 174 K
R~I
~ S I,.,.J I "M_.~~--...504kWi
Propylene cycle
,~337 Re 233 2 K
C2
kW
k,._~-
C3 C4
257
C5
240
C6
120kW
cX
240
513 kW ] 9 '~ ,. ~. 478 k W . f
Ref 3 " ~ J
Ref7
<
Figure 4. Optimal HEN for the cold end retrofit
\
C7 A
237 K C8
1352kW.
268 K
7
~.~ 242 K
256 94 kW
)
<
~
175K
245 K
Rd7'
/
1466 kW R29f75,R
..~ 236 K
C1
Ro~ 63 kW
203 K
238 K
177kW
kW
278 K
258 K (~
~13,867
283K
238 K
Ref4 ~ 555 kW / ~
7335kW
~.45K
281 K
1849kW
~o~6~
176K
>
Ref2
930 kW
~78K
257 K
272 K
606 kW
442 kW (
Refl ~/~
238 K <
247 K
200 kW
_ 500 kW
356 kW~
Ref5 / ~ Ref3 ~
258 K <
289 K
672 kW ~ Ref6
272 K C9 281Kc10 268Kc11
t.-,ono Y cyc,e
200 2 K
~ ~
'1176 k ~ - - ~
k..~" .(j.,~
IR;f31K ~
259 k W / - J ~ ~ ,, \
I 823 kW "~'-~.
'--A" [
Figure 5. Optimal refrigeration system for the cold end retrofit
6. CONCLUTIONS Combining the benefits of thermodynamic analysis and mathematical optimisation, a new method is presented for the overall energy integration and optimal synthesis of a sub-ambient process. The pinch and exergy analysis tools are used in the analysis stage to identify the promising process modifications, which bring the dual benefits of the reduced superstructure and good initial values and bounds for main parameters. In the optimisation stage, the GA is improved to simultaneously synthesize the HEN and refrigeration system, resulting into the best selection of process parameters and structure changes with high heat integration. The methodology has been tested on a case study of the ethylene cold end retrofit, giving a better design with an annual profit of around $1.8" 106 compared to the base case design. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
V. R. Dhole and B. Linnhoff, Computers Chem. Engng., 18 (Suppl, 1994) S 105. T. R. Colmenares and W. D. Seider, Computers Chem. Engng., 13 (1989) 247. S. Vaidyaraman and C. D. Maranas, AIChE J., 45 (1999) 997. B. Linnhoff and V. R. Dhole, Chem. Eng. Sci., 47 (1992) 2081. V. R. Dhole and B. Linnhoff, Computers Chem. Engng., 17 (1993) 549. P. J. Yao, Total Process Energy Integration, Dalian Uni. of Tech. Press, Dalian, 1995. R. Smith, Chemical Process design, McGraw-Hill, New York, 1995. H. R. Ryoo and N. V. Sahinnidis, Computers Chem. Engng., 19 (1995) 551.
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rightsreserved.
1041
Implementation issues for real-time optimization of a crude unit heat exchanger network Tore Lid a, Sigurd Skogestad b * aStatoil Mongstad, N-5954 Mongstad bDepartment of Chemical Engineering, NTNU, N-7491 Trondheim This paper provides a case study on the selection of controlled variables for the implementation of real time optimization results in a crude unit heat exchanger network. Two different control strategies with 22 different control structures are evaluated. The idea is to select the controlled variables that give the best plant economic (smallest loss) when there are disturbances (self-optimizing control). The disturbances are correlated and a simple principal component analysis is used to generate a more realistic set of disturbance variations for evaluation of the different control structures. This analysis shows a large variation of loss for different control structures and that a control structure evaluation is necessary to collect the benefits from a RTO system. 1. I N T R O D U C T I O N A real time optimization system (RTO) can be described as a sequence of three separate functions, White (1997). (1) Data reconciliation and parameter estimation to establish the current operation point. (2) Optimization to find the optimal operation. (3) Implementation of the optimal result as controller setpoints. Estimated parameters and reconciled process variables are the basis for operations optimization. The optimal operation is computed by maximization of some objective subject to the process model and operating constraints. The objective can be a direct measure of the profit or some function of the variables that when maximized drives the process towards the optimal operation. Finally the computed optimal operation is implemented in the process as setpoints in the control system. The selection of these controlled variables is the main focus of this paper. In the RTO "loop" there is a loss related to uncertainty in the process measurements, estimated parameters, model errors, Forbes and Marlin (1996); Zhang and Forbes (2000). Optimal values for operation are computed at regular intervals and implemented as setpoints in the control system. In the period from one optimization run to the next the disturbances will change and the current operation is no longer optimal. In addition uncertainties in the controlled variable measurements causes a operation that deviates from the true optimal operation. This disturbance variation and control error is the source of the disturbance and control loss, Skogestad et al. (1998). These losses depends highly on the control variables selected for implementation of the optimization result. The objective is to select the control variables such that *e-mail:[email protected]
1042 this loss is minimized. If some process constraint is active for all expected variations in the disturbances, this variable should be selected as a controlled variable. This is active constraint control, Maarleveld and Rijnsdorp (1970). The variable is then held at its optimal value for all disturbance variations. If the controlled system has infeasible solutions (constraint violations), with the selected control structure, for normal disturbance variation a back-off from constraints must be computed. The back-off is computed such that the controlled system has feasible solutions for all expected disturbances, Hennin et al. (1994) To simplify the analysis, several assumptions have been made. The controlled variables selection is solely based on steady state considerations and no evaluation of possible dynamic control problems are made. There are no process model error and estimated parameters and process variables (reconciled values) have no uncertainty. By this assumption the computed optimal values, based on reconciled measurements and model parameters, describes the true process optimum. 2. THE OPTIMIZATION PROBLEM
A typical process optimization problem has a linear economic objective function, nonlinear process model and some operational constraints. The optimization problem can be formulated as
maxJ
-
-
x
pTx
st. g(x, do,[5)
=
0
Xmin___
X
___Xmax
(1)
where the process variables are included in x. The objective, J, is typically product price times product flow minus feed price times feed flow and energy price times energy flow. The process model is included as a equality constraint, g(x, do, ~) = 0, where do are the nominal disturbance values 13are the model parameters. Inequality constraints are typically bounds on single process variables e.g. high temperature limits or a low flow limit. In this problem there are n variables (in x), m process equations (g(x, [~)) and md disturbances. The solution, x* (do), to 1 is referred to as the nominal optimum. The solution to the optimization problem in 1, x*, is implemented as setpoints to nf variables using a controller C, where nf is the available number of degrees of freedom. The controller may be included in the system as a set of linear constraints Cx - ro where each row in C has one nonzero element, equal to one, corresponding to the selected controlled variable. The controller setpoints equals the nominal optimum, ro = Cx*. The controlled system has the solution xc(d, ro) and objective Jc(d, ro) - pTxc(d, ro). A requirement on the controller is that the controlled variables are independent such that the the controlled system
[ ~lx,do,~) r
has rank equal to the number of variables, i.e. (rank [
~x
]r
[x*,d0,~CTj
-- n)
3. T H E LOSS F U N C T I O N The disturbance loss function, Skogestad et al. (1998), is defined as the difference of the optimal objective of some disturbance d, J* (d) and the objective achieved by using a control
1043 structure C, with nominal optimal values as setpoints. The loss function can be written as
Ld(d) = J* (d) - Jc(d, ro)
(2)
where J* (d) is the objective of the optimal operation with a known disturbance d and Jc(d, ro) the objective of the controlled system using the nominal optimum as setpoints. The disturbance loss function describes the loss of not re-optimizing, and implement new setpoints when the disturbance d has changed and is different from do. In addition to the loss of a disturbance change there is a loss due to implementation error or control error. The controlled variables varies around the optimal setpoint due to disturbances, measurement inaccuracy and noise. The control error loss function is defined as
Lc(Are) = J* (do) - Jc(do, ro + Are)
(3)
where Are is the control error. This definition of loss gives one loss function for each disturbance. A overall scalar measure, for all disturbances and control errors, can be calculated as the sum of the integrals of the disturbance and control error losses from dn~n...dmax and Aremin...Aremax respectively. With this simplification the loss is calculated along each of the disturbance and control error axis. Other measures, such as the sum of all comer points or the resulting loss of a Monte Carlo simulation could also be used. 4. D I S T U R B A N C E ANALYSIS In the above analysis the aim is to find a controller which minimizes the loss in presence of disturbances. A key issue is to find a good representation of the disturbance variation. The normal range of the disturbance variation should preferably be computed from process measurements. If measured data is unavailable disturbance variations may be estimated based on experience from similar processes and design information. When a RTO updates the optimal setpoints at regular intervals, a average of the disturbance variation for each interval gives a measure of the expected disturbance change from one optimization run to the next. In a real process we often have that the disturbances are correlated. Evaluating the loss of one disturbance at a time will fail to evaluate the loss with the most likely combinations of disturbances. By assuming a linear relation and using simple principal component analysis (PCA), Jackson (1991), the measured disturbances may be transformed into a reduced set of uncorrelated disturbances or principal components. The variation range of the principal components is computed as the average variation within each RTO execution interval. The number of principal components used is selected such that the principal components describes the majority (i.e. 90% or 95%) of the variance in the measured data. This representation of the disturbance data provides a more realistic basis for selection of the minimum loss control structure. 5. CASE STUDY In the crude unit the crude (DCR) is preheated in a heat exchanger network where heat is recovered from the hot products and circulating refluxes. As shown in figure 1 the cold feed is separated into seven parallel streams (A-G). This feed split provides only 5 degrees of freedom, which is used for optimization, since total feed flow and total bottom circulating reflux (BSR)
1044
Fig. 1. Simplified crude unit overview duty is kept constant. Changes in product yields and BSR duty are the main disturbances to the heat exchanger network. The optimization objective is to save energy by recovering as much heat as possible. The heater is the main energy input in the process and heater outlet temperature is held constant. The minimum energy is then achieved by maximizing the heater inlet temperature. A detailed description of the process, steady state model, data reconciliation and optimization is presented in Lid et al. (2001). For simplicity the operating constraints are ignored in the control structure selection. 5.1. Disturbances There are 23 disturbance variables. These are the flows and temperatures of streams flowing into the heat exchanger network. The data used in this analysis are 35 days of 12 minutes averages sampled from normal process operation. The RTO execution interval is one hour. The disturbance measurements where reduced to for principal components using PCA as described in section 4. The standard deviation of the selected principal components averaged for all optimization intervals was computed and used as the disturbance variation range. 5.2. Control structure evaluation There are a large number of possible controllers for implementation of the optimization result. The only controller requirement is that all 5 degrees of freedom in the process must be specified or that the controlled system rank requirement is satisfied. In this case study two control strategies are evaluated. Strategy 1: the optimal result is implemented as setpoints to the flow controllers in each pass (open loop implementation). Strategy 2: the optimal result is implemented as setpoints to pass outlet temperature controllers (closed loop implementation) where the temperature controllers manipulates the corresponding pass flow. The rank requirement for the controller with the open or closed loop implementation strategy may be stated by two simple rules. First, the flow or temperature in pass D and G can not be specified simultaneously since one has to be used to control the total BSR duty. Second, only five of the remaining six flows or temperatures in the seven passes can be specified simultaneously since the total feed flow is to be kept constant. This makes effectively one flow as a dependent variable. In the open loop implementation strategy there exists 11 different control structures which
1045 satisfies the rank requirement. In Table 1 all possible flow control combinations are numbered 1-11 and in Table 2 all possible temperature control combinations are numbered 12-22. For each control structure the disturbance loss, control loss and total loss are computed. The control variable selections in table 1 and 2, are sorted by total loss. The results shows that the best open loop implementation strategy is to select the flow controllers of pass A,B,C,D and E as controlled variables. The setpoints of these controllers is set equal to the current nominal optimum. Pass G is used for total BSR duty control and pass F is used for total flow control. In table 2 the loss functions for different temperature control combinations are listed. The total loss for the best controller is reduced by 57% when the outlet temperature of pass A,B,C,D and E is used as as controlled variables. The selection of pass A,B,C,D and E as controlled variables gives the Table 1 Strategy 1" Flow control No. CV Ld LAre 1 ABCDE 0.013 0.009 4 ACDEF 0.015 0.018 7 ABCEG 0.040 0.010 2 ABCDF 0.021 0.031 6 ABCEF 0.021 0.032 3 A B D E F 0.023 0.031 10 ACEFG 0.053 0.020 5 BCDEF 0.038 0.047 8 ABCFG 0.068 0.034 9 ABEFG 0.080 0.034 11 BCEFG 0.123 0.050
L 0.021 0.034 0.050 0.052 0.053 0.054 0.073 0.084 0.102 0.114 0.173
Table 2 Strategy 2: Temperature control No. CV Ld LAre L 12 ABCDE 0.002 0.007 0.009 15 ACDEF 0.002 0.015 0.017 13 ABCDF 0.005 0.024 0.029 14 ABDEF 0.004 0.025 0.029 17 ABCEF 0.007 0.023 0.030 16 BCDEF 0.006 0.038 0.043 18 ABCEG 0.101 0.054 0.156 21 ACEFG 0.123 0.072 0.195 19 ABCFG 0.183 0.101 0.284 20 ABEFG 0.183 0.105 0.288 22 BCEFG 0.245 0.145 0.390
minimum loss both for the open and closed loop implementation strategy. From table 1 and 2 it is clear that controllers including flow or temperature in pass G and F as controlled variables gives generally a large loss. The difference in loss for the flow control structures may be explained by the fraction of crude flow trough each pass. At the nominal optimum the fractions in pass A-G is [6 15 12 16 10 33 8]% respectively. Pass F has the largest flow and should be used to control the total flow since this will give the smallest relative error in presence of feed flow disturbances. A similar argument applies to the selection of pass E or G to control BSR total duty. The heat transferred from BSR is 4.2MW to pass G and 2.2MW to pass E. The pass receiving the largest duty should be selected to control the total duty in the BSR since this will give the smallest relative change in presence of disturbances. The loss computed using principal components is in general smaller than the loss computed using the disturbances independently. This is explained with the fact that the mass and energy balance in the process is always "zero". If the cold feed flow increases the hot product flows will also increase, if the product yields changes and we have a reduction a hot product flow the product temperature will in general increase. These dependencies in the disturbances seems to cancel some of effect on the total loss. 6. C O N C L U S I O N A method for selection of controlled variables for implementation of real-time optimization results based on self-optimizing control and the loss function, Skogestad et al. (1998),is described. The analysis is solely based on steady state considerations and no evaluation of the
1046 resulting control problem is made. The selection is based on how the controlled process will act in presence of disturbances compared to optimal operation. Some control structures are proposed and evaluated in presence of disturbances and control errors. The minimum loss control structure is achieved by selecting the outlet temperature of pass A,B,C,D and E as controlled variables. The worst case loss, using temperature control, is 0.39~ which is more than 10% of the total RTO potential. This shows that a proper selection of controlled variables is vital for achievement of maximum RTO benefits in presence of disturbances. REFERENCES
Forbes, J. F., Marlin, T. E., 1996. Design cost: A systematic approach to technology selection for model-based real-time optimization systems. Computers & Chemical Engineering 20 (6/7), 717-734. Hennin, S. d., Perkins, J. D., Barton, G. W., 1994. Structural decisions in on-line optimization. Proceedings of PSE'94,297-302. Jackson, J. E., 1991. A user's guide to principal components. Wiley series in probability and mathematical statistics. Applied probability and statistics. John Wiley & Sons, Inc., New York. Lid, T., Strand, S., Skogestad, S., January 2001. On-line optimization of a crude unit heat exchanger network. In: Chemical Process Control - 6. Maarleveld, A., Rijnsdorp, J. E., 1970. Constraint control on distillation columns. Automatica 6, 51-58. Skogestad, S., Halvorsen, I. J., Morud, J. C., 1998. Self-optimizing control: The basic idea and taylor series analysis. Presented at AIChE Annual Meeting, Miami Beach,16-20 Nov;paper 229c. White, D. C., June 1997. Online optimization:what, where and estimating ROI. Hydrocarbon Processing, 43-51. Zhang, Y., Forbes, J. F., 2000. Extended design cost: a performance criterion for real-time optimization systems. Computers & Chemical Engineering 24, 1829-1841.
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rightsreserved.
1047
An Approach to Controllability and Economic Design of Nonlinear Systems with Multiplicity Keming Ma and I. David L. Bogle t Department of Chemical Engineering, University College London, Torrington Place, London WC1E 7JE, UK This paper introduces a generic method to incorporate an aspect of dynamic controllability into steady state process design, that of bifurcation behaviour. This approach consists of identifying the dynamic behaviour of an existing process, and following a design method, which determines a trade-off between good dynamic behaviour and good economic performance. The algorithm determines a sub-optimal solution subject to singular constraints. The economic objective is penalised for poor controllability and the retrofit design will be "bifurcation free" over the disturbance and operating range of interest. This method can provide a design analysis tool to be used to compare or screen alternative conceptual designs at an early stage by introducing quantitative economics and qualitative dynamics. An industrial polymerisation reaction example presented illustrates the application of this approach.
1. I N T R O D U C T I O N In order to understand a process and prevent it from failing to meet the required performance specification due to its inherent characteristics that cannot be overcome by controller designs, it is important to analyse the controllability of a process at an early stage. At this stage modifications are possible and the 'best' process design can be achieved as a trade-off between controllability and economics. Economic optimisation based on the steady state will lead us to build the plant that minimises the cost. However, it is necessary to investigate the flexibility and controllability of the plant. It has been shown that the performance is limited by the inherent behaviour of the plant itself [ 1], and a controller can only ensure the best performance given these limitations. There have been papers published on algorithmic synthesis techniques to balance between economics and dynamic performance in various ways recently. Integrated approaches for dynamic processes have been proposed [2][3][4]. In the literature, these works demonstrated that processes can have good economics and good controllability. However, the indices of controllability are not directly comparable and alternative methods can give different solutions.
* Author to whomcorrespondence should be addressed: email:[email protected]
1048 Nonlinear systems are sensitive to initial conditions and may exhibit multiplicity, or even chaotic behaviour. A small parameter change may cause the system to have a qualitative change in behaviour, i. e. a bifurcation. This presents a challenging control problem. In this work, an algorithmic optimal design method based on singular constraints is presented to eliminate bifurcation problems over the specified disturbance and operating ranges of interest by modifying the design at an early stage in the design process. The method consists of two parts. The first part of the method is to identify the nature of the dynamics of the existing design and to determine how the factors affect controllability over the specified conditions. This is done by investigating the stability of the open-loop steady state and the stability of its inverse in the parameter space simultaneously. We call it the Pseudo-Closed-Loop (PCL) method. The second part is to incorporate dynamics into the economic analysis by using an algorithm to obtain an optimal solution based on the bifurcation constraints. The economic objective is penalised for poor controllability. The modified design will be "bifurcation free" over the specified disturbance and operating range. 2. PSEUDO-CLOSED-LOOP (PCL) APPROACH TO IDENTIFY DYNAMICS The notions of the relative order and zero dynamics have been used in the study of controllability and controller designs in nonlinear systems [5][6]. The zero dynamics express the inverse characteristics of a system. A system with zero dynamics may have hidden unstable modes, which will become unstable when the system is subject to a feedback control law. In this work, the dynamic behaviour of a system is studied by employing bifurcation and singularity theory to identify the inherent properties of the system, i.e. the open-loop stability of a process and the stability of its inverse [7].
2.1. Setting up Pseudo-Closed-Loop (PCL) system Consider an SISO affine nonlinear system with the form: x'= f (x) + g(x)u,
y = h(x)
and transformed into the Byrnes-Isidori normal form [5]: ('= F((1 .... ( n - r , ( n - r + l ..... (n)
(2.1) (2.2)
where x ~ ~1n is a vector of the states, f ~ ~n and g e ~n are smooth function vectors, ( = ~:(x) is a co-ordinate transformation, and r is the relative order. The subsystem of the first n-r equations of equation 2.2 is called the zero dynamics. In the nonlinear setting, the zero dynamics of the system in the new co-ordinates are obtained by using input-output linearization and the derivatives of the outputy, y',y", .... y r , as part of the new state components and setting the output y, Y
=
(n-r+l'
equal to some steady state, y,.~.. The differential equations for the
zero dynamics are described as follows: (i'= Fi((i,Yss,O ..... 0)
(2.3)
where i=l,..,n-r. For the existing system and a set of control inputs, the states must satisfy 0 = Fl.((i,Yss,O ..... 0) (2.4) The zero dynamics of the process is then defined as the dynamic system zi, = Fi(z1.... Zn_r,,Yss), (2.5) which will automatically satisfy
1049 o = Fi(zi,Yss) (2.6) at the steady state, where i= 1.... n-r. Therefore, a new dynamic system is set up by combining the original system (equation 2.2) and its zero dynamics (equation 2.5), giving the form: (i '= Fi((1 .... (n-r,(n-r+l ..... (n) z'j : Fj (z 1.... Zn-r,, Yss)
Y = (n-r+l where i= 1,...n, and j= 1,...,n-r, which is called a Pseudo-Closed-Loop (PCL) system.
(2.7)
2.2. Properties of the PCL system Consider the new system (PCL) with the form 2.7 and assume that the function F can be expanded in a formal power series of 4" and z around the steady state (ss and z.,.~., respectively, and have the following form in which higher order terms are omitted: co'=//co (2.8) where A=
Fz
= ~l(ss,
Fz = ---~lZss' and co = ((,z)T.
The eigenvalues of the PCL system are determined by: det(M-
A) = d e t ( M 1 - Ff) det(M 2 - Fz)
(2.9)
where I, I 1 and 12 are identity matrices. The eigenvalues of the PCL system are exactly the ones of the original system together with its zero dynamics. The stability of the PCL system completely represents the stability of the open-loop steady state of the process and of its zero dynamics. The changes of the stability of the PCL system characterise the bifurcation points. The behaviour of the system is then studied in terms of parameter-dependent branches of steady-state solutions. Furthermore the parameter effects on the behaviour of the system at the bifurcation points are traced out in the parameter space to find the bifurcation regions [8]. 3. CONTROLLABILITY AND ECONOMIC DESIGN APPROACH
With this knowledge of how parameters affect the dynamic behaviour, it is possible to modify the process at the design stage in order for the process to give good economic performance and to exhibit good controllability as well. The undesirable behaviour can be eliminated or minimised by adjusting the design parameters. The economic objective will be penalised for poor controllability. Consider an existing process which is described as: f (u,x,x',|
= o
(3.1)
where u is a control variable, d N , | N a r e the values of the disturbance and design variables at the nominal conditions, respectively. Suppose the ranges of the disturbance and operation are: d L < d < d U and u L < u < u U , respectively. The behaviour of the system is studied in the input space first to detect the bifurcation points by using the method proposed above, and then
1050 branches of bifurcation points are traced out in the input and disturbance parameter space to determine the bifurcation. When the plant is operated with the presence of the disturbances, the control variable (input) has to be adjusted to absorb this disturbance in order to keep the plant at the initial operating point. However, in order to avoid bifurcation problems, the control variable has to be constrained not to breach the bifurcation condition. Once the ability of the control variable is limited, the process cannot necessarily completely reject the disturbance while maintaining the process at the initial conditions. Therefore, other design parameters have to be adjusted to absorb the rest of the disturbance in order for the process to reject the disturbance successfully, which leads to a retrofit design. The optimisation formulation of the retrofit design with the bifurcation constraints is given by: m i n ~ ( u , x , d L (d U ), (9) (P 1) subject to f ( u , x , x ' ,d L (dU),| g (u,x,dL(dU),| u
L
uB
or
=0 <0 uB
U
The objective defines an economic function to minimise the cost. Suppose that the optimal design parameters found in P1 are o , which are the modified design parameters. Therefore, the modified process is expressed as:
f (u,x,x',| *,d N ) =
O,
(3.2)
which will be "bifurcation free" for the specified output inside the range of the input subject to the disturbance. 4. ILLUSTRATIVE EXAMPLE: AN INDUSTRIAL POLYMERISATION REACTION The flee-radical polymerisation of MMA (methyl methacrylate) with AIBN (azo-bisisobutyronide) as initiator, and toluene as solvent takes place in a jacketed CSTR. The reaction is exothermic and a jacket allows heat removal. The model for the process was given by Daoutidis et al [9] to describe the dynamics of the monomer and initiator concentrations (Cm and CI), the reactor and jacket temperature (T and Tj ), and the zeroth and first bulk moments of the molecular weight distribution of the polymer (DO and D 1). The kinetic parameters and system parameter values used were also given by the same authors. The output to be controlled is the Average Molecular Weight Number (MWav) given as the ratio D0/D 1. The control variable is the jacket cooling water flowrate, Fcw. The main disturbance is the inlet reactor temperature, Tin. For the given design, one optimal operation point was given by Lewin and Bogle[10], in which MWav is maintained at 25,000 while minimising the consumption of the expensive initiator, FI. The optimal operating point found in the optimisation is FI=0.00354 m3/h, corresponding to conversion AC = 0.5, and the relevant operation condition is: Fcw=0.168 m3/h, at the nominal value of the disturbance of the inlet feed temperature, Tin=350 K. This point is referred to as the optimal operating point and the given process as the original design.
1051
4.1. Steady-state multiplicity and effects of process design and operation changes. The process exhibits steady-state multiplicity as shown in figures 1 and 2. Two limit points and one Hopf bifurcation point indicated by the solid square are found. The medium state is unstable, indicated by dashed lines in figures 1 and 2. The jacket temperature, exhibits both input and output multiplicity. The obtained optimal operating point is just below one of the limit points located in the lower reaction temperature region. The effects of the inlet feed temperature on the steady states are shown in figure 3 (locus of bifurcation point). It can be seen that the bifurcation values of the input in the face of the disturbance, Tin, are: [Fcw,Tin]=[0.245, 353.8], respectively, as shown in figure 3, around which the jacket temperature will exhibit input multiplicity behaviour while maintaining the process at the initial condition. That means the input multiplicity of the jacket temperature will occur when the input value is beyond the bifurcation value of the input under the change in the feed temperature. The bifurcation values indicate that the original design will be able to reject the +3.8K change in the feed temperature at maximum.
4.2. Retrofit design The modified design aims to make the process reject the worst disturbances successfully by adjusting some design parameters while minimising the cost. The considered range of the disturbances of the feed temperature is (345, 355). The objective of the retrofit design optimisation is to minimise the reactor volume. Therefore, the modified design optimisation formula
MWav vs. Fcw
Tj vs. Fcw
/
iii
",. "'-...
~
03
05
07
Figure l: Steady state: Average Molecular Weight vs feed flowrate showing bifurcation. Fcw vs. Tin
Figure2: Steady state: Feed temperature vs feed flowrate showing bifurcation.
MWav vs. Time (h)
....... --
........ i
Figure 3: Locus of bifurcation for feed flowrate vs feed temperature
/~
....
-
Figure 4: Closed-loop responses of Molecular Weight for +5K degree changes in feed temperature for the original (solid line) and retrofit (dashed line) designs.
1052 tion with bifurcation constraints is described by: min V subject to {MWav=25000; Tin=355; FI=0.00354; Fcw<0.245; and process model equations} The reactor volume found in the optimisation is 0.096 m 3 ( V = 0 . 1 m 3 for the original design). In this particular case, the decrease of the reactor volume can improve the dynamics of the process. The modified design will be "bifurcation-free" over the specified range of the disturbance and operation. The optimisation problems were solved in GAMS/MINOS. The bifurcation analysis was carried out by employing AUTO [8]. 4.3. Dynamic simulation The initial operating point is MWav=25,000 at the steady-state in the lower reaction temperature region. Figure 4 shows the closed-loop (using a regular PI controller) responses of Molecular Weight for the original and retrofit designs. It has been seen that the original process can fail after a +5K degree change in the inlet feed temperature. The modified design rejects the disturbance successfully.
5. CONCLUSIONS It has been shown via simulation that we can identify the inherent nature of the process in the parameter space and determine significant factors which influence controllability. The method is specifically aimed at situations where a bifurcation is known to appear such as in exothermic and polymerisation reactions. The methodology generates a modified process which has good dynamic behaviour and good economic performance, and it is possible to extend the range of control and permit safe operation following a sizeable disturbance. The method can provide a design analysis tool to be used to compare or screen alternative designs by introducing quantitative economics and qualitative dynamics. REFERENCES
[1]. Morari, M., Chem. Eng. Sci., 38, 11 (1983) 1881-1891 [2]. Luyben, M. L. and C. A. Floudas, Comput chem. Engng., 18 (10) (1994) 933-970 [3]. Bahris P. A., J. A. Bandoni and J. A. Romangnoli, AIChEJ., Vol. 43(4) (1997) 997-1015 [4]. Bansal V., R Ross, J. D. Perkins, E, N. Pistikopoulos, J. Proc. Control, 10 (2000) 219-227 [5]. Isidori A., Nonlinear control systems: An introduction, 2nd Edition, Springer-Verlag, Berlin, 1989 [6]. Kravaris C., J. C. Kantor, Ind. Eng. Chem. Res., 29 (1990) 2295-2310 [7]. Iooss G. J. and D. Daniel, Elementary stability and bifurcation theory, 2 nd ed, SpringerVerlag New York, 1990 [8]. E. J. Dodel, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. Sandstede, X. Wang, Auto 97: Concordia University, Montreal, Canada, 1998 [9]. Daoutidis P., M. Soroush, and C. Kravaris, AIChEJ., Vol. 36(10) (1990) 1471-1484 [ 10]. Lewin D.R. and D. Bogle, Comput. chem. Engng., 14, 4/5 (1996) 481-494
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rightsreserved.
1053
Dynamic Data Reconciliation of regenerative Heat Exchangers coupled to a Blast Furnace. Fr6d6ric a
Minet b, Georges Heyen a, Boris Kalitventzeff b, Jos6
Di Puma r Marc Malmendier d
L.A.S.S.C., Universit6 de Li6ge, Sart-Tilman B~timent B6a, B-4000 Li6ge (B), [email protected]
b Belsim s.a., 1 All6e des Noisetiers, B-4031 Angleur - Li6ge (Belgium) c Cockerill-Sambre group Usinor, Ironmaking department, PI. Hauts Fourneaux, B-4102 Ougr6e (B) d Cockerill-Sambre group Usinor, Research&Development, Quai du Halage, 10, B-4400 F16malle (B) To examine the feasibility of dynamic validation, the air preheating section of a blast furnace operated at the Ougr6e facility of Cockerill-Sambre (Usinor group) has been studied. This application shows the benefits and the difficulties of this emerging technology. 1. I N T R O D U C T I O N The data reconciliation technology finds more and more applications in continuous processes. It uses mass and energy conservation laws and statistics (error distribution) to correct measurements. Its benefits are numerous and range from improved measurements layout to increased plant efficiency. The applications of this technique cover the sector of fossil and nuclear power plants as well as the chemical plants. However, all the processes are not continuous. Some of them (or some parts of them) are influenced by their history. In these cases, the data reconciliation technology has to take transients and history into account. 2. P R O C E S S D E S C R I P T I O N Iron manufacturing in blast furnace requires a large flow of air preheated at high temperature (above 1000~ A battery of 3-4 regenerative heat exchangers (stoves), operated
1054
cyclically, is located in the vicinity of the furnace. A stove is a tall, cylindrical vessel filled with refractory (the checkers) used to accumulate and later release energy. The brick geometry (see figure 2) and its properties usually varies in sections, as the upper section operates at much higher temperature than the lower section. During the first part of a cycle (on-gas), the checkers are heated by combustion of coke oven gas, possibly enriched with natural gas (see figure l a). When the temperature is high enough, compressed air (the blast) circulates through the stove in the opposite way (figure lb). During this on-blast operation, a by-pass valve controls the preheat temperature. The hot blast is fed to the furnace until the stove temperature becomes too low. At that moment, the next stove is put into operation and a new cycle is repeated (approximately every 90 minutes). Typically, a stove is on-gas during 50 minutes, and on-blast during 30 minutes. Transition between operation modes requires 5 minutes, needed to switch valves, pressurise or blow-down the stove. The amount of energy stored in the refractory is for sure an important process variable. However it cannot be directly measured. No thermocouple could resist long due to the high temperature levels. A model based evaluation tool is needed to help in process monitoring and planning. In order to achieve this goal, existing steady state data reconciliation software (BELSIM VALI III) has been extended to handle a dynamic model of the stoves. Fluid compression and expansion, mixing and reactions are assumed to be in steady state, since the residence time in the stove is much shorter than the measurements sampling period. Measurements are the air and fuel gas flow rates and temperatures, the fuel composition, the hot blast temperature (several replicates). The only measurements available to characterise the checkers temperature are: the dome temperature, measured by a pyrometer aimed at the top of the checkers (but this measurement is influenced by the gas temperature through which the pyrometer is aimed); some thermocouples are located approximately at mid-height; however thermal expansion of the bricks is such that nobody knows for sure whether their tips are located in the solid or in a gas channel; the temperature of the grid supporting the checkers is also measured. -
-
-
1055 The data reconciliation program handles measurements averaged during 2 minutes. The procedure integrates the heat transfer model using as initial conditions the reconciled profiles obtained in the previous step, and uses this model as a set of constraints to reconcile all available measurements.
3. HEAT TRANSFER MODEL We assumed that the flow of gas through the regenerative heat exchanger is distributed evenly among the parallel channels (approximately 10000 of them). Thus the model will focus on a single channel, and approximate the hexagonal cross section by an equivalent circular section (figure 3). The partial differential equations describing the system will be transformed into ODE's by discretising the space co-ordinates. Assuming plug flow and no axial conduction in the gas leads to a dynamic balance equation (Bird et al., 1960): [OTg
cOTg1
OP oz
4h
(1)
Similarly for the tube wall: p.,Cp,. OT,. " Ot
1 0 rk - k~. " = 0 r Or Or J " Oz 2
(2)
In order to obtain a numerical solution of this equation system, we discretised the solid and the gas into cells of constant geometric and physical properties, each cell being approximately 1 meter long. In the radial direction, the solid was also discretised into layers approximately 1 mm thick (figure 3) exchanging heat by conduction with the neighbouring layers. By symmetry of the system, no heat is transferred from one channel to the adjacent one, so the deepest layer only exchanges heat on one side. An overall heat loss term was added to the gas balance equation to account for exchange with the ambient. The characteristic time for heat transfer in the radial direction was found to be in the order of one minute; this is the same order of magnitude as the phenomena to be monitored in the process, and therefore it may not be ignored. The temperature gradient in the solid in the radial direction can be as high as 1000 K/m, while it is approximately 30 K/m in the axial direction. This is why we neglected the conductive heat transfer in the axial direction. The numerical solution of system dynamic equations was obtained by discretising along the space co-ordinates, and integration using Euler's method to find the variation of the temperature field with time. The integration time step is limited to avoid numerical instability. In order to do so, the equations were rewritten as follows. Gas-solid transfer for a cell On the gas side, an energy balance allows to calculate the heat transfer in a cell: Ho.,
- t1,.
- -Qex~,. - Losses
(3)
where :
Hin : inlet gas enthalpy Hour : outlet gas enthalpy, (Hout = Hin of the next layer) Qexch : heat transferred from the gas to the refractory brickwork The average gas temperature of the layer can be obtained from the heat transfer equation:
Tg =
(Hou t - H m -Losses)
h.S
+T w
(4)
1056 The heat transfer coefficient h is estimated using the standard correlations for pipes. This estimate is multiplied by a correction factor obtained by matching the model predictions with the plant data. This correction factors is needed since the gas channels are slightly tapered, the variation of the section enhancing the heat transfer. A c c u m u l a t i o n a n d c o n d u c t i v e h e a t t r a n s f e r in t h e solid:
For a short time step At, the energy balance for the solid element i surrounded by elements j where all temperatures are considered constant, becomes:
ZK,~ (Tj(O-T~(O)= p, Cp~V, AT, ,
'
where Kij :
(5)
At
thermal conductance between elements i and j (W/K),
For the interior elements exchanging heat by conduction 9Kij = k, A,j Lij where Aij : area of the solid section Lij : thickness of the element For the gas-solid exchange,: Kij = hj Aij An explicit equation can be obtained noting that ATi = Ti(t + A t ) - Ti(t) : T, (t + At) = A t PsCp,sVi
K,jTj(t) +
"
' At
-
K,j T, (t)
(6)
Thus for the solid element in contact with the gas : +
(7)
+
where M : a"
Fourier module = AxV a/At, k~ thermal diffusivity a = 9
Nu:
Nusselt number = h Ax / k.
p.~.Cp,,~,
4. M A J O R RESULTS The data reconciliation algorithm is run 30 times per hour, in order to follow the transient temperature profiles. Based on measured values of the gas and air flow rates, fuel LHV, and previously identified temperature profiles in the checker, the software provides corrected values for all flows and temperatures, and an updated temperature profile for the checkers. However the application of dynamic data reconciliation on such a short period of time has little practical interest, since the results are influenced by : 9 the initial temperature profile; 9 the detection and handling of the transition step (e.g. switch between heating and cooling); 9 the large inertia of the system (low sensitivity of some output variables to a change in input). Figure 4 illustrates this effect. For a similar initial temperature profile in the solid, the blast temperature profile has been simulated for 2 feed temperatures. One notices easily that the hot
1057
T
(~
Gas temperature profile
1400
~
in = 1280 ~
1200 1000 800 60O 4O0 200 0
i
0
5
!
i
i
10
15
20
Cell number
blast temperature is almost independent from the feed temperature, due to the high efficiency of the heat transfer. The dynamic data reconciliation method proves its main efficiency in the long run. Actually, the process has a "long memory effect". This makes it difficult to start a dynamic validation, since the initial temperature profile in the stove is not known in details (only 3 measurements). Any initial error will bias the validation process until the effect of initial conditions dies out. When working on long periods (several cycles), there is a self-adjustment of the profile. A major outcome is the possibility to calculate important process parameters that cannot be measured, e.g. the energy accumulation in the stove. Process trends can be visualised and compared, as shown on figure 5. The diagram in figure 5a shows that the inlet temperature of the cold blast has decreased steadily during a 20-hour period. The long-term result was a corresponding decrease of the dome temperature (linear trend line on figure 5b). When repeated for a long time period, data reconciliation allows detecting errors that would be negligible on short period. For example, the errors on LHV measurement can be negligible if the integration time is short. But if long periods are studied, it clearly appears that the energy balance in the regenerator is biased.
1058 The application of dynamic data reconciliation to the stove system allowed detecting some process malfunctions: 9 Some sensors do not measure what they are expected to do; this is the case for the dome temperature measurement using a pyrometer; the probe is supposed to measure the brick temperature, but it is influenced by the emissivity of the gas through which it is aimed. Since the gas emissivity depends mainly from the partial pressure of CO2 and H20, the bias introduced is not the same when the unit operates on-gas or on-blast. 9 Some sensors are not appropriate (the 02 probe saturates when the blast is enriched with oxygen). 9 Some sensors are not ideally located : e.g. the thermocouples located below the supporting grid do not provide an adequate estimate of the brick temperature when the regenerator operates on blast; in fact the thermocouple is surrounded with cold air, and measures the gas temperature more than the solid temperature. 9 Some measured values were not correctly converted in the process computer, such as the lower heating value for the residual gas used as a fuel. 5. CONCLUSION The main goal of the present study was to better understand the behaviour of the regenerative heat exchanger, in order to improve its control. The model developed allows to carry out a sensitivity analysis and to validate the measurement system. It provided also values of key operating parameters of the plant, which are needed for optimally scheduling of the energy accumulation and delivery. By operating closer to the operating limits and decreasing the heat losses, significant fuel savings are expected (in a recent paper, Muske and al.(2000) developed a similar model as a support for optimal control and they report savings in the range of 5%). This study shows the feasibility and the possibilities of dynamic validation but it also identifies some practical limitations: selection of initial conditions, inertia phenomenon,... 6. SYMBOLS Cp specific heat Dh hydraulic diameter of a channel h heat transfer coefficient k thermal conductivity P pressure
r S T v z
radial co-ordinate gas-solid exchange area in a cell temperature gas velocity axial co-ordinate
p specific gravity Subscripts g s w
gas solid gas-solid interface
7. REFERENCES Bird R.B., Stewart W.E., Lightfoot E.N., Transport Phenomena, Wiley New York (1960) BELSIM, VALI III Users Guide, BELSIM s.a., All6e des Noisetiers 1,4031 Angleur (Belgium) (1999) Cameron D, Fault Detection and Diagnosis, in "Model Based Manufacturing - Consolidated Review of Research and Applications", document available in CAPE.NET web site (http://capenet.chemeng.ucl.ac.uk/) (1999) Muske K.R., Howse J.W., Hansen G.A., Cagliostro D.J, Model-based control of a thermal regenerator:Part 1: dynamic model, Computers & Chemical Engineering 24, 2519-2531 (2000) Part 2: Control and estimation, Computers & Chemical Engineering 24, 2507-2518 (2000)
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1059
New Chemical Process Economic Analysis Methods H i r o s h i O k a d a a a n d T e t s u o Shiraob
aEngineering & IT Department Engineering Division, JGC Corporation 2-3-1, Minato Mirai, Nishi-ku, Yokohama 220-6001, Japan bResearch Planning Department, Mitsubishi Chemical Corporation Yokohama Research Center, 1000, Kamoshida-cho Aoba-ku, Yokohama 227-8502, Japan This report describes New Chemical Process Economic Analysis Methods, including Concept of Chemical Process Analysis and Basic Cost Items, Bridging between Economic Analysis Methods and Process Representation Methods which consists of Classification of Process function and Plant function, and Mapping between them. The Major expected result of this will enable the process industries to reach new quality and productivity levels in designing and operating their plants. This is conducted within a framework of IMS international collaborative program with a financial support provided by IMS Promotion Center, an affiliate of Manufacaning Science and Technology Center in Japan.
1.
INTRODUCTION
GCO Project (99/7-2001/12) is one of the IMS International Projects aiming to construct the integrated environments for chemical process analysis. It is applying Plug & Play based simulation technology of the CAPE-OPEN Project of EU (97/1-99/6) for process design. [1] [2] This project will set interoperability standards for communication between components in process design to provide a solution to overcome interfacing problems of existing proprietary systems such that it shall be able to improve the efficiency and speed of process designing. GCO Scope is the followings: Standards for new classes of Process Model Components : Complex material, Kinetics, Optimisation, Parameter Estimation & Data Reconciliation, Distributed Parameter Modelling, Other Standards Development of CO-compliant software : By companies, universities, software suppliers Enterprise integration aspects : Case studies, Component selection, Work process Additional research on open CAPE: Model structuring, Modeling process, Hybrid and discrete, Lifecycle interface, Real-time and online interfaces. Japanese IMS Project (99/4-2002/3), jointed with the IMS GCO Intemational Project, is concentrating in the other Standards. The major task is to bridge between Economic Analysis Methods and Process Representation Methods by defining Classification of Process function and Plant function, and mapping between them.
1060
Fig. 1. Plug & Play based simulation Technolog 2.
Fig.2. Our Plan on GCO scope
METHODOLOGY AND TARGET
Based on preliminary Standardization work of MITI-IPA funded Chemical Industry CALS [3] and also CAPE- OPEN Methods & Object Oriented UML methodology, the project will be executed. The goal of the project is to establish the Integrated Environment for FS as shown in Fig. 4 below.
Fig. 3 Concept of Object Oriented UML
Fig.4. Integrated Environment for FS
1061 3.
RESEARCH & DEVELOPMENT TOPICS
The topics include the followings: Concepts of plant life cycle, Process Economic Analysis Methods, Process Representation Methods of Japan IMS Research Results. [4] [5] 3.1. Concepts of analysis in plant life cycle assessment Chemical Process Analysis Assessment identifies major critical stages of Plant Life Cycle. (a) Business plan: Sets quality target, sales volume, cost in consideration of timing, investment target (including investment for research and development, plant cost etc) (b) Research and Development (R&D): Establishes a process by the process development and economic analysis mainly based on raw material and fuel costs. (c) Process design: Optimizes Process by Conceptual Process Design and Detailed Process Design: Co-work with Engineering design. Decides a process by economic analysis mainly based on raw material and fuel costs and plant costs by Process design and case study ->Process decision->User requirement specification development. (d) Commercialization decision: Decides the plant location in consideration of off-site conditions and physical distribution based on the analysis of return on investment, profitability, risks (including quality and safety risks). (e) Engineering design and construction: Engineering design: Optimizes plant design by Conceptual Engineering design and Detailed Engineering Design based on user requirement specifications. Construction: Optimizes plant construction and Commissioning (f) Operation and maintenance (Production): Maintain the optimum production in response to changing environmental conditions for the process and plant (by modification, etc.)
Further analysis has been conducted in more detail regarding Conceptual and Detailed Process design. Conceptual Process Design phase includes Process selection, Process synthesis, Process analysis and evaluation. And Process decision will be made in this phase. On the other side, in Detailed Process Design phase, User Requirement Specifications will be produced which is covering Pre-P&ID, Equipment functional design, Control and operability study, Safety & Health & Environmental study and Economic study. After Commercialization decision, Engineering design will start from user Requirement Specifications. Engineering design can be broken down into Conceptual and Detailed Engineering design. After Construction and Commissioning, Operation and maintenance will start. 3.2. Process Economic Analysis Methods Requirements for the economic analysis of a process (which are utilized in respective phases of R&D, designing and production) yielding from what economic analysis required to have, namely fundamental cost factors to complete the task. The economic analysis includes analysis of raw materials and fuel cost analysis of return-on-investment and analysis of product cost used in the optimum process selection has been investigated. This economic analysis will be combined with process representation methods for specific development.
1062 Fundamental cost factors in the economical analysis of a process are as listed below: (utilizable in the life cycle assessment of a plant) (a) Raw materials and fuel cost: Costs for raw materials, energies, utilities, catalysts used at the plant (with by-product value deduction and wastes treatments) (b) Labor cost: Costs for labor requirement at the plant. (c) Plant cost: Costs incurred by the plant (including depreciation expenses, interests on plant investment, fixed assets tax, maintenance costs, etc.) (d) Plant overhead costs: 5-10% equivalent to the total of the raw materials and fuel cost, labor cost and plant cost. Solid products cost relatively more. Overhead expenses (for maintenance, analysis, physical distribution, etc.) which can be directly imposed on a specific product within the plant, as well as expenses (for the management, clerical work, etc. of the plant) commonly distributed to products within the plant. (e) Sales and physical distribution costs: Costs for sales and physical distribution outside the plant: (direct charge + distributed expenses) including costs incurred at stock points. (f) Administrative expenses: 5-10% of the total of the above described costs and expenses. Functional products cost relatively more. Headquarter expenses: Include insurance premiums, patent license fees, R&D expense, and interests on working capital. (g) Production cost: The total of the above described costs and expenses. Table 1: Cooperation between Economic Analysis Methods and Process Representation Methods
Hatching area: Main Estimation D BFD : Rough cost estimation-~ Rough economic analysis D Pre P&ID / P&ID: Design & cost estimation -~ Design and economic analysis (a) BFD phase: Accurate estimation of Raw material and fuel cost can be obtained whilst only rough estimation of plant cost can be calculated approximately before the engineering work has yet done. This holds true with labor cost estimation. Investment effect: Payback period of investment (= Investment / Annual merit from investment) is to be used together with a decision criterion. (b) Pre P&ID/P&ID phase: Accuracy with plant cost will be improved based on engineering work done. Investment effect: Payback period of investment is to be used together with a decision criterion. More precise evaluation of the investment analysis such as Retum on Investment, Payback Period, Retum on Sale based on relevant costs including labor cost and other related ones can be performed. But it still needs to perform quality and safety analysis. (c) Operation Maintenance phase: Under steady state operation, plant cost will be the main concem since only replacement cost due to deterioration is considered. In order to increase the accuracy of the estimation at this stage, detailed construction cost estimation should be conducted. In case of changing operation, case studies using a process simulator should be conducted, moreover
1063 raw material and fuel has to be re-evaluated; thus, a decision on the conditional change has to be made. Using actual cost evaluation figures available, an optimal solution can be calculated 3.3. Process Representation Methods Concepts of process representation methods, as well as classification methods and mapping rules Concepts of process representation methods and classification methods in BFD Functional elements of a process include reaction, separation, blending, heat exchange, storage, transfer, etc. These functional elements are classified as unit operations, and the connecting relationships of the functional elements are defined by strucawal information (stream connection relationship and flow directions) to express the process. Also, based on handled materials and their physical data, stream data, which includes process conditions and material, as well as of material and heat balances can be expressed. Incidentally, known process simulators have built-in models on respective unit operations and physical properties, and perform functions of simulating processes and designing the outline of process specifications. Concepts of functional elements of a process and unit operations Reaction: Reactions of oxidation, reduction, decomposition etc. in a gaseous, liquid or solid, and homogeneous or heterogeneous system to obtain one or more reaction products from one or more starting materials, many of which often require the combination with blending, separation, heat exchange and specially functional unit operations. Separation: Separation of gases, liquids and/or solids, including evaporation, crystallization, condensation, absorption, extraction, adsorption, ion exchange, drying, filtration and compression, precipitating, centrifugal or floating separation, dust collection, etc. Blending: Includes the blending of gaseous, liquid or solid and homogeneous or heterogeneous systems, dissolution, etc. Heat exchange: Heat exchange between gases, liquids and/or solids, including heat exchange between process flows, heating, cooling, etc. Storage: Includes gas, liquid or solid storage, etc. Transfer: includes gas, liquid or solid transfer, pressurization or depressufization, etc. Special unit operations: Mainly classified as the pulverization, granulation and classification of solids, and processing. (Note: Fluidization and heat transfer has been omitted because these are common technologies.) Concepts of methods for process representation and classification methods in PFI) Structural information of a process (the connecting relationship of streams, flow directions, etc.) Stream data (process conditions, physical properties, etc.) inherited from BFD bridges unit operations constituting the process and equipment constituting plants. But because no standardization has been really achieved, there exist extreme variety ranging from BFD to PFD closer to P&ID, depending upon the different philosophy of participating companies. The mapping function of a process (unit operations) and a plant (equipment), which is important for PFD, shall be considered here as the functional representation of PFD. Concepts of functional representation of a plant in P&ID and the classification method (Arrangement of N: M relationship in BFD and P&ID. Simplification of PFD)
1064 BFD: PFD N : M ->1:1 PFD: Pre P&ID N :M -> 1 :N PFD simplification (Defining BFD and Pre P&ID as interface functions) Here, it is necessary in the preparation ofPre P&ID to adopt the best practice as default, and to map the relationship between BFD and Pre P&ID as that close to 1 : 1. Concepts of process representation methods, as well as classification methods and mapping rules the mapping function of a process (unit operations) and a plant (equipment), which is important for PFD, shall be considered here as the functional representation of PFD. P&ID
BFD
PFD
|
I
Process
Plant
Plant
Function
Concept
Function
] ....
,.
] .
I
I
Unit
Equipment
Equipment
Operations
Concept
Fig. 6 Concepts of process representation methods
4.
Fig. 7 Example for mapping functio of a process and a plant
FUTURE WORK
User Requirement of New Economic Analysis has to be composed as well as System Specification. And the next step has to be verified the effectiveness of the System specification over the lifecycle of the process plant. Our Plan is the development of basic specifications by usage of the UML model, which is used, in the intemational joint research to verify the basic specifications. Intemational joint research will be cooperation with additional research project on open CAPE, which covers the life cycle assessment of chemical plants. REFERENCES
[ 1] Joint Symposium ESCAPE-9 and PRES'99 May 31-June 2, 1999, Budapest, Hungary. [2] B. L. Braunschweig, C. Pantelides, H. I. Britt and S. Sama: Open Sottware Architectures for Process Modeling (FORCAPD'99). [3] Information-technology Promotion Agency, Japan (MITI-IPA): Report About Results "Arrangement of Common EC Foundations in Manufacturing, Sales and Physical Distribution in the Chemical Industry" under the Operation for Establishment of Common Grounds for Electronic Commerce issued (edited by Chemical Industry CALS). [4] T. Shirao, A. Ueda(MCC), H. Okada (JGC):Chemical Process Economic Analysis Methods(Conference of Japan IMS Research Results Julyl 1-12 '00) [5] H. Okada(JGC), T. Shirao, N. Dohi(MCC), Chemical Process Representation Methods(Conference of Japan IMS Research Results July 11-12 '00)
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
A Vision of Future Simulation & Control
Needs
and
Capabilities
1065
in Process
Modelling,
J. Pingen Shell International Chemicals B.V., Shell Research and Technology Centre Amsterdam, PO Box 38000, 1030 BN Amsterdam, The Netherlands Process Modelling, Simulation and Control (PMSC) has an important role to play in enhancing the competitiveness of the European process industries. This paper presents an overview of the opportunities for beneficial application of PMSC tools and techniques throughout the process life-cycle, from initial research, through development and design, on into production and finally to decommissioning. The purpose of the document is to articulate the needs of the end user to the wider technology community and to assist the software providers and researchers in the field to understand and share the users' requirements. The main conclusions is, that integration is a key issue. This includes consistency across the life cycle, real time cooperation between departments and organisations and information sharing between different tools and environments.
1. INTRODUCTION The Chemical Process Industry is a large, diverse and currently very successful sector of the European manufacturing industry. The industry is, however, under growing threat from elsewhere and must now respond to a variety of challenges if it is to remain competitive. Computer Aided Process Engineering (CAPE) is a key contributor to the achievement of excellence throughout the life cycle: claims have been made of significant savings in operating costs and capital expenditure and there is general agreement that CAPE technology forms a powerful competitive weapon in all PMSC areas. The CAPRI (Competitive Advantage through PRocess It) cluster, an initiative within CEFIC's SUSTECH programme and CAPE.NET (a thematic network sponsored under the EU Framework IV programme) have jointly developed a preliminary definition of the sector's requirements and a Technical Vision wherein the process model evolves alongside the process itself throughout its life, being enhanced and refined at each stage and contributing to the consistent achievement of excellence. This vision is laid down in an extensive document, of which this paper is an excerpt (Ref. 1).
1066 2. BUSINESS TRENDS
There are a range of business/legislative/technology trends currently in progress which will influence all of this over the next 10-20 years and whose effect, therefore, must be borne in mind whilst developing our vision of the PMSC future. Prominent amongst these are: 9 increasingly volatile and competitive leading to: * shorter product life * improved product differentiation, novel products & processes * more "agile" and flexible performance of multi-product processes (many of the higher-added-value products are specialities, often produced in batch, semi-batch or cyclic processes and involving complex materials) * more flexible and responsive supply, manufacture, and distribution networks * improved and more consistent quality 9 margin compression and intensifying competition: demanding a more rigorous approach to manufacturing excellence. Manufacturing must increasingly become: * highly efficient (ie. optimal conversion of raw materials, energy, etc.), especially in the commodity sectors, where the operating margins are lowest * flexible and responsive to market dynamics (dynamic operations are now becoming normal, rather than exceptional, and there is increasing commonality between campaign-operated continuous plants and batch plants, for example) * safe & clean, both for the workforce and the surrounding countryside and population * more reliable and resilient; improved on-stream factors * consistent in the production of top-quality products.
3. INDUSTRY RESPONSE Responding to these pressures, by the introduction of "new concept" processes and plants which are more flexible, efficient, safe and clean, places very demanding requirements on process development, on process and plant design and on production operations. Plants are becoming more complex and more tightly integrated, both internally within individual plants and on a site-wide basis, via shared utilities, relief systems, waste treatment, etc. Such integrated complexes exhibit more complex operational behaviour and are therefore more difficult to commission and to maintain at optimum performance levels, especially while responding to changing market conditions and ensuring safe and clean operation. Greater demands are, therefore, placed on the operating staff, thus requiring more comprehensive training and new "intelligent" operational aids.
1067 Faced with these challenges and if the current level of success is to be maintained, the industry must achieve and maintain excellence throughout the life cycle. It is clear from the above that a critical issue will be an improved understanding of (and, thereby, an improved ability to predict and control) the fundamental underlying micro-processes and phenomena and how these interact at the macro level to give the overall behaviour of the process. 9 E x c e l l e n c e in R e s e a r c h
Innovation at a fundamental level will be required to achieve the necessary degree of change. However, such research activities need to encompass not just the underpinning chemistry (for example, product innovation & differentiation) but also an exhaustive examination of the opportunities for radically improved new process technologies: innovative integration of thermodynamics & fluid dynamics and of micro-scale phenomena & subprocesses, leading, for example, to inherently cleaner or safer technology, lower energy requirements, multi-functional units, more intense processing, and so on. Integrated development of new products and the processes to produce them will become increasingly important. 9 E x c e l l e n c e in P r o c e s s D e v e l o p m e n t
Process Development is the important link between the often exciting research results and the engineering calculations leading to the hardware of the manufacturing plant. It is especially here, at the earliest stages of the technical design, when apparently simple technical decisions can have far-reaching economic consequences. Critical issues and risks must be identified as early as possible and appropriate interactions with the business decision-making established. Errors or omissions made up to this stage become increasingly difficult and expensive to rectify (both in terms of cost and of schedule delays, caused by major recycling of the design) as the project progresses. It is therefore essential that a good balanced design (ie. one that is flexible and efficient, whilst also being reliable, safe and clean) be developed at the earliest stage possible, taking full account of the plant's required operating behaviour, in order that recycling and revisions are minimised. 9 E x c e l l e n c e in D e s i g n
To achieve manufacturing excellence, the plant must be operated effectively but, in order to make this feasible, the right operating characteristics must have been built into the plant in the first place -consistent design excellence is also required, to ensure that these desired operating characteristics are inherent to the design and do not need to be "bolted on" by later and expensive additions. Such operational characteristics of the plant reflect not only the process itself but also its control, safety and environmental protection systems, its layout and its supervisory and management procedures. These inherent characteristics are effectively frozen very early in the design process (typically during conceptual process development) and our ability to influence them (unless by substantial redesign, excess costs and damaging delays) declines rapidly, once this "front-end design" is completed.
1068 9 Excellence in Manufacturing
Manufacturing excellence is generally taken to mean manufacturing which exhibits a number of desirable characteristics such as efficient, flexible, clean, safe, reliable, etc, though it must also be remembered that these manufacturing facilities exist within supply and distribution chains and that these "extended enterprises" may have characteristics of their own? These characteristics essentially reflect the internal behaviour of the plant and its underlying processes, ie. the material and energy flows, control signals, etc. Thus, radical improvement of these characteristics entails a radical improvement of both the processes themselves and/or the ways in which they are operated and managed. This, in turn, depends on an in-depth understanding of what these processes are and how they behave and so it is clear that process models (and the "deep knowledge" they contain) have the potential to play a substantial role in these improvement strategies.
4. PMSC CONTRIBUTION: THE LIFE-CYCLE APPROACH 9 PMSC models embody knowledge & understanding of the process and its underlying phenomena & behaviour. Depending on the level of complexity, this may include the chemistry, process structure & operating logic, equipment details and so on. Such models are therefore useful throughout the process life cycle. 9 Models are widely used but, currently, the extent of re-use of models is very low: there is extensive duplication and reinvention as the process moves from one phase of its life cycle to the next. 9 Models of the process and its behaviour should evolve alongside the process itself, through the life cycle, so that the knowledge and understanding which they contain can be built upon and re-used and consistency is "guaranteed". 9 An important interaction is that between technical/engineering development and cost. Decisions made at the earliest stages often have the largest cost implications and yet this critical interaction is often missed. A facility for "integrated economic evaluation" would be of significant value throughout the life cycle. 9 Within all of these models, the most basic information is the representation of the underlying physical and chemical behaviour of the materials being processed. Thus, it is critical that data and models for these underpinning aspects are available and that their quality is known. 9 We must not lose sight of the wide range of other technologies which are the primary responsibility of other communities but which, nonetheless, play a potentially critical role within PMSC, including, for example, costing/economic evaluation, advanced information technology, computational chemistry, computational fluid dynamics, etc and, of course, thermodynamics and chemistry itself.
1069 9 A toolbox and/or modelling environment in each of the phases of the life cycle will assist these activities and will contribute to the achievement of excellence. In some cases, a number of tools exist (for example, process simulators) and do make such a contribution but, typically, problems still remain (for example, a lack of integration- see below- and, in some cases, a lack of robustness and effectiveness in non-specialist hands). 9 Such toolboxes will require supporting infrastructures to support the creation of the models and capture and manage the knowledge and understanding, the models themselves, underlying rationale, decisions, etc. Such infrastructures will provide the basis for "design warehouses" and for ensuring that information and models can readily be shared between phases of the life cycle, rather than being re-invented at every stage. 9 A key * * * *
issue is integration and information sharing: between the various tools and environments across the life cycle with other systems and tools, such as CFD, spreadsheets, word processors, etc. across departments and even organisations.
So-called "legacy systems" must also be handled. Such integration and information exchange will involve other initiatives and technologies, such as STEP, PDXI, CORBA, OLE, etc. 9 Some of the "traditional" PMSC areas are addressed by a number of different vendors' products (a particular example is steady-state flowsheet simulation, where the picture is further complicated by a number of in-house simulators in the larger companies). In these areas, each of these systems have particular strengths and weaknesses and the issue of inter-operability amongst these systems must be addressed, so that end-users can "mix & match" amongst models for unit operations, solution methods, thermo-physical properties and so forth, to gain the maximum overall capability. This issue is addressed by the CAPEOPEN project. 9 The incorporation of knowledge based systems technologies into these tools and environments may enable routine tasks to be at least partially automated, thus releasing the engineer's time for making critical decisions and judgements. It is vital, however, to ensure that these "expert" systems remain as aids and advisers to the engineer and that he exercises the final responsibility for the design. There will be a need for interaction with the AI community to explore the appropriate capabilities and technologies. For process modelling the following wish list should be taken up for future development: 9 9 9 9
Stable, robust and bug-free models Efficient error handling and reporting Interactive program execution High speed solvers
1070 9 9 9 9 9 9 9 9 9 9 9 9
Solver optimisation User friendliness (intuitive GUI) Handling of discontinuities and discrete events Suitable for continuous, batch and all combinations Possibilities to add notes Shared models Conditional modelling Model flexibility: simple or rigorous Open structure Use of standardized data Full upward and downward compatibility Possibility to modify standard modules
REFERENCES 1. L. Bolton and T. Perris, A Vision of Future Industrial Needs and Capabilities, CEFIC/SUSTECH, July 1999.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1071
Component-based implementation of a dynamic optimization algorithm using adaptive parameterization * M. Schlegel, T. Binder, A. Cruse, J. Oldenburg, W. Marquardt Lehrstuhl ftir Prozesstechnik, RWTH Aachen, D-52056 Aachen, Germany In this work we present a component software technique applied to a dynamic optimization algorithm based on the sequential approach. The implementation of the algorithm allows the optimization of existing models formulated in recent modeling environments without the need of model transfer or recoding. The numerical algorithm is capable of generating problemdependent, non-uniform discretization grids which might differ for each control variable. The software is used to solve an example problem of industrial relevant size. Based on the experience drawn from the example benefits and drawbacks of this technology are discussed. 1. INTRODUCTION Dynamic optimization nowadays is used for various applications in process design and operations. Examples include the design of trajectories for the optimal operation of batch and semi-batch reactors or for continuous processes during transient phases such as grade transitions, start-up or shut-down. In general, dynamic optimization algorithms require a mathematical model of the process considered. For industrial processes, the development, validation and maintenance of process models often require major financial expenses as well as a substantial amount of engineering experience. For these reasons, the reuse of existing models or parts of them is highly desirable. However, a large variety of proprietary software tools for computer-aided process engineering, especially modeling tools, are used in industry today, whereas model-based numerical algorithms often require the model information in a format, which is typically not compliant to the modeling tools. A transfer of model information between such applications or the conversion to a special forn~at can be cumbersome and error-prone. In certain cases it might be even impossible due to bottlenecks in interoperability and reusability. To overcome such problems, the EC-funded CAPE-OPEN project (Braunschweig et al., 2000) aims at an open standard system of interfaces for information exchange between software tools in process systems engineering. Communication between heterogeneous applications is foreseen to be done by using interoperability standards, e.g. CORBA (Henning and Vinoski, 1999). In order to benefit from the concepts developed in this project, obviously there is the need for application software, which implements those interface definitions for practical use. In this contribution we consider the application of these ideas to the area of dynamic optimization. Applied to industrial processes, dynamic optimization commonly leads to large-scale and *This work has been partially funded by the European Commission under grant G1RD-CT-1999-O0146in the "INCOOP" project.
1072 strongly nonlinear differential-algebraic problems containing path and endpoint constraints. The solution of such problems is still a computationally demanding task. Efficient solution algorithms are important for real-time applications, such as optimization-based monitoring and control on receding horizons. Many algorithms for dynamic optimization have been developed based on the direct methods, which convert the continuous problem into a finite--dimensional nonlinear programming problem (NLP) (e.g. Vassiliadis et al. (1994), Bock et al. (2000), Cervantes and Biegler (1998)). In particular within the sequential or single-shooting approach, this NLP is obtained by parameterization of the control variables only. Accuracy, efficiency, and robustness of the solution of dynamic optimization problems strongly depend on the chosen discretization grids (e.g. Binder et al. (2000), Betts and Huffmann (1998)). An implementation, which combines a dynamic optimization algorithm using a sequential approach and adaptive discretization grid refinement with innovative software technology is presented in this paper. Experiences with the application to a large-scale example problem are shown and benefits and drawbacks are discussed. 2. PROBLEM FORMULATION
We focus on the use of dynamic optimization for optimal trajectory generation and consider an optimal control problem formulation of the following form: ts min
8
~
u,p,tf s.t.
u, P) da;
90 (ts)), f
(P1)
to M•
-
f(x,u,p,t),
0
-
x(to)-xo,
0
>
g(x,u,p,t),
0
>_ e ( x ( t f ) ) .
t C [to, tf],
(1) (2)
t E [to,tf],
(3) (4)
In this formulation, x(t) E I~nx denote the state variables, which can be either of differential or algebraic type, whereas x0 are initial conditions. The variables to be determined by the optimization procedure for minimization of the objective function 9 are the control vector u(t) C IRnu , the unknown time-independent parameters p E R"p, as well as the final time tf. The differentialalgebraic (DAE) model is given by the equation system (1), (2). We only consider DAE systems with an index of less than or equal to one. Furthermore, path constraints (3) can be applied on the states, control variables and time-independent parameters. Finally, endpoint constraints (4) on the state variables can be employed. 3. SINGLF.,-SHOOTING SOLUTION APPROACH In the sequential approach (Vassiliadis et al., 1994) the control profiles ui(t), i = 1, ..., nu have to be approximated, and often piecewise polynomial expansions of the form
tti(t) ~ Ui(Ci,t) -- E Ci,kt~i,k(t)' kEAi
(5)
are used, where Ai denotes the index set of the chosen parameterization functions t~i,k(t ) and the vector ci contains the corresponding parameter vector. For brevity, in this paper we only
1073
1 Vtk <_t <_ tk+l, otherwise ~i,k(t) --0, though considerpiecewise constant functions r an extension to higher-order polynomials is possible. The grid points for each ui are contained in the mesh Aai "-- {tg[k E Ai}. By discretization of the control variables, the dynamic optimization problem (P1) can be reformulated into the following NLP: ty min 9 -- ~0 (x (c, p, tf)) + f fo (x (c, p, I;), ~(c, x), p) dx
c,p,tf
(P2)
to
s.t.
0 0
>_ g(x, c, p, ti) , >__ e ( x ( t f ) ) .
Vti E
AA,
(6) (7)
The path constraints (6) are now evaluated on the unified mesh of all control variables AA : : [..Jinu 1 AAi" The DAE model (1), (2) is not present directly in the NLP problem, rather it is solved by numerical integration in each function evaluation step of the NLP solver to determine x(c, p,t) for given c and p and therefore present implicitly. Algorithms for the solution of this NLP, typically SQP methods, require gradient information of the constraints and the objective function with respect to the decision variables. There are several possibilities to obtain these gradients. Here, we consider the explicit solution of the arising sensitivity equation systems, which is the method of choice in most optimization algorithms (e.g. Vassiliadis et al. (1994)). The sensitivity systems can be solved by numerical integration together with the original DAE system. Although there are efficient algorithms available, which exploit the special properties of the sensitivity system (e.g. Feehery et al. (1997)), still the most significant computational effort is spent on the sensitivity analysis. Since the influence of a decision variable ci,k on the states x is limited to the time region t >__tk, it is sufficient to solve each sensitivity equation system for the determination of si 9_ ~bx on the time interval [tk, tf]. Still, this leads to a computational effort increasing polynomially with the number of decision variables. Hence, it is clearly desirable to keep the number of decision variables as small as possible, without losing much accuracy. This raises the question of an optimal selection of the discretization grids. The formulation (5) offers the choice of separate, non-uniform parameterization grids for each control variable. This fact can be exploited by using adaptive refinement strategies in order to generate efficient, problem-adapted meshes AAi, as explained in the following section. 4. A D A P T I V E
REFINEMENT
ALGORITHM
Problem-adapted possibly non-uniform grids for an efficient approximation of Ui are generated by successive refinement of an initial coarse discretization mesh u~(A~ In each refinement step g the previous solution u e- 1 is inspected by a wavelet-analysis (Binder et al., 2000). Based on this analysis the discretization is refined locally in areas where u e- 1 reveals large variations. (P2) is then resolved on the improved discretization grid A e where the interpolated old solution u e- 1 is used as an initial guess. Hence, (P2) is resolved repeatedly on different meshes with an increasing number of parameterization variables. Consequently all the quantities in (5), (P2), (6), (7) should have the refinement counter g as superscript, but this has been omitted for ease of notation. The refinement is carried out for each control ui such that individual discretization grids for each Ui are obtained.
1074 5. IMPLEMENTATION The numerical concepts presented above have been implemented into the prototype software tool ADOPT. Those parts of the program, which require access to the model information have been coded compliant to the so-called ESO interface definition, as defined in the CAPE-OPEN project (Keeping and Pantelides, 1998). This Equation Set Object (ESO) is an interface definition for communicating all information contained in the DAE model (1) which could be required by numerical algorithms, such as number and values of variables and residuals, structure and values of the model Jacobian matrix etc. Any modeling package, which allows access to numerical model information through the same interface could be used as a model server. To the authors' knowledge, this feature currently is only provided by gPROMS. The prototype is able to access gPROMS as the model server via a CORBA object bus (Henning and Vinoski, 1999), but also any legacy model wrapped with an ESO interface can be used, provided that there are no discontinuities present. Dynamic optimization following the methods presented above can be performed without the need to recode the model in a programming language. The basic structure of the tool is depicted in Figure 1. The refinement loop can be recognized in the left-hand part of the picture. On the fight, the way how the model information is transferred between the dynamic optimizer and the model server is shown. Two of the methods defined in the ESO standard, Set V a r i a b l e s and aetResiduals, exemplarily shown in Figure 1, indicate how the residual values for the current variable set can be obtained from the model server. The CORBA bus enables platform independence and interoperability between operating systems. A drawback of this technology is an overhead in time consumption caused by the communication. In the following section, this will be discussed in more detail. 6. CASE STUDY In this section we present some results obtained by applying this software to an example model. The model describes a batch reactive distillation column for the production of ethyl acetate (Cervantes and Biegler, 1998). We use a modified version with the simplifying assump-
Fig. 1. Basic structure of ADOPT and communication via ESO interface.
1075 80
60
.... ---
50
Iteration 0 Iteration 2 Iteration 4
70
40
60
30
50
Iteration 0 Iteration 2 Iteration 4
40
20 10
.... ---
...... i
0.2
30 0.4
0.6
(a) Refluxratio
0.8
200
0.2
0.4
i'i i 0.6
0.8
1
(b) Vapor stream
Fig. 2. Control variable profiles and corresponding problem-adapted grids. tion of constant molar overflow. The model is available in gPROMS and comprises reboiler, condenser and 10 trays and contains in this form 418 equations and variables, of which 63 are differential. The reflux ratio and the vapor stream leaving the reboiler have been chosen as control variables. The objective is to minimize the energy demand in the reboiler for a process running one hour, under the constraints, that the amount of final distillate should be at least 6.0 kmol, with a purity of the ester of at least 0.46. The optimization was initialized with constant profiles. Figure 2 shows optimal profiles and corresponding grids for the two control variables. For clarity only results from every second iteration in the adaptation loop are shown. The adaptation of the grids to the problem becomes apparent. Table 1 compares computational results from the different refinement iterations with those obtained by using a uniform mesh of comparable accuracy as the one in iteration 4. The objective value decreases, while the CPU time per iteration increases due to the growing number of decision variables caused by the refined grids. With the adaptive approach, results with comparable accuracy (e.g. after iteration 2) can be obtained within significantly smaller computation time as compared to a solution on a uniform mesh. Since each iteration produces a continuously refined intermediate feasible solution, this approach is particularly useful for real-time applications. The additional time consumption, which is required by the CORBA bus, is proportional to the size of the vectors to be transfered and lies in the order of milliseconds. Obviously, the number of calls to the model server plays a crucial role in this context, as well. A typical optimization requires in the order of several hundred thousand calls via the CORBA bus. Therefore, the communication between the optimization software and the model server causes a significant overhead with the current implementation. For this reasons, the applicability to real-time problems is still limited. However, this problem can be overcome by modifications of the software architecture. One option is to rely on future developments and improvements in the CORBA technology. Alternatively, it is conceivable to connect the optimization software and the model server directly "in process" rather than using middleware components. The ESO interface still should be used for consistency. This approach might be the way to proceed for applications, where computation time is the major issue, especially in the real-time area.
1076 Table 1 Solutions on different adapted grids compared with uniform mesh. Iteration g 0 1 2 3 4 No. of dec. vars. Objective value CPU-sec per iter. CPU-sec accum.
8 62.322 34.8 34.8
16 61.655 145.2 180.0
22 61.575 i30.9 310.9
28 61.553 276.1 587.0
34 61.542 493.3 1080.3
[1 Uniform mesh 64 61.564 2474.3 2474.3
7. CONCLUSIONS The implementation of a dynamic optimization algorithm, which adaptively generates problemdependent discretization grids for different control variables in order to increase the efficiency and robustness of the solution has been presented. As a new feature, the access of model information via a CAPE-OPEN interface has been introduced. The functionality of this approach has been proven by applying it to a large-scale problem. The heterogeneous implementation using CORBA as communication middleware appeared to be a practical approach, though there still exists a significant overhead in computation time solely caused by software-related reasons. Future improvements in this area will enable the use of such frameworks in industrial applications.
REFERENCES Betts, J.T. and W.P. Huffmann (1998). Mesh refinement in direct transcription methods for optimal control. Optim. Control Appl. Meth. 19, 1-21. Binder, T., A. Cruse, C. Villas and W. Marquardt (2000). Dynamic optimization using a wavelet based adaptive control vector parameterization strategy. Comp. Chem. Eng. 24, 1201-1207. Bock, H.G., M.M. Diehl, D.B. Leineweber and J.P. Schl6der (2000). A direct multiple shooting method for real-time optimization of nonlinear DAE processes. In: Nonlinear Model Predictive Control (F. Allg6wer and A. Zheng, Eds.). pp. 246-267. Birkh~iuser Verlag. Basel. Braunschweig, B., C. Pantelides, H. Britt and S. Sama (2000). Process modeling: The promise of open software architectures. Chem. Eng. Prog. 96(9), 65-76. Cervantes, A. and L.T. Biegler (1998). Large-scale DAE optimization using a simultaneous NLP formulation. AIChE Journal 44(5), 1038-1050. Feehery, W., J. Tolsma and P.I. Barton (1997). Efficient sensitivity analysis of large-scale differentialalgebraic systems. Appl. Numer. Math. 25, 41-54. Henning, M. and S. Vinoski (1999). Advanced CORBA Programming with C+ +. Addison-Wesley. Reading, MA. Keeping, B. and C. Pantelides (1998). WP4.1 Numerical Solvers Open Interface Specification Draft. Technical Report Internal Draft 8. CPSE, Imperial College. London. Vassiliadis, V.S., R.W.H. Sargent and C.C. Pantelides (1994). Solution of a class of multistage dynamic optimization problems: 2. Problems with path constraints. Ind. Eng. Chem. Res. 33(9), 2123-2133.
European Symposmm on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1077
Integration of available CAPE tools for R e a l T i m e Optimisation systems Sebasti~m Eloy Sequeira, Mois6s Graells, Luis Puigjaner. Chemical Engineering Department, Universitat Polit6nica de Catalunya, Av Diagonal 647 Pav. G, 2~ Spain
Keywords. RTO, Optimisation, CAPE. This paper addresses the communication of present CAPE tools as the way for achieving integrated systems supporting Real Time Optimisation. The work primarily focuses on integration of such tools in order to facilitate the implementation of RTO systems. Scheme and architecture proposed are validated with two case studies and the arising difficulties are discussed. First case study is based in a typical plant scheme where the recycle flow constitutes the decision variable and the plant profit the objective function. The economic model of the system is implemented on a spreadsheet while a commercial process simulator (HYSYS.Plant a) provides the simulation of the physical behaviour of the plant. The more complex second case includes several decision variables. Therefore, a solver function of a mathematics simulator was used (Matlabb). The communication between all these tools is established by means of COM (Component Object Model) and DDE (Dynamic Data Exchange) technologies, thus constituting a hybrid distributed model of the system. Finally, computational advantages of such an aggregated model are discussed as well as the improvement possibility of the optimisation procedures performance. a Hysys.Plant is a trademark of Hyprotech
b Matlab is a trademarkof MathWorks 1. INTRODUCTION Real Time Optimization (RTO) has been receiving increasing attention as communication technologies allow having computer aided decision-making tools connected on-line with the plant or the system to be managed. Software tools and system architectures employed show that RTO systems are designed for providing different information supplying features and for satisfying different goals, Gross Error Detection, Data Reconciliation and Parameter Estimation (PEDR), and Optimisation ([1], [2] and [3]). The main functionality is doubtless the optimisation. This is usually carried out using a simulation model, which incorporates an optimisation tool. Lots of different software tools are currently available for managing some of these features and goals. However, most of these tools are presently used for different activities in an independent way. Hence, the opportunity of using these software tools within an integrated environment should be investigated in order to enhance their performance in a synergetic way, taking advantage of the developments in each specific area and combining them, thus making the best as a whole.
1078 This work introduces a starting point in this direction that is based on the use of commercial software tools (such as simulation and optimisation packages, as well as spreadsheets, which are very familiar to process engineers). Such tools are later integrated into a distributed system with the aid of the presently available communication technologies (COM, DDE). The basic static and dynamic structures are next introduced. 2. SYSTEM ARCHITECTURE Optimisation is the core and main functionality of the RTO system. However, the tasks the system is also expected to manage the data coming from and going back to the plant. Hence, the following modules are proposed: 9 D a t a M a n a g e m e n t : This module receives data from the plant (but previously filtered by the gross error detection and PEDR systems). After optimisation this module sends back the set points (SP' s) to the control system. This should also allow an off-line analysis. 9 O p t i m i s a t i o n : This module consists of the following elements: The process model (constraints), the economic model (objective function) and the search engine (solver). The associated static structure is shown in Figure 1:
Figure 1" Basic static architecture. This is, the Data Manager (DM) is the system's executive, while the remaining elements are interfaced to the DM but not between themselves. The double arrow line in Figure 1 shows that the communication is reciprocal. The process and economic models are basically black box simulators, in the sense that they produce an answer when perturbed by an input, under certain model parameter conditions. On other hand, Figure 2 gives the corresponding dynamic structure: o A ~ , ~ E ~ s ,CONDn',~
I IVI~NAC-~R DATA I '
I
INV(~.ATION
s~"
.,,,..._ r
l
I PR~ MOOEL
I ECONOMIC MODEL
I
For each iteration
INVOCATION RESULTS
E " REsPoNs~ NEW CONDITIONS .=f
OPT OLRPUT
Figure 2" Dynamic Architecture.
INVOCATION RESLLTS
71
1079 Following the scheme in Figure 2, as the DM receives the new plant conditions the solver (OPT) is invoked for optimisation. The OPT will then perturb the model and wait for response the time needed for the corresponding algorithm to converge. This is a non-direct perturbation since the data are sent to the process model (PM) trough the DM. Once the PM gets the results, these results are sent to the economic model (EM) again trough the DM. Off-line analysis is also possible within the same structure. The OPT may be replaced or complemented by any other module since the only requirement for the new component to be plugged into the system is to have the same dynamic behaviour as the OPT module. Thanks to this architecture, the use of different algorithms and modules is extremely easy and helpful. Sensibility analysis is an example of off-line process that may be included in this framework as it will be shown later on. 3. I M P L E M E N T A T I O N Next step is the software election for the Figure 1 functionality. The different choices are next explained: Process Model was developed using the sequential-modular simulation package HYSYS.Plant since it offers very good communication capabilities. On other hand, the Economic Model was implemented on a MSExcel's spreadsheet. Excel c also has an open architecture allowing very easy communication with HYSYS via ActiveX. Additionally, it is a tool that may be considered as a customisable user interface for the system. The economic model implemented includes the investment calculation, the operating cost, and the revenues. The performance measure selected for the problem was the Present Value (PV). In Case A, given that the optimisation involves only one decision variable, the Optimisation algorithm was written in Visual Basic for Applications (VBA), a simple programming language, easily linked with Excel. The algorithm used was the golden section because of its robustness. On other hand, in Case B as the optimisation involves more decision variables no algorithm was written. Instead, the "constr" Matlab function was used. This optimisation algorithm is an implementation of sequential quadratic programming (SQP). Other solvers (public domain, in house, etc.) may be also plugged in this scheme thanks to the modular approach. The Data Manager chosen for this specific implementation was Excel again. It allows the easier communication according the previous choices, and this is the main job of the DM [4]. 4. COMMUNICATION Given all the elements of the system, communication between them is needed. Regarding the tools selected for the specific implementation presented the simplest way is to use VBA and the interfaces available for Excel and HYSYS. In order to communicate the algorithm in Matlab with the DM in Excel, the Dynamic Data Exchange (DDE) service was used. 5. CASE STUDIES For evaluation of the system architecture two scenarios are presented. In the first case study, a typical plant scheme is considered: a plant with a reactor, a separator and a recycle stream. In this case the single decision variable is the recycle flow and the objective function is the plant profit. For the second example more decision variables are contemplated. Both cases are from [5]. c Excel is a trademark of Microsoft
1080 Case A. The problem.
The process under study is the ethyl chloride production according to the process flowsheet shown in Figure 3. A conversion reactor produces a mixed output that is separated into a product stream (P) and the stream ($4) to be recycled. A purge stream (W) prevents the accumulation of inert components in the system. Hence, changes on W flow affect the recycle and thus the process economy. Increasing the recycle increases energy cost, but raw materials cost is reduced instead. Therefore, the flow W constitutes a decision variable, and should be adjusted to produce the best operation point according to the changing feed conditions, market prices and resources cost.
Figure 3: Ethyl Chloride Process Flowsheet. Case A. Results.
System performance is verified varying the feed conditions and the market prices. When input changes occur, the new model response (simulation according to the new plant data) and the last model response are compared. If the changes are significant, the optimisation takes place. After optimisation is done, the DM exposes the new set's points to the control system. Figure 4 shows a screenshot of the DM's main page. Row divisions show to the different input/output sets, input parameters, manipulated variable, objective function and main process model outputs. The first column is for the variable name, next column (Plant) contains the plant data and the simulation results obtained from those data. The last column (Last Model) gives the last optimisation conditions and results. Finally, the centre column gives the difference between the previous values, used for deciding if next optimisation will be done. For time consumption evaluation, random feed conditions were generated for ten scenarios and the average optimisation time was 23.4 seconds (std. deviation 23.5 %) in a PC Pentium Family AT 350 MHz - Ram 128 KB. This is similar to the time required by the HYSYS optimiser, but this approach allows in addition to deal with a more open and flexible model. The same framework may offer other decision-making tools such as off-line sensitivity analysis. Figure 5 shows a screenshot of the page used for this purpose. The chart shows the effect of the W stream flow on the Present Value. This kind of study can be very useful when the calculated optimum can not be achieved for practical reasons not contemplated in the model and for bottleneck identification as well.
1081
Figure 4: DM's main page.
Figure 5: Page for sensibility studies.
Case B. The problem. The system considered next is the separation of a mixture of Hydrogen Chloride (HC1), Benzene (Bz), and MonoChloroBencene (MCB), which is the output of a reactor producing MCB by the chlorination of benzene. The process flowsheet is shown in Figure 6.
Figure 6: Bz-MCB separation process.
Figure 7: Page for sensibility studies.
Consider the influence of temperature in stream S 14, which is given by MCB final cooling. A temperature increase will produce on one hand a better Bz recovery, but on other hand will increase the refrigeration cost. Another interesting trade-off is given by the split fraction of S 14 in separator S-1. As the recycle grows, more Bz will be again recovered, but the energy cost is increased (heating in D-1 and T-l). Finally, another operation decision variable is the Reflux Ratio in tower D-l, which may be adjusted to satisfy the flow and purity of products directly in HYSYS. Hence the problem arising is the determination of the optimum values of these variables once that feed conditions and market prices are given.
1082
Case B. Results. Once again, the system performance is tested varying feed conditions and market prices. The DM structure is the same showed in Case A. Figure 9 shows a screenshot of the page used for sensibility studies. The chart shows the effect of the decision variables on PV: the strong effect of the temperature, and the minor effect of the split fraction. Time consumption was measured in ten scenarios for which random feed conditions were generated. The average optimisation time was 66.7 seconds (std. deviation 19.5 %) in a PC 350 M H z - Ram 128 KB, which is of the same order than that required by HYSYS optimisers. However, the modular approach also allows using different optimisation techniques, which may improve the performance by manipulating the aggregated model. The recycle stream for instance, may be considered at the upper economic model by adding new decision variables and new constraints. This is, the recycle stream may be ignored in HYSYS, while the recycle stream may be considered as manipulated variable. Hence, by adding the recycle convergence as a set of constrains, the solver can optimise the process while converging the flowsheet. Thus, there is no need to wait HYSYS to converge the recycle for each solver iteration. Using this strategy on ten scenarios, the average optimisation time was 132.5 seconds (std. deviation 10.3 %) on the same PC, which is certainly higher than for the original model. However, it will not be the case as more decision variables are implicated. 6. CONCLUSIONS A simple framework for the development and implementation of the essential elements of a RTO system has been introduced. The advantages of the proposed architecture are that known tools for the chemical engineer are the main elements used. Besides, the engineer effort is mainly focused on the system communication and the decision making, but not in the development of its constitutive parts. Experience and development in the areas of optimisation and simulation may be included in the system by communicating the corresponding software packages. Additionally, a modular approach allows the use of plug and play philosophy. This means easily construction and maintenance. Therefore, future work will include the implementation of other components (PEDR, etc.) in the system. Finally, the communication effort is expected to be significantly reduced after GCO standardisation.
Acknowledgements Financial support from the European Community is gratefully acknowledged (project Global-Cape-Open-IMS 26691). Sebastian Eloy Sequeira wishes to acknowledge to Spanish "Ministerio de Educaci6n, Cultura y Deporte" for the financial support (grant FPI). References 1. Z. Zhang, R. W. Pike and T. A. Hertwig, "An approach to on-line optimization of chemical plants", Computers Chem. Engng., vol. 19. Suppl., pp. $30-$310, 1995. 2. S. Yoon, S. Dasgupta and G. Mijares, "Real-Time optimization boosts capacity of Korean olefins plant", Oil & Gas Journal, vol. 94, pp. 36, 1996. 3. K. Chin, "Maximize Profits Plantwide", Chem. Eng., vol. 105-3, pp. 143, 1998. 4. E. M. Rosen, L. R. Partin, "A perspective: The use of the Spreadsheet for Chemical Engineering Computations", Ind. Eng. Chem. Res., vol. 39, pp. 1612-1613., 2000. 5. W. D. Seider, J. D. Seader and D. R. Lewing, "Process Design Principles: Synthesis, Analysis, and Evaluation". John Wiley & Sons, Inc. 1999.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1083
Design and Synthesis of Process Plant Utility Systems under Operational Variations Zhigang Shang ~+ and Antonis Kokossis b aDepartment of Process Integration, UMIST, Manchester M60 1QD, UK bDepartment of Chemical & Process Engineering, University of Surrey, Guildford, Surrey GU2 5XH, UK The paper presents a systematic approach for the optimal design and synthesis of the utility systems under operational variations. The design task is addressed in view of the anticipated variations in the utility demands and the effect of unit capacities and varying loads on the efficiency of the selected units. The proposed approach combines the benefits of pinch analysis, thermodynamic analysis and mathematical optimisation techniques. The numerous design options are reduced with the use of thermodynamics. As a result of the analysis and insights into the formulation, the optimisation effort is simply facilitated by a multi-period MILP model. The approach facilitates the design problem and solves it efficiently. 1. INTRODUCTION In the design of chemical process plant, the performance of the plant utility systems directly influences the operating cost, hence, the optimal design of plant utility systems often leads to significant savings. It is often the case that considerable changes exist in the chemical processes as a result of fluctuating demand and prices of products, feed compositions, ambient temperatures and so on. The changes in the operation of chemical processes result in fluctuating heat and power demands. There is a need to design the plant utility systems in view of the anticipated variations in the process demands. Several works have been presented previously to address the problem of synthesis and design of utility systems. Iyer and Grossmann (1998) have presented a multiperiod MILP approach for the utility systems operating under multiple periods. A simulated annealing algorithm has been used by Maia and Qassim (1997) for synthesis of utility systems with variable utility demands. Most recently, Mavromatis and Kokossis (1998) have presented an MILP approach for the optimal design of steam turbine networks. An MINLP model for the synthesis and design of utility plants has been presented by Bruno et (1998). Wilkendorf et (1998) also proposed an MINLP model for synthesis of complete utility systems. In practice, however, there are several drawbacks for these approaches. If all the candidate options are included in the superstructure, the number of candidate structures should be enormous and the size of the problem is too large to be handled even for moderate problems. The consideration of multiple operation scenarios results in a further increase of the design options to an extent. Secondly, the formulation of utility systems is inherently non-linear efficiencies for the units, which gives rise to complex models. Therefore it is essential to find a systematic
+Currentlywith the Departmentof Process & SystemsEngineering,Cranfield University,Cranfield, MK43 0AL, UK
1084 methodology which can build a superstructure including all the promising alternatives without being too large and develop new modelling methods for the units. 2. NEW METHODOLOGY A new methodology is proposed to address the optimal design of process plant utility systems under operational variations. The methodology is schematically shown in Figure 1. The approach combines the benefits of pinch analysis, thermodynamic analysis and optimisation techniques. The pinch analysis technology is used to screen and identify all possible design options. The thermodynamic analysis is applied to reduce the size of the optimisation problem. The proposed strategy comprises the following five stages: 1. The pinch analysis is used to screen and identify all possible design options 2. The thermodynamic analysis is employed to screen among various design alternatives and identifies the most promising design options that are passed on to the next stage. 3. Based on the promising design options, this stage is to develop the detailed design components for the superstructure that is much smaller than the conventional superstructures. 4. The superstructure is optimised so as to minimise the total cost.
~ Thermodynamican
~--
]
Possibledesignoptions
]
~--
I
Promisingdesignoptions
]
I Superstructuregeneration(reducedsize)I ~--
[
Optimisation I
Figure 1: Schematic showing the configuration design optimisation strategy 2.1. Pinch Analysis There are enormous number of candidate structures which include steam turbine cycles, condensing turbine cycles, simple gas turbine cycles, regenerative gas turbine cycles, combined steam and gas turbine cycles with or without condensing turbine, diesel drivers, and all of their different combinations. The synthesis and design problem is to find a site utility system that satisfies the chemical processes' varying heat and power requirements, subject to minimum energy consumption and capital investment.
Figure 2: Identification of steam turbines of a site by using the SCC
1085 The Site Composite Curves (SCC) (Raissi,1994) reflect on the integration opportunities between chemical production processes and the site utility system. The curves can represent the steam flow in the utility system as well as the heat exchange between site processes and utilities (Figure 2). The enclosed shaded area between the steam levels is proportional to the potential for power cogeneration. The curves also reveal the heat recovery, fuel requirement and cooling utility demands of the site. Therefore, the SCC can be used as a conceptual tool to screen and target the possible design options for the site utility system. Figure 2 illustrates a possible allocation of back-pressure turbines with the use of the SCC. The sizes and positions of the turbines can be identified using the curves.
2.2. Thermodynamic Analysis The objective of the thermodynamic analysis is to screen out the infeasible and inefficient options. As it will be shown later in the paper, the size and complexity of the optimisation problem is reduced dramatically. Heat and power represent energy of different quality. The thermodynamic efficiency is defined by a relationship that determines the ratio of the useful part of the energy to the total fuel input. The thermodynamic efficiency of a typical utility unit is defined by:
Tit =
W+ZQP Qfuet
W is the shaft-work generated, y ' Q P is the sum of the steam heat loads required by the chemical processes at different levels. Qfuel is the net fuel heat input. The thermodynamic efficiency indicates fuel utilisation efficiency. In order to identify the most efficient candidate structures, a Thermodynamic Efficiency Curve (TEC) is constructed. The efficiency relates the possible utility structures. The TEC is presented graphically in Figure 3. The vertical and horizontal axes respectively represent the thermodynamic efficiency and the power requirement. The TEC is constructed by first plotting the efficiency curve of units using surplus heat from chemical processes (SCT in this case), whose length equals the maximum power capacity. The curve follows with the other designs plotted one by one in a step downwards in terms of efficiency until completion of options. In this case, the SCT and the BBPT cycles are the preferable options, the GTWB cycle follows next and the GTWBCT cycle is itself followed by the BCT cycle and the option to import power.
I
boilerand back-pressureturbine cycle G'I~] gasturbine and waste heat boilercycle G'M/3C't"gasturbine, waste heat boiler and condensing turbine cycle BC't"boilerand condendng turbine cycle e. importing power
GIV~ i G]Vk~T BCT
liP [
I
[
i
PB
Pc
PD
PE
Figure 3" The thermodynamic efficiency curve
Power(MW)
1086 The power requirement of the site determines the utility systems to consider. The corresponding sizes of the units are identified and targeted with the use of the TEC; the inefficient options are screened out. The promising options enabled by the TEC form a superstructure for the optimisation that is much simpler than a generic superstructure that includes all possible options. 3. OPTIMISATION Once the promising design options are identified, a superstructure embedding selected options is constructed. Figure 4 illustrates the configuration of the superstructure satisfying varying utility demands for multiple operation scenarios. The components for the superstructure are concerned with back-pressure steam turbine network, condensing steam turbine network, reheat cycles, simple and regenerative gas turbine cycles, boiler network and the auxiliary units. The optimisation of the superstructure is conducted so as to minimise the total cost of the utility systems under variations of utility demands in different scenarios. The total cost consists of capital investment cost and operating cost. The optimum utility systems are determined with respect to type, number, capacity of the operating units as well as their connections. The optimisation model is formulated as a multi-period Mixed Integer Linear Programming (MILP) model that relies on the new mathematical models of the operating units developed in this approach.
Figure 4: The superstructure of process plant utility systems satisfying varying utility demands for different scenarios In order to address the problems of optimisation of utility systems under operational variations, new mathematical models of steam turbines, condensing turbines, boilers and gas turbines are proposed in this approach. All these models account for the variation of efficiency with unit size, load and operating conditions in a simple, yet accurate way. As a result, these models are capable of accounting for the efficiency trends of realistic units. It should be noted, had the conventional models for the units been applied, the use of MINLP formulation would be inevitable. 4. CASE STUDY A case study is selected to illustrate the capabilities of the methodology. The Site Utility Grand Composite Curves (SUGCC) of Figure 5 reflect the steam demand/generation of the site under three operation scenarios. The power demands for the scenarios are given in Table 1. On the basis of the SUGCC, the aim is to find the optimal configuration of the site utility system that satisfies the utility demands and minimises the annual total cost.
1087
T
180
Scenano A
130
T
220
~enano B
"It
9cenano C
A
260
260
300
VAC|
I
Figure 5: The SUGCC of the case Table 1: Power demands of the case Power demand (MW)
Scenario A 22
Scenario B / Scenario C 35 [ 42
By using pinch analysis and thermodynamic analysis, the promising design options are identified and targeted. These options form the superstructure as shown in Figure 6.
Figure 6: The superstructure of the case The superstructure is formulated as an MILP model. The optimisation minimises the total annual cost. The model is developed using GAMS and the optimisation is conducted by employing the OSL solver. The model involves 195 continuous variables, 92 binary variables and 242 constraints. The optimum configuration is given in Figure 7. The selected units include three BP steam turbines, one condensing turbine, one gas turbine, a VHP boiler, an HP waste heat boiler, an MP waste heat boiler and the deaerator. The HP and MP waste heat boilers are selected to produce HP and MP steam respectively. The back-pressure turbines supply power by exploiting the cogeneration potential. The condensing turbine is employed to generate power by using surplus heat from the site processes. The gas turbine is installed to supply the remaining power.
1088
~@
~ MaxO l
32MVV
~,x~o~
MaxSO~.
I ..........
M,x60~h t
*
]
I
Deaerator
Figure 7: The optimal structure of the case 5. CONCLUSIONS A systematic methodology is presented for the design and synthesis of process plant utility systems under operational variations. The methodology combines the benefits of total site analysis, thermodynamic analysis and optimisation techniques. The design task is addressed in view of the anticipated variations in the process demands and the effect of the unit capacities and varying loads on the efficiencies of the selected units. The proposed methodology utilises pinch analysis and thermodynamic analysis to reduce the size and complexity of the design problem. The size of the optimisation problem can be reduced by screening out the uneconomic design options. REFERENCES Bruno, J.C., Femandez, F., Castells F. and Grossmann I.E., 1998, A rigorous MINLP model for the optimal synthesis and operation of utility plants. Chemical Engineering Research & Design 76, 246-258 Iyer, R.R. and Grossmann, I.E., 1998, Synthesis and operational planning of utility systems for multiperiod operation. Computers Chem. Engng 22, 979-993. Maia, L.O.A. and Qassim, R.Y., 1997, Synthesis of utility systems with variable demands using simulated annealing. Computers Chem. Engng 21,947-950. Mavromatic, S.P. and Kokossis, A.C., 1998b, Conceptual optimisation of utility networks for operational variations - 2: Network development and optimisation. Chem. Eng. Sci. 53, 16091630 Raissi, K., 1994, Total site integration. Ph.D. thesis, Dept. of Process Integration, UMIST, Manchester, UK. Wilkendorf, F., Espuna, A. and Puigjaner, L., 1998, Minimization of the annual cost for complete utility systems. Chemical Engineering Research & Design 76, 239-245.
European Symposiumon ComputerAided ProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rightsreserved.
1089
Near independent sub-systems in heat exchanger network design H.K.Shethna a, J.Jezowski b and F.J.L.Castillo a aAEA Technology Engineering Software, Hyprotech Ltd, Suite 800, 707 - 8'h Avenue SW, Calgary, Alberta, CANADA. T2P 3V3. bTechnical University Rzeszow, 35 959 Rzeszow, A1. Powstancow, Warszawy 6, POLAND. An independent sub-system is formed when a subset of process and utility streams and its complement are heat-balanced. Given a balanced set of process and utility streams, a mixedinteger-linear transshipment model is formulated to identify an independent sub-system. For each feasible hot and cold streams, heat exchange is considered between a cold stream in a given temperature interval and a hot stream in the same or higher temperature interval as first proposed by Papoulias and Grossmann (1983). A binary variable is used to denote the existence of a stream in a sub-system. The objective function used minimizes the sum of all binary variables. Special heat-balance constraints are formulated to account for the existence of a stream in a sub-set as well as its complement. All independent sub-systems are found by recursively applying the model to the complementary set of streams. The model incorporates designer-specified heat load tolerances on individual streams into the formulation to determine near-independent sub-systems. The number of sub-systems found helps us to estimate the global minimum number of units more accurately.
1. I N T R O D U C T I O N In a heat exchanger network (HEN), several hot process or utility streams exchange heat with cold process or utility streams. The overall HEN, where the heat load on each utility is completely defined, is heat-balanced. In some cases, the overall HEN problem can be divided into several smaller problems, each termed as a sub-system. Each sub-system has two important characteristics. First, the subset and its complement are both heat-balanced. Second, it should be possible to achieve feasible heat transfer within the subset and its complement for a designer-specified minimum temperature driving force. Identifying a sub-system is necessary to accurately determine the minimum number of heat exchange units. Based on a graph representation of a HEN, Linnhoff et al. (1979) showed, that global minimum number of units (Nglobal) can be determined from:
Nglobal = Ns
-
NI
(1)
where N S is the total number of process and utility streams in the HEN and NI is the number of independent sub-problems. In absence of an appropriate method to estimate NI, it is
1090 generally assumed to be unity. In most cases, this is a good assumption because exact independent sub-systems are rare in most industrial cases. In several industrial cases however, it is quite likely to find a sub-system that is nearly in heat balance within a designer-specified tolerance on the heat load or temperature of streams. In such cases, it is important to identify an exact value of N1 to achieve an accurate target, where N1 now represents the number of near-independent sub-systems. Identifying sub-systems also makes the HEN synthesis process simpler because it divides the HEN problem into more than one problem of smaller size. From an operability point of view, each independent sub-system is easier to operate and control as it has fewer interactions. Moscny and Govind (1984) have proposed a methodology to explicitly identify nearindependent sub-systems subject to a designer specified tolerance. In this method an exhaustive search of all possible (2Nh x 2Nc) permutations of sub-systems is made to identify ones that are heat-balanced. Papoulias and Grossmann (1983) have proposed a MILP model to exactly determine the minimum number of heat exchange units. This model implicitly accounts for independent sub-systems, albeit those that are perfectly heat-balanced. It is however difficult to identify all possible sub-systems from the solution of these models for large problems. In this paper we propose a MILP formulation to identify all near-independent sub-systems subject to a heat load tolerance on each individual stream. In several problems, it is possible to find more than one near-independent sub-system. A methodology is proposed to use the model recursively to find all independent sub-systems that exist in the problem.
2. THEORY 2.1 Definition of independent sub-systems Consider a set S of process and utility streams, such that the heat load on each of the utility streams is just enough to satisfy the heating and cooling requirements of the process streams and no further external utility is required. We will also define a set S, as a subset of S and a set $2 as a complement of S~, such that: S
=
S1 U
$2
(2)
s~ is an independent sub-system if both S~ and S 2 require no further external heating or cooling similar to its parent set S.
2.2 Near independent sub-system It is difficult in large problems to find exactly heat-balanced sub-systems. It is however possible to identify sub-systems if the target temperatures of streams are adjusted by +1 ~ (for example). Such a variation is likely to be acceptable for feasible operation in industrial cases. Such a variation could thus be exploited to identify sub-systems that could lead to reduction in heat exchange units as well as complexity. Referring to Eq. 2, a sub-system $1 is said to be near-independent if both $1 and its complement $2 can be forced to be heatbalanced by adjusting the heat load of the streams in sets SI and $2 within a specified
1091
tolerance. These heat load tolerances have been incorporated into the model to identify nearindependent sub-systems. 3. D E S C R I P T I O N OF T H E M O D E L Consider a set S of process and utility streams that are heat-balanced for a given value of heat recovery approach temperature (HRAT). A set S' that is identical to S is created. S will be used to identify heat-balanced sub-systems and S' will be used to identify its complement. Both S and S' are divided into temperature intervals (TI) for a given value of HRAT. Heat is exchanged by placing matches between a cold stream in each TI and all hot streams that lie in that TI or higher, similar to the utilities targeting model proposed by Papoulias and Grossmann (1983). Unlike the utilities targeting model however, it is necessary that each stream be treated separately. The heat load on every match placed in S and S' is a continuous variable (qijk) that is optimized. Similar to the trans-shipment model of Papoulias and Grossmann (1983), in a given TI, residual heat that is not utilized in a higher temperature interval flows into it, and heat (Rik) that is not utilized by the TI is transferred to the next lower TI. The residual flow into the stream at its first TI is zero. The residual heat flow out of last TI on each stream is of particular interest. In order to find a subset of streams that exist in S that are almost in enthalpy balance it may be necessary to tolerate a small non-zero residual heat flow from the last TI of a hot stream. An additional match is placed on the last interval of each cold stream in S and S'. Physically, this is visualized as a heat exchange between the cold stream and a sufficiently high temperature hot utility. A non-zero heat load on the slack heater (qSj) introduces flexibility in the outlet temperature of the cold streams. The residual heat flow on the last interval of a hot stream and the load on the slack heater on a cold stream are both constrained to a maximum heat load equal to the heat load tolerance specified for the cold stream. Binary variables (Yi) are introduced to define the existence of streams in set S and S'. A stream that exists in S does not exist in S'. One of the objectives of the model would be to minimize the number of streams existing in S that are in heat balance. The value of the residual heat flow on the last TI of a hot stream as well as the load on the slack heater on the cold stream needs to be minimized as well. The model is applied recursively to find all sub-systems in the problem.
3.1 Mathematical equations Sets
Ks {k[ k is a temperature interval for stream s} CS {I"1J is a cold stream} C 's {j'[j is a cold stream} {l[ l is a hot stream that matches with the cold streamj in interval k,j ~ C s, k~ Kh } {m[ m is a cold stream that matches with the hot stream i in interval k, i ~ H s, k ~ Kc } {/1 l is a hot stream that matches with the cold streamj in interval k,j ~ C 's, k~ Kh } {m[m is a cold stream that matches with the hot stream i in interval k, i ~ H 's, k~ K c }
S, S' {s[ s is a stream} /-/s {i[ i is a hot stream} H 's {i[i is a hot stream}
I-ISj, k CSi, k H'Sj, k C'Si, k
1092 Objective function
Ey i
Z
+
tES
R;N
'S
S
min
qJ
RiN
qJc
+ ieH' Z _I7
j~C ~~ k n j
+
jeC '~
(3)
IEH"
Constraints
R,k
-
R,.k_,
+
Rik
-
Ri.k-1 +
~ q,#
= AHi~.y ,
~?~q~k :
Z q,Jk = AH~ . y,
AH,~.(1-y,)
i~HS,k~K,
(4)
i~H'',k~K,
(5)
j ~_C'~,k ~_Ky,k c: N,
(6)
i~H;k
qljk
=
qok
+ qjS
j ~ C"~,k ~ K j , k r N,
AHCk. ( l - y : )
(7)
ieH'jk
= AHjkC "Yj
J ~ C s ,k ~ K j , k
=
Nj
(8)
i~H~k Z
w qi/k
+
'S
q/
=
C
AHjk'(1-Yj)
-6, . y~ < R~k < 6~ . y~
i~H~,k~K,,k=N~
- 6 , . ( 1 - y , ) < Rik < 6~.(1-y,) s -6~ . y~ < q~k < 6j. yj
-6,-(1-y,)
< qi~
J~C'',k~K/,k=N!
i~H"~,k~K~,k=N,
j~C ~,k~K~,k=Nj
< 6;,-(I-y/)
j~C'S,k~K,,k=N,
(9)
(lO) (11)
(12) (13)
4. EXAMPLE
An example of a crude preheat train is used to demonstrate the use of the model (Table 1). In this example we have applied the model recursively to find three independent sub-systems (Table 2). If we are able to tolerate the deviation in outlet temperature of streams, we can see that for this problem we can calculate a minimum units target of 18 (instead of 20) using Eq 1. Further notice that it is possible to simplify the design task since each of the smaller sub-system relatively easily. We will however restrict our attention in this article to the model.
1093
Table 1. Stream information for crude preheat train example, HRAT = 7~ Name
Inlet T
PA 3 Draw COL1 TO PA 3 Return COL1 WasteH20 Main TO Cooled WasteH20 Main Lowtemp_crude_Main_TO_PreheatCrude_Main
319.4 73.2 30.0 85.8 142.8 213.8 263.5 232.2 287.9 347.3 191.9 297.3 195.2 248.0 158.0 73.2 231.8 181.0 167.1 120.7 146.7 126.6 99.9 74.6 226.2 228.7 20.0 181.4 181.9 20.0 138.4 138.9 20.0 138.4 138.9 345.6 20.0 250.0 175.0 1000.0
PA 2 Draw COL1 TO PA 2 Return COL1 PreFlashLiq_Main_TO_HotCrude_Main Residue Main TO Cooled Residue Main AGO Main TO Cooled AGO Main Diesel Main TO Cooled Diesel Main Naphtha_Main_TO_Cooled_Naphtha_Main Kerosene Main TO Cooled Kerosene Main PA 1 Draw COL1 TO PA 1 Return COL1 To_CondenserCOL l_TO_Naphtha_COL 1
KeroS S_ToReb_COL l_TO_KeroS S_BoilUp_COL 1 BottomSteam Main m
DieselSteam Main b
AGOSteam Main
TrimDuty_COL 1 Cooling Water HP Steam MP Steam Fired Heat (1000)
Outlet T 244.1 30.0 85.8 142.8 213.8 232.2 180.1 287.9 343.3 191.9 45.0 195.2 110.0 158.0 50.0 40.0 181.0 120.0 120.7 69.6 126.6 99.9 74.6 73.2 228.7 231.8 181.4 181.9 190.6 138.4 138.9 148.9 138.4 138.9 148.9 351.5 25.0 249.0 174.0 400.0
MCp
Enthalpy
136.2 68.3 327.0 327.0 391.6 482.9 443.0 469.0 503.0 215.8 178.2 21.9 19.3 68.3 58.9 57.7 50.9 46.5 173.1 158.2 225.7 176.3 351.8 91.6 352.3 424.9 4.0 3799.7 2.4 1.6 1626.3 0.8 1.3 1355.3 0.7 1563.0
2451.7 70.5 18038.9
2451.7 12902.8 14272.2 928.1 2988.9 458.1 1296.1 3852.8 4366.7
525.3 613.9
241.4
201.2
2218.6 19283.7 1977.2 224.7 23803.9
1094
Table 2 A solution of the model that yeilds three near-independent sub-systems. Stream Name
Inlet T
~ Naphtha Main TO Cooled Naphtha Main 73.2 To Condenser COL1 TO Naphtha COL1 i46.7 Cooling Water 20.0 HP Steam 250.0 AGO Main TO Cooled AGO Main 297.3 BottomSteam Main 20.0 DieselSteam Main 20.0 . KeroSS_ToReb_COLI_TO_KeroSS_BoilUp_COL1 . 226.2 PA 2 Draw COL1 TO PA 2 Return COL1 263.5 Residue Main TO Cooled Residue Main 347.3 PA 3 Draw COL1 TO PA 3 Return COL1 319.4 Diesel Main TO Cooled Diesel Main 248.0 WasteH20 Main TO Cooled WasteH20 Main 73.2 PA 1 Draw COL1 TO PA 1 Return COL1 167.1 Kerosene Main TO Cooled Kerosene Main 231.8 Fired Heat (1000) 1000.0 AGOSteam Main 20.0 Lowtemp crude Main TO Preheat Crude Main 30.0 TrimDuty COL 1 345.6 9 PreFlashLiq_Main_TO_HotCrude_Main . 232.2
Outlet T ~ 40.0 73.2 25.0 249.0 110.0 190.6 148.9 231.8 180.1 45.0 244.1 50.0 30.0 69.6 120.0 400.0 148.9 232.2 351.5 343.3
Calc. Imbalance Outlet T ~ kW 40.0 -898.9 72.5 25.0 249.0 -83.7 110.3 191.6 148.9 231.8 180.1 1207.0 45.0 244.1 50.0 ! 30.0 69.6 119.6 400.0 148.9 234.2 351.5 343.3 !
5. CONCLUSION A MILP model for the identifying near-independent sub-systems has been presented. The model is applied recursively to find all near-independent sub-systems in a problem. An example of a crude-preheat train is used to demostrate the utility of the model. The model can be used to also simplify the design task since each of the sub-system can be independently designed. REFERENCES
1. Linnhoff, B. (1979) "Thermodynamic Analysis in the design of process networks", Ph.D. Thesis, University of Leeds, U.K. 2. Moscny, D.; Govind R. (1984) "Decomposition Strategy for the Synthesis of MinimumUnit Heat Exchange Networks", AIChE J., 30 (5), 853-856. 3. Papoulias, S. A.; Grossmann, I.E. (1983) " A structural optimization approach in process synthesis - II. Heat recovery networks", Comp. Chem. Engng., 7 (6), 707-721.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1095
Simultaneous MINLP synthesis of heat exchanger networks comprising different exchanger types Aleksander Sor~ak and Zdravko Kravanja FKKT, University of Maribor, Smetanova 17, P.O. Box 219, SI-2000, Maribor, Slovenia This paper describes the simultaneous MINLP sythesis of heat integrated heat exchanger networks comprising different heat exchanger types. The stage-wise superstructure of HEN by Yee and Grossmann [1] is extended to alternative exchanger types. The selection of the types is modeled by disjunctions based on operating limitations and required heat transfer area. Since different types of heat exchangers involve different design geometries, which influence the inlet and outlet temperatures of heat exchangers, additional constraints are specified to provide feasible temperature distribution in HEN. 1. INTRODUCTION In order to obtain the appropriate trade-offs between utility consumption, network topology and exchanger areas, Yee and Grossmann [1] proposed the simultaneous MINLP model of HEN. The model is based on stage-wise superstructure with temperature driving forces as optimization variables, where each hot stream is potentially matched with each cold stream. The heat transfer between hot and cold streams takes place in pure counter flow heat exchangers. In order to upgrade the capabilities of the model also for the selection of different exchanger types, the following extensions have been applied: 9 the superstructure has been extended for the double pipe, shell & tube (U-tubes) and plate & frame (counter flow) exchanger types, 9 exchanger type operating limitations that are modeled by the convex-hull formulation of the corresponding disjunctions are embeded in the model, 9 additional constraints are proposed to avoid infeasible temperature distribution in HEN, since the different exchanger types involve different geometries of heat transfer area, 9 fixed charge investment costs and corrections of the temperature driving force (Ft) for different exchanger types are considered in the modified objective function, 9 since the number of alternative exchanger types drastically increases the size of the extended model, a special synthesis/analysis scheme has been proposed in order to facilitate the synthesis of medium-size problems. The temperature driving force corrections (Ft) are estimated in advance and verified after the optimization. 2. MODELING
2.1 Superstructure of HEN Each cold (]) stream can be potentially matched with each hot stream (i) in several stages (k). In the Yee's model [ 1], heat is potentially transfered from hot to cold stream within pure
1096 counter flow exchangers. HEN topology is specified by the vector of binary variables, which are determined during the MINLP optimization 9 In the extended superstructure (Fig. 1.), each counter flow heat exchanger is now replaced by a match superstructure (Sm, Fig. 2.) comprising the following exchanger types: 9 double pipe (DP) heat exchanger (l=1), 9 plate & frame (PF) heat exchanger (l-2), 9 shell & tube (ST) heat exchanger (U-tubes with an even number of passes, l=3) and 9 bypass (no heat is transferred between the hot and cold streams, l=4). ........ Stage 1
=
&-. . . . . . . . . . - - - i
Double pipe HE
'
F
Stage 2
:
i~
1=1
7
:
Sm
Hi,k+1
Hi,k
Plate & Frame HE
........
tl=
Shell & tube HE (U-tubes) CJ,k+l
I_=3
i i<=1
k=Z
,~
k=3
.....
~1
]
Bypass "=
Fig I. Two-stage superstructure o f H E N with two hot and two cold streams (Sm represents the Match superstructure).
Ill
L=4
)' " -
. . . . .
Fig.2. Match superstructure.
It should be noted that different area, pressure and temperature ranges hold for different exchanger types. Basic usage recommendations for different types [2] are shown in Table 1. In addition, due to the leakage problem, it is not recommended to use the plate & frame heat exchanger when one of the involved streams in the exchanger is toxic. Table 1. Data of considered heat exchanger types. Type
range T[~ -100 .. 600 -200 .. 600 -25 .. 2 5 0
pmxlMPal 30.7 30.7 1.6
Double pipe Shell & tube . Plate & frame
A
lmZl r a n g e
G [$/yrl
Cvl$/(m~ y r ) l _
0.25 .. 200 10 .. 1000 1 .. 1200
3861 21309 3050
178 91 96_
2.2 Objective function The objective function of the extended model defines the annual costs of HEN: Cost = ~ C c" . q,c, + ~ ch" .qjh. + t
J
CfCU y~,, + CvC,. 9
~, ~~
(la)
"
U,
cu
qi ~-~llnTt
cu
+
Cfh,. y~, + Cv h"
Cft'Y"~t+Cv'"Uij
" .ptes'-'~tAlnijkl Ztyk. "
The objective function comprises the following:
qJ "
+
(lb)
hu~-"AlnZ;U U I
(lc)
1097 9 costs for utility consumption (la), 9 annualized investment for coolers and heaters (Ib), and 9 annualized investment for heat exchangers (lc). The charge coefficients (Table 1.) are specified by the linearization of Guthrie's cost function [3]. The cost of plate & frame exchanger has been specified with 70 % [4] of the cost for the shell & tube exchanger. In order to preserve the linearity of the constraints, nonlinear area constraints (eq.2) have been omitted by defining the corresponding charge coefficients: the cost of double pipe exchangers has been specified by the linearization within the area range up to 200 m 2 while the cost of shell & tube exchangers has been specified within the area range from 200 m 2 to 1000 m 2. If the area is less than 200 m 2 then double pipe exchanger will be selected. If the area is greater than 200 m 2 then shell & tube exchanger will be preferred over double pipe one. Since the operating limitations of both types are almost the same, the selection of shell & tube and double pipe exchangers is therefore mostly exclusive.
Amin<
q #t
yki -- U o " "Igtest"ukt
max
"AInT# -<Mvk!
(2)
The temperature driving force is specified by Chen's approximation [5]:
AInT# =I AT# "ATok+l"AT'jk+1AT#+I (1)2
(3)
while the correction of the temperature driving force (Ft#3 is estimated in advance.
2.3 Feasible temperature distribution The original model [1 ] contains constraints for feasible temperature distribution only for pure counter flow exchangers. Since the extended model comprises several different exchanger types, the temperature distribution in HEN that holds for pure counter flow exchanger may become infeasible if the shell & tube exchanger is selected. The problem can be observed if we take a closer look at the temperature distribution of counter flow and shell & tube heat exchangers (Fig. 3.).
Fig.3. Temperature arrangement in (a) counter flow HE and (b) shell & tube HE with U-tubes.
In counter flow heat exchangers the outlet temperature of the cold stream can be higher (Fig.3.a) than in shell & tube heat exchangers (Fig.3.b) because of the geometry of the transfer area. When U-tubes are used, the flow arrangement combines counter and co-current flow. The temperature distribution becomes infeasible when Tc,o>Th,o and a shell & tubes exchanger is selected (Fig.3.b).
To overcome the problem, additional constraints have been specified for the shell & tube exchanger type (l = 3): T~k+~- rjk + r Up. (1 - y,jk3) > 0 (4)
2.4. Disjunctive modeling of operating limitations for selected exchanger types The selection of exchanger types depends on operating temperatures and pressures of involved streams (Table 1.). Because stream pressures are fixed, they can be taken into
1098 account in a prescreening procedure before the optimization. However, all stage temperatures are the optimization variables. Therefore, the temperature ranges of comprised exchanger types have been modeled by the convex-hull disjunctions. For hot streams the following constraints have been specified: Zlk -- Z Tiokl ;
Ziij kt <- zimax " Yijkl ; Zi#kl >- T/min " Y~jkl
(5)
t
The constraints for cold streams: L k "- Z ZJiJkt ; ZJokt <- rlmax " Yijkt ; J
TJ,jM>- T/min" Ydkl
(6)
Similarly, the heat load constraints have been defined:
Qy, = ~ Q,j~a; Qo~a< Q"P "Yyk,
(7)
I
Only one heat exchanger or bypass in the match superstructure should be selected: Y,jk, = l
(8)
I 3. SOLVING SCHEME
In order to facilitate the synthesis of medium-size problems the following synthesis/analysis scheme has been proposed (Fig.4.). The first step in the scheme is the prescreening procedure. It is applied to exclude all impossible matches between the hot and cold streams within the HEN superstructure, due to the operating limitations of exchanger types. Therefore, the model size is reduced before the optimization takes place. The prescreening is followed by the MINLP optimization of the model. Prescreening,
When the optimal solution is found, it has to be verified. The analysis comprises: 9 verification of Ft: the exact values of Ft Update data Solve MINLP HEN determined during corrections for the temperature driving MODEL Simultaneously .~ the analysis force are calculated and compared with NO the estimated ones (estimated values of Ft 1 ., "" "YES should be less than the exact values) Analysis of results ..... "Ft,L :, Ft, L,esi:"..... - Ft,L; correction factor 9 t,<~L,min ~ A _< A L m a x ' > | verification of the heat transfer areas(A) - AL; heat transfer area which have to be within the area ranges .................... ~STOP i Fig.4. The solving scheme. for selected heat exchangers. If the proposed solution is not acceptable, the model data are updated (estimation of Ft, etc.) and the model is optimized again. . . . . . . . . .
&
.................
j",..
4. EXAMPLES
The first example involves two hot and two cold streams (Table 2.) with two-stage HEN superstructure (Fig.l). The example was solved first by the original model [1]. The model contained 12 binary variables and was solved by DICOPT++ [6] in 0.82 sec of CPU time. Only three heat exchangers, one cooler and one heater are required for the entire heat transfer (Table 3.a), with a total annual HEN cost of 362118 S/yr. The cost is underestimated because the required heat transfer area for the match between the streams H1 and C 1 at the stage 2 exceeds 200 m 2 (490.4 m 2) and therefore, requires three heat exchangers. The exact annual HEN cost increases up to 369870 S/yr.
1099
Stage t
:
Stage 2
~.L
Tel,
:
rci 1
[
-
~
!
.o
H2
~
:
~
C1
(U-tubes) I IF
~
-= !
-
Bypass
1,= 1 "! I ~ .... [ 7h~11 k=l
II ....
'
_
< ~
k-'2
cu
!
I
['r~2 -~ r~.ac2
'
The example has been solved again by the extended model (Table 3.b). Now, the model contains 31 binary variables. It has been optimized by DICOPT++ in 3.47 sec of CPU time. The topology of HEN has not been changed (Fig.5), but the annual HEN cost has been decreased to 336130 $/yr due to the selection of different exchanger types. The streams H1 and C1 are matched together within the shell & tube exchanger with Ft of 0.912 (above the estimated value, Ftest=0.9).
k=3
Fig.5. optimal solution - first example.
Table 2. First example data. F C IkW/KI a [kW/(m2K)] Hot streams HI 250 0.6 H2 25 1.0 Cold s t r e a m s C1 240 0.7 C2 13 1.0 Ft for shell & tube e x c h a n g e r t y p e estimation: 0.9 Utility streams a [kW/m2K] Tin [KI Hot (HU) 5.0 510 Cold (CU) 1.0 300 Table 3. Solution of the first example. a. Solution given by the original m o d e l M a t c h (i-j-k) A Im2] No [-1 1-1-2 490.4 3 1-2-1 73.2 1 2-1-2 60.7 1
/t!.T [K] 81.42 24.64 60.00
Tin IKI 470 450
Tout IKI 400 390
p IMPal 2.0 1.0
330 410
390 500
1.0 1.0
Tout [KI 510 321
C [ $/(kW a)l 250 21
b. Solution given by the e x t e n d e d model. At.T [K] Match(i-j-k) A [m2l No l-] 1-1-2 544.9 1 81.42 1-2-1 73.2 1 24.64 2-1-2 60.7 1 60.00
Ft l-I 0.912 1.000 1.000
Type ST DP PF
The second example contains four hot and five cold streams (Table 4.) with four-stage HEN superstructure. The stream data were taken from the HDA process case study. Since the streams contain toxic components, the selection of the plate & frame type is forbidden. The example was solved first by Yee's model [ 1]. The program contained 89 binary variables and was solved by DICOPT++ in 9.83 sec of CPU time. Table 4. Second example data. Hot streams F C lkW/Kl a [kW/(m2K)] H1 49.27 0.15 H2 27.54 0.90 H3 1088.67 0.90 H4 229.17 0.90 Cold s t r e a m s C1 38.92 0.12 C2 14.58 1.00 C3 511.33 1.00 C4 252.60 1.00 C5 236.13 1.00 Ft for shell & tube exchanger type estimation: 0.8 Tin [KI Utility streams a [kW/m2K] Hot (HU) 5.0 850 Cold (CU) 1.0 282
Tin [KI 823.20 330.85 352.32 379.90
Tout IKI 299.89 329.85 349.32 376.9
p IMPal 3.5 3.5 3.5 3.5
330.19 362.95 462.30 376.90 550.60
713.70 463.00 465.30 379.60 553.60
3.5 3.5 3.5 3.5 3.5
Tout [K] 850 290
C [ $/(kW a)] 250 21
1100
Only six exchangers and four coolers were needed (Table 5.a) for the entire heat transfer with annual HEN cost of 867102 S/yr. The cost is underestimated again, because parallel exchangers are needed for the heat transfer within some of the selected matches. The exact cost was 897990 S/yr. The example has been solved by the extended model again. Now the program contains 246 binary variables and has been optimized by DICOPT++ in 478.24 sec of CPU time. Three iterations of the synthesis/analysis scheme have been necessary to obtain the solution, which contains eight exchangers and three coolers (Table 5.b) with the annual cost of 878337 S/yr. The match between the streams H1 and C2 at stage 4 requires two heat exchangers, so the exact HEN cost is increased to 882198 S/yr. Note that the topology of the optimal HEN has been significantly changed. In order to exclude one cooler, two additional exchangers have been selected. Table 5. Solution of the second example.
a. Solution given by the original model Match (i-j-k) A Ira21 No 1-1 1-1-1 1-1-4 1-2-4 1-3-3 1-4-2 1-5-1
1269.8 544.3 154.1 130.0 25.3 49.0
11 3 1 1 1 1
b. Solution given by the extended model.
zl~nTIK1 130.89 105.97 72.59 90.49 199.00 110.81
Match(i-j-k) 1-1-1 1-1-2 1-1-3 1-2-4 1-3-3 1-4-3 1-5-2 4-1-4
A lm2l 742.6 775.3 632.6 306.2 293.0 33.5 55.2 166.8
No I-I 1 1 1 2 1 1 1 1
A~nT IKI 121.24 142.32 139.01 36.52 50.17 150.50 98.38 38.92
Ft l-I 0.953 0.831 0.952 1.000 0.998 1.000 1.000 1.000
Type ST ST ST DP ST DP DP DP
5. CONCLUSIONS Both examples clearly indicate the advantages of the proposed model, which yields not only feasible but also better HEN design with respect to utility consumption and required heat transfer area. The proposed model allows the simultaneous heat integration of HEN and selection of optimal exchanger types due to the operating limitations. NOTATION AtnT --temperature driving force [ K ] C =charge coefficient [ $/yr ] i =set of hot streams [ - ] j =set of cold streams [ - ]
k l y a
=set of superstructure stages [ - ] =set of heat exchanger types [ - ] =binary variables =conv.heat transfer coef.[kW/(m2K)]
REFERENCES
1. T.F. Yee and I.E. Grossmann, Optimization models for heat integration-II, Heat exchanger network synthesis, Computer and Chem. Engng., 14, 1165-1184 (1990). 2. G.F. Hewitt, G.L. Shires and T.R. Bott, Process heat transfer, CRC Press, 155-194 (1994). 3. K.M. Guthrie, Capital cost estimating, Chem.Engng., 76,114 (1969). 4. G. Walker, Industrial heat exchangers, Hemisphere publishing, 93-102 (1990). 5. J.J.J. Chen, Letter to the editors: comments on improvement on a replacement for logaritmic mean, Chem.Engng.Sci., 42, 2488-2489(1987). 6. J. Vishwanathan and I.E. Grossmann, A combined penalty function and outher approximation method for MINLP optimization, Computers and Chem. Engng., 14, 769-782 (1990).
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1101
SIM-OPT: A Computational Architecture to Address Valuable Business Aspects of Research & Development Pipeline Management Dharmashankar Subramanian, Joseph F. Pekny*, and Gintaras V. Reklaitis School of Chemical Engineering, Purdue University, West Lafayette, IN 47907, USA. (dharmash, pekny, reklaiti)@ecn.purdue.edu The R&D Pipeline management problem has far-reaching economic implications for newproduct-development driven industries, such as pharmaceutical, biotechnology, and agrochemical industries. Effective decision-making is required with respect to portfolio selection and project task scheduling in the face of significant uncertainty and an everconstrained resource pool. Recently, Subramanian et al. (2000) described the here-and-now stochastic optimization problem inherent to the management of the R&D Pipeline by viewing it as the control problem of a performance-oriented, resource-constrained, stochastic, discreteevent, dynamic system. They presented a computing architecture, Sim-Opt, which combines mathematical programming and discrete event system simulation to assess the uncertainty present in the pipeline. They introduced the concept of timelines that studies multiple unique realizations of the controlled evolution of the discrete-event pipeline system. This work demonstrates how information may be integrated across the Sim-Opt timelines to obtain insights into the dynamics of the pipeline under uncertainty. It also illustrates how such insights may be applied to plan more effectively from an operational perspective. Lastly, it discusses briefly about other business questions that can be investigated using the Sim-Opt architecture. 1. I N T R O D U C T I O N Process systems engineers are faced with a tremendous opportunity to make an impact not only to the engineering process of making a new product, but also to the business process involving product prioritization, selection and pipeline scheduling (Blau et al., 2000). The R&D pipeline management problem is one such business process. It addresses the issues of a new-product-development pipeline, where several new-product-development projects compete for a limited pool of various resource types. Each project (product) usually involves a precedence-constrained network of testing tasks prior to product commercialization. If the project fails any of these tasks, then all the remaining work on that product is halted and the investment in the previous testing tasks is wasted. It is also possible that failed projects are reconsidered as long-term investments with re-working attempted to overcome the failure. In its most general form, the deterministic R&D pipeline management problem asks the following question: Given a set of research projects, each project containing a set of activities related by generalized precedence constraints, a common pool of limited resources of various (finite) * Author to whom all correspondence should be addressed
1102 kinds and a measurement of performance, what is the best set of projects to pursue, and further, what is the best way to assign resources to activities in the chosen projects, such that the chosen measure of performance is maximized? A more realistic, and practically motivated, problem faced by new-product-pipeline decision-makers is the above question in a stochastic context, i.e., with uncertainty added in terms of task duration, task resource requirements, and task successes and task rewards. Thus, a realistic R&D pipeline management problem is a stochastic optimization problem combining the features of a project selection problem and generalized resource constrained project-scheduling problem. The additional complexities are the task success uncertainty, duration uncertainty, resource requirement uncertainty and project reward forecast uncertainty. Task success uncertainty, a predominant source of uncertainty in the R&D context, has not been adequately addressed in the literature with the noted exception of Schmidt and Grossmann (1996), Honkomp (1998), Jain and Grossmann (1999), Blau et al. (2000), and Subramanian et al. (2000). Subramanian et al. (2000) described the here-and-now stochastic optimization problem inherent to the management of an R&D Pipeline by viewing it as the control problem of a performance-oriented, resource-constrained, stochastic, discreteevent, dynamic system. They presented a computing architecture, Sim-Opt, which combines mathematical programming and discrete event system simulation to assess the uncertainty present in the pipeline. Lastly, they described three policies to for taking actions as the pipeline evolves through its states while traversing the stochastic state space subject to the resource constraints present in the system. Further details about the Sim-Opt architecture can be found in Subramanian et al. (2000). 2. INTEGRATION OF INFORMATION ACROSS TIMELINES The simulation module, in Sim-Opt, marches in time, with indeterminate number of departures to the optimizer whenever a decision-making need is encountered (Subramanian, et. al, 2000). One such controlled walk in time through the stochastic state space constitutes a timeline. The simulation thus experiences different "futures" based upon stochastic realizations, encountered in a Monte-Carlo sense, across timelines. Any single timeline contains information about the temporal coexistence of various feasible tasks, both within and across projects. This information, while being influenced by the nature of control actions exercised, is largely a function of uncertainty, and is not obtainable a priori from a single monolithic stochastic program due to uncertainties present in the task successes which are binary (succeed/fail) in nature, and those present in task processing times and task resource requirements. This coexistence information can be further processed to infer and identify resource types that are binding in the face of uncertainty, and to evaluate the worth of augmenting such resource types. It can also be used to obtain information about the relative tendencies of projects (and tasks) to crowd out other projects (and tasks) due to the associated variations in their resource needs. All the information that is obtained from the framework could be potentially incorporated in a suitable manner into the deterministic optimization formulation to bias it to recognize the same for the purposes of portfolio selection. Finally, multiple such timelines can be explored in a Monte-Carlo fashion to accumulate several unique combinations of realizations of uncertainty. The information mentioned above can be integrated across this accumulation to obtain solutions to the underlying stochastic optimization problem. This will be illustrated on an industrially motivated case study in the next section.
1103
3. CASE STUDY This section demonstrates the integration of information across Sim-Opt timelines using an industrially motivated case study. The Case Study comprises seven projects, as shown in Figure 1. There are two resource types, R1 and R2, which are required in a certain combination for feasibly carrying out any single task. The system limits for R1 and R2 are 16 units and 8 units respectively. There is significant uncertainty with respect to processing duration, resource requirement, and survival probabilities for the tasks in all seven projects, as shown in Table 1. Project reward data is shown in Table 2. Table 1. Case Study Task Data Task
12
P1
14
Duration (weeks), Custom Distribution Value Probability
R1 (units), Custom Distribution Value Probability
1 2 3 4 5 2 3 4 5 6 3 4 5 6
0.295 0.375 0.190 0.110 0.030 0.10 0.18 0.44 0.18 0.10 0.32 0.40 0.18 0.10
4 5 6 7
0.29 0.44 0.21 0.06
1 2 3 4
0.335 0.415 0.166 0.084
3 4 5 6 7
0.123 0.203 0.335 0.228 0.111 0.32 0.44 0.16 0.08 0.26 0.54 0.14 0.06 0.10 0.2O 0.36 0.28 0.06 0.32
4 5 6 7 8 10 11 12 13 14 3 4 5 6 7 4 5 6 7
0.06 0.21 0.46 0.21 0.06 0.06 0.12 0.55 0.21 0.06 0.05 0.20 0.45 0.25 0.05 0.23 0.52 0.20 0.05
10 11 12 13 2 3 4 5 4 5 6 7
0.23 0.55 0.16 0.06 0.23 0.55 0.15 0.07 0.23 0.53 0.17 0.07
12
0.225
P2
2 3 4 5 16
P3
I
I
R2 (units), Custom Distribution Value Probability
Probability Success Triangular Distribution Min
0.28 0.44 0.28
0.74
Most Max Likely . 0.80 0.86
0.22 0.56 0.16 0.06
0.7
0.75
0.8
2 3
2 i 3 4 5
0.29 0.44 J0.21 0.06
0.8
0.85
0.9
0.23 0.52 0.20 0.05
i0.7
0.8
0.85
1
4
i
1 2 3 4
0.23 0.55 0.16 0.06
0.55
0.6
0.65
2 3 4 5 1 2 3 4
0.23 0.53 0.16 0.08 0.23 0.53 0.17 0.07 0.225 0.565 0.155 0.055
0.75
0.8
0.85
0.7
0.8
0.85
0.7
0.75
0.8
0.225
0.85
i i
!
i
i
0.9
0.95
i
(JI
C~
,.~ (JI (JI (JI
~ C ~ O ~ 0 0 0 -.,I .,,,I u,) u,)
~
(.~
~'~ ~'~ . ~
U,) !
I~ h.)
-,.,! ,,..I . ~
h ~ 84
C~h ~r~ . ~
~ ) ~~
~r~ -1~ ~,)
C~ ~rl -1~ , )
~
~
-..,I --,.I u,) ~,)
0
C~
C~ ../1
0
0
0
0
C
~
C~ C~ ~rl
1105
0.284 117
P7
2 3 4 5 3 4 5 6
0.29 0.44 0.21 0.06 0.29 0.44 0.21 0.06
8 9 6 7 8 9 13 14 15
0.17 0.07 0.22 0.54 0.17 0.07 0.20 0.57 0.23
7 8 2 3 4 5 3 4 5 6
0.17 0.07 0.22 0.54 0.17 0.07 0.22 0.54 0.17 0.07
0.3
0.35
0.4
0.45
0.5
0.55
T a b l e 2. Project R e w a r d D a t a Reward $ Project P1 30,000 P2 20,000 P3 15,000 P4 40,000 P5 50,000 P6 40,000 P7 60,000
! !
(D
!
I
j
Figure 1. Case Study Activity on N o d e Graph
A p p l y i n g P o l i c y II described in S u b r a m a n i a n et al. (2000), the distribution o f r e w a r d s and the r e s o u r c e profile b e h a v i o r is as given below in Figures 2 and 3.
1106 Rewards Distribution, Policy II 800
700
15
Mean =36564.74
Average Resource Profiles for R1 and R2, Policy 2
.
.
.
.
IIFrequency
600-t
500 400
300
30[ m / -.-
200
I
System Limit I Desired |
...1
100 o L-_______.~ -2 0
2 4 6 Rewards, Dollars
8
10 x 104
0
2
4
6 8 TIME, WEEKS
10
12
14
Figure 2. Rewards Distribution From Policy II
Figure 3. Resource Profiles From Policy II The plots in Figure 3 show the dynamics of combinatorial interaction between activities across projects in the portfolio. This interaction is due to resource competition. It can be seen that with respect to resource type, R1, the average "desired profile" stays within the system limit of 16 units during the earlier time periods in the planning horizon. But the actual utilization of resource type, R1, is below the desired level, as exhibited by the "utilized profile". While this may appear counter-intuitive, it is because during the corresponding time periods, the average "desired profile" with respect to resource type R2 is well above the system limit of 8 units. This prevents effective resource utilization since the two resource types are required in the right combination. Activities in the pipeline are eligible for active processing only if the resource types R1 and R2 are available in the right combination of amounts. The plots can thus be viewed as the dynamics of interaction between resource types R1 and R2. During earlier time periods, resource type R2 is binding, while at later time periods, resource type R1 becomes binding as well. This combinatorial interaction at the current levels of availability of the resource types leads to poor utilization of the system resource levels. This knowledge can be incorporated to plan more effectively from both a design perspective and an operational perspective. The latter is illustrated with an example in the following section. 4. AN E X A M P L E OF USING THE I N F O R M A T I O N I N T E G R A T E D A C R O S S TIMELINES The resource profile information integrated across the timelines in Figure 3 revealed the under-utilization of the resource types R1 and R2 due their combinatorial interaction and how this interaction evolves in time. Having gained this insight, we can evaluate creative operational decisions, such as accumulating underutilized resource levels of a resource type from lean periods, and utilizing them for tight periods when that resource type becomes binding. This is like under-working a resource type for some time periods, in return for overworking the same resource-type for some other time periods. We implement this operational strategy in Policy IV, which is same as Policy II (Subramanian, et. al., 2000) in all other aspects. In Policy IV, accumulated resource levels are assigned to tasks along with actually present resource levels (if any), only if such an assignment fully satisfies the actual resource
1107 needs of the task for its entire processing duration. In particular, we accumulate underutilized resource levels, corresponding to resource types, R1 and R2, in units of R1-Weeks, and R2Weeks respectively. Figure 4 shows the frequency plot of rewards obtained from 15000 timelines corresponding to the same unique 15000 sets of random numbers, as used before. Rewards Distribution, Policy IV
700
5I | 1 0 I" . . . . .
600 500
Average Resource Profiles for R1 and R2, Policy IV
1
Mean =39522.62 Frequea
R1 t "
400
5
300
3O
200
20[ Rn /~
100 0 -2
0
- - System Limit - , - Desired ~- ..~..~ . . . . . . . _- - - ~ Utilized . . . ~
~'==------ " ' - - "
2
=="==" =" == ======~. == ~ == == ~ . . . . . . .
4
6
. _ . P, =,=-='.-p"- -'
=.. ,m,,=, ,P ,--
8
10
j - ~ " " ' ~ " ....
=.--==
12
-"
=..-.
14
......
.........
10 t . _ . . , . - . - - = " ' " 0
2 4 6 Rewards, Dollars
8
10 x 10"
Figure 4. Rewards Distribution From Policy IV
0
2
4
6 8 TIME, WEEKS
Figure 5. Policy IV
Resource
10
12
Profiles
14
From
CumulativeFrequencyPlotsof Rewardsfrom PolicyII and PolicyIV 1 0.9
0.8Probability
','1
0.7 0.6
0.5 0.4 0.3 0.2 0.1 0 0
2
4 6 Rewards, Dollars
8
10 x 104
Figure 6. Cumulative Frequency Plots of Rewards From Policy II and IV A Cumulative frequency plot of the rewards obtained using Policy IV is shown in Figure 6, along with that corresponding to Policy II for comparison purposes. (Note that Policy IV can be thought as Policy II implemented with an additional operational strategy as described above). Policy IV outperforms Policy II in terms of the mean as well as in terms of the cumulative frequencies. Figure 5 shows the "desired resource profiles" and the "utilized resource profiles" that are accumulated and averaged across the 15000 timelines explored using Policy IV. While the dynamics of combinatorial interaction between the resource types at the current resource levels in the system, continue to prevent effective utilization of the
1108 available resource levels, the extent of under-utilization has improved significantly over what was witnessed in Policy II. This is an example of how the insight obtained with the integration of information across timeline can be utilized effectively to influence and improve the quality of timelines that the pipeline system can witness. 5. CONCLUSIONS The concept of timelines that studies multiple, unique, realizations of the controlled evolution of the discrete-event pipeline system, has been shown to be an effective approach to obtain insights into the dynamics of the pipeline problem under uncertainty. Methods have been presented to integrate information across the timelines in terms of binding resource-types that present bottlenecks. An example has been presented that evaluates operational decisions, such as accumulating underutilized resource levels of a resource type from lean periods, and utilizing them for tight periods when that resource type becomes binding. This is like underworking a resource type for some periods, in retum for over-working the same resource-type for some other periods. Sim-Opt can be an effective architecture for integrating several different kinds of information across timelines. We can evaluate design decisions, such as, the value of acquisition of any resource type(s), the value of entering into outsourcing contracts for binding resource types and estimate the timing of these future contracts, at the here-and-now. Operational decisions such as partially satisfying the resource needs of resource-starved activities, accompanied by a proportional increase in their processing times, can also be studied with the timeline integration within the Sim-Opt architecture. Other investigations that can be carded out using Sim-Opt include analyzing the sensitivity of parameters such as resource levels, cost estimates, technical survival probabilities and commercial estimates of rewards, and answering what-if questions such as addition and generation of new projects. Sim-Opt can also be used to answer questions about the value of investment in research to improve the quality of information as well as the quality of projects and project execution, in terms of investing to improve survival probabilities and processing times. Finally, all such information can be utilized towards more effective decision-making in order to improve the quality of timelines that the pipeline system can witness. REFERENCES 1. Schmidt, C. W and Grossmann, I. E. Optimization models for the scheduling of testing tasks in new product development. Industrial & Engineering Chemistry Research. 35: (10). 3498-3510. Oct 1996. 2. Jain, V. and Grossmann, I. E. Resource-constrained scheduling of tests in new product development. Industrial & Engineering Chemistry Research. 38: (8). 3013-3026. Aug 1999. 3. Honkomp, S. J. Solving mathematical programming planning models subject to stochastic task success. Ph.D. Thesis, School of Chemical Engineering, Purdue University, 1998. 4. Blau, G. E., Mehta, B., Bose, S., Pekny, J. F., Sinclair, G., Kuenker, K. and Bunch, P. Risk Management in the Development of New Products in Highly Regulated Industries. Computers and Chemical Engineering. 24: (2-7). 659-664. July 2000. 5. Subramanian, D., Pekny, J. F. and Reklaitis, G. V., "A Simulation-Optimization Framework for Addressing Combinatorial and Stochastic Aspects of an R&D Pipeline Management Problem", Computers and Chemical Engineering. 24: (2-7). 1005 - 1011. July 2000.
European Symposmm on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1109
Solution of MEN synthesis problems using MINLP: Formulations of the Kremser equation Z. Szitkai, Z. Lelkes, E. Rev, Z. Fonyo Chemical Engineering Department, Budapest University of Technology and Economics, H-1521 Budapest, Hungary One way of solving mass exchange network (MEN) synthesis problems is to formulate the synthesis task as a mixed integer non-linear programming problem. The solution of this optimisation problem delivers the desired MEN. In most cases, the objective function of the optimisation problem is the total annual cost. Assuming linear phase equilibrium relations, capital investment calculations of staged mass exchangers are most commonly based on the Kremser equation. Discontinuous functions, such as the Kremser equation, are difficult to formulate in an MINLP model. A new method is suggested for overcoming the difficulties arising from the discontinuity of the Kremser equation. Our new method is reliable and faster than the big-M, multi-M, simple logic, and convex-hull formulations applied for this particular problem. Our method is tested on a small and a large MEN synthesis problem. The solutions for four well-known MEN synthesis problems are also presented. I. INTRODUCTION A mass exchange network (MEN) is a network of interconnected direct-contact mass-transfer units that employs mass separating agents to selectively remove certain components from different rich phases (E1-Halwagi and Manousiouthakis, 1989). Two main synthesis approaches can be employed to determine the optimal network; structure independent (Pinch analysis) and structure based (mathematical programming). In the case of mathematical programming, the synthesis of MENs is realised by the formulation and solution of a mixed integer non-linear programming (MINLP) optimisation problem (Papalexandri et al., 1994). In most cases, multistage mass exchangers are also used for constructing the mass exchange network. Assuming linear phase equilibrium relations, capital investment calculations of these units are most commonly based on the Kremser equation, which gives the required number of equilibrium stages for a given separation. Generally, discontinuous functions, such as the Kremser equation, are difficult to formulate in MINLP and may cause numerical problems in course of the solutions. Three methods are proposed in the literature for formulating discontinuous functions, namely the conventional big-M, the multi-M (Hui, 1999) and the convex hull (Balas, 1985) formulation. In this article we suggest two new formulations for the Kremser equation and compare them to formulations based on the above mentioned methods. We also discuss the MINLP solution of four MENS problems. Comparison is made between our MINLP solutions and the pinch based
1110 solutions of Hallale (1998). Computation times in this paper concern a SUN Ultra Sparc-1 workstation. 2. FORMULATIONS OF THE KREMSER EQUATION As mentioned above the the Kremser equation gives the required number of equilibrium stages for a given separation, in case of linear phase equilibrium relations. Depending on the value of the removal factor A, the Kremser equation for a given component i has two different forms. Throughout our calculations in GAMS for the case ofAr 1 we used the following form of the Kremser equation (E1-Halwagi, 1997): log
= Yi,out -- m j x j , i n
NTPA=I =
If A=I
N T P A ~ I 9log(A)
-- b j
Where: A =
(1) mjG i
Yi,~,,
- Yi,out
(2)
Y i,out -- m j x j,in -- b j
The removal factors of the units are design variables when solving the MINLP synthesis problem, their values have to be able to vary freely between their phisically imaginable bounds. A can equal 1 or can be less or greater than 1 also. Switching between the two forms of the Kremser equation in an MINLP modelling environment is not a trivial task at all. In GAMS / DICOPT for example it is not possible to build in conditional equations into the model. This is a common attribute of all the available solvers, and origins from the numerical algorithms they use. Using only the first form of the Kremser equation usually leads to a division by zero error or gives solutions that have no physical meaning. Restricting the values of A under or over 1 very likely excludes the real optimal solution from the search space. Without solving this numerical difficulty no reasonable MENs can be expected from the MINLP method. The discontinuity of the Kremser equation can be overcome in the following general way: A binary variable Y has to be defined which equals 1 in the case of A=l, and takes the value of zero when A~I. Then both equations are used to calculate the number of theoretical plates, and Y is used to select the one which corresponds to the value of A in the given mass exchanger.
NW
= r.
+ 0 - r ) . (Nr+'**, )
(3)
For calculating the binary variable Y five methods were examined. The first four formulations below divide the interval of A into three sub intervals with the boundaries of 0.01; 0.99; 1.01; 100, and switch to the formula valid for A=I when A is between 0.99 and 1.01. This interval division is arbitrary.
1111
2.1. Big-M A <0.99+Ml-(1-yl)
;
- A < -0.01 + M 1 9(1 - y~)
; - A < - 0 . 9 9 + M 2 9(1 - Y2) ;
M~ = 99.01 ; M 2 = 98.99
A
A <100+M3-(1-Y3) - A < -0.01 + M 3 9(1 - Y3)
; m 3= 1
Y~ + Y2 + Y3 = 1 ;
Yi
A 1 < 0.99. Yl ;
A2 < 1.01. Y2 ;
A3 < 100- Y3
0.01 _ A _< 100;
;
Y : Y2
- {0,1};
(4-16)
2.2 Convex hull - A1 < - 0 . 0 1 . y 1;
- A 2 _< - 0 . 9 9 . y 2 ;
- A3 < - 1 . 0 1 . y 3
0 < Ai < 100 ;
0.01 < A < 100;
A~ + A 2 + A 3 = A
Yl + Y2 + Y3 = 1;
Yi
Y = Y2
= {0,1};
(17-28)
2.3. Multi-M A <0.99+M1,2.(1-yl)
;
A
A <100+M3,1-(1-y3)
- A <-O.OI+M1,3.(1-Yl);-A < - 0 . 9 9 + M 2 , 3 . ( 1 - Y 2 ) ; - A M1,2 = 99.01 ; M1, 3 = 0 ; M2,1 = 98.99; M2,3 = 0.98; M3,1 = 0.01 < A < 100;
Yl + Y2 + Y3 = 1 ;
<-0.01+M3,z.(1-Y3) 0;
Yi = {0,1};
M3, 2 = 1 (29-44)
Y = Y2
2.4. A simple logical formulation (L-formulation) 0.01- yl + 0.99- Y2 + 1.01. Y3 < A < 0.99. Yl + 1.01. Y2 + 100. Y3 Yl + Y2 + Y3 = 1 ;
y, = {0,1};
(45-48)
Y = Y2
All these methods suffer from applying excess restrictions, introduced by the three arbitrary intervals involving three excess binary variables for just one singularity that could be managed by just one binary variable. Hoping better solution properties, we introduced a simpler method for dealing with this kind o f discontinuity. This method should work with just one discontinuity at some nonzero argument, e.g. the Kremser equation or the logarithmic approach temperature.
2.5. New formulation without intervals for the removal factor A The
equation
(A-l) 2 =c-y
for each
o f the
staged
units
with
the
assumptions:
0.01 _< A _< 100 and 10 -4 _< c _< 9810 is introduced, where c is a continous and y is a binary variable. The binary variable Y = 1 - y
can equal 1 only in the case when A equals 1. This
method applies an arbitrary interval for the removal factor A too. This interval is set up by the lower bound o f c. The main advantage o f this formulation is that instead o f three binary variables it uses only one binary variable per multistage mass exchanger. 3. C O M P A R I S O N The K r e m s e r formulations outlined above have been compared on two test problems. Throughout our calculations, the superstructure and model o f Papalexandri et al. (1994) are used. The pinch solution o f Hallale (1998) serves a starting point to generate initial value for the model variables.
1112
3.1 Small test problem (Example 3.3 of Hallale, 1998) This MEN synthesis problem contains only two rich and two lean streams. Assuming only one subnetwork (Papalexandri et al., 1994) the superstructure contains maximum four staged exchangers. The objective function is the total capital cost of the MEN. Table 1: Different Kremser formulations compared on a small test problem number number of number of solution time of binary single single (seconds) variables equations variables new method 4+4 115 107 2.88 L-formulation 4+ 12 123 111 3.28 convex hull 4+12 143 123 4.91 big-M 4+12 139 111 6.9 multi-M 4+12 139 111 5.11
objective function (thousand USD) 628.851 619.963 632.887 632.887 615.307
It can be seen that the new method is faster that any of the conventional methods.
3.2 Large test problem (adapted from Hallale, 1998) This MEN synthesis problem is identical to example 4.1 of Hallale (1998). It contains five rich and three lean streams. The only difference compared to the original one is that we considered tray columns instead of packed columns. Assuming three subnetworks the superstructure contains maximum 45 staged exchangers. The objective function is the total capital cost of the MEN again. The cost of one stage is taken 20,000 USD/yr. Table 2: Different Kremser formulations compared on a large test problem number number of number of solution time of binary single single (seconds) variables equations variables new method 45+45 1233 2032 145.03 L-formulation 45+135 1323 2077 Over 1800 convex hull 45+ 135 1548 2212 Over 1800 big-M 45+135 1503 2077 Over 1800 multi-M 45+135 1503 2077 249.13
objective function (thousand USD) 552.33 551.78
Among the conventional methods for treating discontinuity, only the multi-M formulation of Hui (1999) gives reasonable result. In case of the large test problem, the other methods did not give solutions in half an hour computation time. The new method proved to be 41.7 % faster than the multi-M formulation. This is probably due to the less number of binary variables and equations used in the model.
1113 4. R E S O L V E D M E N SYNTHESIS P R O B L E M S
Having the new method tested, we tried to solve several industrial examples four of which are partially shown: Example 3.2 from Hallale's PhD thesis; the COG sweetening example of E1Halwagi and Manousiouthakis (1989); the copper recovery problem of E1-Halwagi and Manousiouthakis (1990) and the desulfurisation of rayon wastes problem of E1-Halwagi and Srinivas (1992). These problems cover the areas of wastewater minimisation, MENS with nonuniform exchanger specifications and synthesis of reactive MENs. Our design alternatives are compared to solutions of Hallale (1998) since he got designs with the lowest TAC for the problems mentioned. Table 3: Problems solved by MINLP and the optimal value of their objective function Hallale example type of the task (TAC in USD) wastewater Example 3.2 of Hallale (1998) minimisation 455,000 COG sweetening simple MENS 751,000 MEN with non uniform Copper recovery exchangers 49,000 Rayon wastes desulfurisation
convex REAMEN
Our solution (TAC in USD) 453,302 615,307 50,279
28,000
32,000
[6.49e-4] [0.0621 5.883 stages 13.85 stages
(D1-02e-3(
3e-4
A=0.96 le-4
[9.3465e-51 1.3 stages
[~
2e-4
5.1e-2 [-~0.1
( ~"~3.3939e-2 1.5043e-2
6.154e-4
Stage numbers are rounded up to 28
-~-] 0.9
A=1.42
(~A=2.385
1.862 kg/s 6e-4
7e-2
[0.004991 5.95 stages
~A=8.1s
[~
)
Gi
(kg/s)
( )
1[
Lj (kg/s) 3.097e)2 2.208
3.5e-3
capital cost of this MINLP design: N. Hallale's actual capital cost:
0.225 637 280 USD 751 080 USD
Figure 1" Our MINLP solution for the COG sweetening problem Initial value generation in MINLP is an essential problem. We generated initial values based on approximating pinch solutions. Solutions of the four industrial examples are shown in Table 3
1114 above. The optimal design for the COG sweetening problem, with a solution 17.8 % better that of the pinch, is additionally shown in Figure 1. The advanced targeting method of Hallale and Fraser (2000) is found a well applicable, good approximation but can be improved by MINLP in some cases. Using simple pinch solutions as initial values for the MINLP method proved to be an appropriate choice.
5. CONCLUSION A new method to overcome the difficulties arising from the discontinuity of the Kremser equation used in an MINLP model is introduced and tested. In this particular case, our new method for treating discontinuity results in less computation time than any other methods known in the literature. It is also shown that MENs designed by the method of Hallale (1998) can be improved using MINLP.
REFERENCES
Balas, E. ,(1979), "Disjunctive programming", Discr. Math.,5,pp.3-51 Chi-Wai Hui, (1999), "Optimising Chemical Processes with Discontinuous Function- A Novel Formulation", Comp. Chem. Eng., Vol 23, $479-$482 E1-Halwagi, M.M. and Manousiouthakis,V.,(1990), "Automatic synthesis of Mass Exchange Networks with Single-Component Targets", Chem.Eng.Sci.,45(9),2813-2831 E1-Halwagi, M.M. and Srinivas,B.K.,(1992), "Synthesis of Reactive Mass Exchange Networks", Chem.Eng.Sci.,47(8),2113-2119 E1-Halwagi, M.M. and Manousiouthakis, V., (1989), "Synthesis of Mass Exchange Networks", AIChE J.,3(8), p.1233-1244 E1-Halwagi, M.M.(1997), "Pollution Prevention Through Process Integration", Academic Press 525 B Street, Suite 1900, San Diego, California 92101-4495, USA, (www.apnet.com) Hallale N. and Fraser D.M., (March 2000), "Supertargeting for Mass Exchange Networks, Part III.", Trans IchemE, Vo178, Part A, p.202-216 Hallale N.,(1998), "Capital Cost Targets for the Optimum Synthesis of Mass Exchange Networks", PhD thesis, Dept. of Chemical Engineering, University of Cape Town Papalexandri K.P.et al., (1994), "Mass Exchange Networks for Waste Minimization", Trans IChemE, Vo172, Part A p. 279-293
European Symposium on Computer Aided Process Engineering - I 1 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1115
Modelling Interface for Chemical Plantwide Process Design K. Wang a, G.A. Clark a, P.W.H. Chung b and D. Rossiter a* a Department of Chemical Engineering, Loughborough University, Loughborough, U.K. b Department of Computer Science, Loughborough University, Loughborough, U.K.
In the process engineering industry, the design of a plantwide process is a difficult and time consuming task that requires the sharing of data and knowledge, maintenance of consistency among subdesign and coordination. This paper addresses the problems by taking advantage of significant progress in information technology to develop a modelling interface to support process design. The modelling interface includes a database of process models and recall facility. Moreover for a particular multi-unit process description, it can automatically generate the input code for gPROMS [1 ], a dynamic simulation package. A HDA (hydrodealkylation of toluene) process [2] is used to demonstrate the interface's functionality. 1. INTRODUCTION Over the past three decades, computer-based packages for modelling chemical processes have become more advanced and widely used. For example, equation-oriented dynamic simulators, such as gPROMS [1 ], are useful for simulating plantwide chemical processes as well as single unit operations. However, they rely on the user being proficient in modelling. A useful next step would be to set up a modelling interface containing a database of process models of past models and a recall facility to retrieve them. This would provide design engineers with a library of knowledge to check on first, much as expert engineers use their past experience to guide them through a design. In recent years, several interfaces have been developed to support chemical process design e.g. [3], [4]. Both these approaches are based on data models to capture engineering knowledge and information. These models have been developed for the design and realisation of distinct support systems, which are useful but specific to the software system they were designed for. Hence to be more generally applicable, it is desirable to use standard data representations [5] in building a generic modelling interface. Earlier work by the authors developed a modelling interface namely the Intelligent Modelling Interface (IMIPS) [6]. This interface gives the user access via a database to past models (or cases) already developed and automatic code generation once the model is defined, or retrieved. But IMIPS can only deal with single unit operations. This research "To whom all correspondence should be addressed. Email: D.Rossiter@lboro ac.uk
1116 extends that work and an interface is developed to cope with plantwide processes and also incorporates other changes. Namely, the data model database [7] is based on STEP (Standard for Exchange of Production) [8] and incorporates the protocol for building a database. Also, the indexing terms used within the database are revised and based on AspenPlus [9] terminology. Case Based Reasoning is used to intelligently search through the case modelling database [6]. This is used to assist the less experienced user to have a knowledge base to start their modelling work from, much as experienced engineers base some of their decisions on their past experiences. The index used to compare cases is possibly the most important subsection of the case based reasoning system, otherwise the relevant cases may be missed during retrieval. The index system in the database is as detailed as possible for locating relevant information, retrieving and ranking information that relates to the user's query. In this system, the index system is based on the Aspen Plus Course Notes [9]. Using the modelling interface, the engineer describes the problem, then retrieves the related cases. In the next step, the engineer writes a common mathematical description of the process with the knowledge coming from the past cases. Then, this is entered into the case modelling database. Next, the modelling interface automatically generates the code for gPROMS [1]. The structure of the paper is: section 2 gives the structure of the modelling interface, section 3 focuses on the description of the case modelling database, section 4 presents the gPROMS input file generation. Then, a multi-unit process example is used to demonstrate the ideas in section 5. Finally conclusions and future work are discussed in section 6. 2. STRUCTURE OF MODELLING INTERFACE The system is developed using ACCESS [10]. ACCESS involves creating and manipulating Tables that contain cases in conventional tabular (row-column or spreadsheet) format. The Visual Basic for Application (VBA) programming language is available in ACCESS, which is sufficient to define the procedures needed by applications to run Query, display Form and print Report. As shown in Fig. 1, the core of the modelling interface is the database of the past models (or cases). Using the interface, the users can enter the new cases by filling in the Table, then the modelling data can be visualised by displaying the Form. Similar cases can be used for reference and these are retrieved & reviewed by running a Query. Any process information can be shown by printing the Report, which is generated automatically by the interface. One of the principal incentives for using ACCESS is the ability to display graphics contained or hidden in the database table [10]. In the modelling database, the image of the equipment is contained in the Equipment Table, and the flow sheet of the process is hidden in the Stream Tables. This can be viewed using the visualisation part of ACCESS. The ability to provide a meaningful output report is the ultimate goal of the modelling interface. The report includes all of the case information, which is represented in the form of a gPROMS code file and can be considered as an input file for gPROMS.
1117
Visualisation of modelling data
Cases Retrieval & Review ,k
Que~
Form
Process Information Report, e.g. code for gPROMS Jl Report
Cases Modelling Database Modelling Interface
Open DataBase Connectivity ~ Table
I
Case Input
~r [ IMIPS
Fig. 1. Structure of the Modelling Interface. Earlier work from IMIPS [6] is utilised. IMIPS can only deal with single unit operations, but has a translator to create gPROMS input code. To integrate this translation tool, the Open Data Base Connectivity (ODBC) is used, which is a standard method by which an application communicates with elements of the computer's operation system or environment. 3. CASE M O D E L L I N G DATABASE The database is a store of all of the information through which the user can search for and share the information they are interested in. At the same time, the database has been designed to make the entering of new information easier. The user uses forms to fill in all the relevant information and/or uses queries to retrieve information from the database about the units or multi-units processes. These can then be translated into the simulation package input language. Likewise, past cases that have been amended can be saved in the database, thus increasing the knowledge base for future users of the system. The main interface is where the user can search, view and translate cases stored in the database. 3.1. Modelling Database
The database of process models is represented by six tables which are described as follows:
Process Table: The process table is used to store some general information of the specific process. It has 4 fields. Process Name is the name of the process. The field is Primary Key (PK) in the table, which can be used to identify the data in the table. Operating Mode is a simple choice between continuous, batch and semi-batch. Process State reflects the life-cyclestate of the process: design or operation. Description is used for the description of the Process. Unit Table: The process is composed of some units, which are saved in the Unit Table. It has the 5 fields. Process Name means the unit belongs to which process, which has one-to-many relationship with the Process Name in the Process Table. The one-to-many relationship means that the PK field of Unit ID is unique, but many of them may apply to the same
1118
Process Name in the Process Table. Operating Mode is a simple choice between Continuous, Batch and Semi-batch. Thermal Behaviour is a simple choice between Isothermal, Endothermic, Exothermal and Superheat. Equipment Table: Each unit consists of some equipment, which is shown in the Table. It has the 5 fields. Equipment ID is PK field. Equipment Name is the name of the equipment, which is added since the same equipment may be used in different Unit and Process with different Equipment ID. Unit 1D means the Equipment belongs to which Unit. The field has one-tomany relationship with the Unit ID in the Unit Table. Structure gives the visual description of the Equipment, which contain the image of the equipment. Stream Table: The streams connect to the units to become a process. There are 4 fields in the Table. Stream ID is the PK field. From Unit 1D means the Stream comes from which Unit. The field has one-to-many relationship with the Unit ID in the Unit Table. To Unit 1D means the Stream go to which Unit. The field has also one-to-many relationship with the Unit 1D in the Unit Table. Equation Table: The equations are the main part of the process modeling. There are the 8 fields in the Table. Equation ID is the PK field. Unit ID means the Equation belongs to which Unit. The field has one-to-many relationship with the Unit ID in the Unit Table. Equation Property is the choice among the Boundary, Balance, Constitutive and Conditional. The Differential Algebraic Equations (DAE's) or Integral and Partial Differential Equations (IPDAE's) are shown in the field of Equation Representation. Fixed, Lower and Upper give the description for the Equation. If an equation is only valid at one place in the system, it will be just stated in the Fixed field. Also the user can state Upper and Lower boundary conditions and there can be as many boundaries as necessary with a semi-colon used as a separator for each.
Constant Table: There are 6 fields in the Table. Constant Name is PK field. Constant Value and Constant Unit give the description of the Constant. Value property is the choice between lnteger, Real, Array. Description is used for the short introduction of the Constant. In the process modeling, the constant should be defined for either stream or equipment, so there are two constant tables in database: Equipment Constant and Stream Constant. In the Equipment Constant table, the field of Equipment ID is defined, which means the Constant belongs to which Equipment. The field has one-to-many relationship with the Equipment ID in the Equipment Table. In the Stream Constant table, the field of Stream ID is defined, which means the Constant belongs to which Stream. The field has one-to-many relationship with the Stream 1D in the Stream Table.
Variable Table: There are 6 fields in the Table. Variable Name is the PK field. Unit 1D means the Variable belongs to which Unit. The field has one-to-many relationship with the Unit 1D in the Unit Table. The variable just belongs to stream in any chemical process. Arrays of variables can also be used to simplify some equations, whose dimensions are given in the field of Value Dimensions. Value property is the choice between lnteger, Real and Array. Variable property is the choice of Input variables, State variables and Design variables. Description is used for the short description of the Variable. Since the variables just belong to the streams, the Variable Table is also called Stream Variable Table.
1119
3.2. Case Retrieval and Adaptation Mechanisms The information that belongs to the same process, or unit, is called a 'case'. This is saved in the relational tables. As mentioned above, the cases in the database allow the user to give a high level specification of the problem in a general form. This allows the system to search through these case specifications and finally to translate one case specification selected by the user into an input file for a particular simulator. Case based reasoning ideas have been incorporated to allow intelligent searching through the database of past cases. A query is entered through a form in the system. Each section has a hierarchical index that the user can use to select various key words and to search [11 ] using these keywords. In the database, the Unit Model and Function, Materials Property and Thermodynamic model are used as the index for the search. Once, the relevant cases are retrieved the user can manually adapt the case to fit their process description more closely.
4. gPROMS CODE GENERATION When investigating the performance of an existing, or new, process it would be helpful to have dynamic model(s) of the process available for simulation studies. This can be achieved by using translators in IMIPS [12] to convert the process description to input file format. There are 2 stages. Firstly, gPROMS translator in IMIPS is used to translate the individual unit descriptions to gPROMS input syntax. Then, VBA code is written to combines the individual unit descriptions to create the gPROMS input file.
5. CASE STUDY A typical process, the HDA (hydrodealkylation of toluene) process [2], is used to demonstrate the features of the modelling database. The user inputs the new process name and its description to the Process Table, the names of the units included in the process to the Unit Table, streams of the process to the Stream Table. Each stream connects two units, which are shown in the field of From UnitlD and To UnitlD. Based on the Stream Table, the flowsheet can be drawn automatically by the interface. The constants, variables and equations are shown in Constants, Variables and Equations Table. All of the HDA's information shown in these tables is called a HDA process case. The process case is composed of the type of unit, such as Reactor, Mixer etc, which are called unit cases. All of the fields in the table can be used in Query for case retrieval, as described in $3.2. All of the tables are relational by the PK fields in different tables. If the users want to generated the whole report for the process, firstly, they can get the process general information from Process Table, whose field of Process name is HDA; and then accumulate all of the units and streams which belong to the HDA from Unit and Stream; and then accumulate all of the variables, constants and equations which belong to such units and streams; at last the interface can generate the report as the input file for gPROMS. Then, the gPROMS code can be used for dynamic simulation. However, the gPROMS code may need some manual adjustments to achieve convergence due to issues such as satisfying degrees of freedom criteria, high index problems etc. This is one of the topics of further research.
1120 6. CONCLUSION AND F U T U R E W O R K
In the paper, a modelling interface incorporating a database based on the International Standard Organisation (ISO) Standard for Exchange of Production (STEP) data is described. The interface is composed of tables, queries, forms and reports. At present, the users can enter a new process description. When tackling problems in areas that the user has little, or no experience, they can specify some search conditions, which are listed in the index classification hierarchy, to search for and retrieve similar cases from the past case database for reference. Once the process description is completed and stored as a case in the database. The data can be translated automatically to generate a single unit, and/or multi-unit, process model report. This can then be used as an input file for gPROMS [1 ], a dynamic simulation package. The HDA process has been used to demonstrate the functionality of the interface. Future work will look at the addition of other software translators. Also, more cases need to be entered and saved in the modelling database to make it useful. Plus, the case retrieval and adaptation need to be improved to make the search more intelligent. REFERENCES
1. gPROMS (1998), 'gPROMS Introductory Users Guide'. Process Systems Enterprise Ltd, London, UK. 2. Douglas, J. (1988), Conceptual Design of Chemical Processes, McGraw-Hill, New York. 3. Westerberg, A.& n-dim Group (1997), 'Designing the process design process'. Comput. Chem. Engng., 21, pp. S 1-$9. 4. Batres, R. and Y. Naka (1999), 'Process Plant Ontologies Based on a Multi-dimensional Framework'. 5th FOCAP-D, Breckenridge, Colorado, pp.$434-440. 5. Bayer, B., R. Schneider and W. Marquardt (2000), 'Integration of Data Models for Process Design-First Steps and Experiences', Comput. Chem. Engng., 24, pp. $599-$605. 6. Clark. G.A., D. Rossiter and P.W.H.Chung (2000), 'Intelligent Modelling Interface for Dynamic Simulators'. ChERD, vol. 78, No. A6, pp. 823-839. 7. Lu M.L., A. Yang, H. Li and T. Wada (2000), 'Application Driven Approach for the Development of a Data Model Standard for Process Plant Operation' Comput. Chem. Engng., 24, pp. $463-$469. 8. ISO 10303 (1997), ' Process engineering data: Process design and process specifications of major equipment'. ISO TC 184/SC4/WG3 N740. 9. AspenTech (1998), ' Introduction to Aspen Plus: Course Notes'. Aspen Technology, Inc. 10. Jennings R., M. Harris and S. Lesh (1998), Special Edition, Using Microsoft Access TM 97. Que Corporation. 11. Chung, P.W.H. and M. Jefferson (1998), A fuzzy approach to accessing accident databases, Applied Intelligence, Vol. 9, pp129-137. 12. Clark. G.A. (2000), Intelligent Modelling Interface for Dynamic Simulators, PhD Thesis, in preparation, Department of Chemical Engineering, Loughborough University, Loughborough, LE11 3TU, UK.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
An open thermodynamics environments
server for integrated
1121
process
engineering
A.-D. Yang, L. von Wedel, W. Marquardt Lehrstuhl fOr ProzefStechnik, RWTH, Aachen, Germany {yang,vonwedel,marquardt} @ lfpt.rwth-aachen.de In order to provide CAPE applications unified and improved thermodynamics services, a component-based open thermodynamics server (OpenThermo) has been developed. OpenThermo is able to carry out open and closed-form physical property calculations and to supply and process symbolic thermodynamic model equations. These services are provided based on the integration of individual thermodynamics resources such as physical property databases and physical property packages. Such functionality is achieved with a layered software architecture following a resource/service based classification of the software components that comprise the thermodynamics sever. So far the major functionality of the service components has been implemented, and several resource components have been connected to OpenThermo. 1. INTRODUCTION Computer aided process engineering (CAPE) practice has shown that thermodynamics software tools play a critical role in the process engineering lifecycle including process modeling, simulation, optimization and process design (Adams, 1986; O'Connell and Neurock, 1999). Recent initiatives such as the ICAS (Gani et al., 1997) and the IMPROVE (Nagl and Marquardt, 1997) projects have revealed an emerging trend of developing integrated and open CAPE environments to best deal with all potential challenges faced by process engineers. Aided by such environments, engineers are encouraged to simultaneously work with different applications to perform multiple tasks such as modeling, simulation and optimization during the process design lifecycle (Marquardt, von Wedel and Bayer, 1999). To work in this way, engineers have to use a set of thermodynamics tools to resolve different problems. Unfortunately, most existing thermodynamics tools, including physical property calculation packages and databanks, are not able to appropriately provide support in an integrated environment. The obstacles mainly lie in the following two points: (1) The diversity of the capabilities and interfaces of different tools often leads to repeated work which is not only time consuming but also prone to errors. For example, to use an existing physical property package, usually the users have to spend time on preparing the input file in a specific format, mapping the property values with different units, and getting familiar with specific programming interfaces. When different tools are used for the same material system but support different tasks such as modeling and closed/openform calculations, the same set of information regarding the material system (including e.g. the list of chemical component, selected thermodynamic models, values of model parameters, etc.) has to be repeatedly processed in order to meet the requirements of individual tools.
1122
(2) Still due to the diversity of the interfaces of different tools, it is difficult to have a single implementation of software modules that remedy the weaknesses (e.g. with respect to robustness and efficiency) commonly suffered by those tools. To resolve these problems, rather than making improvements on individual tools, an appropriate integration among individual tools becomes necessary. In this direction, the efforts undertaken in CAPE-OPEN have made significant progress (Braunschweig et aL, 2000). The CAPE-OPEN project, with the component-based software development methodology as its fundamental rationale, has been engaging in defining the standardized interfaces of the software components involved in process simulations, including those for physical property calculations. However, the standard itself only provides the specification of interfaces in order to guide the development of individual tools; it is not intended to design system architectures or to supply software facilities that help to organize the components to be integrated. Based on the above observation, a component-based open thermodynamics server (OpenThermo) has been developed. Following the same technical foundation with CAPEOPEN, OpenThermo is designed to have the following features: (1) it is able to supply and process symbolic thermodynamic model equations as well as to support closed- and openform physical property calculations; (2) in a multiple-task work process, information about the same material system only needs to be specified once and will, thereafter, be shared by different tasks such as modeling and simulation; (3) each type of service, exposed to applications with a unique set of interfaces, is potentially supported by a dynamic collection of existing tools integrated into and managed by OpenThermo; and (4) a set of software modules is available in order to improve the robustness and the efficiency of physical property calculations performed by existing packages that are integrated into OpenThermo. In the rest of this paper, the system architecture, the individual components as well as the implementation of OpenThermo will be introduced sequentially. 2. A COMPONENT-BASED LAYERED SYSTEM ARCHITECTURE The system architecture of OpenThermo is shown in Figure 1. This architecture is designed according to a resource~service based software components classification. In general, the physical property related requirements posed by integrated CAPE environments can be met by components in these two different categories. The first category contains components corresponding to existing individual thermodynamics software tools such as physical property databases and physical property packages, which are defined as resource components in this work. These components are supplied to OpenThermo forming the basis of its capabilities. The components in the second category, called service components, are responsible to directly interact with the users in order to provide them unified and consistent numerical and symbolic services. Generally, resource components are the targets to be organized and/or employed by certain service components. A detailed explanation to each component in the two categories shall be given in Chapter 3. The above classification of components directly leads to the resource layer and the service layer in OpenThermo. The resource layer collects resource components available to OpenThermo and supports the service layer where the service components reside. This way, the service layer is responsible to fulfill various requests posed by users which are not intended to be satisfied by the functionality of certain individual resources but by the functionality of the entire OpenThermo system. In contrast, the resource layer carries out the connection with existing thermodynamics tools. Besides, a presentation layer is designed to
1123
provide graphical user interface elements for presenting the functionalities of service components to human users. For each component in the resource layer and the service layer, a set of interfaces is defined to be accessed by other process engineering applications as well as the graphical interfaces and other components in OpenThermo. The component-based approach enables the arbitrary interchange of alternative implementations of a specific resource or service component (or a part of it) whenever the defined interfaces are supported. This is often referred to as the "plug-and-play" paradigm. Presentation layer
Physical property data service
Symbolic model service
Physical property package-1
Thermodynamic configuration service
Calculation service
Symbolic model store-1
Service layer
Registration service
Physical property package-2
Physical property database-2
Physical property database-1
Resource layer
Figure 1. A three-layer system architecture for OpenThermo 3.THERMODYNAMICS SOFTWARE COMPONENTS 3.1. Resource components
Three kinds of components have been considered as resources in OpenThermo, namely
physical property packages, physical property databases and symbolic model stores. A physical property package is a software component that is capable of performing the calculations of physical properties and/or those of thermodynamic equilibrium. A typical example of a physical property package is IK-CAPE (Fieg et al., 1995). A physical property database stands for a store of a collection of physical property data possibly consisting of experimental data, constants of chemical components, and parameters in thermodynamic models. A physical property database could be a database available for the public such as DIPPR| (Thomson, 1996), or any physical property databases or data files internally used within certain organizations. A symbolic model store contains declarative thermodynamic model equations and supports the operations on equations such as retrieving, adding and modifying equations and related information. An example of this component is ROME, a research prototype of a model repository supporting the life cycle process design activities (von Wedel and Marquardt, 2000). 3.2. Service components
(1) Thermodynamic configuration service. When working with OpenThermo, a user always starts a task with generating a so-called "thermodynamic configuration" - a collection of information specifying the chemical components involved, the selected calculation methods and the required data/parameters. With the configuration service component, a number of thermodynamic configurations can be generated, modified and retrieved. In a
1124 single configuration, more than one physical property database or package might be taken as the resources of certain property data or method implementations. This service component also helps to retrieve predefined configuration information kept by the physical property packages that have been integrated into OpenThermo. (2) Calculation service. This component is responsible for the major functionality of OpenThermo - the calculation of individual thermophysical properties and equilibrium in both closed and open form. A calculation request is fulfilled according to a pre-defined thermodynamic configuration. Besides dispatching requests to specific physical property packages, this service component also contains some common software components which extend the functionality of existing physical property packages to overcome the weaknesses with respect to robustness and efficiency (Yang et al., 2000). (3) Physical property data service. This component provides services for accessing and processing property data to applications (such as the configuration service) and human users in a unified manner. It enables the access to all physical property data resources in OpenThermo in order to get certain data that might be restricted by substance, property, range of physical conditions, and/or quality. In addition, it provides data regression tools that can be applied to the experimental data stored in physical property data resources. (4) Symbolic thermodynamic model service. Occasionally, an application may intend to put thermodynamic model equations together with other parts of process models and then get the entire model solved. In some other cases, users may want to enter their own model equations into OpenThermo for calculating physical properties of certain material systems. Such requirements are supported by this component. (5) Registration service. This component acts as the "central information repository" in OpenThermo. On the one hand, it helps to register any resource component to be integrated into OpenThermo, which is accomplished by retrieving and then recording the specification of the functionality of the resource component. On the other hand, the users of OpenThermo can retrieve the specification of the functionality of the entire thermodynamics server by accessing this component. 4. AN EXAMPLE OF COMPONENT DESIGN In this chapter, the calculation service component is taken as an example to show how interfaces are designed on the component level. The calculation service component, in cooperation with the thermodynamic configuration service component, is responsible to carry out open- and closed-form thermodynamic calculations. To make such functionality available to the clients, interfaces shown in Figure 2 have been designed. MultiplePhaseSystem SinglePhaseSystem ~ ~ ~PhaseSystemFactory calculation service CalculationTask O----- component
------0 ESO
Figure 2. The Calculation Service Component For calculating the physical properties, a user uses PhaseSystemFactory to generate either a SinglePhaseSystem for calculating properties of single phases such as enthalpy, fugacity, density, or MultiplePhaseSystem for performing flash calculations as well as evaluating properties of a system comprising multiple phases such as K-value. Thereafter, a calculation
1125
request described by the specification of the material, the physical conditions (e.g. temperature, pressure and compositions), and the properties to be calculated can be submitted through the phase system object just created. To take advantage of the services for improving robustness and efficiency, a user can specify additional requirements through CalculationTask. Examples include if a meaningful property value should be returned even when the conditions are out of the valid range, and if local models should be used. To support open-form calculations, a CAPE-OPEN equation set object (ESO) interface (CAPE-OPEN, 1999) is provided. In order to get residuals and Jacobian matrices regarding the calculation of certain properties of a specific instance of SinglePhaseSystem or MultiplePhaseSystem, the instance of SinglePhaseSystem or MultiplePhaseSystem should be available at first; then, for this phase system an instance of ESO should be generated. Thereafter, the ESO object can be manipulated by the user for obtaining all the open-form information with respect to the targeted phase system and the properties of interest. 5. IMPLEMENTATION The OpenThermo prototype system has been implemented with C++. The CORBA platform used in this work is omniORB (Lo, 1999). Because of the component-based architecture, the resource components and services components are allowed to be incrementally added to the system. So far, in the service layer, the registration service, the thermodynamic configuration service, and the physical property data service components have been implemented. For the calculation service component, a set of computing facilities for improving the robustness and efficiency of existing physical property package has been developed. An open-form calculation module is under construction in order to support flexible simulation environments such as CHEOPS (von Wedel and Marquardt, 1999). Besides, the exploration of implementing the symbolic model service component with the repository and the functionality of ROME (yon Wedel and Marquadt, 2000) is also on the way. In the resource layer, the physical property package IK-CAPE (Fieg et al., 1995) and a flash module (Bausa and Marquardt, 2000) have been wrapped into two physical property package resource components. Two physical property database components, one based on an internally used pure component data source and one based on a mixture data source, have also been linked to OpenThermo. In the presentation layer, a web-based user interface has been implemented, through which users can perform tasks such as specifying thermodynamic configurations, submitting calculation requests and viewing calculation results for example. 6. CONCLUSIONS AND FUTURE WORK As a component-based open thermodynamics server supporting integrated CAPE environments, OpenThermo is capable of integrating a variety of thermodynamics resources and then providing unified and improved symbolic and numerical thermodynamics services to CAPE applications. This has been achieved by a layered system architecture according to the resource~service based classification of software components, which has proven to be a useful pattern for the component-based legacy system integration. As the focus of the development, the service components in OpenThermo will be further developed in the future. Also, the performance of the service components will be tested and improved by incorporating additional resource components and by supporting various CAPE applications. Besides, for taking advantage of the CAPE-OPEN standard, a wrapper of the calculation service and the thermodynamic configuration service components will be developed in order to enable OpenThermo to provide services through the interfaces defined in CAPE-OPEN.
1126 ACKNOWLEDGEMENTS
This work was partially funded by ABB Automation. The cooperation with members in the Global Cape-Open project, the access to the physical property data source of A. Pfennig and the support of S. Brtiggemann, J. Bausa, and E. Krumrtick are gratefully appreciated. REFERENCES
Adams, J.T. (1986). A review of thermophysical properties in process simulation. Fluid Phase Equilibria, 29, 23-45 Bausa, J. and Marquardt, W. (2000). Shortcut design methods for hybrid membrane / distillation processes for the separation of nonideal multicomponent mixtures. Industrial Engineering Chemistry Research, 39 (6): 1658-1672 Braunschweig, B., Pantelides, C.C., Britt H. I., Sama, S. (2000). Open software architectures for process modeling: Current status and future perspectives. In: M.F. Malone, J.A. Trainham, B. Carnahan (Eds.): "Foundations of Computer-Aided Process Design", AIChE Symp. Set. 323, Vol. 96, 2000, 192-214 CAPE-OPEN (1999). Open interface specification: numerical servers. Available on-line at http://www.global-cape-open.org/ Fieg, G., et al. (1995). A standard interface for use of thermodynamics in process simulation. Comp. Chem. Engng 19, $317-$320 Gani, R., Hytoft, G., Jaksland, C. and Jensen, K.(1997). An integrated computer aided system for integrated design of chemical processes. Comp. Chem. Engng 21, 1135-1146 Lo, S.-L. (1999). The omniORB version 3.0 User's Guide. Available on-line at http://www.uk.research.att.corn/ Marquardt, W., von Wedel, L., Bayer, B. (2000). Perspectives on lifecycle process modeling. In: M.F. Malone, J.A. Trainham, B. Carnahan (Eds.): "Foundations of Computer-Aided Process Design", AIChE Symp. Ser. 323, Vol. 96, 2000, 192-214 Nagl, M., Marquardt, W. (1997). SFB-476: Computer-aided support for overlapping design processes in chemical engineering. In: M. Jarke, K. Pasedach, K. Pohl (Eds.): Informatik '97 - Informatik als Innovationsmotor. Springer-Verlag, Berlin, 1997, 143-154, in German O'Connell, J.P., Neurock, M. (2000). Trends in property estimation for process and product design. In: M.F. Malone, J.A. Trainham, B. Carnahan (Eds.): "Foundations of ComputerAided Process Design", AIChE Syrup. Ser. 323, Vol. 96, 2000, 5-23 OMG (1999). The common object request broker architecture: architecture and specification. The Object Management Group Thomson G.H. (1996). The DIPPR| databases. International Journal of Thermophysics. 17 (1): 223-232 von Wedel, L. and Marquardt, W. (2000). CHEOPS: A component-based approach to process simulation. In: M.F. Malone, J.A. Trainham, B. Carnahan (Eds.): "Foundations of Computer-Aided Process Design", AIChE Symp. Ser. 323, Vol. 96, 2000, 494-498 von Wedel L. and Marquardt, W. (2000). ROME: A repository to support the integration of models over the lifecycle of model-based engineering processes. In: S. Pierucci (Ed.): "European Symposium on Computer Aided Process Engineering-10", Elsevier, 2000, 535541 Yang, A.-D., von Wedel, L., Marquardt, W. (2000). Developing advanced computing facilities to improve the robustness and efficiency in physical property calculations. Submitted to 3r" European Congress of Chemical Engineering (ECCE'3)
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1127
Optimisation of global pharmaceutical syntheses integrating environmental aspects S. Elgue a, M. Cabassud a, L. Prat a, J.M. Le Lann a, G. Casamattaa; J. C6zerac b aLaboratoire de G6nie Chimique, UMR CNRS, INPT-ENSIGC, 18 chemin de la Loge, 31078 Toulouse Cedex 04, France bSanofi-Synthelabo, 45 Chemin de M6t61ine, B.P. 15, 04201 Sisteron, France Pharmaceutical synthesis optimisation, due to its complexity, is often reduced to the optimisation of the reaction step. In this paper, a dynamic model, allowing to simulate the different synthesis steps and particularly start-up and shut-down phases involved, is presented. This model connected to an optimisation method is able to provide the optimal operating conditions satisfying a global objective. The application to an industrial process highlights the benefits of the proposed methodology. 1. INTRODUCTION The syntheses of fine chemicals or pharmaceuticals, widely carried out in batch processes, imply many successive reaction steps. Concentration and purification operations generally follow these reaction steps and are traditionally not considered for the determination of optimal operating conditions. In fact, global synthesis optimisation has often been restricted to the operating parameters tuning of the reaction phase, in order to optimise yield or selectivity [1-3]. Such short cuts can lead, for the global synthesis optimisation, to erroneous conclusions. For example, optimising the conversion of a synthesis for which separation between the desired product and the other reaction products is more difficult than between reactants, will involve an important global operating cost, due to further difficulties in the separation scheme. The necessity to integrate all the process steps in a synthesis optimisation is so clearly demonstrated. Recent issues in the dynamic simulation, associated with efficient optimisation method may be exploited to accomplish this goal [4]. The main objectives of this research were to determine the operating conditions involving the optimisation of the global synthesis, satisfying economical and environmental criteria, in order to develop sustainable methodologies. For this purpose, a framework based on a simulation program, modelling a global synthesis (reaction and separation steps), has been developed. Besides global synthesis treatment, the main features of this work lie in the modelling of batch units and of dynamics, particularly during the start-up and the shut-down involved in the different steps.
2. SIMULATION ENVIRONMENT Because of the step by step structure of global syntheses, the simulation tool developed is based on a hybrid model. The continuous part represents the behaviour of batch equipments, the discontinuous part, the train of the different steps occurring during the entire synthesis of a
1128 product. The modular description adopted for each batch equipment (batch reactor, batch distillation column, overhead condenser), represented by a detailed and complete dynamic model, allows to represent a wide variety of fine chemistry processes : fine chemical reactor, evaporator reactor, reactor with condenser, reactor with overhead column, reactive batch distillation column, ... 2.1. M o d e l e q u a t i o n s
The rigorous description of batch equipment involves modelling aspects resulting on a differential algebraic equations (DAE) system. Differential equations consist of total and component material balances, total energy balances and total continuity equation in the case of steam vapour heating. Algebraic equations are composed of vapour-liquid equilibrium relationships, summation equations, physical property estimations. In order to simplify the model establishment, the following assumptions are applied : - w e l l - m i x i n g on each plate, - v a p o u r - l i q u i d equilibrium with possible introduction of efficiency, - n e g l i g i b l e vapour holdup. Chemical reactions in the vapour phase are not considered, - constant volume of liquid holdup (except for the reactor), - total condenser. Hence, for a generic plate k, just as well representing either the reactor, either the condenser or either a column tray [5], the following equations can be written : dblk = Fk + (l_SVk+l)Vk+l + ( l _ SLk_l)Lk_l _ V k _ L k +Ank/j k + R k L 1 dt
d(Hkx'k) = Fkx f '' + ( 1 - SVk+, )Vk+ly'k+1 + ( 1 - SLk_l )Lk_lX'k_ 1 - Vk y'k - Lkx'k + r/,ttk + ekLlx~ dt
d(b/khk) = F k h [ + (1-- SVk+l)Vk+lHk+l + ( 1 - SLk_l)Lk_lhk_l - VkHk - Lkhk +qkHk + e k L l h l + Qk dt i
i
i
t
Yk+l(1 -- e k) + e k K k x k -- Yk = 0 For the condenser, plate 1 : Q1 = Uc,cjac,cl (Tot - T~ ) - (1 - S V 2 ) A H 1 ,
eI = 1 ,
RI=0
For the reactor, plate n : , e,=l Previous equations constitute the basic mathematical model framework. Summation equations, equations of the thermodynamic models (molar enthalpy, molar holdup, VLE constant), side streams equations and thermal equations complete this mathematical model. Thermal modelling is particularly detailed, in order to allow temperature regulation, notably during constant temperature policy. As an illustration, reactor jacket specific equations, in the case of steam vapour heating and in the case of heat transfer fluid circulation, are respectively given by : Qn=Ur,rwar,rw(Trw-Tn)
d~
,~
v,-y-
-z
-
u ~ , , a ~ , , ( L - T~) + U~,~Ia~,j~(T, - T,I) AHs
1129
dEj dt
= U,.~ja,.~j(T,~ -L)+Uj,jwaj,jw(Tjw -Tj)+ FjCpjp (Tj -Tj) '
"
J
Correlations [6] are used to determined heat transfer coefficients. A complete physical property estimation system with associated data bank is also used [7]. The numerical solution of the developed mathematical model is obtained by a DAE system general solver, based on the Gear method [8] : DISCo [9]. Besides its accuracy and numerical robustness, DISCo allows us, from its events detection facilities, to develop a complex procedure of the dynamic process management. 2.2. Process dynamic
management
The global synthesis treatment requires to simulate the train of different steps and particularly to account for the start-up and shut-down of these steps. Thus, one process step is divided into many modelling sub-steps. As an illustrative example, the batch distillation startup splitting up may be the following one: - r e b o i l e r initial charge heating until bubble point, -column plate heating until bubble point and/or column plate filling, - condenser filling, -total reflux phase and production beginning. A similar procedure is used for the shut-down/cooling-down of the column. Rigorous modelling requires a specific model for each of these steps and so transitions from one to another may lead to discontinuities in the mathematical model structure. The management procedure of this dynamic consists of the following scheme : -detection of the associated event (i.e. for evaporation : temperature reaching the bubble point temperature), -resulting changes in the mathematical model, corresponding to the occurring step, - accurate initialisation of the new mathematical model, -integration of the new mathematical model. The relative heaviness of this procedure is however compensated, in CPU time terms, by the DISCo integration velocity and accuracy, due to the use of operator sparse and due to its automatic initialisation procedure for the new DAE system. 3. SIMULATION VALIDATION Simulation validation has been studied for an industrial pharmaceutical synthesis. During this synthesis, for selectivity and solubility reasons, the solvent has to be changed between two consecutive reaction steps. Because of the impossibility for the product to be dried, the changes of solvent are performed by successive evaporations and new solvent loading successive steps (Fig. 1). During the solvent substitution, evaporation steps end at the admissible minimum volume, defined by agitation or solubility considerations. Substitution ends when the solvent A concentration reaches the purity specification.
1130
Fig. 1. Solvent substitution procedure The solvent substitution considered is a methanol-pyridine changing, in a 2.5 m 3 batch reactor. The initial charge is composed by methanol (744 1) and by reaction products (128 1), which allowed a 1240 1 admissible m i n i m u m volume. The purity specification is a 0.1 percent mass fraction of methanol, in the distillate. It is important to note that because o f the possible products degradation, a 50~ constraint has been defined for the reactor temperature. In table 1, simulation results are compared to the collected industrial process data, in order to verify the model accuracy. Table 1 Solvent removal 9Methanol by Pyridine Simulated process Industrial process Time Temperature (~ Time Temperature (~ Steps l h 15mn Pyridine loading :1218 kg 1 h 20 mn 7 h 39 mn 34.3 Evaporation 7 h 40 mn 34.5 30 mn Pyridine loading : 490 kg 30 mn 2 h 40 mn 32.3 Evaporation 2 h 40 mn 32.0 30 mn Pyridine loading : 490 kg 30 mn 2 h 27 mn 35.4 Evaporation 2 h 25 mn 36.5 30 mn Pyridine loading : 490 kg 30 mn 2 h 19 mn 36.0 Evaporation 2 h 20 mn 40.4 30 mn Pyridine loading : 490 kg 30 mn 2 h 31 mn 37.4 Evaporation 2 h 30 mn 37.9 Operation time 3h 15mn Pyridine loading 3 h 20 mn 17h35 mn Evaporation 17 h 35 mn 24 h 04 mn Total 23 h 55 mn Pyridine = 115.2~ Boiling point (1 atm) : Methanol = 64.8~ 4. O P T I M I S A T I O N In order to consider the whole of synthesis with its different steps, the optimisation problem is defined with an operation global cost as objective function. In fact, only a cost notion allows to gather, on a same criterion, steps with different parameters. Specificities of the different process steps, for example production rate or product purity for reaction step, then are considered as global problem constraints. Control variables of the global p r o b l e m are constituted of the specific control variables of each step : reactor temperature or reactants addition rate for reaction step, reflux ratio for distillation step, ... According to whether control variables are function of time or not, control vector parametrisation (CVP) [10] is
1131 used to transform the dynamic optimisation problem into a non linear programming problem (NLP). A successive quadratic programming (SQP) method [11 ] is then applied to solve the resultant NLP. Methanol-Pyridine substitution has been optimised, with solvent recovery as main purpose. Thus, the objective function, representing the operation global cost, is defined as follow : C : toperanon Cmanpower + mpyridm e Cpyridm e Jr mwast e Cdestructton
Two substitution processes have been studied. On one hand, the current experimental process with number and quantity of pyridine loading as control variables. On the other hand, a batch distillation process, with initial charge and constant reflux ratio as control variables. In table 2, optimisation results are compared to experimental ones. Table 2 Optimal processes comparison Loading / Evaporation process Loading Experimental loading mass (kg) number M 1 M2 M3 M4 M5 M6 5 1218 490 490 490 490
Total 3178
Operation time 24 h 04 mn
Operation cost (FF) 118 343
Loading number 4 5 6
Total 3022 2923 2867
Operation time 21 h 58 mn 21 h 37 mn 21 h 54 mn
Optimal cost (FF) 111 113 108 040 107 023
Operation time 17 h 36 mn
Optimal cost (FF) 61 980
M 1 1135 984 895
M2 632 482 392
Optimal loading mass (kg) M3 M4 M5 629 626 484 487 486 394 395 396
Batch distillation process Loading Optimal loading mass (kg) number M 1 1379
M6
395
Optimal reflux rate R 0.624
For methanol-pyridine substitution, a batch distillation process, with optimal operating conditions, provide a 50 % reduction of the global cost. Experimentally, several solvent substitutions, with different vapour-liquid equilibrium configurations, are performed on the same unit. Such flexibility implies that only a loading/evaporation process can be considered. In this case, profits on the global cost are only 10 %. However that may be, optimisation provide a 10 to 60 % solvent consumption, and so waste generation, reduction. 5. C O N C L U S I O N A reliable framework, for optimisation of all or part of a batch synthesis, has been developed. Benefits link to this framework and its accuracy have been clearly demonstrated in an industrial separation example. Its use in the scope of global syntheses with complex reaction scheme and the resultant advantages actually constitute our main research theme. NOTATION a heat transfer area coefficient c mass or time cost coefficient
H
vapour molar enthalpy
SL liquid
side
stream
K
VLE constant
SV vapour
side
stream
1132 C Cp Cv e E F h
global cost specific heat valve coefficient Murphree efficiency internal energy feed flow rate liquid molar enthalpy
L m P q Q r R
liquid flow rate mass pressure reaction heat rate of heat transfer reaction rate reflux ratio
v p
volume density
t time T temperature U heat transfer coefficient U molar holdup V vapour flow rate x liquid mole fraction y vapour mole fraction
Greek letters AH condensation heat An rate of change in moles
Subscripts c condenser cf condenser cooling fluid j jacket
jw jacket wall k plate index n total number of plates
r reactor rwreactor wall s steam vapour
Superscripts i
component index
f
feed
REFERENCES
1. D. W. T. Rippin, Simulation of single an multiproduct batch chemical plants for optimal design and operation, Comput. Chem. Eng. Vol.7, No. 3, pp. 137-156, 1983 2. S. Marchal-brassely, J. Villermaux, J-L Houzelot, J-L Bamay, Optimal operation of semibatch reactor by self-adaptive models for temperature and feed-rate profile, Chem. Eng. Sci., Vol. 47, No 9-11, pp. 2445-2450, 1992 3. C. Toulouse, Op6rations optimales dans les r6acteurs discontinus. Contraintes de sfiret6 de fonctionnement, Entropie, Vol. 210, pp 29-34, 1998 4. R. M. Wajge, G.V. Relaitis, RBDOPT : a general-purpose object-oriented module for distributed campaign optimization of reactive batch distillation, Chem. Eng. Journal, 75, pp. 57-68, 1999 5. J. Albet, Simulation rigoureuse de colonnes de distillation discontinue ~t s6quences op6ratoires multiples, Ph.D. Thesis, INP, Toulouse, 1992 6. R. F. Dream, Heat transfer in agitated jacketed vessels, Chem. Eng., January, pp. 90-96, 1999 7. Prophy, Manuel utilisateur, Prosim S.A. 8. C.W. Gear, The simultaneous numerical solution of differential-algebraic equations, IEEE Transactions on Circuit Theory, CT-18 (1), pp 89-95, 1971 9. A. Sargousse, J.M. Le Lann, X. Joulia, L. Jourda, DISCo : un nouvel environnement de simulation orient6 objet, MOSIM'99, Annecy- France, 1999 10. J.S. Logsdon, L.T. Biegler, Accurate solution of differential-algebraic optimization problems, Ind. Eng. Chem. Res., 28, pp 1628-1639, 1989 l l.K. Schittkowski, NLPQL : a fortran subroutine solving constrained nonlinear programming problems, Annals of operations research, 5, pp 485-500, 1986
European Symposiumon ComputerAidedProcessEngineering- 11 R. Gani and S.B. Jorgensen(Editors) 9 2001 ElsevierScienceB.V. All rightsreserved.
1133
Genome Wide Functional Annotation Using Mathematical Programming Sanjeev Garg and Luke E. K. Achenie Department of Chemical Engineering, University of Connecticut, Storrs, CT 06269, USA Email: [email protected] and [email protected] Recent advances in cDNA microarray technology have resulted in a surge of gene expression data. Whole genome hybridization results have been reported for many organisms (yeast, worm, etc.) under different experimental conditions of interest. Studies have been reported on various methods for assigning functionality to previously unknown genes. Some of these methods are clustering techniques, self-organizing maps and knowledge based support vector machines (SVMs). These techniques use a similarity measure to associate genes with unknown functionality with genes of known functionality. In this paper we report genome wide functional annotation, for yeast data set, based on an SVM strategy. In our SVM model each gene can potentially be assigned to more than one class. A gene is referred to as a positive sample of a given class if it belongs to that class, otherwise the gene is referred to as a negative sample. We establish the viability of the SVM model using yeast data with several missing experimental measurements. The SVM model correctly annotated the positive samples in all classes. On the other hand the negative genes were not so well annotated. This is attributed to the missing data for the negative gene set. 1. I N T R O D U C T I O N eDNA microarray technology makes it possible to study the gene expression patterns of thousand of genes in parallel [1 ]. This has potential applications in gene discovery, disease diagnosis, drug discovery, toxicogenomics, to name a few. The output of a cDNA microarray experiment is the gene expression data represented by an nxm matrix, where n is the number of genes under study and m is the number of experimental conditions. Each value in this matrix represents the ratio of two experimental expression values, one each for cy3 and cy5, for a unique gene. This ratio is a measure of the expression of gene under the present experimental condition compared to some reference condition. Many experimental studies have been reported in the open literature on genome wide cDNA microarray experiments [2][3][4]. The challenge now is to develop new data mining strategies to analyze and extract knowledge out of these massive data sets. Studies have been reported on various methods, namely, clustering techniques, self-organizing maps, principal component analysis, knowledge based support vector machines and more recently 'gene-shaving' [1][5][6][7][8]. Generally all these techniques use a similarity measure to group together unknown functional genes with the known functional genes. In this paper we use an SVM for genome wide functional annotation of genes based on cDNA microarray gene expression data. SVMs are machine-learning algorithms
1134
based on statistical learning theory [9]. SVMs avoid over-fitting the gene expression data by choosing the maximum margin separating hyper-plane that can separate the subclasses for functional annotation. An SVM can be posed as a mathematical program via a primal-dual decomposition strategy. In many real world problems the genes are inseparable in the original feature space. At the heart of the SVM is the kernel function that is needed to map gene data to higher dimensional feature space. This results in a separation of the genes. We note that linear as well as non-linear kernel functions can be used. In this paper we use polynomial kernel functions. The SVM uses a training set to learn which genes group together and generate the corresponding support vectors that can then be used to classify genes of unknown functionality. 2. F O R M U L A T I O N The problem of annotating the functional class is posed as a binary classification. In the latter a gene is either assigned to a given class or not. This is posed as two convex quadratic programs (QP) with linear constraints. The solution of the QPs result in a classification function for that class. The first QP (primal problem) is a multi-objective mathematical program that minimizes the misclassification errors and maximizes the separation margin. The maximization of the separation margin is achieved by minimizing the norm of the normal vector to the hyperplane [ 10]. Note that a quadratic program has to be solved for each class. The primal problem is formulated as:
s.t.
A~co-ye+y-e>O
(1)
-AZco+ ye+ z - e >_O y>0 z>0 where 0 < 2 < 1 is a constant, and is a measure of the tradeoff between the two objectives. A ~ and A 2 are the positive and negative gene sets for a given functional class in the training set with m l and m2 as their cardinalities (number of genes in the set). The components of 6 are the cardinalities of A 1 and A 2 i.e., 61 = m/ and 62 -- m2. Each component of the vector e is one. Also y and z are the misclassification errors. The resulting separating hyperplane is xco = y , where co is the normal to the hyperplane and 7" is the distance from the origin to the hyperplane. The primal problem (first QP) is solved to determine 7'. The dual problem (second QP) is needed to incorporate the kernel function. The dual problem for (1) is as follows: min
Alru-AZrv iI2 - e
Tu - e T v
u,v
s.t.
eVu-erv=O O - ~ ) 6 , e >_u >_O
(2)
1135 0 - "~')~2e -> V >_ 0 It is convenient to write the nonlinear classification case in summation notation [ 10]. Let A be the total set of all the genes, A 1 and A 2 and let M - ml + m2 be the total number of genes. x, ~ A
,~ti =
dimensional
1
x~ e
feature
. Let t ~
be defined such that for
A 2 . The original data points x are mapped to the higher space
by
the
selected
transformation
function
#(x)" R" --> R"',n' >> n. The dot product of the original vectors xTxj is replaced by the dot product of the transformed vectors ~(x~).~(xj ). Thus the dual can be rewritten as:
min~ - , ~ 1 ' ~ ~ t i t j a i a j (qk(x).qk(x)) a / "-'I, i=l j=l
~ot,t, A4
s.t.
-
-
,=1
ai
(3)
=0
i=1
where 6 is equal to 61 and 62 for the terms corresponding to u and v, respectively. Vapnik's work on SVMs [11] allows one to replace the inner product q~(x,).O(xj ) with the inner product in the Hilbert space by the symmetric kernel function K(x, xi). The classification function is therefore dependent on the kernel function chosen. Example kernel functions are polynomial, radial basis functions and neural networks. It is interesting to note that the number of optimization variables in the dual problem remains constant irrespective of the kemel function chosen. After solving the primal (1) and dual (3) problems the resulting classification function is:
x) I where x e A 1 if f ( x ) > O, else x ~ A 2 .
1136 3. RESULTS AND DISCUSSION
The SVM model discussed above is used for annotating the budding yeast
Saccharomyces cerevisiae data set reported as a set of 2467 genes under 79 different experimental conditions. This data set was used by Eisen et. al. [1 ], for clustering and later by Brown et. al. [7]. The data was generated from microarrays using samples collected at various time points of the diauxic shift, the mitotic cell division cycle, sporulation and temperature and reducing shocks and is available on the web at http://www.rana.stanford.edu/clustering. We use the same functional class definitions made by MYGD and reported in Brown et. al. [7]. We use the SVM model to classify genes into six functional classes, namely tricarboxilic acid (TCA) cycle, respiration (RESP), cytoplasmic ribosomes (RIBO), proteasome (PROT), histones (HIST) and helixturn-helix (HTH) proteins. The first five classes have a biological basis since they exhibit similar expression profiles. HTH class is used as a control group since there is no reason to believe that the genes of this class are similarly regulated. The data has many missing values for the expression ratios. Only a small fraction (25%, 605 out of 2467) of the genes has all the 79 expression ratios reported. This smaller set of genes was used as the training set (see Table 1). A gene is referred to as a positive sample of a given class if it belongs to that class. Otherwise the gene is referred to as a negative sample. We observe that the training set has 2, 13, 49, 8, 5 and 6 positive genes for the respective functional classes. We note further that for all the functional classes, the negative sample size is a small fraction of the total negative gene set (less than 25% in all cases). Thus, the negative gene set might not be a good representation of the genes in the original data and therefore ought not to be used for training. Table 1 Training Set Data Class
PS
NS
PT
TCA RESP RIBO PROT HIST HTH
2 13 49 8 5 6
603 592 556 597 600 599
17 30 121 35 11 16
NT 2450 2437 2346 2432 2456 2451
PS%
NS %
11.76 43.33 40.50 22.86 45.45 37.50
24.61 24.29 23.70 24.55 24.43 24.44
PS: Positive Set, NS: Negative Set, PT: Total Positive Set, TN: Total Negative Set For classification and functional annotation, we employ a polynomial kemel function given as:
1+
(5)
1137 where n = 79, is the number of experimental conditions and d is the degree of the polynomial. We report the results for d = 2. The resulting classification functions are then used to classify the complete set of 2467 genes. It is observed that the support vectors generated for classification are only a small fraction of the training data set. It is also observed that the SVM based classification functions performs very well on the positive samples (see Table 2). An accuracy of nearly 100% is found for the positive samples for the first five classes and a low 81% accuracy is observed for the HTH proteins. The expression pattern within the HTH class is random and explains the low accuracy of positive sample classification. For the negative samples an accuracy rate of 56%-91% is observed. The remaining negative genes (9%-44%) are incorrectly classified as positive genes. This can be attributed to the missing values in the expression ratios and a small fraction of the negatives considered in the training set. It is to be emphasized that although the values are also missing for the positive samples, the support vectors generated are able to classify them correctly as shown Table 2. Table 2 Comparison of error rates for different classes Class
TP
TN
FP
FN
TP%
TN%
FP%
FN%
TCA RESP RIBO PROT HIST HTH
17 30 121 32 11 13
1876 1553 1315 2207 1373 1556
574 884 1031 225 1083 895
0 0 0 3 0 3
100 100 100 91.4 100 81.2
76.57 63.73 56.05 90.75 55.90 63.48
23.43 36.27 43.95 09.25 44.10 36.52
0.0 0.0 0.0 8.6 0.0 18.8
TP: True Positive, TN: True Negative, FP: False Positive, FN: False Negative 4. C O N C L U S I O N S AND O N G O I N G W O R K To improve the existing classification and the functional annotation, missing values can artificially be generated (i.e. imputed) and added to the expression ratio matrix using a data imputation technique such as hot decking, mean or median imputation, and multiple imputation based on EM algorithm [12]. However, such a technique may introduce bias into the existing data. We expect hot decking to be the most appropriate imputation technique for gene expression data imputation. This is because in hot decking, it is assumed that similar genes have similar expression ratios (i.e. similarity by homology) and therefore we can replace the missing value for a gene at the given experiment with the expression ratio of a similar gene at that experimental condition. Work incorporating hot decking imputed data is in progress. We are also currently investigating the use of the radial basis function and the artificial neural network as viable kernel functions. The feasibility of the SVM model for genome wide functional annotation of genes has been established by the present study. The results are very encouraging for the initial
1138 study in the presence of a large number of missing gene expression data. We observed that the positive samples are correctly annotated. The low percentage of true functional annotation for negative genes is attributed to missing values for the negative gene sets. We also noticed that some of the genes were assigned to more than one functional class. We expect that this multi-functional gene annotation feature of the SVM model will eventually lead to a better understanding of the underlying biological complexities in the genetic network of an organism. To the best of our knowledge this is the first study reporting the primal-dual SVM modeling strategy for functional annotation of genes based on microarray data. The ultimate goal of our research is to use this primal-dual SVM model for functional annotation of genes responsible for bone cell differentiation. This is a collaboration with bone cell researchers at the University of Connecticut Health Center researchers. REFERENCES
1. Eisen, MB, Spellman, PT, Brown, PO, Botstein D. Cluster analysis and display of genome-wide expression patterns. (1998) Proc. Natl. Acad. Sci. USA, 95, 1486314868. 2. Goffeau A, Barrell BG, Bussey H, Davis RW, Dujon B, Feldmann H, Galibert F, Hoheisel JD, Jacq C, Johnston M, Louis EJ, Mewes HW, Murakami Y, Philippsen P, Tettelin H, Oliver SG. Life with 6000 genes. (1996) Science 274, 563-567. 3. Consortium TC e S. Genome sequence of the nematode C. elegans: a platform for investigating biology. (1998) Science 282, 2012-2018. 4. Clayton RA, White O, Fraser CM. Findings emerging from complete microbial genome sequences. (1998) Curr. Opin. Microbiol. 1, 562-566. 5. Alon U, Barkai N, Notterman DA, Gish K, Ybarra S, Mack D, Levine AJ. Broad patterns of gene expression revealed by clustering analysis of tumor and normal colon tissues probed by oligonucleotide arrays. (1999) Proc. Natl. Acad. Sci. USA, 96, 6745-6750. 6. Tamayo P, Slonim D, Mesirov J, Zhu Q, Kitareewan S, Dmitrovsky E, Lander ES Golub TR. Interpreting patterns of gene expression with self-organizing maps: Methods and application to hematopoietic differentiation. (1999) Proc. Natl. Acad. Sci. USA, 96, 2907-2912. 7. Brown MPS, Grundy WN, Lin D, Cristianini N, Sugnet CW, Furey TS, Ares M Jr., Haussler D. Knowledge-based analysis of microarray gene expression data by using support vector machines. (2000) Proc. Natl. Acad. Sci. USA, 97, 262-267. 8. Hastie T, Tibshirani R, Eisen MB, Alizadeh A, Levy R, Staudt L, Chan WC, Botstein D, Brown P. 'Gene shaving' as a method for identifying distinct sets of genes with similar expression patterns. (2000) Genome Biology, 1(20), 1-21. 9. Burges CJC. A tutorial on support vector machines for pattern recognition. (1998) Data Mining and Knowledge Discovery, 2, 121-167. 10. Bredensteiner EJ, Bennett KP. Multicategory Classification by Support Vector Machines. (1999) Computational Optimization and Applications, 12, 53-79. 11. Vapnik, V. The Nature of Statistical Learning Theory. Springer-Verlag, New York, 1995. 12. Schafer JL. Analysis of Incomplete Multivariate Data. Chapman & Hall, London, 1997.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1139
On the Optimization of Drug Delivery Devices Michael C. Georgiadis* and Margaritis Kostoglou Centre for Research and T e c h n o l o g y - Hellas, Chemical Process Engineering Research Institute, P.O. Box 361, Thermi 57001, Thessaloniki, Greece. This work presents a novel optimization approach to achieve desired release rates in drug delivery devices using laminated layers. First, a mathematical model is presented to describe the drug release between successive layers laminated together to form matrices with different initial concentrations, drug diffusivities and thickness of each layer. First, an analytical-based optimization approach is performed and results compared with relevant work from the literature. Then, a formal optimal control approach is employed to determine the optimal initial concentration in the layers, along with their thickness and diffusivities, in order to achieve a drug release profile as close to required profile (e.g. constants release) as possible for all times. 1. INTRODUCTION The target of any controlled release system is the transfer of an active material (usually drug) from a reservoir to a target host, in order to maintain a predetermined concentration or emission level of the drug for a specified period of time or desired fraction released. In a large number of medical applications constant release rates are desired for drugs possessing a narrow range of therapeutic index. Diffusion control matrix devices have been among the most widely used drug delivery systems, mainly due to their low manufacturing cost. However, in conventional diffusion controlled devices, where the drug to be released is distributed uniformly through a polymer, the release of a dissolved drug from a homogeneous geometry inherently follows first order diffusion behaviour with an initially high release rate followed by a rapidly declining release rate. In order to achieve zero-order releases and especially to eliminate the initially high release rate, various methods have been proposed such as modification of the geometry of the device and the use of rate-controlling methods [1 ], [2]. An altemative approach which has been extensively investigated experimentally is the use of nonuniform initial concentration profiles as a mechanism for regulating drug release from diffusion controlled and surface erosion controlled matrix systems [3]. Several mathematical models have been presented to describe diffusion controlled systems containing dispersed drug [4],[5]. Other approaches presented mathematical models of simultaneous dissolution and diffusion controlled drug release from swellable viscoelastic matrices [6], [7]. However, limited work has been reported on the simulation and control of drug diffusion when its concentration is below its saturation solubility in the polymer and most of the presented mathematical models were successful in predicting release profiles from known initial concentrations [3]. * Author to whom correspondence should be addressed; email [email protected]
1140 In general, to determine suitable initial parameters to obtain desired release behaviour requires a tedious trial-and-error simulation process. Only recently, optimal initial concentrations were calculated using optimisation techniques [8]. However, no work has been reported on the simultaneous optimisation of initial concentration, layers number and thickness and drug diffusivity (such as variations in the cross linking density) using formal mathematical methods. This can be mainly attributed to the complexity of the underlying mathematical problem. This work presents a novel optimisation approach to calculate rigorously the set of initial concentrations in a multi-layer device along with the thickness of each layer, their number and the drug diffusivities. 2. M O D E L L I N G A polymeric release systems with N layers is depicted in Figure 1. Ci, D i and 6 x i are the drug concentration, diffusion coefficient and thickness in each layer, respectively (i = 1,..,N). The disk has a thickness L and initial drug concentration Ci, o in each layer. It is assumed that drug diffusion is the rate-controlling step rather than swelling or drug dissolution.
x--0
Solventat sink conditions
dmkdiffusion IC Dill I~: Ih,,
x=L 5X1 ~X2
~xN Figure 1: Drug release
Mathematically, this problem is described using Fick's law. In a dimensionless representation the drug transfer between successive layers is described as follows:
OC i _ 1 02Ci c3-----~-(~ci) 2 Di ~0x
Vt > 0, Vi = 1,..,X
Cilx= 1 =Ci+l[x= 0
Vx~(0,1)
Vt>0, Vi=I,..,N-1
(1) (2)
1 Oi(OCiq 1.__~Oi+l(OCi+l~ Vt>0, Vi=I,..,N-1 (3) ~X--T ~-~X J x= I -- ~X i +I k ~X, ) x=O where the diffusion coefficients are assumed constant within each layer. The following boundary and initial conditions are also imposed: OC1 ~
--L-)x=0 Ci(x )
=0
= Ci, o (x)
Vt
V i,
> o, t = O,
Culx=l =0 x ~ (0,1)
Vt > 0
(4) (5)
The flux of drug is given as J = - D N ~ x1N (OCN ~, Ox ! x=l . The objective can be defined as the difference between desired drug release, J*, and the actual release rate. Cost considerations can also be incorporated to define the optimal drug release. For example increasing the
1141 number of layers one would expect to get a better release (closer to the desired one) however the overall system costs would increase due the construction of more layers. In general, there are clear trade-offs between improved release rates on the one hand and increased cost on the other. A general form of the objective function, OF, to be minimised over a total release time, t f , as follows:
OF =
(t)-
(t) dt + WL. N
(6)
o
where WL is a coefficient that represents the effect of number of layers in the objective function. 3. OPTIMIZATION The optimisation approach seeks to determine the optimal values of the available degrees of freedom, i.e. number of layers along with their thickness, initial drug concentration and diffusivities, in order to minimize the above objective function and satisfy certain constraints. A typical constraint may represent a minimum or specified fractional release. Alternatively, a fixed final release time can be imposed. However, determining the optimal values of the control variables is not a trivial tasks especially when considering their synergistic effects. Thus a formal optimisation approach needs to be employed. We consider two cases: optimisation based on an analytical approach utilizing only one control parameter and optimisation using formal optimal control numerical techniques.
3.1. Analytical Approach In the analytical approach the laminated disk is modelled as one layer where the initial drug concentration is distributed along the axial domain. The diffusion coefficient is also assumed uniform. Here, the only available control parameter is the initial concentration, C o (x). The first step is to parameterize the unknown functions by expanded each of them in a K-1 series of K known basis functions with unknown coefficients i.e. Co(x ) Z a i ~ i ( X ) =
i=O The solution of tick' s equation with the boundary conditions is: oo
C(x,t)= ~f'fjcos((j+l/2)~x)e-bjtwhere j=0 expansion coefficients K-1 1
of
f j = ~ 2a i I~oi(x)cos((j + 1/2)nx)dx. i=0
b j = ( ( j + l / 2 ) n ) 2 and
the
initial The
flux
fj
are the Fourier
distribution can
then
given be
as:
expressed
0
asJ(t) = ~ fjkje -b't where kj = n(-1)J(j + 1/2). In order to proceed our analysis one must j=0
choose the basis functions q)i. As a first attempt a set of global, orthogonal and infinitely differentiable functions are tested. The basis functions are % (x)= cos((i + 1/2)nx). After a detailed mathematical analysis the flux can be expressed as:
1142 K-1
J(t) = ~ ajkje -bit and
the
objective
function
takes
the
form
j=0 t~
K-1
F = ~(J*(t)-~ t,
ajkje-bjt)2dt.
In order to minimize F the derivatives
OF
(for
Oai
j=0
i=0,1,..K - 1) are set equal to zero. After performing some analytical integration the following 1in ear sy stem of eq uati o ns gi ves:
K-1 Z aj j=0
kj bi + bj
tl (e -(b~+b~)t' - e -(b~+bj)tl )= ~e-b'tj * (t)dt
(7)
. The above K x K s y s t e m
t,
must be solved for the K expansion coefficients ct0,al,..,aK_ 1. This is an unconstrained minimization procedure that usually give as a result an initial concentration distributions with negative values in some regions. To avoid this type of solutions a term of the following form K-1
is added to the objective function: w i[Co(x)]2 dx = w -1 /~0 a2 where w is an appropriately 0 2.= selected weighting factor. This term has as effect the reduction of oscillations of
C O(x). As w
increases, the exclusion of regions with negative values of
C O(x). Here a typical case where
the flux must be constant
t i = 0to tf = 0.5will be studied.
= 1 during the period from
This case has been numerically optimized in the literature [8]. For this particular case the integration in the right hand side of the equation (7) and objective function can be performed analytically. For K<4 the optimal initial distribution (taken for w=0) does not contain regions with negative values. But for K>4 the initial distribution takes negative values. To overcome the problem of having an optimum distribution without physical meaning, positive values are given to w in order to reduce the oscillations. Due to space limitations the detailed analysis is not presented here. The optimum initial distributions with physical meaning for K=2, 3, 4 and 6 are depicted in Figure 2 along with the corresponding fluxes. Except for the case with K=2, the others exhibit a similar flux behaviour. It is worthwhile to note that the optimal fluxes obtained using our simple analytical approach are almost identical to the ones presented in the literature based on a complicated optimization procedure and utilizing many degrees of freedom [8].
Figure 2: Optimal initial concentration and flux profiles based on an analytical approach
3.2 Dynamic Optimization Approach. A dynamic optimisation approach based on control vector parameterisation (CVP) techniques
1143 is employed to minimize objective function (6) subject to model equations (1) to (3) along with the boundary and initial conditions (4) and (5) and a requirement for a 65% drug release. Other constraints include lower and upper bounds on the time invariant optimisation parameters (e.g. lengths, initial concentrations) are also considered. The form of the model equations after the normalization over fixed domain (from zero to one) allows the direct use of CVP techniques [9]. First, a spatial discretisation approach is employed to eliminate the independent axial distance variable. This leads to an optimisation problem described by differential algebraic equations (DAEs). Nonetheless it remain a complex, infinite dimensional nonlinear optimization problem. The infinite dimensionality arises because the state variables are a function of time rather than scalar quantities. The CVP approach converts the infinite dimensional optimization problem into a finite one and allows the use of standard non-linear programming (NLP) techniques for the solution of the finite-dimensional problem [9]. Due to space limitations the details of this approach are not presented here. The dynamic optimization problem is solved using gOPT, an implementation of the CVP approach in the gPROMS modelling system [10]. The normalized axial domain is descritized using second order orthogonal collocation on 20 finite elements. The equations that describe the model are generated by gPROMS as residual equations with symbolically generated partial derivatives (Jacobian) and used as input to gOPT. The latter employs a sophisticated integrator for the integration the DAEs and a SRQPD nonlinear programming code implementing a reduced non-linear programming code [ 11 ]. Optimal flux profiles for three and seven layers are shown in Figure (4a). For the purpose of our analysis WL has been set to zero. However, the optimal number of layers can easily be determined by solving a number of optimisation problems for different number of layers. We observe, as expected, that as the number of layers increases an improved constant release (closer to the desired release) is obtained. Figure (4b) presents a comparison for a 7-layer matrix for two cases: (i) all optimization variables are utilized and (ii) the diffusivities are kept constant equal to one. The results indicate that case (i) leads to a significantly improved released compared with the case where only two degrees of freedom are utilized. This clearly indicates that there are synergistic effects between the control variables and an integrated formal optimisation approach is clearly desirable. Comparing with the results of the analytical approach presented in the previous section and also with other results from the literature [8] it can be concluded that the proposed dynamic optimisation method favorably compares (almost one order of magnitude better value of the objective) leading to releases which are very close to the desired one. The values of optimization variables are shown in Table 1. It is clear that near to the exit of the device two very thin layers are employed to achieve tighter control of the release. The approach has also been extended to cases of non-constant releases and investigated the optimal number of layers. Due to space limitation results are not presented here. Layer Number D Co 6x
5 6 7
0.001 0.85 0.25 0.08 0.045 0.001
0.001 0.03 0.52 1.5 2.08 1.36 1.23
0.02 0.08 0.23 0.40 0.23 0.02 0.02
Table 1" Values of Optimization Variables for a 7-layer matrix
1144
2.4 2.2 20 m
~" 1.8
._ a
-
' 1.2
~IL. -
1o %~__
"-.2Q-
-.2
0.8 0.8 0.4
0.1
0.2 03 04 Time (Dimensionless)
0.5
Figure 4: (a) Flux Profiles for different layers
0
02 03 04 Time (Dmens~onless)
(b) comparison of optimization approaches
4. C O N C L U D I N G R E M A R K S
This work has considered the exploitation of optimisation approaches to achieve desired release rates in drug delivery devises. The dynamic optimisation method, formally utilizing all degrees of freedom available in the system, leads to significantly improved releases compared with an analytical approach and other simpler approaches from the literature. The proposed optimization approach can be extended to cases where the initial drug is greater than its solubility limit in the release medium and in a different geometry (e.g. polymeric hemispheres). Finally, it is worth pointing out that, because of the local optimization algorithm for the solution of the NLP, the global optimality of any solution obtained with our approach cannot normally be guaranteed. This is a common deficiency of optimization-based design methods than can only be overcome by the adoption of global optimization techniques. REFERENCES
1. E.S. Lee, S.W. Kim, J.R. Cardinal and H. Jacobs, Journal of Membrane Science, 7 (1980) 293. 2. U. Conte, L. Maggi, P. Colombo and A.L. Manna., Journal of Controlled Release, 23 (1993) 39. 3. P.I. Lee, Journal of Controlled Release, 4 (1986) 1. 4. D.R. Paul, Journal of Membrane Science, 23 (1985) 221. 5. B. Narasimhan, and R. Langer., Journal of Controlled Release, 47 (1997) 13. 6. M. Grassi, R. Lapasin and S. Pricl., Chem. Eng. Comm., 169 (1998) 79. 7. M. Grassi, R. Lapasin and S. Pricl., Chem. Eng. Comm., 173 (1999) 147. 8. S. Lu, W.F. Ramirez and K.S. Anseth., AIChE Journal, 44 (1998) 1689. 9. V.S. Vassiliadis, R.W.H. Sargent and C.C. Pantelides, Ind. Eng. Chem. Res., 33 (1994) 2123. 10. gPROMS Advanced User Guide, Process Systems Enterprise Ltd, London (2000). 11. C.L. Chen, and S. Macchietto, Technical Report, Centrefor Process Systems Engineering, Imperial College of Science Technology and Medicine, (1988).
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1145
An Integrated Methodology for Developing Inherently Safer and Environmentally Benign Processes I. Halim, C. Palaniappan and R. Srinivasan* Laboratory for Intelligent Applications in Chemical Engineering, Department of Chemical and Environmental Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 Stringent safety and environmental regulations, and cutthroat competition have challenged the chemical process industries to bring products to market at low life cycle costs without compromising on safety and environmental standards. This has led plant designers to consider inherent safety and waste minimization principles at early stages of the design process. Tools and methods are available for developing inherently safer process and carrying out waste minimization analysis individually without taking into account the close coupling between them. This results in an incomplete and inaccurate analysis. In this paper, we present a systematic methodology for the integrated safety and waste minimization analysis during process design. This is done using material-centric approach, which brings out the similarities between issues and prevention strategies related to inherent safety and waste minimization. The integrated methodology is discussed and illustrated on an industrial process involving acrylic acid production process. 1. INTRODUCTION Conventional design of chemical plants has been primarily driven by factors related to economics and engineering. Issues concerning safety and environment are usually addressed at the later stages of design. This approach often leads to extensive end-of-pipe treatment and add-on safety features to reduce the consequences of acute hazards and chronic effects of a release. Intense competition, demand for consistently high product quality and more stringent safety and environmental regulations have challenged process designers to develop inherently safer and environmentally benign process. The term inherently safer implies that the process is safe by its very nature and not due to the use of add-on safety systems and devices. This is normally accomplished by reducing the use of hazardous materials and unsafe operations, minimizing inventory, moderation of operating conditions and by designing a simpler plant. The concept of waste minimization incorporates any technique, process or activity, which avoids, eliminates or reduces a waste at its source, or allows reuse or recycling of the waste. Both inherently safer process development and waste minimization techniques share the same basic philosophy, i.e., eliminating undesirable traits of a process. This often leads to synergies, for example substitution of material with safer and environmentally benign material. Nevertheless, inherently safer processes are not necessarily environmentally benign even though inherent safety concepts address certain environmental issues. For example, the use of CFCs as refrigerants is inherently safer with respect to fire, explosion and acute ,
Author to whom correspondence should be addressed (email: [email protected])
1146
toxicity hazards as compared to altematives such as propane. However, from the environmental perspective, propane is more desirable since CFCs cause ozone depletion. Thus, there are times when tradeoffs between developing inherently safer and environmentally benign process have to be reconciled in order to design an all-round better process. The need for an integrated methodology to analyze both safety and environmental issues and their interactions has been emphasized in literature [ 1,2]. Despite the growing need and obvious importance of such a design approach, its adoption into practice has been quite slow. This is mainly due to factors such as time and cost constraints during the design, conservatism in design, and lack of supporting tools. Research on developing an intelligent tool for waste minimization analysis and development of inherently safer chemical processes has been ongoing in our group. An intelligent waste minimization tool called ENVOPExpert has been developed and successfully tested on several industrial case studies [3, 4]. An inherent safety analysis tool, called/Safe, which can assist plant designers by identifying safety issues and recommending solutions is also being developed [5]. These tools highlight the issues and offer recommendations considering safety or pollution individually without taking into account interactions between them. In this paper, we present an integrated methodology for inherent safety and waste minimization analysis. The task for the integrated analysis can be defined as follows: Given the details about materials involved, process chemistry, flowsheet and the reaction and separation schemes of a chemical process plant, the goal is to identify opportunities to minimize the hazards and wastes in that process by evaluating the synergies and tradeoffs between the two. The organization of this paper is as follows: in the next section, the methodology and intelligent system for integrated analysis based on a material-centric approach is proposed. These are illustrated in Section 3 on an acrylic acid production case study; finally, in Section 4 the overall conclusion about the approach is presented. 2. M E T H O D O L O G Y FOR SAFETY & POLLUTION PREVENTION ANALYSIS The integrated methodology for developing inherently safer and environmentally benign processes is based on a material-centric view of a process. Inherent safety principles address the prevention of unintended effects of materials while waste minimization principles deal with the minimization of their release to the environment. Process materials form a focal hub and mold the hazards and wastes occurring in a process. Hazards as well as wastes can be therefore recognized and remedied by focusing on process materials. Figure 1 describes this material-centric view of safety and waste-related issues. Here, both issues have been organized based on their source as due to (1) individual materials and their properties (2) interactions between two or more materials, for example through reaction (3) interaction between material and process conditions, and (4) interaction between material and process unit. Figure l a shows some examples of safety issues that originate from these four sources. Similarly, Figure l b shows examples of pollution issues from them. The organization of safety and waste issues along this material-centric scheme brings out the essential similarities between the sources of hazards and pollution. A comprehensive inherent safety analysis would consider reduction of material inventory, substitution of material with a safer one, moderating the conditions at which a material is processed, use of simpler design to minimize the possibility of material release, etc. The material-centric view also lends itself to representing the different inherent safety principles used to develop a safer plant. Figure l c shows the five common inherent safety principles- substitution, attenuation, intensification,
1147 simplification and toleration- arranged in the material-process unit-process condition mold. A waste minimization study would focus on elimination or reduction of waste generation at the source by altering the process conditions, or substituting with an environmentally friendly material, recycling of material, etc. Thus, the waste minimization principles also can be cast in the material-centric view as shown in Figure 1d. The strong parallels between hazard and pollution prevention strategies are also clearly brought out in this material-centric view thus facilitating an integrated hazard and waste analysis. Figure 2a illustrates that the sources of environmental and safety issues can be merged into an integrated material-centric cast. The similarities in the philosophy of waste-minimization and inherent safety principles in tackling the hazards and pollution due to each of the four sources is depicted in Figure 2b.
( ~Reaction
Was(~ Ozonedepletion i generation~tential
T o x i c ~ ards I
n
~
em -- " 0
Fug.ltlve L ~ , ~
Fig l a. Safety Issues
M: Material P : Process conditions U : Process Unit
Fig lb. Environmental Issues
.
~
(•ecycle
Elimination stitution
rSeducfion urce ~ s t i t u t i o n S
Fig lc. Inherent Safety Principles
Dust Pollution
i
m
p
l
~
, i~
Fig ld. Waste Minimization Principles
Figure 1" Material-centric Approach to Identify and Resolve Safety and Pollution Issues The safety and environmental issues related to a material can be identified by evaluating its properties such as flash point, threshold limit value, ozone depletion potential, etc. Issues such as decomposition at high temperature and dust pollution due to small particle size arising from interaction of materials with process conditions are identified by evaluating changes in physical and chemical aspects of the material. Hazards and pollution arising from materialmaterial interaction such as waste generation, run away reaction, etc are identified by evaluating intended and tmintended reactions occurring in the process. Issues due to materialunit interaction such as leak of toxic chemicals from storage, fugitive emissions from flanges, etc are identified by evaluating failure modes of the equipment. Once the issues have been identified, suggestions to rectify the issues can be proposed using common keywords derived from inherent safety and waste minimization principles. Table 1
1148 shows examples of such keywords and variables applied to material, process conditions and process unit along with the suggestions derived using them. For example, in order to rectify all safety and pollution issues arising from a solid, the suggestion "Modify the particle size distribution of the solid - "can be proposed. Similarly, in order to improve the absorption of useful material, the suggestion "Increase the pressure in the absorber" could be recommended. For a reactor unit that produces waste, the suggestion "Optimize the operating conditions of reactor" can be proposed. The synergies and tradeoffs from the safety and pollution perspective can then be highlighted to the designer. The synergies between the two suggestions are self-evident when making the same change made to the item (material, unit or process condition) results in improved safety and waste performance. For example, replacing benzene solvent with water would improve both safety and environmental performance. Similarly, self-evident trade-offs occur if the change has conflicting effects on different issues. The example of refrigerant CFCs versus propane described earlier falls in this category. Both the synergies and trade-offs can be found by comparing the effects of each recommendation on issues in the following order:
- (material, process conditions, unit), (temperature, phase) and (change, optimize, recycle) respectively. When the item and variable match, depending on whether the keywords concur or conflict, the tradeoffs and synergies can be identified. Common safety and environmental indices can be used as yardsticks for measuring inherent safeness and environmental friendliness of a process. Fig 2a. Integrated identification of Safety Pollution Issues
Material-Material interactions Material-Unit interactions
Material
condition interactions
&
Fig 2b. Integrated Inherent Safety Waste Minimization Solutions
EliminateRecycl~sS~ e
reduction
Simplify ~
ubstitute
&
Intensify
Synergies ~ ~ / ~ ~ Tradeoffs Figure 2" Commonalties between safety and environmental issues and solutions In order to automate the methodology described above, the tool for combined approach must be capable of identifying hazards and pollution issues that arise due to material and its interaction with process based on ~.nformation on materials, process chemistry, and reactionseparation scheme of a process. We have earlier developed an intelligent system called ENVOPExpertthat uses P-graph models along with digraphs and functional models for waste minimization [4]. An expert system-/Safe - that uses the same models for inherent safety analysis is currently under development [5]. In both systems, P-graph models represent the cause and effect relationship of materials, reactions and separations involved in the process. Similarly, digraph and functional models represent the relation between process variables and
1149 the issues related to safety and pollution. The knowledge bases of the two systems are being merged based on the common material-centric view and implemented in G2 as an intelligent expert system for integrated safety and waste minimization analysis. Table 1" Keywords for generating suggestions
3. CASE STUDY: ACRYLIC ACID PROCESS
We have performed a combined inherently safety and waste minimization analysis using our integrated methodology on an acrylic acid case study obtained from literature [6]. Figure 3 shows the flowsheet of the process. Acrylic acid is produced by partial oxidation of propylene in a fluidized-bed-catalytic-reactor. Reaction products are quenched immediately using cold quench recycle and off-gas is absorbed using deionized water. The quenched stream is sent to extraction column in which diisopropylether is used as solvent to separate the products from waste streams. Synergy with respect to change in operating conditions in reactor and trade-off with respect to change in operating pressure in off-gas absorber in order to improve safety and environmental performance have been identified. Two suggestions derived using our integrated approach along with the synergies and tradeoffs are shown in Table 2. 4. CONCLUSIONS Development of inherently safer and environmentally benign process is of prime importance in today's business environment. In this paper, we have discussed the need for an integrated methodology for safety and environmental impact analysis. A material-centric approach to identify hazards and pollution issues has been developed. A guideword based solution generation technique, which meshes the material-centric view with inherent safety and waste minimization principles, has also been proposed. We have discussed an intelligent system for automating the integrated analysis and illustrated the methodology using an industrial case study. REFERENCES
1. R. D. Tumey, "Designing plants for 1990 and beyond: Procedures for the control of safety, health and environmental hazards in the design of chemical plant," Trans. 2. I ChemE, vol. 68, pp. 12 - 16, 1990.
1150 3. K. Lien and Perris, T., "Future directions for CAPE research: Perceptions of industrial needs and opportunities," Computers and chemical engineering, vol. 20, pp. S1551-S1557, 1996. 4. I. Halim and Srinivasan, R., "An Intelligent System for Identifying Waste Minimization Opportunities in Chemical Processes," presented at European Symposium on Computer Aided Process Engineering - 10, 2000. 5. I. Halim and Srinivasan, R., "A Hybrid Qualitative-Quantitative Approach of Waste Minimization Analysis in Chemical Processes," presented at AIChE Annual Meeting, Los Angeles, Paper No. 233r, 2000. 6. C. Palaniappan, Srinivasan, R. and Tan, R.B.H., "An Intelligent Support System for Design of Inherently Safer Process Flowsheets," presented at AIChE Annual Meeting, Los Angeles, Paper No. 249b, 2000. 7. R. Turton, Bailie, R.C., Whiting, W.B. and Shaeiwitz, J.A., Analysis, Synthesis and Design of Chemical Processes. New Jersey: Prentice Hall, 1998.
Table 2: Synergies and Tradeoffs for Safety and Pollution for Acrylic acid Case study
European Symposium on Computer Aided Process Engineering - 11 R. Gan] and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1151
Particle Size Distribution by Design Priscilla J. Hill a and Ka M. Ng b a
Dept. of Chemical Engineering, University of Minnesota Duluth, Duluth, MN 55812, USA
b Dept. of Chemical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong This paper presents a strategy for designing solids processes to produce particles with a desired size distribution. Heuristics, and generic flowsheet structures are used to guide decision making. Computer simulations based on discretized population equations along with suitable solids processing models are used to meet the particle size distribution target. 1. INTRODUCTION It is well recognized that the product quality of many solids products depends on the product's particle size distribution (PSD). For example, the PSD of the powder in a tablet is carefully controlled in the manufacture process because the particle size has a significant effect on the dissolution and absorption rate in the gastrointestinal tract. Much work has been done on predicting PSD in individual unit operations such as crystallizers. However, the PSD may still change considerably as the crystals migrate through the downstream filters, washers, and dryers [1 ]. Similar situations are present in bulk solids systems in which the PSD changes due to breakage and granulation [2]. Thus, one must investigate the effect one unit has on other units in the entire system. To meet the need to track changes in the PSD as solids flow through a process, a modular simulation code based on discretized population balance equations was developed [3-6]. Although PSD tracking is necessary to evaluate a given process, it does not provide guidance in synthesizing a process that will produce a product with the desired product PSD. In this paper, we discuss the strategy that has been developed for process synthesis with PSD.
2. PARTICLE SIZE DISTRIBUTION BY DESIGN This hierarchical, multi-scale approach considers issues ranging from flowsheet structure to fundamentals such as the choice of the functions used to model particle breakage [5]. We start out with a general flowsheet and gradually add details as necessary. This method consists of a series of steps (Table 1). The first step is to gather all available information on the process. This includes material properties, chemical reactions, and crystallization kinetics as well as the process feed conditions, the design specifications (particularly the PSD specifications), and any design constraints.
1152 The second step is to develop a flowsheet for the process. One way of doing this is to start with a generic solids flowsheet such as the one for a crystallizer-filter-dryer train (Figure 1). Using the information gathered in the first step, the generic flowsheet is modified to produce a general flowsheet for the process. Heuristics are used to help in decision making. For example, if the crystals from the crystallization system meet the purity requirement, then obviously recrystallization is not included in the process flowsheet. Several alternatives may be generated in this step. Table 1. Strategy for particle size distribution by design Step 1. Input Information Product specifications, feed conditions, solid-liquid equilibrium data, crystallization kinetics, process constraints, etc. Step 2. Selection of Functional Structures Identification of the primary functions to be performed to meet the PSD target. Step 3. Selection of Equipment for Functional Structure Specification of equipment types and connections among the various streams. Step 4. Evaluation of Process Alternatives Based on Discretized PBEs Determination of process feasibility as well as the best process alternative. A different solids subsystem has a different generic flowsheet For example, a bulk solids system would involve screens, blenders, conveyors, crushers, granulators, etc Generic flowsheets for bulk solids have been reported [2, 7] Liquid Recycle ! T
Reaction, Extraction, Feed(s)--~Pretreatment ~-i~ and/or Dissolution
Crystallization System
Solid/Liquid Separation
1
d
Liquid Recycle ......................................................................
"J~IL
T
Recrystallization System
Solid/Liquid Separation
Postprocessing
Product with Desired PSD
Figure 1 Generic flowsheet for a crystallizer-solid/liquid separation train
1153 The third step is to select equipment or equipment items for each block on the flowsheet. This requires a knowledge of equipment capabilities and an understanding of the objective of each process operation. A short list of guidelines for choosing equipment systems is given in Table 2. Since there is more than one equipment choice for some of the steps, more process alternatives will be produced in this step. Table 2. Guidelines for selection of equipment for functional structure 1. If the desired PSD falls within the range of the PSD in the effluent of the particle generation unit (a crystallizer in this case) 9 Use hydrocyclones to remove the fines and large particles. Recycle these particles to the crystallizer feed. 9 If the amounts of fines and large particles are relatively small, leave them in the main processing train. Use screens after the dryer to obtain the desired PSD. Recycle the oversized particles to a crusher and the undersized particles to the crystallizer upstream. Alternatively, send the undersized particles to a granulator (see Figure 3). 2. 9 9 9 9 9
If particles from the particle generation unit are not sufficiently large for the desired PSD Avoid using reactive crystallization which tends to produce small particles. Change crystallizer operating conditions to produce larger particles, if possible. If recrystallization is necessary, switch to another solvent that produces larger particles. If all of the above does not work, use an agglomeration system after the dryer. Consider the use of spray dryers that may lead to larger particles.
3. If the particles are too large for the desired PSD 9 Change crystallizer operating conditions to produce smaller particles. 9 Use hydrocyclones to remove the oversized particles and return them to the crystallizer feed after dissolution. 9 Use a crusher after the dryer.
The fourth step is to evaluate the alternatives. Once the flowsheet and equipment units are chosen, simulation can be used to evaluate the altematives. There are still many decisions to be made at this stage. These include determining the significant phenomena which must by included in modeling the unit operations, the functional forms used in the population balance equations, the appropriate design variables, and the operating conditions. The particle tracking simulations are used as a tool to evaluate the altematives. From the fourth step, feasible operating conditions can be determined for each process alternative. If an economic analysis is performed, the feasible operating conditions can be evaluated to determine the more economical options. 3. SOFTWARE CONSIDERATIONS To implement this strategy, one must have the appropriate software. The computer simulation programs must include adequate methods for solving the equations. Discretized population balance equations are used in our code because they are very
1154 robust and require relatively modest calculational effort. They must also be able to incorporate the fundamental chemistry and physics needed to model the unit operations. They should also be able to include cost estimation and economic analysis. The software platform should be flexible enough to allow models from different platforms. For example, the main interface can be written in a general code which allows users to connect to commercial process simulators, commercial thermodynamics packages, spreadsheets, experimental data, and user supplied routines in FORTRAN or C. 4. CASE S T U D Y - SALT PRODUCTION
To illustrate the design procedure, the step by step development of a process for the manufacture of NaC1 salt is shown. Step 1: The objective is to crystallize NaC1 from a liquid brine at a rate of 105 ton/yr with no more than 5 mass % fines less than 160 ~tm. This process does not have any chemical reactions and recrystallization is not required for purification. Other data is given in Table 3. To meet the production rate, we will use a continuous process. Table 3. Selected input data for the salt plant Production Rate Feed Concentration Liquid Fraction in Feed Solubility of NaC1 kg NaC1/kg H20 Crystallizer Operating Temperature Agitator Speed, N Growth Rate Nucleation Rate MT 105.8 kg/m 3 Hydrocyclone cx~r 0.2 Rut* 0.15 Filter Rotation Speed, co Angle of Submergence, Vacuum Level, Ap Filter Cake Porosity, ~0
105 tons/yr = 185.4 kg/min 214.6 kg NaC1/m3 solution 8f = 1.0 SN = 0.35712 + 8.48x10 -5 T +3.17x10 6 T 2
50~ 500 rpm 6 ~trn/min 0.32 N 2 G 2 MT no./min/m 3
5 rpm 145~ 3500 N ] m 2 0.412
~rcx is the ratio of solids in the overflow to the solids in the underflow :~ Rut is the underflow to thoughput ratio Step 2: Based on Step 1 and Figure 1, we can generate a general flowsheet. Since preprocessing, reaction, and recrystallization are not needed, they are not included. Step 3: In this step we define the blocks in the generic diagram and determine the recycle stream connections. In this case we have chosen an evaporative crystallizer for removing excess water from the brine as well as crystallizing the product. The solid/liquid
1155 separation block could be represented by a single filter. However, a block is not limited to a single equipment unit; it can be a system. A second alternative is to use a hydrocyclone to separate undersized particles from the crystallizer effluent, followed by a filter. Different types of filters could be chosen for this operation including a centrifuge or a rotary vacuum drum filter. For the evaluation step, we have chosen a rotary vacuum drum filter. Another alternative to consider is the liquid stream exiting the solid/liquid separation unit. It could exit the process as a waste stream or it could be recycled to the crystallizer. Since more salt could be recovered by recycling the liquid stream back to the crystallizer, this would probably be the better option. One danger with this is that impurities could build up in the system, but this could be controlled by taking a purge stream off of the recycle stream. Step 4: Now that there are alternative flowsheets to evaluate, let us consider the flowsheet shown in Figure 2. In this flowsheet the salt production rate is P and the quantity of solids recycled from the hydrocyclone to the crystallizer is ~P, where ot is a function of the cutoff value, ds0, in the hydrocyclone. The cutoff value is the particle size where half of the particles exit in the underflow stream and the other half exit in the overflow stream of the hydrocyclone. To meet the product specifications of no more than 5 mass % fines less than 160 ~tm, we can adjust ds0. A series of simulations were performed in which ds0 was varied and the magma density was held constant at the same temperature and supersaturation. The residence time was allowed to vary to keep the magma density constant. Figure 3 shows the PSD in the actual product stream. As ds0 is increased, the fines decrease and the dominant particle size increases. A cutoff value of 141 ~tm or larger must be used to meet the fines constraint.
Water 1 Brine ~ ~
~P
I
Crystallizer
.J Filter S~t P Product Figure 2. Salt manufacturing process.
1156
85o --,~- 2 1 6 141 --~ 5 8
%
j,; 0
200
400
600
800
1000
Particle size, ~ m Figure 3. Effect of dso on product stream PSD. 5. CONCLUSIONS A four step strategy for designing solids processes to produce a specified PSD is presented. The method consists of gathering information, developing general flowsheets, specifying equipment units or systems, and using simulation with evaluation to determine feasible flowsheets and operating conditions. An accompanying computer code has been developed to facilitate such design efforts.
REFERENCES 1. 2. 3. 4. 5. 6. 7.
Chang, W.-C., and K. M. Ng, AIChE J., 44, 2240 (1998). Wibowo, C., and K. M. Ng, AIChE J., 45, 1629 (1999). Hill, P. J., and K. M. Ng, AIChEJ., 41, 1204 (1995). Hill, P. J., and K. M. Ng, AIChE J., 42, 727 (1996). Hill, P. J., and K. M. Ng, AIChE J., 42, 1600 (1996). Hill, P. J., and K. M. Ng, AIChEJ., 43, 715 (1997). Gruhn, G., J. Rosenkranz, J. Werther, and J. C. Toebermann, Computers Chem. Engng., 21, S 187 (1997).
European Symposium on Computer Aided Process Engineering - 11 R. Ganl and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1157
Optimization in Molecular Design and Bioinformatics Costas D. Maranas a aDepartment of Chemical Engineering, The Pennsylvania State University This work is an exposition on the application of optimization tools to problems in molecular design and bioinformatics. The specific areas addressed by the author include the design of polymers, surfactants, refrigerants, and enzymes. The goal is to systematically design molecules for the given application with desired performance characteristics. The performance measures of interest in polymer design are mechanical, electrical and thermophysical properties. In case of surfactants properties such as the HLB, emulsivity, detergency, and foaming stability influence the performance significantly. The performance measure in refrigerant selection and cycle synthesis is the balance between operating costs related to energy input and the investment costs. The performance measure in enzyme design is the probability of achieving a given nucleotide sequence target. The role of optimization is to "systematically" search through the alternatives. The research results in each of the applications mentioned above are presented. 1. INTRODUCTION The competitive edge and market share of many chemical industries manufacturing polymers, refrigerants, solvents, surfactants, enzymes, and biomaterials are ultimately intertwined with the identification of "new" and "better" products. Though the vast number of alternatives presents a designer with an opportunity to find a better product, it also poses the challenge of systematically searching through the alternatives. With the rapid growth in optimization theory, algorithm development and high-performance computing, exciting and unprecedented research opportunities are emerging in molecular design to assist in this endeavor. Research results in polymer design, surfactant design, refrigerant selection and enzyme design are discussed in this work. Previous work include the computer-aided design of molecular products such as polymers [9,7,5], solvents [5] and refrigerants [4,5] to name a few. The employed search algorithms include enumeration techniques, knowledge-based strategies, genetic algorithms and mathematical programming based methods. A comprehensive review of prior work can be found in Camarda and Maranas [3]. The objective is to find a molecule for a given application which optimally satisfies the desired performance targets.
2. POLYMER DESIGN In polymer design the problem of identifying the polymer repeat unit architecture so that a performance objective that is a function of mechanical, electrical and/or physicochemical properties is addressed. Since the molecular design problem is posed within an optimization
1158 framework, a quantitative representation of the molecule and a quantitative structure-property relation is required. Group contribution methods (GCM) provide popular, versatile and relatively accurate ways for estimating properties based on the number and type of molecular groups participating in a molecule or repeat unit. (GCM) are based on the additivity principle of the groups constituting the molecule under investigation and have been extensively utilized in the estimation of a wide spectrum of polymeric properties including volumetric, calorimetric, thermophysical, optical, electromagnetic and mechanical properties. An extensive compilation of these estimation methods along with the corresponding parameters can be found in van Krevelen [11]. The use of (GCM) makes adequate the molecular representation using n=(nl ,n2,... ,nN) where ni are the number of groups of type i present in the molecule. The problem of identifying the best molecule based on some measure of performance can be expressed as the following mixed-integer nonlinear optimization problem.
min
MP(pj(n)) pjL _< pj (n) _< p~
subject to ni
(OMD)
C
{nL, n L + l , . . . , n U } ,
i=l,...,N
The following two most widely used measures of performance are considered in this study [7]: (1) Minimization of the maximum scaled deviation of properties from some target values (property matching (PM)), 1
min M P - max ~ I p j ( n ) - P~I
J
Pj
where p~ is the target for property j and p} the corresponding scale. (2) Minimization/maximization of a single property j* (property optimization (PO)), min / max M P = p j. (n). To maintain structural feasibility of the molecule a number of linear constraints on n must be included in the problem (OMD). These structural feasibility constraints define the necessary conditions under which a set of molecular groups can be interconnected so that there is no shortage or excess of free attachments. The estimation of most properties pertinent to engineering design is given by the ratio of two linear expressions in ni. Though the above formulation is a mixed integer nonlinear program (MINLP) in general, the underlying mathematical functionalities of the above property estimation model are utilized to reformulate and solve the problem as a mixed integer linear program (MILP). One of the limitations of group contribution estimation is that the internal molecular structure of the polymer repeat unit is only partially taken into account. For example, both polypropylene -CHzCH(CH3)CH2CH(CH3)- and head to head polypropylene-CHzCH(CH3)CH(CH3)CH2-, have the same molecular group representation. These shortcomings are alleviated with the use of property correlations involving topological indices as structural descriptors. These indices are numerical values which uniquely identify the polymer repeat unit and contain information about the atomic and electronic structure. Specifically, Bicerano [ 1] used the zeroth- and firstorder molecular connectivity indices to correlate a wide range of polymer properties, including
1159 density, glass transition temperature, bulk modulus, and heat capacity. The functional form of the topological indices used are given in Camarda and Maranas [3]. The following additive property predictive form is utilized: (Property Prediction) = (Basic Group Contribution) + (Connectivity Indices Contribution) Though in general the above problem is a nonconvex MINLP, it is reformulated and solved as a convex MINLP utilizing the mathematical functionality of the connectivity indices. So far it has been assumed that the properties are uniquely determined by the types of groups present in the molecule and their interconnectivity. However, in reality there are discrepancies between predicted and observed values. These can be reconciled by recognizing that the parameters of the property model vary around their nominal values. This can be expressed mathematically by utilizing probability distributions to describe the likelihood of different realizations for the model parameters. The probabilistic description of performance objectives and constraints is described in Maranas [6]. This formulation involves probability terms whose evaluation for each realization of the deterministic variables requires the integration of multivariate probability density distributions. This is accomplished without resorting to computationally intensive explicit or implicit multivariate integration. This is done by transforming the stochastic constraints into equivalent deterministic ones. Furthermore, it is shown that for probabilities of interest this formulation is a convex MINLP which can be solved to global optimality using commercial packages. The objective of using this formulation is to construct a trade-off curve between performance target and the probability of meeting the target. This aids the designer in choosing the optimal level of risk in selecting the molecule. Next, the surfactant design problem is briefly discussed. 3. SURFACTANT DESIGN The design of surfactant solutions is an important problem in many industries since they are extensively utilized in diverse applications such as detergents, emulsifiers, and to ensure film coating and waterproofing. In the design of surfactant solutions the performance measures of interest are HLB, emulsivity, detergency, and foaming stability. Though this problem is also addressed within the general molecular design paradigm discussed previously, this problem presents additional unique challenges. The macroscopic properties of interest are related to structural descriptors of surfactants through fundamental solution properties such as critical micelle concentration (CMC) and area of a surfactant molecule within a micelle. Though this has the same flavor as relating property of polymers to connectivity of the molecule through topological indices there is an important difference. In polymer design, connectivity indices could be determined from the connectivity by simple evaluation. In the case of surfactants, determination of fundamental solution properties involves the minimization of free energy. Therefore the problem of identifying the molecular structure of a surfactant with optimal values for the desired macroscopic properties is posed as a two-stage optimization problem [2]. The inner stage identifies the CMC and other micellar properties by minimizing the free energy ~tg, while the outer stage optimizes over the surfactant structural descriptors. A conceptual optimization formulation of the problem is as follows: max / min subject to
f(macroscopic properties)
1160 macroscopic properties ) fundamental properties )
=
g(fundamental properties) arg min /lg (structural) descriptors
This formulation is solved using a truncated newton method. Since this problem may possess multiple local minima the problem is solved with multiple starting points. The structural descriptors include the number of carbon atoms in the surfactant tail nc, the cross-sectional area of the head ah, the charge separation for an ionic head group 5, and the dipole separation for dipolar surfactants d. These descriptors provide a concise description of the surfactant molecular topology and polarity. They are theoretically related to fundamental solution properties determining the shape, size and concentration of the surfactant micelles. The fundamental solution properties include the equilibrium area per molecule in a micelle a, the micellar shape, and the concentration at which micelles form (known as the critical micellar concentration or CMC). These properties are related through local regression models to macroscopic surfactant properties characterizing the suitability and effectiveness of the surfactant for a particular application (e.g., hydrophilic-lipophilic balance number (HLB)). Details of the functional relation of free energy to surfactant molecular structure and solution properties is given in Camarda et.al. [2]. This methodology is applied to identifying a nonionic surfactant with hydrophiliclipophilic balance (HLB) of 13.8. HLB is a widely used measure of the emulsifying ability of a surfactant. High value for HLB implies high water solubility, and suitability for detergent or emulsifier. A local regression model is constructed which relates HLB to CMC as follows: l n H L B -- 2.76 + 0.04 l n C M C
The truncated-Newton algorithm was started from a number of initial starting points, and in each case, the algorithm converged to the same optimal solution involving a head cross-sectional area of 0.54977 nm and 5.997 carbons in a straight-chain tail. The CMC for this surfactant was found to be 0.034 mM. A search over tabulated surfactant properties reveals that a surfactant with a dimethyl phosphene oxide head group and a six carbon tail is compatible with those structural descriptors. 4. R E F R I G E R A N T S E L E C T I O N AND C Y C L E SYNTHESIS The focus now shifts from designing a molecule (refrigerant) to selecting a molecule from a prepostulated set of potential candidates. This still poses a challenge when placed within the context of synthesizing refrigeration cycles. The combinatorial problem of appropriately assigning refrigerants to different locations in the refrigeration cycles requires the use of optimization tools. The problem addressed is stated as follows [ 10]: Given a set of process cooling loads, heat sinks at different temperatures and a set of available pure refrigerants, find the refrigeration cycle topology, operating conditions and refrigerants, selected from the list, that optimize a weighted sum of the investment and operating costs for the refrigeration system. The proposed model involves a superstructure representation for both the synthesis and the refrigerant selection problems. The model allows for the identification of the number of stages, their operating temperature ranges, the type of refrigerant participating in a stage, the temperature where a switch between two refrigerants occurs, the use of economizers, presaturators
1161 (31OK) (294 K) (278K) Provane Refrigerant
(263K) (247K) ( 2 3 2 K ) f----(236K) ~_
Refrigerant Switch
(218K) Ethane Refrigerant
(201K) (186K) (190K)
~
Process Stream
Fig. 1. Vertical Cascade for pure refrigerant system
or heat exchangers between intermediate stages. The objective to be optimized considers both investment and operating costs. These alternatives are compactly represented as a network. The operating temperature range of each potential refrigerant is discretized and these discretized levels are the nodes of the network. The alternatives corresponding to (i) operation of vapor compression cycle between temperature levels of a particular refrigerant (ii) heat intake from a cooling load (iii) switch between refrigerants are represented by the arcs of the network. The process configuration is obtained once the optimal energy flows in the network are identified. The optimization problem is solved as an MILE An example of the optimal configuration generated by this procedure for pumping 100kW of heat from 190K to 31 OK using a ethane-propane refrigeration system is shown in Figure 1. Examples demonstrating the advantage of simultaneous refrigerant selection and cycle synthesis over a sequential approach are given in Vaidyaraman and Maranas [ 10]. 5. E N Z Y M E DESIGN
DNA recombination techniques provide the backbone of directed evolution experiments for engineering improved proteins and enzymes. The setup of directed evolution experiments is vital to the rapid and economical production of enhanced enzymes since screening a large number of proteins for the desired property is expensive and time consuming. The goal is to develop predictive models for quantifying the outcome of DNA recombination employed in directed evolution experiments for the generation of novel enzymes. Specifically, predictive models are outlined for (i) tracking the DNA fragment size distribution after random fragmentation and subsequent assembly into genes of full length and (ii) estimating the fraction of the assembled full length sequences matching a given nucleotide target. Based on these quantitative models, optimization formulations are constructed which are aimed at identifying the optimal recombinatory length and parent sequences for maximizing the assembly of a sought after sequence target [8]. A flowchart of DNA shuffling is shown in Figure 2. First an initial set of parent DNA sequences is selected for recombination. The parent sequences undergo random fragmentation, typically by DNase I digestion. The fragment length distribution QO, which describes the fraction of fragments of length L found in the reaction mixture after fragmentation is calculated to
1162 be as follows
aO _
Pcutexp(-PcutL) for 1 < L < B - 1 exp(-PcutB) for L - B
Next, the double-stranded fragments within a particular size range (i.e., 50-200 base pairs) are isolated and reassembled by the Polymerase Chain Reaction (PCR) without added primers. This step is quantified using a fragment assembly model that tracks the fragment length distribution through a given number of annealing/extension steps. This is used to estimate how many shuffling cycles will be needed before full length genes are assembled. A sequence matching model is developed to aid in the goal of optimizing experimental parameters to maximize the probability of obtaining a desired sequence. This model quantitatively predicts the probability of having a randomly chosen full length sequence, assembled through DNA shuffling, match the given nucleotide sequence target. This model recursively calculates the probability Pi of a reassembled sequence matching the target sequence from position i to position B (length of parent sequence). The probability P1 represents assembly of the entire target sequence. The recursive expression for evaluating Pi is shown below. 1,
i>B
zxs.B
i -- B
K L-1
QO ~, AL-V,L
ei--
L=L1 •
V'-Vmin
(Ai'i+L-V-1)pi+L_V, 1
In the above expression, K is the number of parent sequences, (L1,L2) are the range of fragment lengths retained for shuffling, QO is the probability that a fragment is of length L, AL-v,L is the probability that a fragment of length L will anneal with an overlap V, and Ai,i+L-V-1 is the number of parent sequences that match the target between the positions i and i + L - V - 1. The above predictive sequence matching model enables the formulation of mathematical programs for optimizing the recombinatory fragment length and parent sequence set. The objective of the desired mathematical program is the maximization of the probability of matching the target sequence (P1). Two variations of this model are considered. In the first variation, the parent sequences which are chosen occur in equal relative concentration. This problem is combinatorial in nature due to the need to choose the optimum set of parent sequences. The binary decision variables are Yk denoting if parent k is chosen and xL indicating if the recombinatory
Fig. 2. The three steps of DNA shuffling.
1163
Fig. 3. Illustrative example for the effects of fragment length and parent set selection
length is L. This results in a MILP formulation. The second variation allows all the parent sequences to be present but optimizes the relative concentration Ck of each parent sequence. This is solved as a bilinear NLP once for each L in the range being considered to find the optimal recombinatory fragment length. The importance of using the MILP and bilinear formulations are illustrated through an example. The goal is to shuffle six parent sequences each with a variety of mutations and to produce a sequence containing all twelve of the mutations. The sequences are B = 151 nucleotides long and are shown in Figure 3(a). First, when all parents are selected for recombination (achieved by fixing all Yk = 1), the optimal recombination probability P1 of 0.0105% for a fragment length L of 37 nucleotides is confirmed. However, when the complete MILP is solved for both xL and Yk, the subset of parent sequences 1, 3 and 6 is revealed to be the optimum recombinatory choice with a recombination probability of 0.0294%, an almost three-fold improvement. Note that the new optimal length is L = 70 nucleotides, almost twice the length of the previous optimum implying that the selection of the optimal fragment length L strongly depends on the selection of the parent set. A plot of P1 versus L for different parent sequence recombination sets is shown in Figure 3(b). These results suggest a surprising complexity in the shape and form of the P1 versus L plots for different parent choices. Specifically, the multimodal characteristics of these curves reveal narrow fragment length regions for which favorable recombination results are obtained. Next, the bilinear formulation is solved, producing the result shown in Figure 3(b). The optimal recombination probability is equal to 0.0297% at L - 71. The optimal parent sequence concentrations for this fragment length are C1 = 0.362, Ca = 0.339, C6 - 0.299, with all other Ck -- 0, which are fairly close to the equal relative concentration solution. These results indicate that utilizing these formulations can produce a substantial increase in recombination probability.
1164 6. S U M M A R Y
This paper discussed the application of optimization techniques to the area of molecular design and bioinformatics. Key issues in the area of polymer design, surfactant design, refrigerant selection, and enzyme design were identified. In each case it was shown how the use of optimization techniques helped in "homing in" on the desired alternatives in a systematic way as opposed to time and labor intensive trial and error approach. REFERENCES
1. J. Bicerano, Prediction of Polymer Properties. Marcel Dekker, New York, 1996. 2. K.V. Camarda, B. W. Bonnell, C. D. Maranas, and R. Nagarajan, Design of Surfactant Solutions with Optimal Macroscopic Properties. Comput. Chem. Eng., Suppl:S467, 1999. 3. K.V. Camarda and C. D. Maranas, Optimization in Polymer Design Using Connectivity Indices, Ind. Eng. Chem. Res., 38(5): 1884, 1999. 4. A.P. Duvedi and L. E. K. Achenie, Designing Environmentally Safe Refrigerants Using Mathematical Programming, Chemical Engineering Science, 51:3727, 1996. 5. R. Gani, B. Nielsen, and A. Fredenslund, A Group Contribution Approach to ComputerAided Molecular Design, AIChE Journal, 37(9):1318, 1991 6. C.D. Maranas, Optimal Molecular Design under Property Prediction Uncertainty, AIChE Journal, 43(5):1250, 1997. 7. C.D. Maranas, Optimal Computer-Aided Molecular Design: A Polymer Design Case Study, Ind. Chem. Eng. Res., 35(10):3403, 1996. 8. G.L. Moore, C. D. Maranas, K. R. Gutshall, and J. E. Brenchley, Modeling and Optimization of DNA Recombination, Comput. Chem. Eng., 24(2/7):693, 2000. 9. R. Vaidyanathan and M. E1-Halwagi, Computer-aided design of high performance polymers, Journal of Elastomers and Plastics, 26(3):277, 1994. 10. S. Vaidyaraman and C. D. Maranas, Optimal Synthesis of Refrigeration Cycles and Selection of Refrigerants. AIChE Journal, 45(5):997, 1999. 11. D. W. van Krevelen, Properties of Polymers: their correlation with chemical structure; their numerical estimation and prediction from additive group contributions, Elsevier, Amsterdam, 3rd edition, 1990.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1165
Designing Efficient, Economic and Environmentally Friendly Chemical Processes R. L. Smith a, T. M. Mata b, D. M. Young a, H. Cabezas a, and C. A. V. Costa b aNational Risk Management Research Laboratory U.S. Environmental Protection Agency 26 W. Martin Luther King Drive Cincinnati, OH 45268 USA bLaboratory of Processes, Environment and Energy Engineering Faculty of Engineering, University of Porto Rua Dr. Roberto Frias, 4200-465 Porto, Portugal A catalytic reforming process has been studied using hierarchical design and simulation calculations. Approximations for the fugitive emissions indicate which streams allow the most value to be lost and which have the highest potential environmental impact. One can use this information to focus attention on these high value and/or high potential environmental impact streams. It was also found that increased recycling leads to higher potential environmental impact. The effect of larger recycle flows is an increase in fugitive emissions, which leads to larger impacts. 1. INTRODUCTION In the design of chemical processes one can consider aspects of economics, flexibility, safety, controllability, and environmental friendliness to prepare for possible future circumstances. While no one can predict the future, it is prudent to take precautions when making long-term decisions- like those involving chemical process design. The precautionary principle invokes a determination to act so that mistakes are made on the side of being too safe or too environmentally friendly. This work examines a process for which it is assumed that all of the open emissions to air, water, etc. are prevented (i.e., not released to the environment). Improving the environmental friendliness above the level of preventing open emissions means adding elements of precaution and thoroughness to a design. The critical aspect in such a case is that fugitive emissions become a dominant source of environmental impact. Therefore, this work examines the possible fugitive emissions from the catalytic reformer of a refinery, considering how one could design such a process economically and with the least potential environmental impact (PEI).
1166 2. BACKGROUND Designing chemical processes involves many people and techniques. To get an initial perspective on design alternatives the methods of Douglas (1988) can be employed. This hierarchical method leads to the synthesis of design alternatives based on various chemistries, pieces of equipment, and/or operating conditions. A hierarchical approach for waste minimization has been suggested by Rossiter and Klee (1995). The resulting flowsheets and evaluations can quickly eliminate some designs and point to others for further analysis. Once a superstructure of process elements is developed, one can use optimization methods to further specify a process (e.g., Pistikopoulos et al., 1994). One method to further analyze design alternatives is to use a process simulator (ASPEN PLUS, CHEMCAD, HYSYS, or PRO/II for example). These software programs can do a more rigorous analysis than the simpler short-cut methods often used to initially synthesize a flowsheet. The results of such an analysis can more easily include detailed reaction kinetics, separations with more complex splits, rigorous recycle streams, etc. While this further level of detailed analysis takes longer to set up and calculate, the resulting flowsheets can increase the confidence in a design. When designing processes with a view towards the environment one can use the Waste Reduction (WAR) algorithm (Young and Cabezas, 1999), which allows the user to quantify the environmental concerns of a chemical process. WAR has made available a method to simply evaluate processes with a library of approximately 1600 chemicals. (For related references, Cano-Ruiz and McRae (1998) have reviewed the environmental design literature.) The WAR algorithm applies a balance equation around the process of interest, and the potential environmental impact for a stream entering or leaving is defined as the impact of dumping that stream directly into the environment. The local impacts considered by this method include human toxicity by ingestion and by dermal/inhalation routes, terrestrial toxicity, and aquatic toxicity. Regional impacts evaluated by WAR are photo-chemical oxidation and acid rain, while global impacts include global warming and ozone depletion. Each of these categories has scores that have been normalized within the category, while weighting factors (currently all set equal to 1.0) are used to balance between the categories. Before this work, WAR has always been applied to streams that were openly emitted to the environment. These open emissions of chemicals are decreasing due to laws, penalties and increased community awareness. As a result less chemicals are allowed to enter the environment. However, fugitive emissions still escape from processes and create environmental impacts. Siegell (1997) describes the distribution of VOC's from point and non-point sources, with the majority (86%) of non-point sources being due to fugitive emissions. 3. H I E R A R C H I C A L DESIGN OF PROCESSES W I T H O U T WASTE STREAMS By controlling emissions that are openly emitted to the environment, considerable impacts are avoided. Even so, fugitive emissions do escape from processes and create environmental impacts. Here, we examine possible fugitive emissions, where an assumption is made that 0.1% of each stream becomes a fugitive emission. The expectation is that this is not a very accurate assumption, but rather something that can be refined later (e.g., by the methods described in EPA, 1993).
1167 The process under consideration is the catalytic reformer, where the reformer takes a naphtha fraction (mostly paraffins) and reacts it into higher octane gasoline (isoparaffins, naphthenes and aromatics), light gases, and hydrogen. (In some versions of the process the aromatics generated are further separated and processed.) The reactions and kinetics used in our model are described in Padmavathi and Chaudhuri (1997). Shown in Figure 1 is a diagram of the process, which consists of a reactor, a separation system (in this case a flash and distillation tower), heat exchangers, a valve for decreasing stream pressure, and sometimes a recycle stream (with an additional compressor and heat exchanger). Various operating conditions can be considered: reactor temperatures of 460-540~ reactor pressures of 500-3000 kPa, and hydrogen recycles of 0-80 percent (equivalent to defining a hydrogen to hydrocarbon molar ratio for the reactor feed). These various operating conditions are important because of their affects on the process. For instance, reactor temperature will affect the product distribution, and percent hydrogen recycled will affect the size of the recycle stream (and associated equipment).
l REACTOR
Fig. 1. The catalytic reforming process, shown without flash overhead recycle. Through spreadsheet calculations based on a hierarchical design procedure as described in Douglas (1988), a fiowsheet as shown in Figure 1 can be developed. This flowsheet is based on modeling of 26 components and 35 reactions (integrating 21 independent extents of reaction for an isothermal plug flow reactor). These reactions do not include cyclopentanes for the hierarchical design calculations. For a reactor temperature and pressure of 460~ and 500 kPa with no recycle, the results shown in Figures 2a and 2b were obtained. Figure 2a shows that the economic potential, EP, decreases with the addition of reactor and separation system costs in levels three and four of the hierarchy, respectively. (Level one is where a decision between batch and continuous operation is made.) Level two of the hierarchy considers only product values and feed and fugitive emission costs. With details added at each level, more streams are added to the flowsheet, and the amount of fugitive emissions increase. This represents an increased number of valves, flanges, and other equipment that can leak. Thus, in a manner similar to the economic potential only decreasing with additional detail, the potential environmental impact from fugitive emissions only increases with additional detail. This is a major result of this work, as it allows one to eliminate designs with too high of a potential environmental impact early in the hierarchical design procedure (i.e., with the least amount of work). In the case of Figure 2a, the potential environmental impact increases with more streams and their associated fugitive emissions, while PEI decreases with the conversion of n-paraffins at each level. The reason for the decreasing PEI with conversion is shown in Figure 2b, which
1168 shows that the reactor products and associated down-stream flows have lower PEI values due to the conversion of reactants into potentially less harmful products. Based on the economic potential of Figure 2a one would choose either low or high conversions for operation (i.e., not around 0.65), while the potential environmental impact points towards using higher conversions. Note that the increase in economic potential at higher conversions is due to an increasing octane number that leads to higher value for the products. This increase in value becomes more dramatic as the octane level increases, and so the really desirable products are made at higher conversions. The results displayed in Figure 2b depict the fugitive emissions in terms of their value (lost) and the potential environmental impacts from each stream. The figure shows that the fugitive emissions from the feed have the highest value with reactor products, tower feed and tower bottoms close in value. The tower tops (butane and smaller molecules) have less value, and the flash overhead has even less value (not depicted here). Thus, from this figure one can see that more attention should be concentrated on certain streams and less to others, unless of course, one knows that a stream has very high emissions. For example, if the tower tops leaked 10 times more than the other streams, it would be of higher significance in terms of economic potential. 1.4 l0 s-
4.0 10'
1.2 1 oS- :" - .-'-~. _.~.
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4.0 lO 4-
-
-
- Tower Topq
2.0 104-
.--"
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I
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.
. | 0.80
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Fig. 2. a) Total economic potential and potential environmental impact for various levels of the design hierarchy, b) Stream-specific economic potentials and potential environmental impacts of fugitive emissions.
1169 The potential environmental impacts by stream shown in Figure 2b follow the economic potentials rather closely. However, one can see that the feed impact is relatively more important at higher conversions, rather than the reactor products and tower feed streams which are larger at the lowest conversions. Again, this figure directs attention to the streams with potentially higher environmental impacts. 4. SIMULATION FOR E N V I R O N M E N T A L PROCESS DESIGN
After considering aspects of hierarchical design in paper and spreadsheet form as described above, it is often valuable to consider simulating the process. The simulations described offer more details of the process than from using shorthand calculations. In particular, the model of the process has been extended to 43 species and 91 reactions. For these simulations an isothermal reactor has been assumed, where normally the reactor is adiabatic with interstage heating (e.g., Rachford and Gilsdorf, 1994). A simulation at 500~ 3000 kPa, and 50% hydrogen recycle leads to a conversion of nparaffins of 0.90 and a research octane number of 96. This compares favorably to the model used in the hierarchical design to create Figure 2, where the maximum octane number was 90. This points to the need to include the various cyclopentane species in the model. Without the cyclopentane to cyclohexane route which continues on to produce aromatics, the paraffins predominantly react to form cracked products. Thus, simulating with a more complete model can be an important aspect of achieving meaningful results. An interesting result obtained from simulating identical reactors at 3000 kPa was found for various reactor temperatures and percent recycles. The results are shown in Table 1. For all temperatures the potential environmental impact increased with the amount of recycle. Since larger recycles mean larger flowrates through much of the process, it follows that the fugitive emissions are larger. With the increasing fugitive emissions the potential environmental impact increases. However, the impact does not increase proportionately to the amount of recycle. For instance, 80% recycle does not have a PEI close to four times that for 20% recycle. There are two reasons for this effect: one is that not all the process streams participate in the recycle loop, and two is that the recycle composition is mostly hydrogen which has zero environmental impact. Even with the dominance of hydrogen in the recycle, the potential environmental impact due to fugitive emissions increases with recycle. One could speculate that for other processes where the dominant recycle species is less environmentally friendly, that the affect of recycle on environmental impacts would be more substantial. Table 1. Potential environmental impact per year from fugitive emissions. Temperature (~
Percent Flash Overhead Recycled 0
20
50
80
460
6499
7352
7593
8093
500
6192
7082
7347
8169
540
6046
7115
7651
9158
1170 5. CONCLUSIONS From hierarchical design and simulation calculations performed on catalytic reformer models, fugitive emission results have been obtained. The fugitive emissions indicate that emissions from certain streams have more value than others, and also that emissions from certain streams have higher potential environmental impacts. With this knowledge one can place more attention on these streams. In addition, more recycling of chemicals through the process increases the potential environmental impact. The effect of recycle on the potential environmental impact is less pronounced when hydrogen dominates the recycle flow, although one would expect it to be more important when other chemicals dominate the recycle. REFERENCES
Cano-Ruiz, J. A. and McRae, G. J. (1998) "Environmentally Conscious Chemical Process Design," Annu. Rev. Energy Environ., 23,499-536. Douglas, J. M. (1988) Conceptual Design of Chemical Processes. New York: McGraw-Hill. EPA 453/R-93-026 (1993) Protocol for Equipment Leak Emission Estimates. Office of Air Quality Planning and Standards, Research Triangle Park, NC. Padmavathi, G. and Chaudhuri, K. K. (1997) "Modelling and Simulation of Commercial Catalytic Naphtha Reformers," Can. J. Chem. Engng., 75, 930-937. Pistikopoulos, E. N., Stefanis, S. K. and Livingston, A.G. (1994) "A Methodology for Minimum Environmental Impact Analysis," in E1-Halwagi, M. M. and Petrides, D. P. (eds.), Pollution Prevention via Process and Product Modifications, AIChE Symp. Ser., No. 303, Vol. 90, 139150. Rachford, R. H. and Gilsdorf, N. L. (1994) "Reforming Processes for Fuel Production," in McKetta, J. J. (ed.), Encyclopedia of Chemical Processing and Design, New York: Marcel Dekker, 47, 133-151. Rossiter, A. P. and Klee, H. (1995) "Hierarchical Process Review for Waste Minimization," in Rossiter, A. P. (ed.), Waste Minimization through Process Design, New York: McGraw-Hill. Siegell, J. H. (1997) "Improve VOC Emission Predictions," Hydrocarbon Processing, April, 119-121. Young, D. M. and Cabezas H. (1999) "Designing Sustainable Processes with Simulation: The Waste Reduction (WAR) Algorithm," Comput. Chem. Engng., 23, 1477-1491.
European Symposium on Computer Aided Process Engineering - 11 R. Gani and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1171
Study on lifecycle and agility of process industry Xin' an Xie a, Ben Hua a, Qinglin Chen a, Ren Liangb, Mingang Zeng a aSouth China University of Technology, Guangzhou, 510641, RR.China bGuangdong College of Pharmacy, Guangzhou, 510224, RR.China Based on the characters of process industry, the life cycle of process enterprise and its agility were researched and analyzed. According to the concept of product life cycle, (a) the model for the life cycle of process enterprise was developed, (b) agile variables of process enterprise change were proposed, and (c) the process enterprise agility model oriented to the life cycle of process enterprise was presented. 1. INTRODUCTION Because of the economic globality, market dynamic characters and the intensification of market competition, the lifecycle of products and process technology in the process industry becomes shorter and shorter. To meet the changeful market demands, there are a series of problems to be solved. Such as: (a) how to speed up development of new products, new process technology and advanced equipment; (b) how to control production, so as to improve the agility of the process industry and so on. Therefor, many new concepts, thoughts and approaches in discrete manufacturing industry were put forward, like FMS(Flexible Manufacturing Systems), CIMS(Computer Integrated Manufacturing Systems) and LPS(Lean Production Systems). These technologies and ideas emphasize particularly on information integration of an enterprise, but neglect integration of information, technology and human resources. With the development of CIMS in discrete manufacturing systems, computer Integrated Process Systems (CIPS) was put forward based on the concept of CIMS. In recent years, CIPS has been succeeded in applying to many petrochemical complexes and chemical plants. The application of CIPS in current can be reduced to integrated application of information technology, it just stay at the step of integration of enterprise infra information. There is no consideration of lifecycle integration including R&D, production and marketing. [11 Base on above-mentioned causes, the concurrent engineering (CE) which the main idea is the life cycle of a product and agile manufacturing (AM) which evaluates the agility of a product in the life cycle were brought forward. Agile manufacturing is a dynamic integration of enterprises from lifecycle. It requires incorporating flexible process systems and human resources in order to get maximum benefits in the long run. The target of agile manufacturing is global optimization of social, economic, resources and environment. 2. LIFECYCLE OF PROCESS INDUSTRY ENTERPRISES Lifecycle is a new design thought for shortening time of development and production of a product, improving production flexibility.
1172 The product's lifecycle includes raw material extraction and processing, manufacturing, using, recycling, and ultimate disposal, so the every phase of a product's lifecycle should be considered from its initial conceptual design to disposition or regeneration after being used. Phases of a product's lifecycle are shown in Fig. 1. Fig.1 shows that a product come out was through many phases to be considered such as market demands, design, development, production and so on. These phases are determined by many factors in the outside circle above mentioned, like working condition, corporation's policy, utilizing resources and product's characters, etc.
Environment fmanufacture f ~ characters / " ' ~ ' ~ / "
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] utilizing /
~oduct's ~ lifecqg.,v.deI
Fig. 1. Phases in the lifecycle of a product In virtue of the idea of a product's lifecycle in discrete manufacturing systems, the lifecycle oriented model of a continuous process industry enterprise will be set up based on the characters of continuous process industry. The raw materials and products in continuous process industry are not subjects assembled by several parts, but homogenous materials that are characterized by such chemical and physical aspects as purity and viscosity. There are many differences from discrete manufacturing systems as follows: (a) The production of continuous process industry is continuous or batch with a large scale. Material flow and energy flow is continuous and relative stable. Flowsheets are relative constant and no pause is permitted among operation units during the long production cycle. However, multi-products and discontinuous are the characteristics of discrete manufacturing systems. They have much flexibility by revising their assembling lines. (b) With waste emission during production, environment protection is a heavy duty to continuous process industry. Many units were operated under such critical environment as high temperature, high pressure, easy to flame, to explode and to poison. One unit being out of work will effect global system in no time. Safety and reliability require rigorous automated control. Therefore, fault diagnosis is very difficult but necessary. Operations in discrete manufacturing systems are usually under common environment. Fault diagnosis is relatively easy since units operated independently. (c) Continuous process industry is usually energy intensive. Especially in China, it contributes more than 50% of national energy consumption, only chemical industry, metallurgy and construction materials departments contribute 40.6%. Many products of
1173 chemical industry and metallurgy cost 1.5 or double energy in average than that in developed countries It1. Furthermore, energy intensive means high pollution and low energy-use efficiency in some tense. Therefore, energy conservation has great significance in continuous process industry and is the key to energy resources sustainability. It offers great potential profits both in economy and environment. (d) The optimization of continuous process systems aims to safety, stable, balance, long operating period, precise control and no pollution, through planning, scheduling and keeping operational parameters around optimized set points. Depending on the characteristics of continuous process enterprises, the hierarchical structure model of process industry system was put forward, as shown in the Fig.2. [21 Hard Technology
Soft Technology
~ . . . . . . . . + CapitalFlow DCS/APC~_-.-------'~ . . . . . . ~........ Maintenance Stable Pr Accidents / / . - - " ~ / / [ \Human Ware Fixed F~ed Stock I Safety _ . .- ~ ~ I ~nformation Network Product~ ] On-line M o n ~ / / ] / Flowsh]eets, and [ Fault D i a ~ o s i s ~ ~ I Enviro~lme~ Systemeval~);rrf ~.('~M~rtrmlno~",)l~urehasing | k, uperatlOn~ Debottlene~g ~ ~ ~oduct Marketing Operatipn O p i 7 " - - 7 ~ OperationO p t . ~~~~ [ t/evises Sutmlv Changl~d Feed[ Planning&ScheduyJJJ [ / Stock, k Product I Process S y n t ~ ~ l ~ " / / I / & E n v i r o n ~ Proc~/egrfit/jafi~ ...~---l.----...~J Fle~i/hitity/~ Decision ~ Expansion O p e r a ~ y ~ Making ..// Investment Reproduction: Expah~ ~ ~ -"'----'-"/ MarketPrediction Simple Sustai'na~ ~ Retrofit Expansion New Product R&D *DCS( Discrete Control Systems),APC(Advanced Process Control),IC(IntelligenceControl) Fig.2 The hierarchical structure model of process industry systems This model consists of two parts of the hardware technique and software technique of process industry systems. Two parts includes six levels such as design, operation, control and decision making, marketing, management. It shows clearly the relationship among of them and points out that process optimization is not only control optimization or operation optimization, but global optimization of integrated all activities from all levels. From the point of process industry systems lifecycle, the hierarchical structure model of process industry systems may be modified as the lifecycle oriented model of a continuous process industry enterprise. It is shown in Fig.3. Fig.3 shows that the lifecycle oriented model of a continuous process industry enterprise include an inner level and an outer level. The inner level is important in process industry systems because about 85%of the lifecycle cost is determined by the decisions made in this level [2 ]. The whole process from a plant design, control, operation to redesign forms the lifecycle of process industry enterprises. The outer level contains three phases made of management, marketing and decision making. The phases in the outer level are decision-making variables of the inner in the process system lifecycle. The change of each phase of lifecycle in process industry system is result of response to outer factors directly.
1174 Fig.3 indicates the relationship between the inner level and the outer clearly.
Fig.3 the lifecycle oriented model of a continuous process industry enterprise
Therefore, the model gives a study framework for choosing qualitative factors and quantitative factors, studying the relationship between the inner factors and the outer factors and researching the agility of a process enterprise. 3. AGILITY AND ITS MEASUREMENTS OF PROCESS ENTERPRISE
3.1. Enterprise agility Enterprise agility was put forward first in discrete manufacturing systems. Agile manufacturing is a dynamic integration of enterprises from lifecycle. It requires incorporating flexible manufacturing systems and human resources in order to get maximum benefits and responding quickly to uncertain market. The figure of two dimensions describing the enterprise agility was shown in Fig.4 based on the ability that a plant grasps chances and possesses technique innovation [3'4l.
Reactive viability
Opportunistic -Fragile
//)~ ~ g i l e ~/f Innovative
Proactive leadershin Fig.4 Figure of two dimensions describing the enterprise agility
Fig.4 shows that the reactive viability and the proactive leadership are abilities to complete agile changes in a plant, and they structure the figure of two dimensions describing the enterprise agility. Reactive viability is the abilities to manufacture products with high quality, high reliability and low price when an enterprise encounters a chance. Proactive leadership is the abilities to forecast the future tendency of market changing and develop new products through adjusting planing and designing. Assessing the enterprise agility, determining the position in Fig.4 and comparing the agility between enterprises, the enterprise agility must be measured. Studying the agile degree of each phase in a process enterprise lifecycle responding to a market, every factor impacting the agility must be found out and measured.
1175
3.2 Integration items of estimating the enterprise agility The enterprise agility is measured by four items, the items consist of cost(C), time (T), robustness(R) and scope of change (S). They are called CTRS for short [5]. The cost is the money needed that the enterprise completes once agile change, it correlates with the cost that a new product comes into market, and with the design and manufacture in the process. The time is hours spent that the enterprise completes once agile change. The robustness is obdurability and stability during completing an agile change. Scope of change is the ability to complete an agile change in a plant. The CTRS are restricted and associated. Therefore, the proportion of the CTRS must be taken into account if the global optimization is achieved in a plant. In other words, the quick change without cost is meaningless. The change without robustness is not agile for a plant under the limitation of the cost and the time. So is the scope of change. Hence the synergy among the cost, time, robustness and scope of change must be taken into account in studying the agility of process systems. 3.3 Agility variables of a process enterprise Agility variables of discrete manufacturing systems are lumped into the communication connectedness, interorganization participation, production flexibility, management involvement, and employee empowerment. These variables are called CIPME for short in discrete manufacturing systems. They stand for 5 aspects of discrete manufacturing systems, and to be used to describe the manufacturing systems agility. Continuous process systems are different from discrete manufacturing systems in many places. There are characters in continuous process systems. Therefore, variables impacting process systems agility differ from the discrete manufacturing systems. Based on the lifecycle model of process systems in Fig.3 described, agility variables of process systems are placed in the every phase in the lifecycle of process systems. Because the model in Fig.3 is continual, agility variables are not easy to be decomposed. The model in Fig.3 is modified as the "discrete" lifecycle model of process systems. It is shown in Fig.5. The basic structure of process enterprises ,,..--
Management
o i-.
o o
Marketing
Decision making
Design Control Operation
.,.,
.1
Redesign
Fig.5 the lifecycle two dimensions model with agility variables of process systems
The model is expressed with two dimensions table. These intersections (little panes) in Fig. 5 illustrate relationship of agility variables against themselves functions in each phase in process systems lifecycle. Because there are more intersections in the model i.e. there are more agility variables, it results in troublesome in setting up and solving the model. Depending on the characters of process systems, the agility variables are lumped into 6 types for solving the model easily. The agility variables of process systems are: material flow
1176 variable (M), energy flow variable (E), information flow variable (I), humanware flow variable (H), cost flow variable (C) and workpiece flow variable (W). They are called MEIHCW agility variables for short. The MEIHCW variables vary with different phase in the lifecycle of process systems, hence they are dynamic variables. 4. THE LIFECYCLE MATHEMATIC MODEL WITH AGILITY VARIABLES OF PROCESS SYSTEMS The lifecycle mathematic model with agility variables of process systems would be used to evaluate agility degree and find out the key step or the "blunt point" responded to change of process systems in different phase of whole lifecycle, so as to guide decision-making for an enterprise. Taking the MEIHCW agility variables as independent variable of measuring the CTRS synthetically, the function relationships are expressed as follows. T l=fl (M, E, I, H, C, W) T2=f2 (M, E, I, H, C, W) T3=f3 (M, E, I, H, C, W) Ta=f4 (M, E, I, H, C, W) Where flDf2Df3Dand f4r refer to the needed time, cost, robustness and scope of change respectively when the process enterprise completed an agile change. 5. CONCLUSIONS AND SUGGESTIONS This paper discusses the lifecycle and the agility in the process industry enterprise, and brings forward: (1) Lifecycle model of process industry enterprise, (2) The agility variables of process industry enterprise---MEIHCW, (3) The agility framework model of process industry enterprise, Because the concepts of lifecycle and agility were introduced in disperse manufacture, they are in studying and developing, so lifecycle model with agility variables of process systems should be more studied and developed from a few aspects as follows: (1)The lifecycle of process systems should be more researched so as to increase the meaning of the lifecycle. (2)The concepts about the material flow variable, energy flow variable, information flow variable, humanware flow variable, cost flow variable and workpiece flow variable should be analyzed clearly. The relationship among the "flows" should be made certain. ACKNOWLEDGEMENTS Financial support from the Chinese NSFC (No. 7993100) and GDSFC (No. 990638) is greatly acknowledged. REFERENCES
1. Cheng, Siwei, Proceedings of Seminar on Process System Engineering and Intensivism, Jiangxi, (1998) 1. 2. Hua, Ben, EHan, Proceedings of Seminar on Process System Engineering and Intensivism, Jiangxi, (1998)46. 3. Roger Nagel, Rick Dove. 21 st Century Manufacturing Enterprise Strategy. Iacocca institute, Lehigh University, 1991. 4. Rick Dove. Production Magazine, 9(1995). 5. Rick Dove. Production Magazine, 6(1995).
European Symposium on Computer Aided Process Engineering - 11 R. Gain and S.B. Jorgensen (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
1177
Dynamic Assessment of Induced Thermal Stresses on the Semiconductor Packaging Substrates in a Radiation Belt Furnace by Computational Fluid Dynamics Thomas C.-K. Yang *a, Hsien-Sheng Lin a , Sea-Fue Wangb, and M.-C. Cheng a a Department of Chemical Engineering, [email protected] b Department of Materials and Resource Engineering National Taipei University of Technology, Taipei, TAIWAN This paper is studying the energy dissipation, transport, and the temperature distribution of the packaging substrates along the moving belt during the heating stage in the time basis. After
achieving
the
temperature
profiles,
coefficients
of thermal
expansion
and
temperature-dependent thermophysical properties are used as parameters for further structural analysis. Transient thermal stress, which is highly sensitive to thermal history and process time, was calculated at each grid point along the belt. As a result, the subsequent stress-time profile integrated all over the heating surface can then be used to investigate the thermal response and product quality. This information will be very useful to the process engineers and furnace designers. 1. INTRODUCTION Residual stresses of films and substrates induced during the thin film deposition or thick film fabrication play an important role in determining the reliability of electronic devices. As the temperature profiles are distributed unevenly on the heated substrate, thermal stresses will develop in the films due to a mismatch in the coefficient of thermal expansion between film and substrate. As the temperature gradient keeps increase, large stresses are generated, which may induce plastic deformation and creep deformation in the film. As a result, breakage of interconnecting line structures and short circuits between adjacent lines occur based on this stress migration [1]. These phenomena cause the defects of VLSI devices. Therefore, knowledge on the thermal stress in film structures during the heating process is very essential to understand the mechanism of formation of such faults. The radiation belt furnace has the intrinsic advantages over a conventional batch-type furnace in the aspects of a continuous operation, high ramping rates, fast response time [2,3]. Therefore, it is widely adopted for the semiconductor applications such as the curing of solder
1178 masks, thin film deposition and thick film cofiring. This paper is studying the energy dissipation, transport, and the temperature distribution of the packaging substrates along the moving belt during the heating stage in the time basis. After achieving the temperature profiles, coefficients of thermal expansion and elasticity modulus measured by the dynamic mechanical analyzer will be used as parameters for further structural analysis. The calculated stress will be compared with the maximum yield stress, which will lead the plastic deformation. This information will be very useful to the process engineers and furnace designers. 2. ANALYTICAL SYSTEM Figure 1 illustrates the firing processes of ceramic packaging substrates or thick films in a belt furnace. The sliding substrates in the entrance zone initially encounter a hot surface as it moves down the tube to the burnout zone at the temperature of 3 0 0 - 400 ~
for binder
removal. After the completeness of the polymer pyrolysis, propagating substrates were then carried into a higher temperature zone for sintering, where the densification of ceramic powders occurred. As the sintering stage completed, substrates were lead to a cooling section for annealing to room temperature [4].
Figure 1. the cross section of a belt furnace. 2.1 Grid Generation
For establishing a computation domain, an unstructured grid method (6) was used to define the geometric dimensions of the furnace including the heating elements, conveyor belt and the substrates. The furnace entrance has san inside dimension of 25.4 cm by 35.6 cm with the length of 610.0 cm. The geometry of the substrate is defined as 25.0 cm by 20.0 cm with a thickness of 0.6 cm. A thermal profile of 12 heating zones on the furnace wall is also given according to the normal operation setting programs. The inlet for the purging gases and the outlet for the evolution of organic vapors were all placed and defined in appropriate locations. Accordingly, the inlet velocity and outlet gas volumetric flow rates are given as 84.1 cm/min and 5.1x105 cm3/min. In summary, a three-dimensional flow model accompanied with radiation/convection heat transfer was imposed onto a belt furnace. About 105 grids were generated to undergo the finite-volume calculation. Various numbers of grids have been tested and no significantly different results were observed.
1179 2.2 Numerical Scheme for Moving Mesh To take the belt conveying speed into consideration, the sliding grid technique was utilized, instead of using the moving wall boundary, to generate the moving mesh for the belt and the substrate as shown in Figure 2. To facilitate the moving mesh, an unsteady state computation mode was applied [5]. The time step [~t of the system was selected so that the mesh was moved the grid spacing within that interval. This simulates the relative motion of the substrate as well as the belt to the wall. At each time step, the thermophysical properties of the leading edge of the substrate are changed from the fluid to substrate. In addition, for every one physical second, it uses 20 iterations for finite-volume computation.
Figure 2. Diagram of the computing domain. 2.3 Calculation of Thermal Stresses In order to estimate the magnitude of the heat transfer for substrates sliding along the wall, thermal conduction, force convection and thermal radiation modes were all considered This simulation uses the Discrete Ordinate method to account for the radiative property of non-gray body, where the emission energy varies with frequency (e=0.8 for 4~tm <X< 8~m, e=0 for both ~<4 pm and E >8 9m). The force convection effect was considered in the computation since the purging gasses were facilitated to remove the pyrolysis gasses during the firing process. After the temperature distributions in the solution domain were computed, thermal stresses then can be evaluated based on the transient temperature gradients. The governing equation for stress analysis is the thermoelastic equation:
Ao- = nC,,A~/(a. ~') where T is the absolute temperature, D the thermal expansion coefficient, Cp the heat capacity and M7? the thermal stresses. All the material properties are evaluated as a function of the temperature. As a result, the thermal stress at each grid point can be calculated by applying the calculated temperature gradients (such as
OT 0x
of the substrate.
aT and ~ ) with thermophysical properties @
1180 3. RESULTS AND DISCUSSIONS Figure 3 illustrates the velocity profiles and path-lines of purging gases as well as the evolving gasses. The arrows on the streamlines show the directions of the flows and obviously the chimney performs a strong suction force toward all the gasses. The expression of multicolor in the vector plots and streamlines indicates the temperature profiles. Blue line shows the temperature of 300K whereas the red line symbolizes the highest temperature in the furnace wall (1200K). Apparently, the substrates are successfully heated to l150K near the end of the furnace. This verifies that the heat transfer between substrates, furnace wall, surrounding fluid and moving belt all comes to a thermal equilibrium through the mode of conduction, convection and radiation. Figure 4 shows the temperature profile of substrates at different process time. When examined carefully in the feature figure, one can see clearly that the belt accompanied with substrates was moving continuously toward the exit direction. In addition, it takes substrates at least 50 seconds to reach the thermal equilibrium as the belt slides in a speed of 0.17 cm/s. The feature figure also shows the success usage of the sliding grid technique in the moving boundary for the dynamic system. In addition, the substrate in the last heating zone near the exit showed a gradual heating as the time elapsed and it finally goes to a stable temperature after the thermal equilibrium is achieved. Unsteady-state stress arises as a result of transient temperature gradients in a substrate as well as the mismatch between the corresponding thermal expansion and geometrically incompatible displacement within the substrate. Thermal stress and strains when below some critical values, will continuously adjust themselves to make the internal forces in the substrate self-equilibrating until a compatible displacements. However, this reversible deformation will be continuously accumulated as long as the thermal impact exists until the principal stress exceeds the mechanical limitation of the material. Figure 5 shows that the maximum thermal stress occurred at the heating zones of 10 and 11 corresponding to the highest heating temperature in the wall. The red flooded area of the substrate reads the stress intensity of 994 Mpa and the stress magnitude of the green flooded area reads to 500 Mpa. And these heavily induced stresses were built up near the four edges of the substrate. 4. CONCLUSIONS The energy dissipation, transport, and the temperature distribution of the packaging substrates along the moving belt during the heating stage was obtained. After achieving the temperature
profiles,
coefficients
of thermal
expansion
and temperature-dependent
thermophysical properties were used as parameters for further structural analysis. Transient thermal stress, which is highly sensitive to thermal history and process time, was calculated at each grid point along the belt. As a result, the subsequent stress-time profile integrated all
1181 over the heating surface can then be used to investigate the thermal response and product quality. This information will be very useful to the process engineers and furnace designers. ACKNOWLEDGEMENT Authors would like to thank Dr. Eric S.-Y. Cheng of Industrial Technology Research Institute for providing the detailed information on the belt fumace. Partial financial support form the national science council of Republic of China under the grant no.of 89-2214-E-027-003 was also appreciated. REFERENCES 1. A. G. Evans and J. W. Hutchinson, 'The Thermomechanical Integrity of the Thin Films and Multilayers', Acta Metal. Mater. 43 (1995) 2507. 2.D.K. Flattery, 'Rapid Binder Removal From Thick Film Composite', Radiant Tech. Corp., Cerritos, CA, U.S.A., 1995 3. C. R. S. Needs, 'Infrared Firing Studies of Thick Film Material System for High Volume Hybrid Microcircuits', The International Journal for Hybrid Microelectronics, 7 (1984) 1. 4. D. R. Barbour, 'Multichip Module Technology', Ceram. Eng. Sci. Proc., 9 (1988) 1549. 5. Fluent User's Guide, Fluent Inc., New Hampshire, U.S.A., 1999
Figure 3. Velocity profiles of evolving gasses and temperature distribution of the substrates as well as wall of furnace.
1182
Figure 4. Transient temperature profiles of the substrates along the moving belt.
Figure 5. Stress profiles of the substrates along the moving belt.
1183
Author Index Aartovaara, M. 597 Achenie, L.E.K. 585,1133 Afonso, A. 841 Agachi, S. 711 Agrawal, R. 339 Akay, B. 603 Aldrich, C. 75,81 Alexandridis, A.P. 69 Alfadala, H.E. 943 Allgower, F. 711 Alloula, K. 967 Alpbaz, M. 603 An, W. 773 Andersen, T.R. 627 Aoyama, A. 87,829 Aziz, N. 609 Azzaro-Pantel, C. 117,949 Badell, M. 835 Bafas, G.V. 69 Balakrishna, S. 401 Balogh, S. 381 Banares-Alcantara, R. 955 Bandoni, J.A. 615 Bansal, V. 273,961 Baptiste, D.R. 773 Barbosa-P6voa, A.P.F.D. 847 Barnard, J.P. 75,81 Barros, A.A.C. 321 Barton, G.W. 823 Barton, P.I. 309,767 Basualdo, M.S. 731 Batres, R. 87,829 Batzias F.A. 451 Baur, R. 93 Bayer, B. 345 Bek-Pedersen, E. 517 Belaud, J.-P. 967 Berezowski, M. 99 Bertok, B. 351 Bildea, C.S. 973 Binder, T. 1071 Binet, G. 651 Biscaia Jr., E.C. 123 Bj6rn, I.N. 105 Blanco, A.M. 615
Bliek, A. 463 Bock, H.G. 711 Bogle, I.D.L. 1047 Bok, J.-K. 767 Bonn6, D. 621 Boudouvis, A.G. 69 Bozmis, A. 979 Brauner, N. 291 Brignole, E.A. 375 Brown, S. 675 Biihner, C. 357 Bursali, N. 603 Caballero, J.A. 363 Cabassud, M. 817,1127 Cabezas, H. 1165 Camarda, K.V. 369 Cameron, D.B. 111 Cameron, I.T. 195 Canton, J. 841 Casamatta, G. 817,1127 Castillo, F.J.L. 591,1089 Castro, P. 847 Cerd6, J. 693 Cezerac, J. 1127 Chang, T.S. 645 Charles, A.S. 117 Chen, H.-Z. 1035 Chen, Q. 1171 Cheng, M.-C. 1177 Chiu, M.-S. 333 Choi, D.-K. 427 Chung, P.W.H. 1115 Cismondi, M. 375 Clark, G.A. 1115 Cofer, D. 1 Coffey, D.P. 627 Confinillo, G. 135 Cordiner, J. 27 Cortese, F. 723 Corvahln, S.M. 129 Costa, C.A.V. 1165 Costa, J. 413 Costa Jr., E.F. 123 CresciteUi, S. 225 Cruse, A. 1071 Csukas, B. 381
Cutlip, M.B. 291 Dash, S. 853 Davin, A. 949 de Deugd, R.M. 699 de Jong, P. 681 Dechechi, E.C. 633 Devereux, B.M. 243 Di Puma, J. 1053 Diehl, M. 711 Dimian, A.C. 463,973 Domenech, S. 117,949 Domingues, A. 805 Douglas, D. 249 Dua, V. 979 Duarte, B. 847 Dunn, R.F. 985 E1-Halwagi, M.M. 943 Elgue, S. 1127 Eliceche, A.M. 129,387 Ender, L. 639 Engl, G. 433 Eo, S.Y. 645 Ertunc, S. 603 Espuna, A. 841 Ettedgui, E. 817 Fan, L.T. 351 Faraoni, V. 135 Farza, M. 651 Feliu, J.A. 687 Feng, G. 351 Fernandez-Anaya, G. 755 Figueroa, J.L. 615 Findeisen, R. 711 Fischer, U. 573 Floquet, P. 117,949 Flores-Tlacuahuac, A. 755 Floudas, C . A . 153,393 Fonyo, Z. 553,1109 Fraga, E.S. 955 Fraser, D.M. 991 Freitas Jr., B.B. 279 Friedl, A. 535 Friedler, F. 351,541,547 Gabbar, H.A. 859 Gal6n, O. 141 Gani, R. 559,773,997
1184 Garea, A. 147 Garg, S. 1133 Gatica, G. 865 Gehan, O. 651 GeorgiaNs, M.C. 997,1139 Gerbaud, V. 499 Gesthuisen, R. 183 Gilbert, R.G. 823 Gioia, F. 225 Glavic, P. 529 Glende, E. 111 Gomes, V.G. 823 G6rak, A. 285 Govatsmark, M.S. 657 Graells, M. 841,913,1077 Grancharova, A. 1003 Gren, U. 105 Grof, Z. 177 Grosman, B. 663 Grossmann, I.E. 363,487, 877 Grzywacz, R. 99 Gudi, R.D. 243 Giimiis, Z.H. 393 Gupta, A. 871 Halim, I. 1145 Hallale, N. 445 Hangos, K.M. 195,787 Hapoglu, H. 603 Harding, N. 991 Harding, S.T. 153 Harjunkoski, I. 877 Harper, P. 925 Haug-Warberg, T. 297 Hechavarria, T.L. 147 Heckl, I. 541 Henning, G.P. 883 Henriksen, J.P. 1009 Heppert, j. 369 Herron, D.M. 339 Hertwig, T.A. 1017 Hertzberg, T. 219 Hfitreux, G. 889 Heyen, G. 1053 Hill, pJ. 1151 Hirao, M. 1023 Hjertager, B.H. 159 Hjertager, L.K. 159
Hoch, P.M. 387 Hofbauer, H. 535 Hopper, J.R. 1017 Hostrup, M. 401,517 Hua, B. 1171 Huang, Y. 669 Hui, C.-W. 1029 Hungerbfihler, K. 573 Hurrne, M. 469 Iedema, P.D. 973 Ierapetritou, M.G. 407 Irabien, A. 147 Ishikawa, T. 559 acobsen, E.W. 99,207,737 arke, M. 345 laume, D. 117 ezowksi, J. 1089 im~nez, L. 413,731 iobson, M. 505,511 [ordache, C. 675 Foulia, X. 201,499,967 Jorgensen, S.B. 189,621, 627,773,787 Kahn, D. 419 ICalitventzeff, B. 457,1053 ICantharao, S. 853 ICapteijn, F. 699 Karim, N.M. 761 ICaylor, J.M. 773 Kenig, E.Y. 285 Kheawhom, S. 1023 Kim, D. 895 Kirn, M. 427 Kint, E. 681 Kiparissides, C. 327,705 Ko, D. 427 K6hler, R. 165 Kohout, M. 171 Kokossis, A.C. 451,493, 541,919,1083 559 Kolar, P. 177 Kosek, J. 1139 Kostoglou, M. 249 Kozel, L. 183 Kr~imer, S. 439 Kraslawski, A. 1095 Kravanja, Z. 93 Krishna, R.
Ktistensen, N.R. 189,787 Kr6ner, A. 433 Kronseder, Th. 433 Kubicek, M. 171 Kwok, Y.-Y. 1029 Lakner, R. 195 Lapham, D.S. 773 201,967, Le Lann, J.M. 1127 Le Lann, M.-V. 717,817 645 Lee, B. Lee, T.-y. 547 Lee, Y. 895 Lee, Y. 895 553,1109 Lelkes, Z. Lenz, D.H. 773 663 Lewm, D.R. 1035 Li, B.-H. 1035 Li, H.-Q. 439 Li, X.-N.L. 1035 Li, Z.-H. 1171 Liang, R. 1041 Lid, T. 249 Lira, P.P.S. 201 Lim, Y.I. 1177 Lm, H.-S. 493 Linke, P. 445 Liu, F. 207 Liu, Y. 237 Lopez, G.D. 213 Lucia, A. 811 Lukasse, LJ:S. 901 Luo, K. 773 Lymbumer, C.J. 219 Lovik, I. 1047 Ma, K. 687 Macias, j.j. 279 Maciel, M.R.W. Maciel Filho. R. 279,633, 639,805 189 Madsen, H. 743 Madvadhan, K.P. 225 Maffettone, P.L. 937 Majozi, T. 1053 Malmendier, M. 723 Manca, D. 135,225 Mancusi, E. 871,1157 Maranas, C.D.
1185 Marchetti, M. 231 Marcoulaki, E.C. 451 Marechal, F. 457 Marek, M. 177 Marquardt, W. 345,1071, 1121 Marqu&, J.A. 147 Martmez, E.C. 237 Martini, R.F. 567 Mata, T.M. 1165 Mathisen, K.W. 297 Matos, H. 847 Matthews, C. 991 Meleiro, L.A.C. 279,633 M~ndez, C.A. 693 Meyer, X.M. 201,267 Michiel Meeuse, F. 699, 799 Mmet, F. 1053 Mockus, L. 901 Modi, A. 419 Moharir, A.S. 243 Molma, A. 731 Moon, I. 427,895 Morris, A.J. 327,705 Morton, W. 249 Mourikas, G. 705 M'Saad, M. 651 Mu, F. 925,931 Mujtaba, I.M. 609 Mussone, P. 723 Nagy, A.B. 1017 Nagy, Z. 711 Naka, Y. 87,829 Ng, K.M. 41,523,1151 Niclout, N. 481 Novak, A. 177 Okada, H. 1059 Oldenburg, J. 1071 Oliveira, N.M.C. 749 Olsvik, O. 219 Omota, F. 463 Ortiz, I. 129,387 Ortiz-G6mez, A. 907 O'Young, L. 523 Pajula, E. 469 Palaniappan, C. 1145 Palazoglu, A. 141
Paloschi, J.R. Pantelides, C.C. Papageorgiou, L.G.
255 15 475, 865 Parisse, B. 731 Park, S. 547 Patsiatzis, D.I. 475 Pekny, J.F. 1101 Perez-Cisneros, E. 517 Perkins, J.D. 273,961 Perret, J. 889 Perris, T. 955 Pibouleau, L. 117,949 Pike, R.W. 1017 Pingaud, H. 889 Pingen, J. 1065 pmsky, M. 773 Pinto, J.M. 579 Pistikopoulos, E.N. 273, 961,979,997 Plapp, R. 419 Prat, L. 1127 Preisig, H.A. 261,811 Preuss, K. 717 Prevost, M. 267 Puigjaner, L. 835,841,913, 1077 Rao, A. 231 Reklaitis, G.V. 669,1101 Reneaume, J.-M. 481 Rengaswamy, R. 853 Rev, E. 553,1109 Reyes-Labarta, J.A. 487 Richalet, J. 731 Rico-Ramirez, V. 907 Rieber, J. 165 Rigopoulos, S. 493 Rodriguez-Donis, I. 499 Romagnoli, J.A. 141,823 Rong, B.-G. 439 Ross, R. 273 Rossiter, D. 1115 Rouzineau, D. 267 Rovaglio, M. 723 Ruiz, C. 731 Ruiz, D. 835 Russel, B . M . 925,1009 Russo, L. 135
Ronnekleiv, M. 219 Sakizlis, V. 273 Samad, T. 1 Samanta, A. 505,511 Samyudia, Y. 681 Sanchez Daza, O. 517 Santana, P.L. 279 Saraiva, P.M. 315 Sarimveis, H.K. 49 Scheffer, R. 279,639 Schembecker, G. 357 Schenk, M. 997 Schlegel, M. 1071 Schloder, J.P. 711 Schmidt, H. 737 Schmitt, W.A. 55 Schneider, R. 285 Schreiber, I. 171 Schroer, J.W. 523 Secchi, A.R. 123 Seferlis, P. 705 Sequeira, S.E. 913,1077 Seuranen, T. 469 Shacham, M. 291 Shah, N. 865 Shah, S.S. 243,743 Shang, Z. 1083 Shethna, H.K. 1089 Shimada, Y. 859 Shin, D. 645 Shirao, T. 1059 Siddhaye, S. 369 Siepmann, V. 297 Siettos, C.I. 69 Silva, D.C.M. 749 Silva-Beard, A. 755 Simon, L. 761 Singer, A.B. 767 Skogestad, S. 657,1041 Skotte, R. 773 Smets, I.Y.M. 781 Smith, R.L. 1165 Soares, C. 321 Sobocan, G. 529 Solberg, T. 159 Sorsak, A. 1095 Srinivasan, R. 1145 Stembach, W. 535
1186 Stepanek, F. 177 Stephanopoulos, G. 55 Strouvalis, A.M. 541,919 Subramanian, D. 1101 Suh, M.-h. 547 Sunderesan, P. 369 Sunol, A.K. 943 Suppes, G.J. 369 Suzuki, K. 859 Svensson, F. 105 Szederk6nyi, G. 787 Szitkai, Z. 553,1109 Sorensen, E. 303 Takano, K. 559 Taylor, R. 93 Teoh, H.K. 303 Ternet, D. 675 Tilhac, F. 117 Titchener-Hooker, N. 303 Tolsma, J.E. 309 Torgashov, A.Yu. 793 Torres Alvarez, M.E. 567 Tousam, R.L. 799 Tresmondi, A. 805 Turner, M. 303
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